TPTP Problem File: SLH0458^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00059_002458__11646258_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1606 ( 758 unt; 328 typ; 0 def)
% Number of atoms : 3465 (2275 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10059 ( 525 ~; 105 |; 406 &;7902 @)
% ( 0 <=>;1121 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 5 avg)
% Number of types : 31 ( 30 usr)
% Number of type conns : 972 ( 972 >; 0 *; 0 +; 0 <<)
% Number of symbols : 301 ( 298 usr; 27 con; 0-3 aty)
% Number of variables : 3550 ( 160 ^;3082 !; 308 ?;3550 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:53:31.430
%------------------------------------------------------------------------------
% Could-be-implicit typings (30)
thf(ty_n_t__Option__Ooption_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
option_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_option_list_nat: $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__Option__Ooption_It__List__Olist_It__Option__Ooption_I_Eo_J_J_J,type,
option_list_option_o: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_It__Option__Ooption_I_Eo_J_J_J,type,
list_option_option_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
option_list_list_o: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
list_option_list_o: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Option__Ooption_I_Eo_J_J_J,type,
list_list_option_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Option__Ooption_I_Eo_J_J_J,type,
set_list_option_o: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
list_list_list_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
set_list_list_o: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J,type,
option_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
list_option_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__Option__Ooption_It__Option__Ooption_I_Eo_J_J,type,
option_option_o: $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__Option__Ooption_I_Eo_J_J,type,
list_option_o: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
set_option_o: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_I_Eo_J_J,type,
list_list_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
set_list_o: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $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__Option__Ooption_I_Eo_J,type,
option_o: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (298)
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001_Eo,type,
bNF_Greatest_Succ_o: set_list_o > list_o > set_o ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_I_Eo_J,type,
bNF_Gr2388622420913600185list_o: set_list_list_o > list_list_o > set_list_o ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr3053708287304744325st_nat: set_list_list_nat > list_list_nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Option__Ooption_I_Eo_J,type,
bNF_Gr5056813354227606905tion_o: set_list_option_o > list_option_o > set_option_o ).
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_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_I_Eo_J,type,
if_list_o: $o > list_o > list_o > list_o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001_Eo,type,
append_o: list_o > list_o > list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_I_Eo_J,type,
append_list_o: list_list_o > list_list_o > list_list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
append_list_list_o: list_list_list_o > list_list_list_o > list_list_list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
append_list_list_nat: list_list_list_nat > list_list_list_nat > list_list_list_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_It__Option__Ooption_I_Eo_J_J,type,
append_list_option_o: list_list_option_o > list_list_option_o > list_list_option_o ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Option__Ooption_I_Eo_J,type,
append_option_o: list_option_o > list_option_o > list_option_o ).
thf(sy_c_List_Obind_001_Eo_001_Eo,type,
bind_o_o: list_o > ( $o > list_o ) > list_o ).
thf(sy_c_List_Obind_001_Eo_001t__List__Olist_I_Eo_J,type,
bind_o_list_o: list_o > ( $o > list_list_o ) > list_list_o ).
thf(sy_c_List_Obind_001_Eo_001t__List__Olist_It__Nat__Onat_J,type,
bind_o_list_nat: list_o > ( $o > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001_Eo_001t__Nat__Onat,type,
bind_o_nat: list_o > ( $o > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001_Eo_001t__Option__Ooption_I_Eo_J,type,
bind_o_option_o: list_o > ( $o > list_option_o ) > list_option_o ).
thf(sy_c_List_Obind_001t__List__Olist_I_Eo_J_001_Eo,type,
bind_list_o_o: list_list_o > ( list_o > list_o ) > list_o ).
thf(sy_c_List_Obind_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
bind_list_o_list_o: list_list_o > ( list_o > list_list_o ) > list_list_o ).
thf(sy_c_List_Obind_001t__List__Olist_I_Eo_J_001t__Nat__Onat,type,
bind_list_o_nat: list_list_o > ( list_o > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001_Eo,type,
bind_nat_o: list_nat > ( nat > list_o ) > list_o ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
bind_nat_list_o: list_nat > ( nat > list_list_o ) > list_list_o ).
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__Option__Ooption_I_Eo_J_001_Eo,type,
bind_option_o_o: list_option_o > ( option_o > list_o ) > list_o ).
thf(sy_c_List_Obind_001t__Option__Ooption_I_Eo_J_001t__Nat__Onat,type,
bind_option_o_nat: list_option_o > ( option_o > list_nat ) > list_nat ).
thf(sy_c_List_Obutlast_001_Eo,type,
butlast_o: list_o > list_o ).
thf(sy_c_List_Obutlast_001t__List__Olist_I_Eo_J,type,
butlast_list_o: list_list_o > list_list_o ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Option__Ooption_I_Eo_J,type,
butlast_option_o: list_option_o > list_option_o ).
thf(sy_c_List_Oconcat_001_Eo,type,
concat_o: list_list_o > list_o ).
thf(sy_c_List_Oconcat_001t__List__Olist_I_Eo_J,type,
concat_list_o: list_list_list_o > list_list_o ).
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__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__Option__Ooption_I_Eo_J,type,
concat_option_o: list_list_option_o > list_option_o ).
thf(sy_c_List_Odistinct__adj_001_Eo,type,
distinct_adj_o: list_o > $o ).
thf(sy_c_List_Odrop_001_Eo,type,
drop_o: nat > list_o > list_o ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ofind_001_Eo,type,
find_o: ( $o > $o ) > list_o > option_o ).
thf(sy_c_List_Ofind_001t__List__Olist_I_Eo_J,type,
find_list_o: ( list_o > $o ) > list_list_o > option_list_o ).
thf(sy_c_List_Ofind_001t__List__Olist_It__Nat__Onat_J,type,
find_list_nat: ( list_nat > $o ) > list_list_nat > option_list_nat ).
thf(sy_c_List_Ofind_001t__Nat__Onat,type,
find_nat: ( nat > $o ) > list_nat > option_nat ).
thf(sy_c_List_Ofind_001t__Option__Ooption_I_Eo_J,type,
find_option_o: ( option_o > $o ) > list_option_o > option_option_o ).
thf(sy_c_List_Ofold_001_Eo_001t__List__Olist_I_Eo_J,type,
fold_o_list_o: ( $o > list_o > list_o ) > list_o > list_o > list_o ).
thf(sy_c_List_Ofold_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
fold_list_o_list_o: ( list_o > list_o > list_o ) > list_list_o > list_o > list_o ).
thf(sy_c_List_Ofoldr_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
foldr_list_o_list_o: ( list_o > list_o > list_o ) > list_list_o > list_o > list_o ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__List__Olist_I_Eo_J_J_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
foldr_4660670129333661223list_o: ( list_list_o > list_list_o > list_list_o ) > list_list_list_o > list_list_o > list_list_o ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
foldr_7810631890432470921st_nat: ( list_list_nat > list_list_nat > list_list_nat ) > list_list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldr_6871341030409798377st_nat: ( list_nat > list_nat > list_nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Option__Ooption_I_Eo_J_J_001t__List__Olist_It__Option__Ooption_I_Eo_J_J,type,
foldr_7481994381061804839tion_o: ( list_option_o > list_option_o > list_option_o ) > list_list_option_o > list_option_o > list_option_o ).
thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Oinsert_001_Eo,type,
insert_o: $o > list_o > list_o ).
thf(sy_c_List_Oinsert_001t__List__Olist_I_Eo_J,type,
insert_list_o: list_o > list_list_o > list_list_o ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Option__Ooption_I_Eo_J,type,
insert_option_o: option_o > list_option_o > list_option_o ).
thf(sy_c_List_Olast_001_Eo,type,
last_o: list_o > $o ).
thf(sy_c_List_Olast_001t__List__Olist_I_Eo_J,type,
last_list_o: list_list_o > list_o ).
thf(sy_c_List_Olast_001t__List__Olist_It__Nat__Onat_J,type,
last_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Option__Ooption_I_Eo_J,type,
last_option_o: list_option_o > option_o ).
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_I_Eo_J,type,
cons_list_o: list_o > list_list_o > list_list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
cons_list_list_o: list_list_o > list_list_list_o > list_list_list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
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_It__Option__Ooption_I_Eo_J_J,type,
cons_list_option_o: list_option_o > list_list_option_o > list_list_option_o ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Option__Ooption_I_Eo_J,type,
cons_option_o: option_o > list_option_o > list_option_o ).
thf(sy_c_List_Olist_ONil_001_Eo,type,
nil_o: list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_I_Eo_J,type,
nil_list_o: list_list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
nil_list_list_o: list_list_list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
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_It__Option__Ooption_I_Eo_J_J,type,
nil_list_option_o: list_list_option_o ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Option__Ooption_I_Eo_J,type,
nil_option_o: list_option_o ).
thf(sy_c_List_Olist_ONil_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
nil_option_list_o: list_option_list_o ).
thf(sy_c_List_Olist_ONil_001t__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J,type,
nil_option_list_nat: list_option_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Option__Ooption_It__Nat__Onat_J,type,
nil_option_nat: list_option_nat ).
thf(sy_c_List_Olist_ONil_001t__Option__Ooption_It__Option__Ooption_I_Eo_J_J,type,
nil_option_option_o: list_option_option_o ).
thf(sy_c_List_Olist_Ohd_001_Eo,type,
hd_o: list_o > $o ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
map_o_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_Olist_Omap_001_Eo_001t__List__Olist_I_Eo_J,type,
map_o_list_o: ( $o > list_o ) > list_o > list_list_o ).
thf(sy_c_List_Olist_Omap_001_Eo_001t__List__Olist_It__Nat__Onat_J,type,
map_o_list_nat: ( $o > list_nat ) > list_o > list_list_nat ).
thf(sy_c_List_Olist_Omap_001_Eo_001t__Nat__Onat,type,
map_o_nat: ( $o > nat ) > list_o > list_nat ).
thf(sy_c_List_Olist_Omap_001_Eo_001t__Option__Ooption_I_Eo_J,type,
map_o_option_o: ( $o > option_o ) > list_o > list_option_o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_I_Eo_J_001_Eo,type,
map_list_o_o: ( list_o > $o ) > list_list_o > list_o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
map_list_o_list_o: ( list_o > list_o ) > list_list_o > list_list_o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_I_Eo_J_001t__List__Olist_It__Nat__Onat_J,type,
map_list_o_list_nat: ( list_o > list_nat ) > list_list_o > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_I_Eo_J_001t__Nat__Onat,type,
map_list_o_nat: ( list_o > nat ) > list_list_o > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_I_Eo_J_J_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
map_li5464752787020143300list_o: ( list_list_o > list_list_o ) > list_list_list_o > list_list_list_o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li2855073862107769254st_nat: ( list_list_nat > list_list_nat ) > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li1587113026813354006st_nat: ( list_list_nat > list_nat ) > list_list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
map_list_nat_o: ( list_nat > $o ) > list_list_nat > list_o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_I_Eo_J,type,
map_list_nat_list_o: ( list_nat > list_o ) > list_list_nat > list_list_o ).
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_It__Option__Ooption_I_Eo_J_J_001t__List__Olist_It__Option__Ooption_I_Eo_J_J,type,
map_li7029392385944470980tion_o: ( list_option_o > list_option_o ) > list_list_option_o > list_list_option_o ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001_Eo,type,
map_nat_o: ( nat > $o ) > list_nat > list_o ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
map_nat_list_o: ( nat > list_o ) > list_nat > list_list_o ).
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_001t__Option__Ooption_I_Eo_J,type,
map_nat_option_o: ( nat > option_o ) > list_nat > list_option_o ).
thf(sy_c_List_Olist_Omap_001t__Option__Ooption_I_Eo_J_001_Eo,type,
map_option_o_o: ( option_o > $o ) > list_option_o > list_o ).
thf(sy_c_List_Olist_Omap_001t__Option__Ooption_I_Eo_J_001t__Nat__Onat,type,
map_option_o_nat: ( option_o > nat ) > list_option_o > list_nat ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
set_list_o2: list_list_o > set_list_o ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Osize__list_001_Eo,type,
size_list_o: ( $o > nat ) > list_o > nat ).
thf(sy_c_List_Olist_Otl_001_Eo,type,
tl_o: list_o > list_o ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist__ex1_001_Eo,type,
list_ex1_o: ( $o > $o ) > list_o > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_I_Eo_J,type,
list_ex1_list_o: ( list_o > $o ) > list_list_o > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Option__Ooption_I_Eo_J,type,
list_ex1_option_o: ( option_o > $o ) > list_option_o > $o ).
thf(sy_c_List_Olist__ex_001_Eo,type,
list_ex_o: ( $o > $o ) > list_o > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_I_Eo_J,type,
list_ex_list_o: ( list_o > $o ) > list_list_o > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Nat__Onat_J,type,
list_ex_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Option__Ooption_I_Eo_J,type,
list_ex_option_o: ( option_o > $o ) > list_option_o > $o ).
thf(sy_c_List_Olist__update_001_Eo,type,
list_update_o: list_o > nat > $o > list_o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001_Eo_001_Eo,type,
map_tailrec_rev_o_o: ( $o > $o ) > list_o > list_o > list_o ).
thf(sy_c_List_Omap__tailrec__rev_001_Eo_001t__List__Olist_I_Eo_J,type,
map_ta3905426649961562107list_o: ( $o > list_o ) > list_o > list_list_o > list_list_o ).
thf(sy_c_List_Omap__tailrec__rev_001_Eo_001t__Nat__Onat,type,
map_ta6980961046827203379_o_nat: ( $o > nat ) > list_o > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__List__Olist_I_Eo_J_001_Eo,type,
map_ta8539203598313750511st_o_o: ( list_o > $o ) > list_list_o > list_o > list_o ).
thf(sy_c_List_Omap__tailrec__rev_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
map_ta2047065599740801025_nat_o: ( list_nat > $o ) > list_list_nat > list_o > list_o ).
thf(sy_c_List_Omap__tailrec__rev_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_ta3159490279798172583at_nat: ( list_nat > nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001_Eo,type,
map_ta860260371080612881_nat_o: ( nat > $o ) > list_nat > list_o > list_o ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Option__Ooption_I_Eo_J_001_Eo,type,
map_ta2677093932182474799on_o_o: ( option_o > $o ) > list_option_o > list_o > list_o ).
thf(sy_c_List_Omap__tailrec__rev_001t__Option__Ooption_I_Eo_J_001t__Nat__Onat,type,
map_ta909136395571573497_o_nat: ( option_o > nat ) > list_option_o > list_nat > list_nat ).
thf(sy_c_List_Omaps_001_Eo_001_Eo,type,
maps_o_o: ( $o > list_o ) > list_o > list_o ).
thf(sy_c_List_Omaps_001_Eo_001t__List__Olist_I_Eo_J,type,
maps_o_list_o: ( $o > list_list_o ) > list_o > list_list_o ).
thf(sy_c_List_Omaps_001_Eo_001t__List__Olist_It__Nat__Onat_J,type,
maps_o_list_nat: ( $o > list_list_nat ) > list_o > list_list_nat ).
thf(sy_c_List_Omaps_001_Eo_001t__Nat__Onat,type,
maps_o_nat: ( $o > list_nat ) > list_o > list_nat ).
thf(sy_c_List_Omaps_001_Eo_001t__Option__Ooption_I_Eo_J,type,
maps_o_option_o: ( $o > list_option_o ) > list_o > list_option_o ).
thf(sy_c_List_Omaps_001t__List__Olist_I_Eo_J_001_Eo,type,
maps_list_o_o: ( list_o > list_o ) > list_list_o > list_o ).
thf(sy_c_List_Omaps_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
maps_list_o_list_o: ( list_o > list_list_o ) > list_list_o > list_list_o ).
thf(sy_c_List_Omaps_001t__List__Olist_I_Eo_J_001t__Nat__Onat,type,
maps_list_o_nat: ( list_o > list_nat ) > list_list_o > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001_Eo,type,
maps_nat_o: ( nat > list_o ) > list_nat > list_o ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
maps_nat_list_o: ( nat > list_list_o ) > list_nat > list_list_o ).
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__Option__Ooption_I_Eo_J_001_Eo,type,
maps_option_o_o: ( option_o > list_o ) > list_option_o > list_o ).
thf(sy_c_List_Omaps_001t__Option__Ooption_I_Eo_J_001t__Nat__Onat,type,
maps_option_o_nat: ( option_o > list_nat ) > list_option_o > list_nat ).
thf(sy_c_List_Omember_001_Eo,type,
member_o: list_o > $o > $o ).
thf(sy_c_List_Omember_001t__List__Olist_I_Eo_J,type,
member_list_o: list_list_o > list_o > $o ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Option__Ooption_I_Eo_J,type,
member_option_o: list_option_o > option_o > $o ).
thf(sy_c_List_Onth_001_Eo,type,
nth_o: list_o > nat > $o ).
thf(sy_c_List_Onth_001t__List__Olist_I_Eo_J,type,
nth_list_o: list_list_o > nat > list_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__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Oord_Olexordp__eq_001_Eo,type,
lexordp_eq_o: ( $o > $o > $o ) > list_o > list_o > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001_Eo,type,
ord_lexordp_eq_o: list_o > list_o > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Nat__Onat,type,
ord_lexordp_eq_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oproduct__lists_001_Eo,type,
product_lists_o: list_list_o > list_list_o ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_I_Eo_J,type,
product_lists_list_o: list_list_list_o > list_list_list_o ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
produc6783906451316923569st_nat: list_list_list_nat > list_list_list_nat ).
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_001t__Option__Ooption_I_Eo_J,type,
produc5165959316431261517tion_o: list_list_option_o > list_list_option_o ).
thf(sy_c_List_Oreplicate_001_Eo,type,
replicate_o: nat > $o > list_o ).
thf(sy_c_List_Oreplicate_001t__List__Olist_I_Eo_J,type,
replicate_list_o: nat > list_o > list_list_o ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Orev_001_Eo,type,
rev_o: list_o > list_o ).
thf(sy_c_List_Orev_001t__List__Olist_I_Eo_J,type,
rev_list_o: list_list_o > list_list_o ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001_Eo,type,
rotate1_o: list_o > list_o ).
thf(sy_c_List_Orotate1_001t__List__Olist_I_Eo_J,type,
rotate1_list_o: list_list_o > list_list_o ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__Nat__Onat_J,type,
rotate1_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Option__Ooption_I_Eo_J,type,
rotate1_option_o: list_option_o > list_option_o ).
thf(sy_c_List_Orotate_001_Eo,type,
rotate_o: nat > list_o > list_o ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Osorted__wrt_001_Eo,type,
sorted_wrt_o: ( $o > $o > $o ) > list_o > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osubseqs_001_Eo,type,
subseqs_o: list_o > list_list_o ).
thf(sy_c_List_Osubseqs_001t__List__Olist_I_Eo_J,type,
subseqs_list_o: list_list_o > list_list_list_o ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
subseqs_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Option__Ooption_I_Eo_J,type,
subseqs_option_o: list_option_o > list_list_option_o ).
thf(sy_c_List_Otake_001_Eo,type,
take_o: nat > list_o > list_o ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Othose_001_Eo,type,
those_o: list_option_o > option_list_o ).
thf(sy_c_List_Othose_001t__List__Olist_I_Eo_J,type,
those_list_o: list_option_list_o > option_list_list_o ).
thf(sy_c_List_Othose_001t__List__Olist_It__Nat__Onat_J,type,
those_list_nat: list_option_list_nat > option_list_list_nat ).
thf(sy_c_List_Othose_001t__Nat__Onat,type,
those_nat: list_option_nat > option_list_nat ).
thf(sy_c_List_Othose_001t__Option__Ooption_I_Eo_J,type,
those_option_o: list_option_option_o > option_list_option_o ).
thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
size_s2710708370519433104list_o: list_list_o > 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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
size_s9086679804311813456list_o: option_list_o > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Option_Ooption_ONone_001_Eo,type,
none_o: option_o ).
thf(sy_c_Option_Ooption_ONone_001t__List__Olist_I_Eo_J,type,
none_list_o: option_list_o ).
thf(sy_c_Option_Ooption_ONone_001t__List__Olist_It__Nat__Onat_J,type,
none_list_nat: option_list_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
none_nat: option_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_I_Eo_J,type,
none_option_o: option_option_o ).
thf(sy_c_Option_Ooption_OSome_001_Eo,type,
some_o: $o > option_o ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_I_Eo_J,type,
some_list_o: list_o > option_list_o ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
some_list_list_o: list_list_o > option_list_list_o ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
some_list_list_nat: list_list_nat > option_list_list_nat ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__Nat__Onat_J,type,
some_list_nat: list_nat > option_list_nat ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__Option__Ooption_I_Eo_J_J,type,
some_list_option_o: list_option_o > option_list_option_o ).
thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
some_nat: nat > option_nat ).
thf(sy_c_Option_Ooption_Osize__option_001t__List__Olist_I_Eo_J,type,
size_option_list_o: ( list_o > nat ) > option_list_o > nat ).
thf(sy_c_Option_Ooption_Othe_001t__List__Olist_I_Eo_J,type,
the_list_o: option_list_o > list_o ).
thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
the_nat: option_nat > nat ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_I_Eo_J,type,
max_list_o: ( list_o > list_o > $o ) > list_o > list_o > list_o ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
max_list_list_o: ( list_list_o > list_list_o > $o ) > list_list_o > list_list_o > list_list_o ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
max_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__Nat__Onat_J,type,
max_list_nat: ( list_nat > list_nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__Option__Ooption_I_Eo_J_J,type,
max_list_option_o: ( list_option_o > list_option_o > $o ) > list_option_o > list_option_o > list_option_o ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_I_Eo_J,type,
min_list_o: ( list_o > list_o > $o ) > list_o > list_o > list_o ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
min_list_list_o: ( list_list_o > list_list_o > $o ) > list_list_o > list_list_o > list_list_o ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
min_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__Nat__Onat_J,type,
min_list_nat: ( list_nat > list_nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__Option__Ooption_I_Eo_J_J,type,
min_list_option_o: ( list_option_o > list_option_o > $o ) > list_option_o > list_option_o > list_option_o ).
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_Eo,type,
ord_less_eq_o: $o > $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_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
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_Prefix__Free__Code__Combinators_Oopt__prefix_001_Eo,type,
prefix8824957607401505554efix_o: option_list_o > option_list_o > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oopt__prefix_001t__Nat__Onat,type,
prefix9059114240934526678ix_nat: option_list_nat > option_list_nat > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Sublist_OLongest__common__prefix_001_Eo,type,
longes2916941589570675010efix_o: set_list_o > list_o ).
thf(sy_c_Sublist_Olongest__common__prefix_001_Eo,type,
longes218265413398645026efix_o: list_o > list_o > list_o ).
thf(sy_c_Sublist_Olongest__common__prefix_001t__List__Olist_I_Eo_J,type,
longes802630962832238888list_o: list_list_o > list_list_o > list_list_o ).
thf(sy_c_Sublist_Olongest__common__prefix_001t__List__Olist_It__Nat__Onat_J,type,
longes3715614165048771798st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Sublist_Olongest__common__prefix_001t__Nat__Onat,type,
longes266370323089874118ix_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Sublist_Olongest__common__prefix_001t__Option__Ooption_I_Eo_J,type,
longes1709640641647806696tion_o: list_option_o > list_option_o > list_option_o ).
thf(sy_c_Sublist_Oprefix_001_Eo,type,
prefix_o: list_o > list_o > $o ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_I_Eo_J,type,
prefix_list_o: list_list_o > list_list_o > $o ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__Nat__Onat_J,type,
prefix_list_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Option__Ooption_I_Eo_J,type,
prefix_option_o: list_option_o > list_option_o > $o ).
thf(sy_c_Sublist_Oprefixes_001_Eo,type,
prefixes_o: list_o > list_list_o ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_I_Eo_J,type,
prefixes_list_o: list_list_o > list_list_list_o ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__Nat__Onat_J,type,
prefixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__Option__Ooption_I_Eo_J,type,
prefixes_option_o: list_option_o > list_list_option_o ).
thf(sy_c_Sublist_Ostrict__prefix_001_Eo,type,
strict_prefix_o: list_o > list_o > $o ).
thf(sy_c_Sublist_Ostrict__prefix_001t__List__Olist_I_Eo_J,type,
strict_prefix_list_o: list_list_o > list_list_o > $o ).
thf(sy_c_Sublist_Ostrict__prefix_001t__List__Olist_It__Nat__Onat_J,type,
strict6380099159608944142st_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Sublist_Ostrict__prefix_001t__Nat__Onat,type,
strict_prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Ostrict__prefix_001t__Option__Ooption_I_Eo_J,type,
strict4666211328587260336tion_o: list_option_o > list_option_o > $o ).
thf(sy_c_Sublist_Ostrict__suffix_001t__Nat__Onat,type,
strict_suffix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Osublist_001_Eo,type,
sublist_o: list_o > list_o > $o ).
thf(sy_c_Sublist_Osublist_001t__Nat__Onat,type,
sublist_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Osublists_001_Eo,type,
sublists_o: list_o > list_list_o ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_I_Eo_J,type,
sublists_list_o: list_list_o > list_list_list_o ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_It__Nat__Onat_J,type,
sublists_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__Option__Ooption_I_Eo_J,type,
sublists_option_o: list_option_o > list_list_option_o ).
thf(sy_c_Sublist_Osuffix_001_Eo,type,
suffix_o: list_o > list_o > $o ).
thf(sy_c_Sublist_Osuffix_001t__Nat__Onat,type,
suffix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Osuffixes_001_Eo,type,
suffixes_o: list_o > list_list_o ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_I_Eo_J,type,
suffixes_list_o: list_list_o > list_list_list_o ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__Nat__Onat_J,type,
suffixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Option__Ooption_I_Eo_J,type,
suffixes_option_o: list_option_o > list_list_option_o ).
thf(sy_c_member_001_Eo,type,
member_o2: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o2: list_o > set_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
member_list_list_o: list_list_o > set_list_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Option__Ooption_I_Eo_J_J,type,
member_list_option_o: list_option_o > set_list_option_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_I_Eo_J,type,
member_option_o2: option_o > set_option_o > $o ).
thf(sy_v_x,type,
x: option_list_o ).
thf(sy_v_x_H____,type,
x2: list_o ).
thf(sy_v_y,type,
y: option_list_o ).
thf(sy_v_y_H____,type,
y2: list_o ).
thf(sy_v_z,type,
z: option_list_o ).
thf(sy_v_z_H____,type,
z2: list_o ).
% Relevant facts (1272)
thf(fact_0_a_I3_J,axiom,
( z
= ( some_list_o @ z2 ) ) ).
% a(3)
thf(fact_1_a_I2_J,axiom,
( y
= ( some_list_o @ y2 ) ) ).
% a(2)
thf(fact_2_a_I1_J,axiom,
( x
= ( some_list_o @ x2 ) ) ).
% a(1)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_H_Ay_H_Az_H_O_A_092_060lbrakk_062x_A_061_ASome_Ax_H_059_Ay_A_061_ASome_Ay_H_059_Az_A_061_ASome_Az_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ? [X: list_o] :
( x
= ( some_list_o @ X ) )
=> ( ? [Y: list_o] :
( y
= ( some_list_o @ Y ) )
=> ! [Z: list_o] :
( z
!= ( some_list_o @ Z ) ) ) ) ).
% \<open>\<And>thesis. (\<And>x' y' z'. \<lbrakk>x = Some x'; y = Some y'; z = Some z'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_same__prefix__prefix,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ ( append_list_o @ Xs @ Ys ) @ ( append_list_o @ Xs @ Zs ) )
= ( prefix_list_o @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_5_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_6_same__prefix__prefix,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Xs @ Zs ) )
= ( prefix_o @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_7_prefix__order_Oorder__refl,axiom,
! [X2: list_nat] : ( prefix_nat @ X2 @ X2 ) ).
% prefix_order.order_refl
thf(fact_8_prefix__order_Oorder__refl,axiom,
! [X2: list_o] : ( prefix_o @ X2 @ X2 ) ).
% prefix_order.order_refl
thf(fact_9_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_10_prefix__order_Odual__order_Orefl,axiom,
! [A: list_o] : ( prefix_o @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_11_append_Oassoc,axiom,
! [A: list_list_o,B: list_list_o,C: list_list_o] :
( ( append_list_o @ ( append_list_o @ A @ B ) @ C )
= ( append_list_o @ A @ ( append_list_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_12_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_13_append_Oassoc,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( append_o @ ( append_o @ A @ B ) @ C )
= ( append_o @ A @ ( append_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_14_append__assoc,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( append_list_o @ ( append_list_o @ Xs @ Ys ) @ Zs )
= ( append_list_o @ Xs @ ( append_list_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_15_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_16_append__assoc,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( append_o @ ( append_o @ Xs @ Ys ) @ Zs )
= ( append_o @ Xs @ ( append_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_17_append__same__eq,axiom,
! [Ys: list_list_o,Xs: list_list_o,Zs: list_list_o] :
( ( ( append_list_o @ Ys @ Xs )
= ( append_list_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_18_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_19_append__same__eq,axiom,
! [Ys: list_o,Xs: list_o,Zs: list_o] :
( ( ( append_o @ Ys @ Xs )
= ( append_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_20_same__append__eq,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_21_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_22_same__append__eq,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( append_o @ Xs @ Ys )
= ( append_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_23_prefixE,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( prefix_list_o @ Xs @ Ys )
=> ~ ! [Zs2: list_list_o] :
( Ys
!= ( append_list_o @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_24_prefixE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ~ ! [Zs2: list_nat] :
( Ys
!= ( append_nat @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_25_prefixE,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ~ ! [Zs2: list_o] :
( Ys
!= ( append_o @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_26_prefixI,axiom,
! [Ys: list_list_o,Xs: list_list_o,Zs: list_list_o] :
( ( Ys
= ( append_list_o @ Xs @ Zs ) )
=> ( prefix_list_o @ Xs @ Ys ) ) ).
% prefixI
thf(fact_27_prefixI,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( Ys
= ( append_nat @ Xs @ Zs ) )
=> ( prefix_nat @ Xs @ Ys ) ) ).
% prefixI
thf(fact_28_prefixI,axiom,
! [Ys: list_o,Xs: list_o,Zs: list_o] :
( ( Ys
= ( append_o @ Xs @ Zs ) )
=> ( prefix_o @ Xs @ Ys ) ) ).
% prefixI
thf(fact_29_prefix__def,axiom,
( prefix_list_o
= ( ^ [Xs2: list_list_o,Ys2: list_list_o] :
? [Zs3: list_list_o] :
( Ys2
= ( append_list_o @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_30_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
? [Zs3: list_nat] :
( Ys2
= ( append_nat @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_31_prefix__def,axiom,
( prefix_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
? [Zs3: list_o] :
( Ys2
= ( append_o @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_32_prefix__append,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ Xs @ ( append_list_o @ Ys @ Zs ) )
= ( ( prefix_list_o @ Xs @ Ys )
| ? [Us: list_list_o] :
( ( Xs
= ( append_list_o @ Ys @ Us ) )
& ( prefix_list_o @ Us @ Zs ) ) ) ) ).
% prefix_append
thf(fact_33_prefix__append,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ Zs ) )
= ( ( prefix_nat @ Xs @ Ys )
| ? [Us: list_nat] :
( ( Xs
= ( append_nat @ Ys @ Us ) )
& ( prefix_nat @ Us @ Zs ) ) ) ) ).
% prefix_append
thf(fact_34_prefix__append,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ Xs @ ( append_o @ Ys @ Zs ) )
= ( ( prefix_o @ Xs @ Ys )
| ? [Us: list_o] :
( ( Xs
= ( append_o @ Ys @ Us ) )
& ( prefix_o @ Us @ Zs ) ) ) ) ).
% prefix_append
thf(fact_35_assms,axiom,
prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ x @ y ) @ z ).
% assms
thf(fact_36_append__eq__append__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o,Ts: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs @ Ts ) )
= ( ? [Us: list_list_o] :
( ( ( Xs
= ( append_list_o @ Zs @ Us ) )
& ( ( append_list_o @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_o @ Xs @ Us )
= Zs )
& ( Ys
= ( append_list_o @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_37_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 ) )
= ( ? [Us: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us ) )
& ( ( append_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_38_append__eq__append__conv2,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o,Ts: list_o] :
( ( ( append_o @ Xs @ Ys )
= ( append_o @ Zs @ Ts ) )
= ( ? [Us: list_o] :
( ( ( Xs
= ( append_o @ Zs @ Us ) )
& ( ( append_o @ Us @ Ys )
= Ts ) )
| ( ( ( append_o @ Xs @ Us )
= Zs )
& ( Ys
= ( append_o @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_39_append__eq__appendI,axiom,
! [Xs: list_list_o,Xs1: list_list_o,Zs: list_list_o,Ys: list_list_o,Us2: list_list_o] :
( ( ( append_list_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_o @ Xs1 @ Us2 ) )
=> ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_40_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us2 ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_41_append__eq__appendI,axiom,
! [Xs: list_o,Xs1: list_o,Zs: list_o,Ys: list_o,Us2: list_o] :
( ( ( append_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_o @ Xs1 @ Us2 ) )
=> ( ( append_o @ Xs @ Ys )
= ( append_o @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_42_prefix__same__cases,axiom,
! [Xs_1: list_nat,Ys: list_nat,Xs_2: list_nat] :
( ( prefix_nat @ Xs_1 @ Ys )
=> ( ( prefix_nat @ Xs_2 @ Ys )
=> ( ( prefix_nat @ Xs_1 @ Xs_2 )
| ( prefix_nat @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_43_prefix__same__cases,axiom,
! [Xs_1: list_o,Ys: list_o,Xs_2: list_o] :
( ( prefix_o @ Xs_1 @ Ys )
=> ( ( prefix_o @ Xs_2 @ Ys )
=> ( ( prefix_o @ Xs_1 @ Xs_2 )
| ( prefix_o @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_44_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_nat,A: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_45_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_o,A: list_o] :
( ( prefix_o @ B @ A )
=> ( ( prefix_o @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_46_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [A2: list_nat,B2: list_nat] :
( ( prefix_nat @ B2 @ A2 )
& ( prefix_nat @ A2 @ B2 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_47_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y2: list_o,Z2: list_o] : ( Y2 = Z2 ) )
= ( ^ [A2: list_o,B2: list_o] :
( ( prefix_o @ B2 @ A2 )
& ( prefix_o @ A2 @ B2 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_48_prefix__order_Odual__order_Otrans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ C @ B )
=> ( prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_49_prefix__order_Odual__order_Otrans,axiom,
! [B: list_o,A: list_o,C: list_o] :
( ( prefix_o @ B @ A )
=> ( ( prefix_o @ C @ B )
=> ( prefix_o @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_50_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( B = C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_51_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( prefix_o @ A @ B )
=> ( ( B = C )
=> ( prefix_o @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_52_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_53_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( A = B )
=> ( ( prefix_o @ B @ C )
=> ( prefix_o @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_54_prefix__order_Oorder__antisym,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( prefix_nat @ X2 @ Y3 )
=> ( ( prefix_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% prefix_order.order_antisym
thf(fact_55_prefix__order_Oorder__antisym,axiom,
! [X2: list_o,Y3: list_o] :
( ( prefix_o @ X2 @ Y3 )
=> ( ( prefix_o @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% prefix_order.order_antisym
thf(fact_56_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( prefix_nat @ X3 @ Y4 )
& ( prefix_nat @ Y4 @ X3 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_57_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y2: list_o,Z2: list_o] : ( Y2 = Z2 ) )
= ( ^ [X3: list_o,Y4: list_o] :
( ( prefix_o @ X3 @ Y4 )
& ( prefix_o @ Y4 @ X3 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_58_prefix__order_Oantisym__conv,axiom,
! [Y3: list_nat,X2: list_nat] :
( ( prefix_nat @ Y3 @ X2 )
=> ( ( prefix_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% prefix_order.antisym_conv
thf(fact_59_prefix__order_Oantisym__conv,axiom,
! [Y3: list_o,X2: list_o] :
( ( prefix_o @ Y3 @ X2 )
=> ( ( prefix_o @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% prefix_order.antisym_conv
thf(fact_60_prefix__order_Oorder__trans,axiom,
! [X2: list_nat,Y3: list_nat,Z3: list_nat] :
( ( prefix_nat @ X2 @ Y3 )
=> ( ( prefix_nat @ Y3 @ Z3 )
=> ( prefix_nat @ X2 @ Z3 ) ) ) ).
% prefix_order.order_trans
thf(fact_61_prefix__order_Oorder__trans,axiom,
! [X2: list_o,Y3: list_o,Z3: list_o] :
( ( prefix_o @ X2 @ Y3 )
=> ( ( prefix_o @ Y3 @ Z3 )
=> ( prefix_o @ X2 @ Z3 ) ) ) ).
% prefix_order.order_trans
thf(fact_62_prefix__order_Oeq__refl,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( X2 = Y3 )
=> ( prefix_nat @ X2 @ Y3 ) ) ).
% prefix_order.eq_refl
thf(fact_63_prefix__order_Oeq__refl,axiom,
! [X2: list_o,Y3: list_o] :
( ( X2 = Y3 )
=> ( prefix_o @ X2 @ Y3 ) ) ).
% prefix_order.eq_refl
thf(fact_64_prefix__order_Oantisym,axiom,
! [A: list_nat,B: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_65_prefix__order_Oantisym,axiom,
! [A: list_o,B: list_o] :
( ( prefix_o @ A @ B )
=> ( ( prefix_o @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_66_prefix__order_Oeq__iff,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [A2: list_nat,B2: list_nat] :
( ( prefix_nat @ A2 @ B2 )
& ( prefix_nat @ B2 @ A2 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_67_prefix__order_Oeq__iff,axiom,
( ( ^ [Y2: list_o,Z2: list_o] : ( Y2 = Z2 ) )
= ( ^ [A2: list_o,B2: list_o] :
( ( prefix_o @ A2 @ B2 )
& ( prefix_o @ B2 @ A2 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_68_prefix__order_Otrans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_69_prefix__order_Otrans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( prefix_o @ A @ B )
=> ( ( prefix_o @ B @ C )
=> ( prefix_o @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_70_append__prefixD,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ ( append_list_o @ Xs @ Ys ) @ Zs )
=> ( prefix_list_o @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_71_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_72_append__prefixD,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ Zs )
=> ( prefix_o @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_73_prefix__prefix,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( prefix_list_o @ Xs @ Ys )
=> ( prefix_list_o @ Xs @ ( append_list_o @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_74_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_75_prefix__prefix,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( prefix_o @ Xs @ ( append_o @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_76_opt__append_Osimps_I1_J,axiom,
! [X2: list_o,Y3: list_o] :
( ( prefix5314359684614007693append @ ( some_list_o @ X2 ) @ ( some_list_o @ Y3 ) )
= ( some_list_o @ ( append_o @ X2 @ Y3 ) ) ) ).
% opt_append.simps(1)
thf(fact_77_option_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( some_nat @ X22 )
= ( some_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_78_option_Oinject,axiom,
! [X22: list_o,Y22: list_o] :
( ( ( some_list_o @ X22 )
= ( some_list_o @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_79_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X3: option_list_o,Y4: option_list_o] : ( prefix454693708527911765comp_o @ Y4 @ X3 ) ) ) ).
% opt_comp_sym
thf(fact_80_opt__prefix_Osimps_I1_J,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( prefix9059114240934526678ix_nat @ ( some_list_nat @ X2 ) @ ( some_list_nat @ Y3 ) )
= ( prefix_nat @ X2 @ Y3 ) ) ).
% opt_prefix.simps(1)
thf(fact_81_opt__prefix_Osimps_I1_J,axiom,
! [X2: list_o,Y3: list_o] :
( ( prefix8824957607401505554efix_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y3 ) )
= ( prefix_o @ X2 @ Y3 ) ) ).
% opt_prefix.simps(1)
thf(fact_82_opt__prefix_Oelims_I2_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat] :
( ( prefix9059114240934526678ix_nat @ X2 @ Xa )
=> ~ ! [X4: list_nat] :
( ( X2
= ( some_list_nat @ X4 ) )
=> ! [Y5: list_nat] :
( ( Xa
= ( some_list_nat @ Y5 ) )
=> ~ ( prefix_nat @ X4 @ Y5 ) ) ) ) ).
% opt_prefix.elims(2)
thf(fact_83_opt__prefix_Oelims_I2_J,axiom,
! [X2: option_list_o,Xa: option_list_o] :
( ( prefix8824957607401505554efix_o @ X2 @ Xa )
=> ~ ! [X4: list_o] :
( ( X2
= ( some_list_o @ X4 ) )
=> ! [Y5: list_o] :
( ( Xa
= ( some_list_o @ Y5 ) )
=> ~ ( prefix_o @ X4 @ Y5 ) ) ) ) ).
% opt_prefix.elims(2)
thf(fact_84_opt__append_Oelims,axiom,
! [X2: option_list_o,Xa: option_list_o,Y3: option_list_o] :
( ( ( prefix5314359684614007693append @ X2 @ Xa )
= Y3 )
=> ( ! [X4: list_o] :
( ( X2
= ( some_list_o @ X4 ) )
=> ! [Y5: list_o] :
( ( Xa
= ( some_list_o @ Y5 ) )
=> ( Y3
!= ( some_list_o @ ( append_o @ X4 @ Y5 ) ) ) ) )
=> ( ( ( X2 = none_list_o )
=> ( Y3 != none_list_o ) )
=> ~ ( ( Xa = none_list_o )
=> ( Y3 != none_list_o ) ) ) ) ) ).
% opt_append.elims
thf(fact_85_same__prefix__nil,axiom,
! [Xs: list_option_o,Ys: list_option_o] :
( ( prefix_option_o @ ( append_option_o @ Xs @ Ys ) @ Xs )
= ( Ys = nil_option_o ) ) ).
% same_prefix_nil
thf(fact_86_same__prefix__nil,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( prefix_list_o @ ( append_list_o @ Xs @ Ys ) @ Xs )
= ( Ys = nil_list_o ) ) ).
% same_prefix_nil
thf(fact_87_same__prefix__nil,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_list_nat ) ) ).
% same_prefix_nil
thf(fact_88_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_89_same__prefix__nil,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ Xs )
= ( Ys = nil_o ) ) ).
% same_prefix_nil
thf(fact_90_list__ex__append,axiom,
! [P: list_o > $o,Xs: list_list_o,Ys: list_list_o] :
( ( list_ex_list_o @ P @ ( append_list_o @ Xs @ Ys ) )
= ( ( list_ex_list_o @ P @ Xs )
| ( list_ex_list_o @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_91_list__ex__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
( ( list_ex_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( list_ex_nat @ P @ Xs )
| ( list_ex_nat @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_92_list__ex__append,axiom,
! [P: $o > $o,Xs: list_o,Ys: list_o] :
( ( list_ex_o @ P @ ( append_o @ Xs @ Ys ) )
= ( ( list_ex_o @ P @ Xs )
| ( list_ex_o @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_93_longest__common__prefix__max__prefix,axiom,
! [Ps: list_nat,Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Ps @ Xs )
=> ( ( prefix_nat @ Ps @ Ys )
=> ( prefix_nat @ Ps @ ( longes266370323089874118ix_nat @ Xs @ Ys ) ) ) ) ).
% longest_common_prefix_max_prefix
thf(fact_94_longest__common__prefix__max__prefix,axiom,
! [Ps: list_o,Xs: list_o,Ys: list_o] :
( ( prefix_o @ Ps @ Xs )
=> ( ( prefix_o @ Ps @ Ys )
=> ( prefix_o @ Ps @ ( longes218265413398645026efix_o @ Xs @ Ys ) ) ) ) ).
% longest_common_prefix_max_prefix
thf(fact_95_longest__common__prefix__prefix2,axiom,
! [Xs: list_nat,Ys: list_nat] : ( prefix_nat @ ( longes266370323089874118ix_nat @ Xs @ Ys ) @ Ys ) ).
% longest_common_prefix_prefix2
thf(fact_96_longest__common__prefix__prefix2,axiom,
! [Xs: list_o,Ys: list_o] : ( prefix_o @ ( longes218265413398645026efix_o @ Xs @ Ys ) @ Ys ) ).
% longest_common_prefix_prefix2
thf(fact_97_mem__Collect__eq,axiom,
! [A: list_o,P: list_o > $o] :
( ( member_list_o2 @ A @ ( collect_list_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o2 @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A3: set_list_o] :
( ( collect_list_o
@ ^ [X3: list_o] : ( member_list_o2 @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A3: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o2 @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_103_longest__common__prefix__prefix1,axiom,
! [Xs: list_nat,Ys: list_nat] : ( prefix_nat @ ( longes266370323089874118ix_nat @ Xs @ Ys ) @ Xs ) ).
% longest_common_prefix_prefix1
thf(fact_104_longest__common__prefix__prefix1,axiom,
! [Xs: list_o,Ys: list_o] : ( prefix_o @ ( longes218265413398645026efix_o @ Xs @ Ys ) @ Xs ) ).
% longest_common_prefix_prefix1
thf(fact_105_append_Oright__neutral,axiom,
! [A: list_option_o] :
( ( append_option_o @ A @ nil_option_o )
= A ) ).
% append.right_neutral
thf(fact_106_append_Oright__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ A @ nil_list_o )
= A ) ).
% append.right_neutral
thf(fact_107_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_108_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_109_append_Oright__neutral,axiom,
! [A: list_o] :
( ( append_o @ A @ nil_o )
= A ) ).
% append.right_neutral
thf(fact_110_append__Nil2,axiom,
! [Xs: list_option_o] :
( ( append_option_o @ Xs @ nil_option_o )
= Xs ) ).
% append_Nil2
thf(fact_111_append__Nil2,axiom,
! [Xs: list_list_o] :
( ( append_list_o @ Xs @ nil_list_o )
= Xs ) ).
% append_Nil2
thf(fact_112_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_113_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_114_append__Nil2,axiom,
! [Xs: list_o] :
( ( append_o @ Xs @ nil_o )
= Xs ) ).
% append_Nil2
thf(fact_115_append__self__conv,axiom,
! [Xs: list_option_o,Ys: list_option_o] :
( ( ( append_option_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_option_o ) ) ).
% append_self_conv
thf(fact_116_append__self__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_o ) ) ).
% append_self_conv
thf(fact_117_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_118_append__self__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_nat ) ) ).
% append_self_conv
thf(fact_119_append__self__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_o ) ) ).
% append_self_conv
thf(fact_120_self__append__conv,axiom,
! [Y3: list_option_o,Ys: list_option_o] :
( ( Y3
= ( append_option_o @ Y3 @ Ys ) )
= ( Ys = nil_option_o ) ) ).
% self_append_conv
thf(fact_121_self__append__conv,axiom,
! [Y3: list_list_o,Ys: list_list_o] :
( ( Y3
= ( append_list_o @ Y3 @ Ys ) )
= ( Ys = nil_list_o ) ) ).
% self_append_conv
thf(fact_122_self__append__conv,axiom,
! [Y3: list_nat,Ys: list_nat] :
( ( Y3
= ( append_nat @ Y3 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_123_self__append__conv,axiom,
! [Y3: list_list_nat,Ys: list_list_nat] :
( ( Y3
= ( append_list_nat @ Y3 @ Ys ) )
= ( Ys = nil_list_nat ) ) ).
% self_append_conv
thf(fact_124_self__append__conv,axiom,
! [Y3: list_o,Ys: list_o] :
( ( Y3
= ( append_o @ Y3 @ Ys ) )
= ( Ys = nil_o ) ) ).
% self_append_conv
thf(fact_125_append__self__conv2,axiom,
! [Xs: list_option_o,Ys: list_option_o] :
( ( ( append_option_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_option_o ) ) ).
% append_self_conv2
thf(fact_126_append__self__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_o ) ) ).
% append_self_conv2
thf(fact_127_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_128_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_129_append__self__conv2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_o ) ) ).
% append_self_conv2
thf(fact_130_self__append__conv2,axiom,
! [Y3: list_option_o,Xs: list_option_o] :
( ( Y3
= ( append_option_o @ Xs @ Y3 ) )
= ( Xs = nil_option_o ) ) ).
% self_append_conv2
thf(fact_131_self__append__conv2,axiom,
! [Y3: list_list_o,Xs: list_list_o] :
( ( Y3
= ( append_list_o @ Xs @ Y3 ) )
= ( Xs = nil_list_o ) ) ).
% self_append_conv2
thf(fact_132_self__append__conv2,axiom,
! [Y3: list_nat,Xs: list_nat] :
( ( Y3
= ( append_nat @ Xs @ Y3 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_133_self__append__conv2,axiom,
! [Y3: list_list_nat,Xs: list_list_nat] :
( ( Y3
= ( append_list_nat @ Xs @ Y3 ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_134_self__append__conv2,axiom,
! [Y3: list_o,Xs: list_o] :
( ( Y3
= ( append_o @ Xs @ Y3 ) )
= ( Xs = nil_o ) ) ).
% self_append_conv2
thf(fact_135_Nil__is__append__conv,axiom,
! [Xs: list_option_o,Ys: list_option_o] :
( ( nil_option_o
= ( append_option_o @ Xs @ Ys ) )
= ( ( Xs = nil_option_o )
& ( Ys = nil_option_o ) ) ) ).
% Nil_is_append_conv
thf(fact_136_Nil__is__append__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( nil_list_o
= ( append_list_o @ Xs @ Ys ) )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% Nil_is_append_conv
thf(fact_137_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_138_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys ) )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_139_Nil__is__append__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( nil_o
= ( append_o @ Xs @ Ys ) )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% Nil_is_append_conv
thf(fact_140_append__is__Nil__conv,axiom,
! [Xs: list_option_o,Ys: list_option_o] :
( ( ( append_option_o @ Xs @ Ys )
= nil_option_o )
= ( ( Xs = nil_option_o )
& ( Ys = nil_option_o ) ) ) ).
% append_is_Nil_conv
thf(fact_141_append__is__Nil__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= nil_list_o )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% append_is_Nil_conv
thf(fact_142_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_143_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_144_append__is__Nil__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= nil_o )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% append_is_Nil_conv
thf(fact_145_prefix__code_I1_J,axiom,
! [Xs: list_option_o] : ( prefix_option_o @ nil_option_o @ Xs ) ).
% prefix_code(1)
thf(fact_146_prefix__code_I1_J,axiom,
! [Xs: list_list_o] : ( prefix_list_o @ nil_list_o @ Xs ) ).
% prefix_code(1)
thf(fact_147_prefix__code_I1_J,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% prefix_code(1)
thf(fact_148_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_149_prefix__code_I1_J,axiom,
! [Xs: list_o] : ( prefix_o @ nil_o @ Xs ) ).
% prefix_code(1)
thf(fact_150_prefix__bot_Oextremum__unique,axiom,
! [A: list_option_o] :
( ( prefix_option_o @ A @ nil_option_o )
= ( A = nil_option_o ) ) ).
% prefix_bot.extremum_unique
thf(fact_151_prefix__bot_Oextremum__unique,axiom,
! [A: list_list_o] :
( ( prefix_list_o @ A @ nil_list_o )
= ( A = nil_list_o ) ) ).
% prefix_bot.extremum_unique
thf(fact_152_prefix__bot_Oextremum__unique,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
= ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_153_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_154_prefix__bot_Oextremum__unique,axiom,
! [A: list_o] :
( ( prefix_o @ A @ nil_o )
= ( A = nil_o ) ) ).
% prefix_bot.extremum_unique
thf(fact_155_prefix__Nil,axiom,
! [Xs: list_option_o] :
( ( prefix_option_o @ Xs @ nil_option_o )
= ( Xs = nil_option_o ) ) ).
% prefix_Nil
thf(fact_156_prefix__Nil,axiom,
! [Xs: list_list_o] :
( ( prefix_list_o @ Xs @ nil_list_o )
= ( Xs = nil_list_o ) ) ).
% prefix_Nil
thf(fact_157_prefix__Nil,axiom,
! [Xs: list_list_nat] :
( ( prefix_list_nat @ Xs @ nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% prefix_Nil
thf(fact_158_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_159_prefix__Nil,axiom,
! [Xs: list_o] :
( ( prefix_o @ Xs @ nil_o )
= ( Xs = nil_o ) ) ).
% prefix_Nil
thf(fact_160_not__Some__eq,axiom,
! [X2: option_nat] :
( ( ! [Y4: nat] :
( X2
!= ( some_nat @ Y4 ) ) )
= ( X2 = none_nat ) ) ).
% not_Some_eq
thf(fact_161_not__Some__eq,axiom,
! [X2: option_list_o] :
( ( ! [Y4: list_o] :
( X2
!= ( some_list_o @ Y4 ) ) )
= ( X2 = none_list_o ) ) ).
% not_Some_eq
thf(fact_162_not__None__eq,axiom,
! [X2: option_nat] :
( ( X2 != none_nat )
= ( ? [Y4: nat] :
( X2
= ( some_nat @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_163_not__None__eq,axiom,
! [X2: option_list_o] :
( ( X2 != none_list_o )
= ( ? [Y4: list_o] :
( X2
= ( some_list_o @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_164_list__ex__simps_I2_J,axiom,
! [P: $o > $o] :
~ ( list_ex_o @ P @ nil_o ) ).
% list_ex_simps(2)
thf(fact_165_list__ex__simps_I2_J,axiom,
! [P: option_o > $o] :
~ ( list_ex_option_o @ P @ nil_option_o ) ).
% list_ex_simps(2)
thf(fact_166_list__ex__simps_I2_J,axiom,
! [P: list_o > $o] :
~ ( list_ex_list_o @ P @ nil_list_o ) ).
% list_ex_simps(2)
thf(fact_167_list__ex__simps_I2_J,axiom,
! [P: nat > $o] :
~ ( list_ex_nat @ P @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_168_list__ex__simps_I2_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex_list_nat @ P @ nil_list_nat ) ).
% list_ex_simps(2)
thf(fact_169_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu: list_option_o] :
( ( longes1709640641647806696tion_o @ Uu @ nil_option_o )
= nil_option_o ) ).
% longest_common_prefix.simps(3)
thf(fact_170_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu: list_list_o] :
( ( longes802630962832238888list_o @ Uu @ nil_list_o )
= nil_list_o ) ).
% longest_common_prefix.simps(3)
thf(fact_171_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu: list_nat] :
( ( longes266370323089874118ix_nat @ Uu @ nil_nat )
= nil_nat ) ).
% longest_common_prefix.simps(3)
thf(fact_172_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu: list_list_nat] :
( ( longes3715614165048771798st_nat @ Uu @ nil_list_nat )
= nil_list_nat ) ).
% longest_common_prefix.simps(3)
thf(fact_173_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu: list_o] :
( ( longes218265413398645026efix_o @ Uu @ nil_o )
= nil_o ) ).
% longest_common_prefix.simps(3)
thf(fact_174_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_option_o] :
( ( longes1709640641647806696tion_o @ nil_option_o @ Uv )
= nil_option_o ) ).
% longest_common_prefix.simps(2)
thf(fact_175_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_list_o] :
( ( longes802630962832238888list_o @ nil_list_o @ Uv )
= nil_list_o ) ).
% longest_common_prefix.simps(2)
thf(fact_176_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_nat] :
( ( longes266370323089874118ix_nat @ nil_nat @ Uv )
= nil_nat ) ).
% longest_common_prefix.simps(2)
thf(fact_177_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_list_nat] :
( ( longes3715614165048771798st_nat @ nil_list_nat @ Uv )
= nil_list_nat ) ).
% longest_common_prefix.simps(2)
thf(fact_178_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_o] :
( ( longes218265413398645026efix_o @ nil_o @ Uv )
= nil_o ) ).
% longest_common_prefix.simps(2)
thf(fact_179_opt__comp__def,axiom,
( prefix454693708527911765comp_o
= ( ^ [X3: option_list_o,Y4: option_list_o] :
( ( prefix8824957607401505554efix_o @ X3 @ Y4 )
| ( prefix8824957607401505554efix_o @ Y4 @ X3 ) ) ) ) ).
% opt_comp_def
thf(fact_180_opt__prefix_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
~ ( prefix8824957607401505554efix_o @ none_list_o @ Uv ) ).
% opt_prefix.simps(2)
thf(fact_181_opt__prefix_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
~ ( prefix8824957607401505554efix_o @ Uu @ none_list_o ) ).
% opt_prefix.simps(3)
thf(fact_182_opt__append_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
( ( prefix5314359684614007693append @ none_list_o @ Uv )
= none_list_o ) ).
% opt_append.simps(2)
thf(fact_183_opt__append_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
( ( prefix5314359684614007693append @ Uu @ none_list_o )
= none_list_o ) ).
% opt_append.simps(3)
thf(fact_184_combine__options__cases,axiom,
! [X2: option_list_o,P: option_list_o > option_nat > $o,Y3: option_nat] :
( ( ( X2 = none_list_o )
=> ( P @ X2 @ Y3 ) )
=> ( ( ( Y3 = none_nat )
=> ( P @ X2 @ Y3 ) )
=> ( ! [A4: list_o,B3: nat] :
( ( X2
= ( some_list_o @ A4 ) )
=> ( ( Y3
= ( some_nat @ B3 ) )
=> ( P @ X2 @ Y3 ) ) )
=> ( P @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_185_combine__options__cases,axiom,
! [X2: option_nat,P: option_nat > option_list_o > $o,Y3: option_list_o] :
( ( ( X2 = none_nat )
=> ( P @ X2 @ Y3 ) )
=> ( ( ( Y3 = none_list_o )
=> ( P @ X2 @ Y3 ) )
=> ( ! [A4: nat,B3: list_o] :
( ( X2
= ( some_nat @ A4 ) )
=> ( ( Y3
= ( some_list_o @ B3 ) )
=> ( P @ X2 @ Y3 ) ) )
=> ( P @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_186_combine__options__cases,axiom,
! [X2: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
( ( ( X2 = none_nat )
=> ( P @ X2 @ Y3 ) )
=> ( ( ( Y3 = none_nat )
=> ( P @ X2 @ Y3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( X2
= ( some_nat @ A4 ) )
=> ( ( Y3
= ( some_nat @ B3 ) )
=> ( P @ X2 @ Y3 ) ) )
=> ( P @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_187_combine__options__cases,axiom,
! [X2: option_list_o,P: option_list_o > option_list_o > $o,Y3: option_list_o] :
( ( ( X2 = none_list_o )
=> ( P @ X2 @ Y3 ) )
=> ( ( ( Y3 = none_list_o )
=> ( P @ X2 @ Y3 ) )
=> ( ! [A4: list_o,B3: list_o] :
( ( X2
= ( some_list_o @ A4 ) )
=> ( ( Y3
= ( some_list_o @ B3 ) )
=> ( P @ X2 @ Y3 ) ) )
=> ( P @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_188_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X5: option_nat] : ( P2 @ X5 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_189_split__option__all,axiom,
( ( ^ [P2: option_list_o > $o] :
! [X5: option_list_o] : ( P2 @ X5 ) )
= ( ^ [P3: option_list_o > $o] :
( ( P3 @ none_list_o )
& ! [X3: list_o] : ( P3 @ ( some_list_o @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_190_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X5: option_nat] : ( P2 @ X5 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_191_split__option__ex,axiom,
( ( ^ [P2: option_list_o > $o] :
? [X5: option_list_o] : ( P2 @ X5 ) )
= ( ^ [P3: option_list_o > $o] :
( ( P3 @ none_list_o )
| ? [X3: list_o] : ( P3 @ ( some_list_o @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_192_option_Oexhaust,axiom,
! [Y3: option_nat] :
( ( Y3 != none_nat )
=> ~ ! [X23: nat] :
( Y3
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_193_option_Oexhaust,axiom,
! [Y3: option_list_o] :
( ( Y3 != none_list_o )
=> ~ ! [X23: list_o] :
( Y3
!= ( some_list_o @ X23 ) ) ) ).
% option.exhaust
thf(fact_194_option_OdiscI,axiom,
! [Option: option_nat,X22: nat] :
( ( Option
= ( some_nat @ X22 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_195_option_OdiscI,axiom,
! [Option: option_list_o,X22: list_o] :
( ( Option
= ( some_list_o @ X22 ) )
=> ( Option != none_list_o ) ) ).
% option.discI
thf(fact_196_option_Odistinct_I1_J,axiom,
! [X22: nat] :
( none_nat
!= ( some_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_197_option_Odistinct_I1_J,axiom,
! [X22: list_o] :
( none_list_o
!= ( some_list_o @ X22 ) ) ).
% option.distinct(1)
thf(fact_198_append__Nil,axiom,
! [Ys: list_option_o] :
( ( append_option_o @ nil_option_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_199_append__Nil,axiom,
! [Ys: list_list_o] :
( ( append_list_o @ nil_list_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_200_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_201_append__Nil,axiom,
! [Ys: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_202_append__Nil,axiom,
! [Ys: list_o] :
( ( append_o @ nil_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_203_append_Oleft__neutral,axiom,
! [A: list_option_o] :
( ( append_option_o @ nil_option_o @ A )
= A ) ).
% append.left_neutral
thf(fact_204_append_Oleft__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ nil_list_o @ A )
= A ) ).
% append.left_neutral
thf(fact_205_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_206_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_207_append_Oleft__neutral,axiom,
! [A: list_o] :
( ( append_o @ nil_o @ A )
= A ) ).
% append.left_neutral
thf(fact_208_eq__Nil__appendI,axiom,
! [Xs: list_option_o,Ys: list_option_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_option_o @ nil_option_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_209_eq__Nil__appendI,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_o @ nil_list_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_210_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_211_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_212_eq__Nil__appendI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_o @ nil_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_213_prefix__bot_Obot__least,axiom,
! [A: list_option_o] : ( prefix_option_o @ nil_option_o @ A ) ).
% prefix_bot.bot_least
thf(fact_214_prefix__bot_Obot__least,axiom,
! [A: list_list_o] : ( prefix_list_o @ nil_list_o @ A ) ).
% prefix_bot.bot_least
thf(fact_215_prefix__bot_Obot__least,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_216_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_217_prefix__bot_Obot__least,axiom,
! [A: list_o] : ( prefix_o @ nil_o @ A ) ).
% prefix_bot.bot_least
thf(fact_218_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_option_o] :
( ( prefix_option_o @ A @ nil_option_o )
=> ( A = nil_option_o ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_219_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_list_o] :
( ( prefix_list_o @ A @ nil_list_o )
=> ( A = nil_list_o ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_220_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
=> ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_221_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_222_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_o] :
( ( prefix_o @ A @ nil_o )
=> ( A = nil_o ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_223_Nil__prefix,axiom,
! [Xs: list_option_o] : ( prefix_option_o @ nil_option_o @ Xs ) ).
% Nil_prefix
thf(fact_224_Nil__prefix,axiom,
! [Xs: list_list_o] : ( prefix_list_o @ nil_list_o @ Xs ) ).
% Nil_prefix
thf(fact_225_Nil__prefix,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% Nil_prefix
thf(fact_226_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_227_Nil__prefix,axiom,
! [Xs: list_o] : ( prefix_o @ nil_o @ Xs ) ).
% Nil_prefix
thf(fact_228_opt__prefix_Oelims_I3_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat] :
( ~ ( prefix9059114240934526678ix_nat @ X2 @ Xa )
=> ( ! [X4: list_nat] :
( ( X2
= ( some_list_nat @ X4 ) )
=> ! [Y5: list_nat] :
( ( Xa
= ( some_list_nat @ Y5 ) )
=> ( prefix_nat @ X4 @ Y5 ) ) )
=> ( ( X2 != none_list_nat )
=> ( Xa = none_list_nat ) ) ) ) ).
% opt_prefix.elims(3)
thf(fact_229_opt__prefix_Oelims_I3_J,axiom,
! [X2: option_list_o,Xa: option_list_o] :
( ~ ( prefix8824957607401505554efix_o @ X2 @ Xa )
=> ( ! [X4: list_o] :
( ( X2
= ( some_list_o @ X4 ) )
=> ! [Y5: list_o] :
( ( Xa
= ( some_list_o @ Y5 ) )
=> ( prefix_o @ X4 @ Y5 ) ) )
=> ( ( X2 != none_list_o )
=> ( Xa = none_list_o ) ) ) ) ).
% opt_prefix.elims(3)
thf(fact_230_opt__prefix_Oelims_I1_J,axiom,
! [X2: option_list_nat,Xa: option_list_nat,Y3: $o] :
( ( ( prefix9059114240934526678ix_nat @ X2 @ Xa )
= Y3 )
=> ( ! [X4: list_nat] :
( ( X2
= ( some_list_nat @ X4 ) )
=> ! [Y5: list_nat] :
( ( Xa
= ( some_list_nat @ Y5 ) )
=> ( Y3
= ( ~ ( prefix_nat @ X4 @ Y5 ) ) ) ) )
=> ( ( ( X2 = none_list_nat )
=> Y3 )
=> ~ ( ( Xa = none_list_nat )
=> Y3 ) ) ) ) ).
% opt_prefix.elims(1)
thf(fact_231_opt__prefix_Oelims_I1_J,axiom,
! [X2: option_list_o,Xa: option_list_o,Y3: $o] :
( ( ( prefix8824957607401505554efix_o @ X2 @ Xa )
= Y3 )
=> ( ! [X4: list_o] :
( ( X2
= ( some_list_o @ X4 ) )
=> ! [Y5: list_o] :
( ( Xa
= ( some_list_o @ Y5 ) )
=> ( Y3
= ( ~ ( prefix_o @ X4 @ Y5 ) ) ) ) )
=> ( ( ( X2 = none_list_o )
=> Y3 )
=> ~ ( ( Xa = none_list_o )
=> Y3 ) ) ) ) ).
% opt_prefix.elims(1)
thf(fact_232_those_Osimps_I1_J,axiom,
( ( those_option_o @ nil_option_option_o )
= ( some_list_option_o @ nil_option_o ) ) ).
% those.simps(1)
thf(fact_233_those_Osimps_I1_J,axiom,
( ( those_list_o @ nil_option_list_o )
= ( some_list_list_o @ nil_list_o ) ) ).
% those.simps(1)
thf(fact_234_those_Osimps_I1_J,axiom,
( ( those_nat @ nil_option_nat )
= ( some_list_nat @ nil_nat ) ) ).
% those.simps(1)
thf(fact_235_those_Osimps_I1_J,axiom,
( ( those_list_nat @ nil_option_list_nat )
= ( some_list_list_nat @ nil_list_nat ) ) ).
% those.simps(1)
thf(fact_236_those_Osimps_I1_J,axiom,
( ( those_o @ nil_option_o )
= ( some_list_o @ nil_o ) ) ).
% those.simps(1)
thf(fact_237_prefix__bot_Omax__bot,axiom,
! [X2: list_option_o] :
( ( max_list_option_o @ prefix_option_o @ nil_option_o @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_238_prefix__bot_Omax__bot,axiom,
! [X2: list_list_o] :
( ( max_list_list_o @ prefix_list_o @ nil_list_o @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_239_prefix__bot_Omax__bot,axiom,
! [X2: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ nil_list_nat @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_240_prefix__bot_Omax__bot,axiom,
! [X2: list_nat] :
( ( max_list_nat @ prefix_nat @ nil_nat @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_241_prefix__bot_Omax__bot,axiom,
! [X2: list_o] :
( ( max_list_o @ prefix_o @ nil_o @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_242_prefix__bot_Omin__bot,axiom,
! [X2: list_option_o] :
( ( min_list_option_o @ prefix_option_o @ nil_option_o @ X2 )
= nil_option_o ) ).
% prefix_bot.min_bot
thf(fact_243_prefix__bot_Omin__bot,axiom,
! [X2: list_list_o] :
( ( min_list_list_o @ prefix_list_o @ nil_list_o @ X2 )
= nil_list_o ) ).
% prefix_bot.min_bot
thf(fact_244_prefix__bot_Omin__bot,axiom,
! [X2: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ nil_list_nat @ X2 )
= nil_list_nat ) ).
% prefix_bot.min_bot
thf(fact_245_prefix__bot_Omin__bot,axiom,
! [X2: list_nat] :
( ( min_list_nat @ prefix_nat @ nil_nat @ X2 )
= nil_nat ) ).
% prefix_bot.min_bot
thf(fact_246_prefix__bot_Omin__bot,axiom,
! [X2: list_o] :
( ( min_list_o @ prefix_o @ nil_o @ X2 )
= nil_o ) ).
% prefix_bot.min_bot
thf(fact_247_prefix__bot_Omax__bot2,axiom,
! [X2: list_option_o] :
( ( max_list_option_o @ prefix_option_o @ X2 @ nil_option_o )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_248_prefix__bot_Omax__bot2,axiom,
! [X2: list_list_o] :
( ( max_list_list_o @ prefix_list_o @ X2 @ nil_list_o )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_249_prefix__bot_Omax__bot2,axiom,
! [X2: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ X2 @ nil_list_nat )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_250_prefix__bot_Omax__bot2,axiom,
! [X2: list_nat] :
( ( max_list_nat @ prefix_nat @ X2 @ nil_nat )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_251_prefix__bot_Omax__bot2,axiom,
! [X2: list_o] :
( ( max_list_o @ prefix_o @ X2 @ nil_o )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_252_prefix__bot_Omin__bot2,axiom,
! [X2: list_option_o] :
( ( min_list_option_o @ prefix_option_o @ X2 @ nil_option_o )
= nil_option_o ) ).
% prefix_bot.min_bot2
thf(fact_253_prefix__bot_Omin__bot2,axiom,
! [X2: list_list_o] :
( ( min_list_list_o @ prefix_list_o @ X2 @ nil_list_o )
= nil_list_o ) ).
% prefix_bot.min_bot2
thf(fact_254_prefix__bot_Omin__bot2,axiom,
! [X2: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ X2 @ nil_list_nat )
= nil_list_nat ) ).
% prefix_bot.min_bot2
thf(fact_255_prefix__bot_Omin__bot2,axiom,
! [X2: list_nat] :
( ( min_list_nat @ prefix_nat @ X2 @ nil_nat )
= nil_nat ) ).
% prefix_bot.min_bot2
thf(fact_256_prefix__bot_Omin__bot2,axiom,
! [X2: list_o] :
( ( min_list_o @ prefix_o @ X2 @ nil_o )
= nil_o ) ).
% prefix_bot.min_bot2
thf(fact_257_list__ex1__simps_I1_J,axiom,
! [P: $o > $o] :
~ ( list_ex1_o @ P @ nil_o ) ).
% list_ex1_simps(1)
thf(fact_258_list__ex1__simps_I1_J,axiom,
! [P: option_o > $o] :
~ ( list_ex1_option_o @ P @ nil_option_o ) ).
% list_ex1_simps(1)
thf(fact_259_list__ex1__simps_I1_J,axiom,
! [P: list_o > $o] :
~ ( list_ex1_list_o @ P @ nil_list_o ) ).
% list_ex1_simps(1)
thf(fact_260_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_261_list__ex1__simps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_262_prefix__snoc,axiom,
! [Xs: list_option_o,Ys: list_option_o,Y3: option_o] :
( ( prefix_option_o @ Xs @ ( append_option_o @ Ys @ ( cons_option_o @ Y3 @ nil_option_o ) ) )
= ( ( Xs
= ( append_option_o @ Ys @ ( cons_option_o @ Y3 @ nil_option_o ) ) )
| ( prefix_option_o @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_263_prefix__snoc,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Y3: list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) )
= ( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) )
| ( prefix_list_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_264_prefix__snoc,axiom,
! [Xs: list_list_o,Ys: list_list_o,Y3: list_o] :
( ( prefix_list_o @ Xs @ ( append_list_o @ Ys @ ( cons_list_o @ Y3 @ nil_list_o ) ) )
= ( ( Xs
= ( append_list_o @ Ys @ ( cons_list_o @ Y3 @ nil_list_o ) ) )
| ( prefix_list_o @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_265_prefix__snoc,axiom,
! [Xs: list_nat,Ys: list_nat,Y3: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_266_prefix__snoc,axiom,
! [Xs: list_o,Ys: list_o,Y3: $o] :
( ( prefix_o @ Xs @ ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
= ( ( Xs
= ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
| ( prefix_o @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_267_bind__simps_I1_J,axiom,
! [F: $o > list_o] :
( ( bind_o_o @ nil_o @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_268_bind__simps_I1_J,axiom,
! [F: $o > list_nat] :
( ( bind_o_nat @ nil_o @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_269_bind__simps_I1_J,axiom,
! [F: nat > list_o] :
( ( bind_nat_o @ nil_nat @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_270_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_271_bind__simps_I1_J,axiom,
! [F: $o > list_option_o] :
( ( bind_o_option_o @ nil_o @ F )
= nil_option_o ) ).
% bind_simps(1)
thf(fact_272_bind__simps_I1_J,axiom,
! [F: $o > list_list_o] :
( ( bind_o_list_o @ nil_o @ F )
= nil_list_o ) ).
% bind_simps(1)
thf(fact_273_bind__simps_I1_J,axiom,
! [F: $o > list_list_nat] :
( ( bind_o_list_nat @ nil_o @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_274_bind__simps_I1_J,axiom,
! [F: option_o > list_o] :
( ( bind_option_o_o @ nil_option_o @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_275_bind__simps_I1_J,axiom,
! [F: option_o > list_nat] :
( ( bind_option_o_nat @ nil_option_o @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_276_bind__simps_I1_J,axiom,
! [F: list_o > list_o] :
( ( bind_list_o_o @ nil_list_o @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_277_find_Osimps_I1_J,axiom,
! [Uu: $o > $o] :
( ( find_o @ Uu @ nil_o )
= none_o ) ).
% find.simps(1)
thf(fact_278_find_Osimps_I1_J,axiom,
! [Uu: option_o > $o] :
( ( find_option_o @ Uu @ nil_option_o )
= none_option_o ) ).
% find.simps(1)
thf(fact_279_find_Osimps_I1_J,axiom,
! [Uu: nat > $o] :
( ( find_nat @ Uu @ nil_nat )
= none_nat ) ).
% find.simps(1)
thf(fact_280_find_Osimps_I1_J,axiom,
! [Uu: list_nat > $o] :
( ( find_list_nat @ Uu @ nil_list_nat )
= none_list_nat ) ).
% find.simps(1)
thf(fact_281_find_Osimps_I1_J,axiom,
! [Uu: list_o > $o] :
( ( find_list_o @ Uu @ nil_list_o )
= none_list_o ) ).
% find.simps(1)
thf(fact_282_member__rec_I2_J,axiom,
! [Y3: $o] :
~ ( member_o @ nil_o @ Y3 ) ).
% member_rec(2)
thf(fact_283_member__rec_I2_J,axiom,
! [Y3: option_o] :
~ ( member_option_o @ nil_option_o @ Y3 ) ).
% member_rec(2)
thf(fact_284_member__rec_I2_J,axiom,
! [Y3: list_o] :
~ ( member_list_o @ nil_list_o @ Y3 ) ).
% member_rec(2)
thf(fact_285_member__rec_I2_J,axiom,
! [Y3: nat] :
~ ( member_nat @ nil_nat @ Y3 ) ).
% member_rec(2)
thf(fact_286_member__rec_I2_J,axiom,
! [Y3: list_nat] :
~ ( member_list_nat @ nil_list_nat @ Y3 ) ).
% member_rec(2)
thf(fact_287_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( ( some_nat @ ( the_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_288_option_Ocollapse,axiom,
! [Option: option_list_o] :
( ( Option != none_list_o )
=> ( ( some_list_o @ ( the_list_o @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_289_list_Oinject,axiom,
! [X21: $o,X222: list_o,Y21: $o,Y222: list_o] :
( ( ( cons_o @ X21 @ X222 )
= ( cons_o @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_290_list_Oinject,axiom,
! [X21: list_o,X222: list_list_o,Y21: list_o,Y222: list_list_o] :
( ( ( cons_list_o @ X21 @ X222 )
= ( cons_list_o @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_291_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_292_Cons__prefix__Cons,axiom,
! [X2: list_o,Xs: list_list_o,Y3: list_o,Ys: list_list_o] :
( ( prefix_list_o @ ( cons_list_o @ X2 @ Xs ) @ ( cons_list_o @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
& ( prefix_list_o @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_293_Cons__prefix__Cons,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
& ( prefix_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_294_Cons__prefix__Cons,axiom,
! [X2: $o,Xs: list_o,Y3: $o,Ys: list_o] :
( ( prefix_o @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
& ( prefix_o @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_295_list__ex__simps_I1_J,axiom,
! [P: $o > $o,X2: $o,Xs: list_o] :
( ( list_ex_o @ P @ ( cons_o @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_o @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_296_list__ex__simps_I1_J,axiom,
! [P: list_o > $o,X2: list_o,Xs: list_list_o] :
( ( list_ex_list_o @ P @ ( cons_list_o @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_list_o @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_297_list__ex__simps_I1_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_ex_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_nat @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_298_append1__eq__conv,axiom,
! [Xs: list_option_o,X2: option_o,Ys: list_option_o,Y3: option_o] :
( ( ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) )
= ( append_option_o @ Ys @ ( cons_option_o @ Y3 @ nil_option_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_299_append1__eq__conv,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys: list_list_nat,Y3: list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) )
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_300_append1__eq__conv,axiom,
! [Xs: list_list_o,X2: list_o,Ys: list_list_o,Y3: list_o] :
( ( ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) )
= ( append_list_o @ Ys @ ( cons_list_o @ Y3 @ nil_list_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_301_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_302_append1__eq__conv,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y3: $o] :
( ( ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) )
= ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_303_bind__simps_I2_J,axiom,
! [X2: $o,Xs: list_o,F: $o > list_o] :
( ( bind_o_o @ ( cons_o @ X2 @ Xs ) @ F )
= ( append_o @ ( F @ X2 ) @ ( bind_o_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_304_bind__simps_I2_J,axiom,
! [X2: $o,Xs: list_o,F: $o > list_list_o] :
( ( bind_o_list_o @ ( cons_o @ X2 @ Xs ) @ F )
= ( append_list_o @ ( F @ X2 ) @ ( bind_o_list_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_305_bind__simps_I2_J,axiom,
! [X2: $o,Xs: list_o,F: $o > list_nat] :
( ( bind_o_nat @ ( cons_o @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_o_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_306_bind__simps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o,F: list_o > list_o] :
( ( bind_list_o_o @ ( cons_list_o @ X2 @ Xs ) @ F )
= ( append_o @ ( F @ X2 ) @ ( bind_list_o_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_307_bind__simps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o,F: list_o > list_list_o] :
( ( bind_list_o_list_o @ ( cons_list_o @ X2 @ Xs ) @ F )
= ( append_list_o @ ( F @ X2 ) @ ( bind_list_o_list_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_308_bind__simps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o,F: list_o > list_nat] :
( ( bind_list_o_nat @ ( cons_list_o @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_list_o_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_309_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_o] :
( ( bind_nat_o @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_o @ ( F @ X2 ) @ ( bind_nat_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_310_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_list_o] :
( ( bind_nat_list_o @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_list_o @ ( F @ X2 ) @ ( bind_nat_list_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_311_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_312_not__Cons__self2,axiom,
! [X2: $o,Xs: list_o] :
( ( cons_o @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_313_not__Cons__self2,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( cons_list_o @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_314_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_315_member__rec_I1_J,axiom,
! [X2: $o,Xs: list_o,Y3: $o] :
( ( member_o @ ( cons_o @ X2 @ Xs ) @ Y3 )
= ( ( X2 = Y3 )
| ( member_o @ Xs @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_316_member__rec_I1_J,axiom,
! [X2: list_o,Xs: list_list_o,Y3: list_o] :
( ( member_list_o @ ( cons_list_o @ X2 @ Xs ) @ Y3 )
= ( ( X2 = Y3 )
| ( member_list_o @ Xs @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_317_member__rec_I1_J,axiom,
! [X2: nat,Xs: list_nat,Y3: nat] :
( ( member_nat @ ( cons_nat @ X2 @ Xs ) @ Y3 )
= ( ( X2 = Y3 )
| ( member_nat @ Xs @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_318_transpose_Ocases,axiom,
! [X2: list_list_option_o] :
( ( X2 != nil_list_option_o )
=> ( ! [Xss: list_list_option_o] :
( X2
!= ( cons_list_option_o @ nil_option_o @ Xss ) )
=> ~ ! [X4: option_o,Xs3: list_option_o,Xss: list_list_option_o] :
( X2
!= ( cons_list_option_o @ ( cons_option_o @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_319_transpose_Ocases,axiom,
! [X2: list_list_list_nat] :
( ( X2 != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X4: list_nat,Xs3: list_list_nat,Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ ( cons_list_nat @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_320_transpose_Ocases,axiom,
! [X2: list_list_list_o] :
( ( X2 != nil_list_list_o )
=> ( ! [Xss: list_list_list_o] :
( X2
!= ( cons_list_list_o @ nil_list_o @ Xss ) )
=> ~ ! [X4: list_o,Xs3: list_list_o,Xss: list_list_list_o] :
( X2
!= ( cons_list_list_o @ ( cons_list_o @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_321_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X4: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_322_transpose_Ocases,axiom,
! [X2: list_list_o] :
( ( X2 != nil_list_o )
=> ( ! [Xss: list_list_o] :
( X2
!= ( cons_list_o @ nil_o @ Xss ) )
=> ~ ! [X4: $o,Xs3: list_o,Xss: list_list_o] :
( X2
!= ( cons_list_o @ ( cons_o @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_323_find_Osimps_I2_J,axiom,
! [P: $o > $o,X2: $o,Xs: list_o] :
( ( ( P @ X2 )
=> ( ( find_o @ P @ ( cons_o @ X2 @ Xs ) )
= ( some_o @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( find_o @ P @ ( cons_o @ X2 @ Xs ) )
= ( find_o @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_324_find_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( some_nat @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( find_nat @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_325_find_Osimps_I2_J,axiom,
! [P: list_o > $o,X2: list_o,Xs: list_list_o] :
( ( ( P @ X2 )
=> ( ( find_list_o @ P @ ( cons_list_o @ X2 @ Xs ) )
= ( some_list_o @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( find_list_o @ P @ ( cons_list_o @ X2 @ Xs ) )
= ( find_list_o @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_326_list__nonempty__induct,axiom,
! [Xs: list_option_o,P: list_option_o > $o] :
( ( Xs != nil_option_o )
=> ( ! [X4: option_o] : ( P @ ( cons_option_o @ X4 @ nil_option_o ) )
=> ( ! [X4: option_o,Xs3: list_option_o] :
( ( Xs3 != nil_option_o )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_option_o @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_327_list__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X4: list_nat] : ( P @ ( cons_list_nat @ X4 @ nil_list_nat ) )
=> ( ! [X4: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_nat @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_328_list__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X4: $o] : ( P @ ( cons_o @ X4 @ nil_o ) )
=> ( ! [X4: $o,Xs3: list_o] :
( ( Xs3 != nil_o )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_o @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_329_list__nonempty__induct,axiom,
! [Xs: list_list_o,P: list_list_o > $o] :
( ( Xs != nil_list_o )
=> ( ! [X4: list_o] : ( P @ ( cons_list_o @ X4 @ nil_list_o ) )
=> ( ! [X4: list_o,Xs3: list_list_o] :
( ( Xs3 != nil_list_o )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_o @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_330_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_331_list__induct2_H,axiom,
! [P: list_o > list_o > $o,Xs: list_o,Ys: list_o] :
( ( P @ nil_o @ nil_o )
=> ( ! [X4: $o,Xs3: list_o] : ( P @ ( cons_o @ X4 @ Xs3 ) @ nil_o )
=> ( ! [Y5: $o,Ys3: list_o] : ( P @ nil_o @ ( cons_o @ Y5 @ Ys3 ) )
=> ( ! [X4: $o,Xs3: list_o,Y5: $o,Ys3: list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X4 @ Xs3 ) @ ( cons_o @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_332_list__induct2_H,axiom,
! [P: list_o > list_nat > $o,Xs: list_o,Ys: list_nat] :
( ( P @ nil_o @ nil_nat )
=> ( ! [X4: $o,Xs3: list_o] : ( P @ ( cons_o @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys3: list_nat] : ( P @ nil_o @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( ! [X4: $o,Xs3: list_o,Y5: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_333_list__induct2_H,axiom,
! [P: list_nat > list_o > $o,Xs: list_nat,Ys: list_o] :
( ( P @ nil_nat @ nil_o )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_o )
=> ( ! [Y5: $o,Ys3: list_o] : ( P @ nil_nat @ ( cons_o @ Y5 @ Ys3 ) )
=> ( ! [X4: nat,Xs3: list_nat,Y5: $o,Ys3: list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_o @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_334_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( ! [X4: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_335_list__induct2_H,axiom,
! [P: list_option_o > list_o > $o,Xs: list_option_o,Ys: list_o] :
( ( P @ nil_option_o @ nil_o )
=> ( ! [X4: option_o,Xs3: list_option_o] : ( P @ ( cons_option_o @ X4 @ Xs3 ) @ nil_o )
=> ( ! [Y5: $o,Ys3: list_o] : ( P @ nil_option_o @ ( cons_o @ Y5 @ Ys3 ) )
=> ( ! [X4: option_o,Xs3: list_option_o,Y5: $o,Ys3: list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_option_o @ X4 @ Xs3 ) @ ( cons_o @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_336_list__induct2_H,axiom,
! [P: list_list_nat > list_o > $o,Xs: list_list_nat,Ys: list_o] :
( ( P @ nil_list_nat @ nil_o )
=> ( ! [X4: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X4 @ Xs3 ) @ nil_o )
=> ( ! [Y5: $o,Ys3: list_o] : ( P @ nil_list_nat @ ( cons_o @ Y5 @ Ys3 ) )
=> ( ! [X4: list_nat,Xs3: list_list_nat,Y5: $o,Ys3: list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_nat @ X4 @ Xs3 ) @ ( cons_o @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_337_list__induct2_H,axiom,
! [P: list_option_o > list_nat > $o,Xs: list_option_o,Ys: list_nat] :
( ( P @ nil_option_o @ nil_nat )
=> ( ! [X4: option_o,Xs3: list_option_o] : ( P @ ( cons_option_o @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys3: list_nat] : ( P @ nil_option_o @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( ! [X4: option_o,Xs3: list_option_o,Y5: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_option_o @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_338_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs: list_list_nat,Ys: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X4: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys3: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( ! [X4: list_nat,Xs3: list_list_nat,Y5: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_339_list__induct2_H,axiom,
! [P: list_o > list_option_o > $o,Xs: list_o,Ys: list_option_o] :
( ( P @ nil_o @ nil_option_o )
=> ( ! [X4: $o,Xs3: list_o] : ( P @ ( cons_o @ X4 @ Xs3 ) @ nil_option_o )
=> ( ! [Y5: option_o,Ys3: list_option_o] : ( P @ nil_o @ ( cons_option_o @ Y5 @ Ys3 ) )
=> ( ! [X4: $o,Xs3: list_o,Y5: option_o,Ys3: list_option_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X4 @ Xs3 ) @ ( cons_option_o @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_340_list__induct2_H,axiom,
! [P: list_o > list_list_nat > $o,Xs: list_o,Ys: list_list_nat] :
( ( P @ nil_o @ nil_list_nat )
=> ( ! [X4: $o,Xs3: list_o] : ( P @ ( cons_o @ X4 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y5: list_nat,Ys3: list_list_nat] : ( P @ nil_o @ ( cons_list_nat @ Y5 @ Ys3 ) )
=> ( ! [X4: $o,Xs3: list_o,Y5: list_nat,Ys3: list_list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X4 @ Xs3 ) @ ( cons_list_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_341_neq__Nil__conv,axiom,
! [Xs: list_option_o] :
( ( Xs != nil_option_o )
= ( ? [Y4: option_o,Ys2: list_option_o] :
( Xs
= ( cons_option_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_342_neq__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
= ( ? [Y4: list_nat,Ys2: list_list_nat] :
( Xs
= ( cons_list_nat @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_343_neq__Nil__conv,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
= ( ? [Y4: $o,Ys2: list_o] :
( Xs
= ( cons_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_344_neq__Nil__conv,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
= ( ? [Y4: list_o,Ys2: list_list_o] :
( Xs
= ( cons_list_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_345_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_346_remdups__adj_Ocases,axiom,
! [X2: list_option_o] :
( ( X2 != nil_option_o )
=> ( ! [X4: option_o] :
( X2
!= ( cons_option_o @ X4 @ nil_option_o ) )
=> ~ ! [X4: option_o,Y5: option_o,Xs3: list_option_o] :
( X2
!= ( cons_option_o @ X4 @ ( cons_option_o @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_347_remdups__adj_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [X4: list_nat] :
( X2
!= ( cons_list_nat @ X4 @ nil_list_nat ) )
=> ~ ! [X4: list_nat,Y5: list_nat,Xs3: list_list_nat] :
( X2
!= ( cons_list_nat @ X4 @ ( cons_list_nat @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_348_remdups__adj_Ocases,axiom,
! [X2: list_o] :
( ( X2 != nil_o )
=> ( ! [X4: $o] :
( X2
!= ( cons_o @ X4 @ nil_o ) )
=> ~ ! [X4: $o,Y5: $o,Xs3: list_o] :
( X2
!= ( cons_o @ X4 @ ( cons_o @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_349_remdups__adj_Ocases,axiom,
! [X2: list_list_o] :
( ( X2 != nil_list_o )
=> ( ! [X4: list_o] :
( X2
!= ( cons_list_o @ X4 @ nil_list_o ) )
=> ~ ! [X4: list_o,Y5: list_o,Xs3: list_list_o] :
( X2
!= ( cons_list_o @ X4 @ ( cons_list_o @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_350_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X4: nat] :
( X2
!= ( cons_nat @ X4 @ nil_nat ) )
=> ~ ! [X4: nat,Y5: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X4 @ ( cons_nat @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_351_min__list_Ocases,axiom,
! [X2: list_list_nat] :
( ! [X4: list_nat,Xs3: list_list_nat] :
( X2
!= ( cons_list_nat @ X4 @ Xs3 ) )
=> ( X2 = nil_list_nat ) ) ).
% min_list.cases
thf(fact_352_min__list_Ocases,axiom,
! [X2: list_o] :
( ! [X4: $o,Xs3: list_o] :
( X2
!= ( cons_o @ X4 @ Xs3 ) )
=> ( X2 = nil_o ) ) ).
% min_list.cases
thf(fact_353_min__list_Ocases,axiom,
! [X2: list_list_o] :
( ! [X4: list_o,Xs3: list_list_o] :
( X2
!= ( cons_list_o @ X4 @ Xs3 ) )
=> ( X2 = nil_list_o ) ) ).
% min_list.cases
thf(fact_354_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X4: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X4 @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_355_list_Oexhaust,axiom,
! [Y3: list_option_o] :
( ( Y3 != nil_option_o )
=> ~ ! [X212: option_o,X223: list_option_o] :
( Y3
!= ( cons_option_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_356_list_Oexhaust,axiom,
! [Y3: list_list_nat] :
( ( Y3 != nil_list_nat )
=> ~ ! [X212: list_nat,X223: list_list_nat] :
( Y3
!= ( cons_list_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_357_list_Oexhaust,axiom,
! [Y3: list_o] :
( ( Y3 != nil_o )
=> ~ ! [X212: $o,X223: list_o] :
( Y3
!= ( cons_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_358_list_Oexhaust,axiom,
! [Y3: list_list_o] :
( ( Y3 != nil_list_o )
=> ~ ! [X212: list_o,X223: list_list_o] :
( Y3
!= ( cons_list_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_359_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y3
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_360_list_OdiscI,axiom,
! [List: list_option_o,X21: option_o,X222: list_option_o] :
( ( List
= ( cons_option_o @ X21 @ X222 ) )
=> ( List != nil_option_o ) ) ).
% list.discI
thf(fact_361_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X222: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X222 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_362_list_OdiscI,axiom,
! [List: list_o,X21: $o,X222: list_o] :
( ( List
= ( cons_o @ X21 @ X222 ) )
=> ( List != nil_o ) ) ).
% list.discI
thf(fact_363_list_OdiscI,axiom,
! [List: list_list_o,X21: list_o,X222: list_list_o] :
( ( List
= ( cons_list_o @ X21 @ X222 ) )
=> ( List != nil_list_o ) ) ).
% list.discI
thf(fact_364_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_365_list_Odistinct_I1_J,axiom,
! [X21: option_o,X222: list_option_o] :
( nil_option_o
!= ( cons_option_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_366_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_367_list_Odistinct_I1_J,axiom,
! [X21: $o,X222: list_o] :
( nil_o
!= ( cons_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_368_list_Odistinct_I1_J,axiom,
! [X21: list_o,X222: list_list_o] :
( nil_list_o
!= ( cons_list_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_369_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_370_Cons__eq__appendI,axiom,
! [X2: list_o,Xs1: list_list_o,Ys: list_list_o,Xs: list_list_o,Zs: list_list_o] :
( ( ( cons_list_o @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_o @ Xs1 @ Zs ) )
=> ( ( cons_list_o @ X2 @ Xs )
= ( append_list_o @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_371_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_372_Cons__eq__appendI,axiom,
! [X2: $o,Xs1: list_o,Ys: list_o,Xs: list_o,Zs: list_o] :
( ( ( cons_o @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_o @ Xs1 @ Zs ) )
=> ( ( cons_o @ X2 @ Xs )
= ( append_o @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_373_append__Cons,axiom,
! [X2: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( append_list_o @ ( cons_list_o @ X2 @ Xs ) @ Ys )
= ( cons_list_o @ X2 @ ( append_list_o @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_374_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_375_append__Cons,axiom,
! [X2: $o,Xs: list_o,Ys: list_o] :
( ( append_o @ ( cons_o @ X2 @ Xs ) @ Ys )
= ( cons_o @ X2 @ ( append_o @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_376_option_Osel,axiom,
! [X22: nat] :
( ( the_nat @ ( some_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_377_option_Osel,axiom,
! [X22: list_o] :
( ( the_list_o @ ( some_list_o @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_378_option_Oexpand,axiom,
! [Option: option_list_o,Option2: option_list_o] :
( ( ( Option = none_list_o )
= ( Option2 = none_list_o ) )
=> ( ( ( Option != none_list_o )
=> ( ( Option2 != none_list_o )
=> ( ( the_list_o @ Option )
= ( the_list_o @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_379_prefix__order_Omin__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= A ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= B ) ) ) ).
% prefix_order.min_def
thf(fact_380_prefix__order_Omin__def,axiom,
! [A: list_o,B: list_o] :
( ( ( prefix_o @ A @ B )
=> ( ( min_list_o @ prefix_o @ A @ B )
= A ) )
& ( ~ ( prefix_o @ A @ B )
=> ( ( min_list_o @ prefix_o @ A @ B )
= B ) ) ) ).
% prefix_order.min_def
thf(fact_381_prefix__order_Omax__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= B ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= A ) ) ) ).
% prefix_order.max_def
thf(fact_382_prefix__order_Omax__def,axiom,
! [A: list_o,B: list_o] :
( ( ( prefix_o @ A @ B )
=> ( ( max_list_o @ prefix_o @ A @ B )
= B ) )
& ( ~ ( prefix_o @ A @ B )
=> ( ( max_list_o @ prefix_o @ A @ B )
= A ) ) ) ).
% prefix_order.max_def
thf(fact_383_rev__induct,axiom,
! [P: list_option_o > $o,Xs: list_option_o] :
( ( P @ nil_option_o )
=> ( ! [X4: option_o,Xs3: list_option_o] :
( ( P @ Xs3 )
=> ( P @ ( append_option_o @ Xs3 @ ( cons_option_o @ X4 @ nil_option_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_384_rev__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [X4: list_nat,Xs3: list_list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X4 @ nil_list_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_385_rev__induct,axiom,
! [P: list_list_o > $o,Xs: list_list_o] :
( ( P @ nil_list_o )
=> ( ! [X4: list_o,Xs3: list_list_o] :
( ( P @ Xs3 )
=> ( P @ ( append_list_o @ Xs3 @ ( cons_list_o @ X4 @ nil_list_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_386_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X4 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_387_rev__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ( P @ nil_o )
=> ( ! [X4: $o,Xs3: list_o] :
( ( P @ Xs3 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X4 @ nil_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_388_rev__exhaust,axiom,
! [Xs: list_option_o] :
( ( Xs != nil_option_o )
=> ~ ! [Ys3: list_option_o,Y5: option_o] :
( Xs
!= ( append_option_o @ Ys3 @ ( cons_option_o @ Y5 @ nil_option_o ) ) ) ) ).
% rev_exhaust
thf(fact_389_rev__exhaust,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ~ ! [Ys3: list_list_nat,Y5: list_nat] :
( Xs
!= ( append_list_nat @ Ys3 @ ( cons_list_nat @ Y5 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_390_rev__exhaust,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
=> ~ ! [Ys3: list_list_o,Y5: list_o] :
( Xs
!= ( append_list_o @ Ys3 @ ( cons_list_o @ Y5 @ nil_list_o ) ) ) ) ).
% rev_exhaust
thf(fact_391_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys3: list_nat,Y5: nat] :
( Xs
!= ( append_nat @ Ys3 @ ( cons_nat @ Y5 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_392_rev__exhaust,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ~ ! [Ys3: list_o,Y5: $o] :
( Xs
!= ( append_o @ Ys3 @ ( cons_o @ Y5 @ nil_o ) ) ) ) ).
% rev_exhaust
thf(fact_393_Cons__eq__append__conv,axiom,
! [X2: option_o,Xs: list_option_o,Ys: list_option_o,Zs: list_option_o] :
( ( ( cons_option_o @ X2 @ Xs )
= ( append_option_o @ Ys @ Zs ) )
= ( ( ( Ys = nil_option_o )
& ( ( cons_option_o @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_option_o] :
( ( ( cons_option_o @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_option_o @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_394_Cons__eq__append__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( append_list_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_nat )
& ( ( cons_list_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_list_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_395_Cons__eq__append__conv,axiom,
! [X2: list_o,Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( cons_list_o @ X2 @ Xs )
= ( append_list_o @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_o )
& ( ( cons_list_o @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_list_o] :
( ( ( cons_list_o @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_list_o @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_396_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 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_397_Cons__eq__append__conv,axiom,
! [X2: $o,Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( cons_o @ X2 @ Xs )
= ( append_o @ Ys @ Zs ) )
= ( ( ( Ys = nil_o )
& ( ( cons_o @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_o] :
( ( ( cons_o @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_o @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_398_append__eq__Cons__conv,axiom,
! [Ys: list_option_o,Zs: list_option_o,X2: option_o,Xs: list_option_o] :
( ( ( append_option_o @ Ys @ Zs )
= ( cons_option_o @ X2 @ Xs ) )
= ( ( ( Ys = nil_option_o )
& ( Zs
= ( cons_option_o @ X2 @ Xs ) ) )
| ? [Ys4: list_option_o] :
( ( Ys
= ( cons_option_o @ X2 @ Ys4 ) )
& ( ( append_option_o @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_399_append__eq__Cons__conv,axiom,
! [Ys: list_list_nat,Zs: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( ( append_list_nat @ Ys @ Zs )
= ( cons_list_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_nat )
& ( Zs
= ( cons_list_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_list_nat] :
( ( Ys
= ( cons_list_nat @ X2 @ Ys4 ) )
& ( ( append_list_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_400_append__eq__Cons__conv,axiom,
! [Ys: list_list_o,Zs: list_list_o,X2: list_o,Xs: list_list_o] :
( ( ( append_list_o @ Ys @ Zs )
= ( cons_list_o @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_o )
& ( Zs
= ( cons_list_o @ X2 @ Xs ) ) )
| ? [Ys4: list_list_o] :
( ( Ys
= ( cons_list_o @ X2 @ Ys4 ) )
& ( ( append_list_o @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_401_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 ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_402_append__eq__Cons__conv,axiom,
! [Ys: list_o,Zs: list_o,X2: $o,Xs: list_o] :
( ( ( append_o @ Ys @ Zs )
= ( cons_o @ X2 @ Xs ) )
= ( ( ( Ys = nil_o )
& ( Zs
= ( cons_o @ X2 @ Xs ) ) )
| ? [Ys4: list_o] :
( ( Ys
= ( cons_o @ X2 @ Ys4 ) )
& ( ( append_o @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_403_rev__nonempty__induct,axiom,
! [Xs: list_option_o,P: list_option_o > $o] :
( ( Xs != nil_option_o )
=> ( ! [X4: option_o] : ( P @ ( cons_option_o @ X4 @ nil_option_o ) )
=> ( ! [X4: option_o,Xs3: list_option_o] :
( ( Xs3 != nil_option_o )
=> ( ( P @ Xs3 )
=> ( P @ ( append_option_o @ Xs3 @ ( cons_option_o @ X4 @ nil_option_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_404_rev__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X4: list_nat] : ( P @ ( cons_list_nat @ X4 @ nil_list_nat ) )
=> ( ! [X4: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X4 @ nil_list_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_405_rev__nonempty__induct,axiom,
! [Xs: list_list_o,P: list_list_o > $o] :
( ( Xs != nil_list_o )
=> ( ! [X4: list_o] : ( P @ ( cons_list_o @ X4 @ nil_list_o ) )
=> ( ! [X4: list_o,Xs3: list_list_o] :
( ( Xs3 != nil_list_o )
=> ( ( P @ Xs3 )
=> ( P @ ( append_list_o @ Xs3 @ ( cons_list_o @ X4 @ nil_list_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_406_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X4 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_407_rev__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X4: $o] : ( P @ ( cons_o @ X4 @ nil_o ) )
=> ( ! [X4: $o,Xs3: list_o] :
( ( Xs3 != nil_o )
=> ( ( P @ Xs3 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X4 @ nil_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_408_prefix__code_I2_J,axiom,
! [X2: option_o,Xs: list_option_o] :
~ ( prefix_option_o @ ( cons_option_o @ X2 @ Xs ) @ nil_option_o ) ).
% prefix_code(2)
thf(fact_409_prefix__code_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
~ ( prefix_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat ) ).
% prefix_code(2)
thf(fact_410_prefix__code_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] :
~ ( prefix_list_o @ ( cons_list_o @ X2 @ Xs ) @ nil_list_o ) ).
% prefix_code(2)
thf(fact_411_prefix__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_412_prefix__code_I2_J,axiom,
! [X2: $o,Xs: list_o] :
~ ( prefix_o @ ( cons_o @ X2 @ Xs ) @ nil_o ) ).
% prefix_code(2)
thf(fact_413_prefix__Cons,axiom,
! [Xs: list_option_o,Y3: option_o,Ys: list_option_o] :
( ( prefix_option_o @ Xs @ ( cons_option_o @ Y3 @ Ys ) )
= ( ( Xs = nil_option_o )
| ? [Zs3: list_option_o] :
( ( Xs
= ( cons_option_o @ Y3 @ Zs3 ) )
& ( prefix_option_o @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_414_prefix__Cons,axiom,
! [Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( cons_list_nat @ Y3 @ Ys ) )
= ( ( Xs = nil_list_nat )
| ? [Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y3 @ Zs3 ) )
& ( prefix_list_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_415_prefix__Cons,axiom,
! [Xs: list_list_o,Y3: list_o,Ys: list_list_o] :
( ( prefix_list_o @ Xs @ ( cons_list_o @ Y3 @ Ys ) )
= ( ( Xs = nil_list_o )
| ? [Zs3: list_list_o] :
( ( Xs
= ( cons_list_o @ Y3 @ Zs3 ) )
& ( prefix_list_o @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_416_prefix__Cons,axiom,
! [Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) )
= ( ( Xs = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_417_prefix__Cons,axiom,
! [Xs: list_o,Y3: $o,Ys: list_o] :
( ( prefix_o @ Xs @ ( cons_o @ Y3 @ Ys ) )
= ( ( Xs = nil_o )
| ? [Zs3: list_o] :
( ( Xs
= ( cons_o @ Y3 @ Zs3 ) )
& ( prefix_o @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_418_not__prefix__cases,axiom,
! [Ps: list_option_o,Ls: list_option_o] :
( ~ ( prefix_option_o @ Ps @ Ls )
=> ( ( ( Ps != nil_option_o )
=> ( Ls != nil_option_o ) )
=> ( ! [A4: option_o,As: list_option_o] :
( ( Ps
= ( cons_option_o @ A4 @ As ) )
=> ! [X4: option_o,Xs3: list_option_o] :
( ( Ls
= ( cons_option_o @ X4 @ Xs3 ) )
=> ( ( X4 = A4 )
=> ( prefix_option_o @ As @ Xs3 ) ) ) )
=> ~ ! [A4: option_o] :
( ? [As: list_option_o] :
( Ps
= ( cons_option_o @ A4 @ As ) )
=> ! [X4: option_o] :
( ? [Xs3: list_option_o] :
( Ls
= ( cons_option_o @ X4 @ Xs3 ) )
=> ( X4 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_419_not__prefix__cases,axiom,
! [Ps: list_list_nat,Ls: list_list_nat] :
( ~ ( prefix_list_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_list_nat )
=> ( Ls != nil_list_nat ) )
=> ( ! [A4: list_nat,As: list_list_nat] :
( ( Ps
= ( cons_list_nat @ A4 @ As ) )
=> ! [X4: list_nat,Xs3: list_list_nat] :
( ( Ls
= ( cons_list_nat @ X4 @ Xs3 ) )
=> ( ( X4 = A4 )
=> ( prefix_list_nat @ As @ Xs3 ) ) ) )
=> ~ ! [A4: list_nat] :
( ? [As: list_list_nat] :
( Ps
= ( cons_list_nat @ A4 @ As ) )
=> ! [X4: list_nat] :
( ? [Xs3: list_list_nat] :
( Ls
= ( cons_list_nat @ X4 @ Xs3 ) )
=> ( X4 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_420_not__prefix__cases,axiom,
! [Ps: list_list_o,Ls: list_list_o] :
( ~ ( prefix_list_o @ Ps @ Ls )
=> ( ( ( Ps != nil_list_o )
=> ( Ls != nil_list_o ) )
=> ( ! [A4: list_o,As: list_list_o] :
( ( Ps
= ( cons_list_o @ A4 @ As ) )
=> ! [X4: list_o,Xs3: list_list_o] :
( ( Ls
= ( cons_list_o @ X4 @ Xs3 ) )
=> ( ( X4 = A4 )
=> ( prefix_list_o @ As @ Xs3 ) ) ) )
=> ~ ! [A4: list_o] :
( ? [As: list_list_o] :
( Ps
= ( cons_list_o @ A4 @ As ) )
=> ! [X4: list_o] :
( ? [Xs3: list_list_o] :
( Ls
= ( cons_list_o @ X4 @ Xs3 ) )
=> ( X4 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_421_not__prefix__cases,axiom,
! [Ps: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A4: nat,As: list_nat] :
( ( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X4: nat,Xs3: list_nat] :
( ( Ls
= ( cons_nat @ X4 @ Xs3 ) )
=> ( ( X4 = A4 )
=> ( prefix_nat @ As @ Xs3 ) ) ) )
=> ~ ! [A4: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X4: nat] :
( ? [Xs3: list_nat] :
( Ls
= ( cons_nat @ X4 @ Xs3 ) )
=> ( X4 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_422_not__prefix__cases,axiom,
! [Ps: list_o,Ls: list_o] :
( ~ ( prefix_o @ Ps @ Ls )
=> ( ( ( Ps != nil_o )
=> ( Ls != nil_o ) )
=> ( ! [A4: $o,As: list_o] :
( ( Ps
= ( cons_o @ A4 @ As ) )
=> ! [X4: $o,Xs3: list_o] :
( ( Ls
= ( cons_o @ X4 @ Xs3 ) )
=> ( ( X4 = A4 )
=> ( prefix_o @ As @ Xs3 ) ) ) )
=> ~ ! [A4: $o] :
( ? [As: list_o] :
( Ps
= ( cons_o @ A4 @ As ) )
=> ! [X4: $o] :
( ? [Xs3: list_o] :
( Ls
= ( cons_o @ X4 @ Xs3 ) )
=> ( X4 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_423_not__prefix__induct,axiom,
! [Ps: list_option_o,Ls: list_option_o,P: list_option_o > list_option_o > $o] :
( ~ ( prefix_option_o @ Ps @ Ls )
=> ( ! [X4: option_o,Xs3: list_option_o] : ( P @ ( cons_option_o @ X4 @ Xs3 ) @ nil_option_o )
=> ( ! [X4: option_o,Xs3: list_option_o,Y5: option_o,Ys3: list_option_o] :
( ( X4 != Y5 )
=> ( P @ ( cons_option_o @ X4 @ Xs3 ) @ ( cons_option_o @ Y5 @ Ys3 ) ) )
=> ( ! [X4: option_o,Xs3: list_option_o,Y5: option_o,Ys3: list_option_o] :
( ( X4 = Y5 )
=> ( ~ ( prefix_option_o @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_option_o @ X4 @ Xs3 ) @ ( cons_option_o @ Y5 @ Ys3 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_424_not__prefix__induct,axiom,
! [Ps: list_list_nat,Ls: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ~ ( prefix_list_nat @ Ps @ Ls )
=> ( ! [X4: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X4 @ Xs3 ) @ nil_list_nat )
=> ( ! [X4: list_nat,Xs3: list_list_nat,Y5: list_nat,Ys3: list_list_nat] :
( ( X4 != Y5 )
=> ( P @ ( cons_list_nat @ X4 @ Xs3 ) @ ( cons_list_nat @ Y5 @ Ys3 ) ) )
=> ( ! [X4: list_nat,Xs3: list_list_nat,Y5: list_nat,Ys3: list_list_nat] :
( ( X4 = Y5 )
=> ( ~ ( prefix_list_nat @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_nat @ X4 @ Xs3 ) @ ( cons_list_nat @ Y5 @ Ys3 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_425_not__prefix__induct,axiom,
! [Ps: list_list_o,Ls: list_list_o,P: list_list_o > list_list_o > $o] :
( ~ ( prefix_list_o @ Ps @ Ls )
=> ( ! [X4: list_o,Xs3: list_list_o] : ( P @ ( cons_list_o @ X4 @ Xs3 ) @ nil_list_o )
=> ( ! [X4: list_o,Xs3: list_list_o,Y5: list_o,Ys3: list_list_o] :
( ( X4 != Y5 )
=> ( P @ ( cons_list_o @ X4 @ Xs3 ) @ ( cons_list_o @ Y5 @ Ys3 ) ) )
=> ( ! [X4: list_o,Xs3: list_list_o,Y5: list_o,Ys3: list_list_o] :
( ( X4 = Y5 )
=> ( ~ ( prefix_list_o @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_o @ X4 @ Xs3 ) @ ( cons_list_o @ Y5 @ Ys3 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_426_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( ( X4 != Y5 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( ! [X4: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( ( X4 = Y5 )
=> ( ~ ( prefix_nat @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_427_not__prefix__induct,axiom,
! [Ps: list_o,Ls: list_o,P: list_o > list_o > $o] :
( ~ ( prefix_o @ Ps @ Ls )
=> ( ! [X4: $o,Xs3: list_o] : ( P @ ( cons_o @ X4 @ Xs3 ) @ nil_o )
=> ( ! [X4: $o,Xs3: list_o,Y5: $o,Ys3: list_o] :
( ( X4 != Y5 )
=> ( P @ ( cons_o @ X4 @ Xs3 ) @ ( cons_o @ Y5 @ Ys3 ) ) )
=> ( ! [X4: $o,Xs3: list_o,Y5: $o,Ys3: list_o] :
( ( X4 = Y5 )
=> ( ~ ( prefix_o @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X4 @ Xs3 ) @ ( cons_o @ Y5 @ Ys3 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_428_longest__common__prefix_Oelims,axiom,
! [X2: list_option_o,Xa: list_option_o,Y3: list_option_o] :
( ( ( longes1709640641647806696tion_o @ X2 @ Xa )
= Y3 )
=> ( ! [X4: option_o,Xs3: list_option_o] :
( ( X2
= ( cons_option_o @ X4 @ Xs3 ) )
=> ! [Y5: option_o,Ys3: list_option_o] :
( ( Xa
= ( cons_option_o @ Y5 @ Ys3 ) )
=> ~ ( ( ( X4 = Y5 )
=> ( Y3
= ( cons_option_o @ X4 @ ( longes1709640641647806696tion_o @ Xs3 @ Ys3 ) ) ) )
& ( ( X4 != Y5 )
=> ( Y3 = nil_option_o ) ) ) ) )
=> ( ( ( X2 = nil_option_o )
=> ( Y3 != nil_option_o ) )
=> ~ ( ( Xa = nil_option_o )
=> ( Y3 != nil_option_o ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_429_longest__common__prefix_Oelims,axiom,
! [X2: list_list_nat,Xa: list_list_nat,Y3: list_list_nat] :
( ( ( longes3715614165048771798st_nat @ X2 @ Xa )
= Y3 )
=> ( ! [X4: list_nat,Xs3: list_list_nat] :
( ( X2
= ( cons_list_nat @ X4 @ Xs3 ) )
=> ! [Y5: list_nat,Ys3: list_list_nat] :
( ( Xa
= ( cons_list_nat @ Y5 @ Ys3 ) )
=> ~ ( ( ( X4 = Y5 )
=> ( Y3
= ( cons_list_nat @ X4 @ ( longes3715614165048771798st_nat @ Xs3 @ Ys3 ) ) ) )
& ( ( X4 != Y5 )
=> ( Y3 = nil_list_nat ) ) ) ) )
=> ( ( ( X2 = nil_list_nat )
=> ( Y3 != nil_list_nat ) )
=> ~ ( ( Xa = nil_list_nat )
=> ( Y3 != nil_list_nat ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_430_longest__common__prefix_Oelims,axiom,
! [X2: list_list_o,Xa: list_list_o,Y3: list_list_o] :
( ( ( longes802630962832238888list_o @ X2 @ Xa )
= Y3 )
=> ( ! [X4: list_o,Xs3: list_list_o] :
( ( X2
= ( cons_list_o @ X4 @ Xs3 ) )
=> ! [Y5: list_o,Ys3: list_list_o] :
( ( Xa
= ( cons_list_o @ Y5 @ Ys3 ) )
=> ~ ( ( ( X4 = Y5 )
=> ( Y3
= ( cons_list_o @ X4 @ ( longes802630962832238888list_o @ Xs3 @ Ys3 ) ) ) )
& ( ( X4 != Y5 )
=> ( Y3 = nil_list_o ) ) ) ) )
=> ( ( ( X2 = nil_list_o )
=> ( Y3 != nil_list_o ) )
=> ~ ( ( Xa = nil_list_o )
=> ( Y3 != nil_list_o ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_431_longest__common__prefix_Oelims,axiom,
! [X2: list_nat,Xa: list_nat,Y3: list_nat] :
( ( ( longes266370323089874118ix_nat @ X2 @ Xa )
= Y3 )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X4 @ Xs3 ) )
=> ! [Y5: nat,Ys3: list_nat] :
( ( Xa
= ( cons_nat @ Y5 @ Ys3 ) )
=> ~ ( ( ( X4 = Y5 )
=> ( Y3
= ( cons_nat @ X4 @ ( longes266370323089874118ix_nat @ Xs3 @ Ys3 ) ) ) )
& ( ( X4 != Y5 )
=> ( Y3 = nil_nat ) ) ) ) )
=> ( ( ( X2 = nil_nat )
=> ( Y3 != nil_nat ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y3 != nil_nat ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_432_longest__common__prefix_Oelims,axiom,
! [X2: list_o,Xa: list_o,Y3: list_o] :
( ( ( longes218265413398645026efix_o @ X2 @ Xa )
= Y3 )
=> ( ! [X4: $o,Xs3: list_o] :
( ( X2
= ( cons_o @ X4 @ Xs3 ) )
=> ! [Y5: $o,Ys3: list_o] :
( ( Xa
= ( cons_o @ Y5 @ Ys3 ) )
=> ~ ( ( ( X4 = Y5 )
=> ( Y3
= ( cons_o @ X4 @ ( longes218265413398645026efix_o @ Xs3 @ Ys3 ) ) ) )
& ( ( X4 = (~ Y5) )
=> ( Y3 = nil_o ) ) ) ) )
=> ( ( ( X2 = nil_o )
=> ( Y3 != nil_o ) )
=> ~ ( ( Xa = nil_o )
=> ( Y3 != nil_o ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_433_longest__common__prefix_Osimps_I1_J,axiom,
! [X2: option_o,Y3: option_o,Xs: list_option_o,Ys: list_option_o] :
( ( ( X2 = Y3 )
=> ( ( longes1709640641647806696tion_o @ ( cons_option_o @ X2 @ Xs ) @ ( cons_option_o @ Y3 @ Ys ) )
= ( cons_option_o @ X2 @ ( longes1709640641647806696tion_o @ Xs @ Ys ) ) ) )
& ( ( X2 != Y3 )
=> ( ( longes1709640641647806696tion_o @ ( cons_option_o @ X2 @ Xs ) @ ( cons_option_o @ Y3 @ Ys ) )
= nil_option_o ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_434_longest__common__prefix_Osimps_I1_J,axiom,
! [X2: list_nat,Y3: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( X2 = Y3 )
=> ( ( longes3715614165048771798st_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) )
= ( cons_list_nat @ X2 @ ( longes3715614165048771798st_nat @ Xs @ Ys ) ) ) )
& ( ( X2 != Y3 )
=> ( ( longes3715614165048771798st_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) )
= nil_list_nat ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_435_longest__common__prefix_Osimps_I1_J,axiom,
! [X2: list_o,Y3: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( ( X2 = Y3 )
=> ( ( longes802630962832238888list_o @ ( cons_list_o @ X2 @ Xs ) @ ( cons_list_o @ Y3 @ Ys ) )
= ( cons_list_o @ X2 @ ( longes802630962832238888list_o @ Xs @ Ys ) ) ) )
& ( ( X2 != Y3 )
=> ( ( longes802630962832238888list_o @ ( cons_list_o @ X2 @ Xs ) @ ( cons_list_o @ Y3 @ Ys ) )
= nil_list_o ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_436_longest__common__prefix_Osimps_I1_J,axiom,
! [X2: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
( ( ( X2 = Y3 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( cons_nat @ X2 @ ( longes266370323089874118ix_nat @ Xs @ Ys ) ) ) )
& ( ( X2 != Y3 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= nil_nat ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_437_longest__common__prefix_Osimps_I1_J,axiom,
! [X2: $o,Y3: $o,Xs: list_o,Ys: list_o] :
( ( ( X2 = Y3 )
=> ( ( longes218265413398645026efix_o @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y3 @ Ys ) )
= ( cons_o @ X2 @ ( longes218265413398645026efix_o @ Xs @ Ys ) ) ) )
& ( ( X2 = (~ Y3) )
=> ( ( longes218265413398645026efix_o @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y3 @ Ys ) )
= nil_o ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_438_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option
= ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_439_option_Oexhaust__sel,axiom,
! [Option: option_list_o] :
( ( Option != none_list_o )
=> ( Option
= ( some_list_o @ ( the_list_o @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_440_maps__simps_I1_J,axiom,
! [F: $o > list_o,X2: $o,Xs: list_o] :
( ( maps_o_o @ F @ ( cons_o @ X2 @ Xs ) )
= ( append_o @ ( F @ X2 ) @ ( maps_o_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_441_maps__simps_I1_J,axiom,
! [F: $o > list_list_o,X2: $o,Xs: list_o] :
( ( maps_o_list_o @ F @ ( cons_o @ X2 @ Xs ) )
= ( append_list_o @ ( F @ X2 ) @ ( maps_o_list_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_442_maps__simps_I1_J,axiom,
! [F: $o > list_nat,X2: $o,Xs: list_o] :
( ( maps_o_nat @ F @ ( cons_o @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_o_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_443_maps__simps_I1_J,axiom,
! [F: list_o > list_o,X2: list_o,Xs: list_list_o] :
( ( maps_list_o_o @ F @ ( cons_list_o @ X2 @ Xs ) )
= ( append_o @ ( F @ X2 ) @ ( maps_list_o_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_444_maps__simps_I1_J,axiom,
! [F: list_o > list_list_o,X2: list_o,Xs: list_list_o] :
( ( maps_list_o_list_o @ F @ ( cons_list_o @ X2 @ Xs ) )
= ( append_list_o @ ( F @ X2 ) @ ( maps_list_o_list_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_445_maps__simps_I1_J,axiom,
! [F: list_o > list_nat,X2: list_o,Xs: list_list_o] :
( ( maps_list_o_nat @ F @ ( cons_list_o @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_list_o_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_446_maps__simps_I1_J,axiom,
! [F: nat > list_o,X2: nat,Xs: list_nat] :
( ( maps_nat_o @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_o @ ( F @ X2 ) @ ( maps_nat_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_447_maps__simps_I1_J,axiom,
! [F: nat > list_list_o,X2: nat,Xs: list_nat] :
( ( maps_nat_list_o @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_o @ ( F @ X2 ) @ ( maps_nat_list_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_448_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_449_insert__Nil,axiom,
! [X2: option_o] :
( ( insert_option_o @ X2 @ nil_option_o )
= ( cons_option_o @ X2 @ nil_option_o ) ) ).
% insert_Nil
thf(fact_450_insert__Nil,axiom,
! [X2: list_nat] :
( ( insert_list_nat @ X2 @ nil_list_nat )
= ( cons_list_nat @ X2 @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_451_insert__Nil,axiom,
! [X2: $o] :
( ( insert_o @ X2 @ nil_o )
= ( cons_o @ X2 @ nil_o ) ) ).
% insert_Nil
thf(fact_452_insert__Nil,axiom,
! [X2: list_o] :
( ( insert_list_o @ X2 @ nil_list_o )
= ( cons_list_o @ X2 @ nil_list_o ) ) ).
% insert_Nil
thf(fact_453_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_454_prefixes__snoc,axiom,
! [Xs: list_option_o,X2: option_o] :
( ( prefixes_option_o @ ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) ) )
= ( append_list_option_o @ ( prefixes_option_o @ Xs ) @ ( cons_list_option_o @ ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) ) @ nil_list_option_o ) ) ) ).
% prefixes_snoc
thf(fact_455_prefixes__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( prefixes_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( append_list_list_nat @ ( prefixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_456_prefixes__snoc,axiom,
! [Xs: list_list_o,X2: list_o] :
( ( prefixes_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) )
= ( append_list_list_o @ ( prefixes_list_o @ Xs ) @ ( cons_list_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) @ nil_list_list_o ) ) ) ).
% prefixes_snoc
thf(fact_457_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_458_prefixes__snoc,axiom,
! [Xs: list_o,X2: $o] :
( ( prefixes_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= ( append_list_o @ ( prefixes_o @ Xs ) @ ( cons_list_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ nil_list_o ) ) ) ).
% prefixes_snoc
thf(fact_459_ord_Omax__def,axiom,
( max_list_o
= ( ^ [Less_eq: list_o > list_o > $o,A2: list_o,B2: list_o] : ( if_list_o @ ( Less_eq @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% ord.max_def
thf(fact_460_ord_Omin__def,axiom,
( min_list_o
= ( ^ [Less_eq: list_o > list_o > $o,A2: list_o,B2: list_o] : ( if_list_o @ ( Less_eq @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% ord.min_def
thf(fact_461_ord_Omax_Ocong,axiom,
max_list_o = max_list_o ).
% ord.max.cong
thf(fact_462_ord_Omin_Ocong,axiom,
min_list_o = min_list_o ).
% ord.min.cong
thf(fact_463_butlast__snoc,axiom,
! [Xs: list_option_o,X2: option_o] :
( ( butlast_option_o @ ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_464_butlast__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( butlast_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_465_butlast__snoc,axiom,
! [Xs: list_list_o,X2: list_o] :
( ( butlast_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_466_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_467_butlast__snoc,axiom,
! [Xs: list_o,X2: $o] :
( ( butlast_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_468_prefixes__not__Nil,axiom,
! [Xs: list_o] :
( ( prefixes_o @ Xs )
!= nil_list_o ) ).
% prefixes_not_Nil
thf(fact_469_prefixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( prefixes_nat @ Xs )
!= nil_list_nat ) ).
% prefixes_not_Nil
thf(fact_470_butlast_Osimps_I1_J,axiom,
( ( butlast_o @ nil_o )
= nil_o ) ).
% butlast.simps(1)
thf(fact_471_butlast_Osimps_I1_J,axiom,
( ( butlast_option_o @ nil_option_o )
= nil_option_o ) ).
% butlast.simps(1)
thf(fact_472_butlast_Osimps_I1_J,axiom,
( ( butlast_list_o @ nil_list_o )
= nil_list_o ) ).
% butlast.simps(1)
thf(fact_473_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_474_butlast_Osimps_I1_J,axiom,
( ( butlast_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% butlast.simps(1)
thf(fact_475_prefixeq__butlast,axiom,
! [Xs: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs ) @ Xs ) ).
% prefixeq_butlast
thf(fact_476_prefixeq__butlast,axiom,
! [Xs: list_o] : ( prefix_o @ ( butlast_o @ Xs ) @ Xs ) ).
% prefixeq_butlast
thf(fact_477_prefixes_Osimps_I1_J,axiom,
( ( prefixes_option_o @ nil_option_o )
= ( cons_list_option_o @ nil_option_o @ nil_list_option_o ) ) ).
% prefixes.simps(1)
thf(fact_478_prefixes_Osimps_I1_J,axiom,
( ( prefixes_list_o @ nil_list_o )
= ( cons_list_list_o @ nil_list_o @ nil_list_list_o ) ) ).
% prefixes.simps(1)
thf(fact_479_prefixes_Osimps_I1_J,axiom,
( ( prefixes_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_480_prefixes_Osimps_I1_J,axiom,
( ( prefixes_o @ nil_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% prefixes.simps(1)
thf(fact_481_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_482_butlast_Osimps_I2_J,axiom,
! [Xs: list_option_o,X2: option_o] :
( ( ( Xs = nil_option_o )
=> ( ( butlast_option_o @ ( cons_option_o @ X2 @ Xs ) )
= nil_option_o ) )
& ( ( Xs != nil_option_o )
=> ( ( butlast_option_o @ ( cons_option_o @ X2 @ Xs ) )
= ( cons_option_o @ X2 @ ( butlast_option_o @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_483_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= nil_list_nat ) )
& ( ( Xs != nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ X2 @ ( butlast_list_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_484_butlast_Osimps_I2_J,axiom,
! [Xs: list_o,X2: $o] :
( ( ( Xs = nil_o )
=> ( ( butlast_o @ ( cons_o @ X2 @ Xs ) )
= nil_o ) )
& ( ( Xs != nil_o )
=> ( ( butlast_o @ ( cons_o @ X2 @ Xs ) )
= ( cons_o @ X2 @ ( butlast_o @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_485_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_o,X2: list_o] :
( ( ( Xs = nil_list_o )
=> ( ( butlast_list_o @ ( cons_list_o @ X2 @ Xs ) )
= nil_list_o ) )
& ( ( Xs != nil_list_o )
=> ( ( butlast_list_o @ ( cons_list_o @ X2 @ Xs ) )
= ( cons_list_o @ X2 @ ( butlast_list_o @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_486_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_487_butlast__append,axiom,
! [Ys: list_option_o,Xs: list_option_o] :
( ( ( Ys = nil_option_o )
=> ( ( butlast_option_o @ ( append_option_o @ Xs @ Ys ) )
= ( butlast_option_o @ Xs ) ) )
& ( ( Ys != nil_option_o )
=> ( ( butlast_option_o @ ( append_option_o @ Xs @ Ys ) )
= ( append_option_o @ Xs @ ( butlast_option_o @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_488_butlast__append,axiom,
! [Ys: list_list_o,Xs: list_list_o] :
( ( ( Ys = nil_list_o )
=> ( ( butlast_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( butlast_list_o @ Xs ) ) )
& ( ( Ys != nil_list_o )
=> ( ( butlast_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( append_list_o @ Xs @ ( butlast_list_o @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_489_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_490_butlast__append,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( ( Ys = nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( butlast_list_nat @ Xs ) ) )
& ( ( Ys != nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_list_nat @ Xs @ ( butlast_list_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_491_butlast__append,axiom,
! [Ys: list_o,Xs: list_o] :
( ( ( Ys = nil_o )
=> ( ( butlast_o @ ( append_o @ Xs @ Ys ) )
= ( butlast_o @ Xs ) ) )
& ( ( Ys != nil_o )
=> ( ( butlast_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ Xs @ ( butlast_o @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_492_maps__simps_I2_J,axiom,
! [F: $o > list_o] :
( ( maps_o_o @ F @ nil_o )
= nil_o ) ).
% maps_simps(2)
thf(fact_493_maps__simps_I2_J,axiom,
! [F: $o > list_nat] :
( ( maps_o_nat @ F @ nil_o )
= nil_nat ) ).
% maps_simps(2)
thf(fact_494_maps__simps_I2_J,axiom,
! [F: nat > list_o] :
( ( maps_nat_o @ F @ nil_nat )
= nil_o ) ).
% maps_simps(2)
thf(fact_495_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_496_maps__simps_I2_J,axiom,
! [F: $o > list_option_o] :
( ( maps_o_option_o @ F @ nil_o )
= nil_option_o ) ).
% maps_simps(2)
thf(fact_497_maps__simps_I2_J,axiom,
! [F: $o > list_list_o] :
( ( maps_o_list_o @ F @ nil_o )
= nil_list_o ) ).
% maps_simps(2)
thf(fact_498_maps__simps_I2_J,axiom,
! [F: $o > list_list_nat] :
( ( maps_o_list_nat @ F @ nil_o )
= nil_list_nat ) ).
% maps_simps(2)
thf(fact_499_maps__simps_I2_J,axiom,
! [F: option_o > list_o] :
( ( maps_option_o_o @ F @ nil_option_o )
= nil_o ) ).
% maps_simps(2)
thf(fact_500_maps__simps_I2_J,axiom,
! [F: option_o > list_nat] :
( ( maps_option_o_nat @ F @ nil_option_o )
= nil_nat ) ).
% maps_simps(2)
thf(fact_501_maps__simps_I2_J,axiom,
! [F: list_o > list_o] :
( ( maps_list_o_o @ F @ nil_list_o )
= nil_o ) ).
% maps_simps(2)
thf(fact_502_prefixes__eq__snoc,axiom,
! [Ys: list_option_o,Xs: list_list_option_o,X2: list_option_o] :
( ( ( prefixes_option_o @ Ys )
= ( append_list_option_o @ Xs @ ( cons_list_option_o @ X2 @ nil_list_option_o ) ) )
= ( ( ( ( Ys = nil_option_o )
& ( Xs = nil_list_option_o ) )
| ? [Z4: option_o,Zs3: list_option_o] :
( ( Ys
= ( append_option_o @ Zs3 @ ( cons_option_o @ Z4 @ nil_option_o ) ) )
& ( Xs
= ( prefixes_option_o @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_503_prefixes__eq__snoc,axiom,
! [Ys: list_list_nat,Xs: list_list_list_nat,X2: list_list_nat] :
( ( ( prefixes_list_nat @ Ys )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) ) )
= ( ( ( ( Ys = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( append_list_nat @ Zs3 @ ( cons_list_nat @ Z4 @ nil_list_nat ) ) )
& ( Xs
= ( prefixes_list_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_504_prefixes__eq__snoc,axiom,
! [Ys: list_list_o,Xs: list_list_list_o,X2: list_list_o] :
( ( ( prefixes_list_o @ Ys )
= ( append_list_list_o @ Xs @ ( cons_list_list_o @ X2 @ nil_list_list_o ) ) )
= ( ( ( ( Ys = nil_list_o )
& ( Xs = nil_list_list_o ) )
| ? [Z4: list_o,Zs3: list_list_o] :
( ( Ys
= ( append_list_o @ Zs3 @ ( cons_list_o @ Z4 @ nil_list_o ) ) )
& ( Xs
= ( prefixes_list_o @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_505_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 ) )
| ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z4 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_506_prefixes__eq__snoc,axiom,
! [Ys: list_o,Xs: list_list_o,X2: list_o] :
( ( ( prefixes_o @ Ys )
= ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) )
= ( ( ( ( Ys = nil_o )
& ( Xs = nil_list_o ) )
| ? [Z4: $o,Zs3: list_o] :
( ( Ys
= ( append_o @ Zs3 @ ( cons_o @ Z4 @ nil_o ) ) )
& ( Xs
= ( prefixes_o @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_507_suffixes__eq__snoc,axiom,
! [Ys: list_option_o,Xs: list_list_option_o,X2: list_option_o] :
( ( ( suffixes_option_o @ Ys )
= ( append_list_option_o @ Xs @ ( cons_list_option_o @ X2 @ nil_list_option_o ) ) )
= ( ( ( ( Ys = nil_option_o )
& ( Xs = nil_list_option_o ) )
| ? [Z4: option_o,Zs3: list_option_o] :
( ( Ys
= ( cons_option_o @ Z4 @ Zs3 ) )
& ( Xs
= ( suffixes_option_o @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_508_suffixes__eq__snoc,axiom,
! [Ys: list_list_nat,Xs: list_list_list_nat,X2: list_list_nat] :
( ( ( suffixes_list_nat @ Ys )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) ) )
= ( ( ( ( Ys = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( Xs
= ( suffixes_list_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_509_suffixes__eq__snoc,axiom,
! [Ys: list_list_o,Xs: list_list_list_o,X2: list_list_o] :
( ( ( suffixes_list_o @ Ys )
= ( append_list_list_o @ Xs @ ( cons_list_list_o @ X2 @ nil_list_list_o ) ) )
= ( ( ( ( Ys = nil_list_o )
& ( Xs = nil_list_list_o ) )
| ? [Z4: list_o,Zs3: list_list_o] :
( ( Ys
= ( cons_list_o @ Z4 @ Zs3 ) )
& ( Xs
= ( suffixes_list_o @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_510_suffixes__eq__snoc,axiom,
! [Ys: list_o,Xs: list_list_o,X2: list_o] :
( ( ( suffixes_o @ Ys )
= ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) )
= ( ( ( ( Ys = nil_o )
& ( Xs = nil_list_o ) )
| ? [Z4: $o,Zs3: list_o] :
( ( Ys
= ( cons_o @ Z4 @ Zs3 ) )
& ( Xs
= ( suffixes_o @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_511_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 ) )
| ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( Xs
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_512_sublists_Osimps_I1_J,axiom,
( ( sublists_option_o @ nil_option_o )
= ( cons_list_option_o @ nil_option_o @ nil_list_option_o ) ) ).
% sublists.simps(1)
thf(fact_513_sublists_Osimps_I1_J,axiom,
( ( sublists_list_o @ nil_list_o )
= ( cons_list_list_o @ nil_list_o @ nil_list_list_o ) ) ).
% sublists.simps(1)
thf(fact_514_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_515_sublists_Osimps_I1_J,axiom,
( ( sublists_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% sublists.simps(1)
thf(fact_516_sublists_Osimps_I1_J,axiom,
( ( sublists_o @ nil_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% sublists.simps(1)
thf(fact_517_product__lists_Osimps_I1_J,axiom,
( ( produc5165959316431261517tion_o @ nil_list_option_o )
= ( cons_list_option_o @ nil_option_o @ nil_list_option_o ) ) ).
% product_lists.simps(1)
thf(fact_518_product__lists_Osimps_I1_J,axiom,
( ( product_lists_list_o @ nil_list_list_o )
= ( cons_list_list_o @ nil_list_o @ nil_list_list_o ) ) ).
% product_lists.simps(1)
thf(fact_519_product__lists_Osimps_I1_J,axiom,
( ( produc6783906451316923569st_nat @ nil_list_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_520_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_521_product__lists_Osimps_I1_J,axiom,
( ( product_lists_o @ nil_list_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% product_lists.simps(1)
thf(fact_522_append__butlast__last__id,axiom,
! [Xs: list_option_o] :
( ( Xs != nil_option_o )
=> ( ( append_option_o @ ( butlast_option_o @ Xs ) @ ( cons_option_o @ ( last_option_o @ Xs ) @ nil_option_o ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_523_append__butlast__last__id,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( append_list_nat @ ( butlast_list_nat @ Xs ) @ ( cons_list_nat @ ( last_list_nat @ Xs ) @ nil_list_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_524_append__butlast__last__id,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
=> ( ( append_list_o @ ( butlast_list_o @ Xs ) @ ( cons_list_o @ ( last_list_o @ Xs ) @ nil_list_o ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_525_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_526_append__butlast__last__id,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( ( append_o @ ( butlast_o @ Xs ) @ ( cons_o @ ( last_o @ Xs ) @ nil_o ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_527_concat__eq__append__conv,axiom,
! [Xss2: list_list_option_o,Ys: list_option_o,Zs: list_option_o] :
( ( ( concat_option_o @ Xss2 )
= ( append_option_o @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_option_o )
=> ( ( Ys = nil_option_o )
& ( Zs = nil_option_o ) ) )
& ( ( Xss2 != nil_list_option_o )
=> ? [Xss1: list_list_option_o,Xs2: list_option_o,Xs4: list_option_o,Xss22: list_list_option_o] :
( ( Xss2
= ( append_list_option_o @ Xss1 @ ( cons_list_option_o @ ( append_option_o @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_option_o @ ( concat_option_o @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_option_o @ Xs4 @ ( concat_option_o @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_528_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( concat_list_o @ Xss2 )
= ( append_list_o @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_list_o )
=> ( ( Ys = nil_list_o )
& ( Zs = nil_list_o ) ) )
& ( ( Xss2 != nil_list_list_o )
=> ? [Xss1: list_list_list_o,Xs2: list_list_o,Xs4: list_list_o,Xss22: list_list_list_o] :
( ( Xss2
= ( append_list_list_o @ Xss1 @ ( cons_list_list_o @ ( append_list_o @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_o @ ( concat_list_o @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_list_o @ Xs4 @ ( concat_list_o @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_529_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_list_nat )
=> ( ( Ys = nil_list_nat )
& ( Zs = nil_list_nat ) ) )
& ( ( Xss2 != nil_list_list_nat )
=> ? [Xss1: list_list_list_nat,Xs2: list_list_nat,Xs4: list_list_nat,Xss22: list_list_list_nat] :
( ( Xss2
= ( append_list_list_nat @ Xss1 @ ( cons_list_list_nat @ ( append_list_nat @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_nat @ ( concat_list_nat @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_list_nat @ Xs4 @ ( concat_list_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_530_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,Xs2: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_531_concat__eq__append__conv,axiom,
! [Xss2: list_list_o,Ys: list_o,Zs: list_o] :
( ( ( concat_o @ Xss2 )
= ( append_o @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_o )
=> ( ( Ys = nil_o )
& ( Zs = nil_o ) ) )
& ( ( Xss2 != nil_list_o )
=> ? [Xss1: list_list_o,Xs2: list_o,Xs4: list_o,Xss22: list_list_o] :
( ( Xss2
= ( append_list_o @ Xss1 @ ( cons_list_o @ ( append_o @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_o @ ( concat_o @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_o @ Xs4 @ ( concat_o @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_532_suffixes_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( suffixes_o @ ( cons_o @ X2 @ Xs ) )
= ( append_list_o @ ( suffixes_o @ Xs ) @ ( cons_list_o @ ( cons_o @ X2 @ Xs ) @ nil_list_o ) ) ) ).
% suffixes.simps(2)
thf(fact_533_suffixes_Osimps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( suffixes_list_o @ ( cons_list_o @ X2 @ Xs ) )
= ( append_list_list_o @ ( suffixes_list_o @ Xs ) @ ( cons_list_list_o @ ( cons_list_o @ X2 @ Xs ) @ nil_list_list_o ) ) ) ).
% suffixes.simps(2)
thf(fact_534_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_535_subseqs_Osimps_I1_J,axiom,
( ( subseqs_option_o @ nil_option_o )
= ( cons_list_option_o @ nil_option_o @ nil_list_option_o ) ) ).
% subseqs.simps(1)
thf(fact_536_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_o @ nil_list_o )
= ( cons_list_list_o @ nil_list_o @ nil_list_list_o ) ) ).
% subseqs.simps(1)
thf(fact_537_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_538_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_539_subseqs_Osimps_I1_J,axiom,
( ( subseqs_o @ nil_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% subseqs.simps(1)
thf(fact_540_snoc__eq__iff__butlast,axiom,
! [Xs: list_option_o,X2: option_o,Ys: list_option_o] :
( ( ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) )
= Ys )
= ( ( Ys != nil_option_o )
& ( ( butlast_option_o @ Ys )
= Xs )
& ( ( last_option_o @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_541_snoc__eq__iff__butlast,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) )
= Ys )
= ( ( Ys != nil_list_nat )
& ( ( butlast_list_nat @ Ys )
= Xs )
& ( ( last_list_nat @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_542_snoc__eq__iff__butlast,axiom,
! [Xs: list_list_o,X2: list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) )
= Ys )
= ( ( Ys != nil_list_o )
& ( ( butlast_list_o @ Ys )
= Xs )
& ( ( last_list_o @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_543_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_544_snoc__eq__iff__butlast,axiom,
! [Xs: list_o,X2: $o,Ys: list_o] :
( ( ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) )
= Ys )
= ( ( Ys != nil_o )
& ( ( butlast_o @ Ys )
= Xs )
& ( ( last_o @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_545_concat__eq__appendD,axiom,
! [Xss2: list_list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( concat_list_o @ Xss2 )
= ( append_list_o @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_list_o )
=> ? [Xss12: list_list_list_o,Xs3: list_list_o,Xs5: list_list_o,Xss23: list_list_list_o] :
( ( Xss2
= ( append_list_list_o @ Xss12 @ ( cons_list_list_o @ ( append_list_o @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_list_o @ ( concat_list_o @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_list_o @ Xs5 @ ( concat_list_o @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_546_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,Xs3: list_nat,Xs5: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_547_concat__eq__appendD,axiom,
! [Xss2: list_list_o,Ys: list_o,Zs: list_o] :
( ( ( concat_o @ Xss2 )
= ( append_o @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_o )
=> ? [Xss12: list_list_o,Xs3: list_o,Xs5: list_o,Xss23: list_list_o] :
( ( Xss2
= ( append_list_o @ Xss12 @ ( cons_list_o @ ( append_o @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_o @ ( concat_o @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_o @ Xs5 @ ( concat_o @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_548_last__prefixes,axiom,
! [Xs: list_o] :
( ( last_list_o @ ( prefixes_o @ Xs ) )
= Xs ) ).
% last_prefixes
thf(fact_549_last__prefixes,axiom,
! [Xs: list_nat] :
( ( last_list_nat @ ( prefixes_nat @ Xs ) )
= Xs ) ).
% last_prefixes
thf(fact_550_last__suffixes,axiom,
! [Xs: list_nat] :
( ( last_list_nat @ ( suffixes_nat @ Xs ) )
= Xs ) ).
% last_suffixes
thf(fact_551_last__appendL,axiom,
! [Ys: list_option_o,Xs: list_option_o] :
( ( Ys = nil_option_o )
=> ( ( last_option_o @ ( append_option_o @ Xs @ Ys ) )
= ( last_option_o @ Xs ) ) ) ).
% last_appendL
thf(fact_552_last__appendL,axiom,
! [Ys: list_list_o,Xs: list_list_o] :
( ( Ys = nil_list_o )
=> ( ( last_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( last_list_o @ Xs ) ) ) ).
% last_appendL
thf(fact_553_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_554_last__appendL,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( Ys = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_555_last__appendL,axiom,
! [Ys: list_o,Xs: list_o] :
( ( Ys = nil_o )
=> ( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( last_o @ Xs ) ) ) ).
% last_appendL
thf(fact_556_last__appendR,axiom,
! [Ys: list_option_o,Xs: list_option_o] :
( ( Ys != nil_option_o )
=> ( ( last_option_o @ ( append_option_o @ Xs @ Ys ) )
= ( last_option_o @ Ys ) ) ) ).
% last_appendR
thf(fact_557_last__appendR,axiom,
! [Ys: list_list_o,Xs: list_list_o] :
( ( Ys != nil_list_o )
=> ( ( last_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( last_list_o @ Ys ) ) ) ).
% last_appendR
thf(fact_558_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_559_last__appendR,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( Ys != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_560_last__appendR,axiom,
! [Ys: list_o,Xs: list_o] :
( ( Ys != nil_o )
=> ( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( last_o @ Ys ) ) ) ).
% last_appendR
thf(fact_561_concat__append,axiom,
! [Xs: list_list_list_o,Ys: list_list_list_o] :
( ( concat_list_o @ ( append_list_list_o @ Xs @ Ys ) )
= ( append_list_o @ ( concat_list_o @ Xs ) @ ( concat_list_o @ Ys ) ) ) ).
% concat_append
thf(fact_562_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_563_concat__append,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( concat_o @ ( append_list_o @ Xs @ Ys ) )
= ( append_o @ ( concat_o @ Xs ) @ ( concat_o @ Ys ) ) ) ).
% concat_append
thf(fact_564_last__snoc,axiom,
! [Xs: list_option_o,X2: option_o] :
( ( last_option_o @ ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) ) )
= X2 ) ).
% last_snoc
thf(fact_565_last__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( last_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_566_last__snoc,axiom,
! [Xs: list_list_o,X2: list_o] :
( ( last_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) )
= X2 ) ).
% last_snoc
thf(fact_567_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_568_last__snoc,axiom,
! [Xs: list_o,X2: $o] :
( ( last_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= X2 ) ).
% last_snoc
thf(fact_569_suffixes__not__Nil,axiom,
! [Xs: list_o] :
( ( suffixes_o @ Xs )
!= nil_list_o ) ).
% suffixes_not_Nil
thf(fact_570_suffixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( suffixes_nat @ Xs )
!= nil_list_nat ) ).
% suffixes_not_Nil
thf(fact_571_concat_Osimps_I2_J,axiom,
! [X2: list_list_o,Xs: list_list_list_o] :
( ( concat_list_o @ ( cons_list_list_o @ X2 @ Xs ) )
= ( append_list_o @ X2 @ ( concat_list_o @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_572_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_573_concat_Osimps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( concat_o @ ( cons_list_o @ X2 @ Xs ) )
= ( append_o @ X2 @ ( concat_o @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_574_last__ConsR,axiom,
! [Xs: list_option_o,X2: option_o] :
( ( Xs != nil_option_o )
=> ( ( last_option_o @ ( cons_option_o @ X2 @ Xs ) )
= ( last_option_o @ Xs ) ) ) ).
% last_ConsR
thf(fact_575_last__ConsR,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_576_last__ConsR,axiom,
! [Xs: list_o,X2: $o] :
( ( Xs != nil_o )
=> ( ( last_o @ ( cons_o @ X2 @ Xs ) )
= ( last_o @ Xs ) ) ) ).
% last_ConsR
thf(fact_577_last__ConsR,axiom,
! [Xs: list_list_o,X2: list_o] :
( ( Xs != nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X2 @ Xs ) )
= ( last_list_o @ Xs ) ) ) ).
% last_ConsR
thf(fact_578_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_579_last__ConsL,axiom,
! [Xs: list_option_o,X2: option_o] :
( ( Xs = nil_option_o )
=> ( ( last_option_o @ ( cons_option_o @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_580_last__ConsL,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_581_last__ConsL,axiom,
! [Xs: list_o,X2: $o] :
( ( Xs = nil_o )
=> ( ( last_o @ ( cons_o @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_582_last__ConsL,axiom,
! [Xs: list_list_o,X2: list_o] :
( ( Xs = nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_583_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_584_last_Osimps,axiom,
! [Xs: list_option_o,X2: option_o] :
( ( ( Xs = nil_option_o )
=> ( ( last_option_o @ ( cons_option_o @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_option_o )
=> ( ( last_option_o @ ( cons_option_o @ X2 @ Xs ) )
= ( last_option_o @ Xs ) ) ) ) ).
% last.simps
thf(fact_585_last_Osimps,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_586_last_Osimps,axiom,
! [X2: $o,Xs: list_o] :
( ( last_o @ ( cons_o @ X2 @ Xs ) )
= ( ( ( Xs = nil_o )
=> X2 )
& ( ( Xs != nil_o )
=> ( last_o @ Xs ) ) ) ) ).
% last.simps
thf(fact_587_last_Osimps,axiom,
! [Xs: list_list_o,X2: list_o] :
( ( ( Xs = nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X2 @ Xs ) )
= ( last_list_o @ Xs ) ) ) ) ).
% last.simps
thf(fact_588_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_589_concat_Osimps_I1_J,axiom,
( ( concat_option_o @ nil_list_option_o )
= nil_option_o ) ).
% concat.simps(1)
thf(fact_590_concat_Osimps_I1_J,axiom,
( ( concat_list_o @ nil_list_list_o )
= nil_list_o ) ).
% concat.simps(1)
thf(fact_591_concat_Osimps_I1_J,axiom,
( ( concat_list_nat @ nil_list_list_nat )
= nil_list_nat ) ).
% concat.simps(1)
thf(fact_592_concat_Osimps_I1_J,axiom,
( ( concat_o @ nil_list_o )
= nil_o ) ).
% concat.simps(1)
thf(fact_593_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_594_last__append,axiom,
! [Ys: list_option_o,Xs: list_option_o] :
( ( ( Ys = nil_option_o )
=> ( ( last_option_o @ ( append_option_o @ Xs @ Ys ) )
= ( last_option_o @ Xs ) ) )
& ( ( Ys != nil_option_o )
=> ( ( last_option_o @ ( append_option_o @ Xs @ Ys ) )
= ( last_option_o @ Ys ) ) ) ) ).
% last_append
thf(fact_595_last__append,axiom,
! [Ys: list_list_o,Xs: list_list_o] :
( ( ( Ys = nil_list_o )
=> ( ( last_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( last_list_o @ Xs ) ) )
& ( ( Ys != nil_list_o )
=> ( ( last_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( last_list_o @ Ys ) ) ) ) ).
% last_append
thf(fact_596_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_597_last__append,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( ( Ys = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Xs ) ) )
& ( ( Ys != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_598_last__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( ( ( Ys = nil_o )
=> ( last_o @ Xs ) )
& ( ( Ys != nil_o )
=> ( last_o @ Ys ) ) ) ) ).
% last_append
thf(fact_599_longest__common__suffix,axiom,
! [Xs: list_option_o,Ys: list_option_o] :
? [Ss: list_option_o,Xs5: list_option_o,Ys5: list_option_o] :
( ( Xs
= ( append_option_o @ Xs5 @ Ss ) )
& ( Ys
= ( append_option_o @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_option_o )
| ( Ys5 = nil_option_o )
| ( ( last_option_o @ Xs5 )
!= ( last_option_o @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_600_longest__common__suffix,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
? [Ss: list_list_o,Xs5: list_list_o,Ys5: list_list_o] :
( ( Xs
= ( append_list_o @ Xs5 @ Ss ) )
& ( Ys
= ( append_list_o @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_list_o )
| ( Ys5 = nil_list_o )
| ( ( last_list_o @ Xs5 )
!= ( last_list_o @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_601_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs5: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Xs5 @ Ss ) )
& ( Ys
= ( append_nat @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( last_nat @ Xs5 )
!= ( last_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_602_longest__common__suffix,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
? [Ss: list_list_nat,Xs5: list_list_nat,Ys5: list_list_nat] :
( ( Xs
= ( append_list_nat @ Xs5 @ Ss ) )
& ( Ys
= ( append_list_nat @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_list_nat )
| ( Ys5 = nil_list_nat )
| ( ( last_list_nat @ Xs5 )
!= ( last_list_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_603_longest__common__suffix,axiom,
! [Xs: list_o,Ys: list_o] :
? [Ss: list_o,Xs5: list_o,Ys5: list_o] :
( ( Xs
= ( append_o @ Xs5 @ Ss ) )
& ( Ys
= ( append_o @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_o )
| ( Ys5 = nil_o )
| ( ( last_o @ Xs5 )
!= ( last_o @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_604_suffixes_Osimps_I1_J,axiom,
( ( suffixes_option_o @ nil_option_o )
= ( cons_list_option_o @ nil_option_o @ nil_list_option_o ) ) ).
% suffixes.simps(1)
thf(fact_605_suffixes_Osimps_I1_J,axiom,
( ( suffixes_list_o @ nil_list_o )
= ( cons_list_list_o @ nil_list_o @ nil_list_list_o ) ) ).
% suffixes.simps(1)
thf(fact_606_suffixes_Osimps_I1_J,axiom,
( ( suffixes_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_607_suffixes_Osimps_I1_J,axiom,
( ( suffixes_o @ nil_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% suffixes.simps(1)
thf(fact_608_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_609_sublists_Osimps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( sublists_list_o @ ( cons_list_o @ X2 @ Xs ) )
= ( append_list_list_o @ ( sublists_list_o @ Xs ) @ ( map_li5464752787020143300list_o @ ( cons_list_o @ X2 ) @ ( prefixes_list_o @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_610_sublists_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( sublists_o @ ( cons_o @ X2 @ Xs ) )
= ( append_list_o @ ( sublists_o @ Xs ) @ ( map_list_o_list_o @ ( cons_o @ X2 ) @ ( prefixes_o @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_611_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_612_rotate1_Osimps_I2_J,axiom,
! [X2: option_o,Xs: list_option_o] :
( ( rotate1_option_o @ ( cons_option_o @ X2 @ Xs ) )
= ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) ) ) ).
% rotate1.simps(2)
thf(fact_613_rotate1_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( rotate1_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_614_rotate1_Osimps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( rotate1_list_o @ ( cons_list_o @ X2 @ Xs ) )
= ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) ) ).
% rotate1.simps(2)
thf(fact_615_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_616_rotate1_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( rotate1_o @ ( cons_o @ X2 @ Xs ) )
= ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) ) ).
% rotate1.simps(2)
thf(fact_617_concat__conv__foldr,axiom,
( concat_option_o
= ( ^ [Xss3: list_list_option_o] : ( foldr_7481994381061804839tion_o @ append_option_o @ Xss3 @ nil_option_o ) ) ) ).
% concat_conv_foldr
thf(fact_618_concat__conv__foldr,axiom,
( concat_list_o
= ( ^ [Xss3: list_list_list_o] : ( foldr_4660670129333661223list_o @ append_list_o @ Xss3 @ nil_list_o ) ) ) ).
% concat_conv_foldr
thf(fact_619_concat__conv__foldr,axiom,
( concat_nat
= ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).
% concat_conv_foldr
thf(fact_620_concat__conv__foldr,axiom,
( concat_list_nat
= ( ^ [Xss3: list_list_list_nat] : ( foldr_7810631890432470921st_nat @ append_list_nat @ Xss3 @ nil_list_nat ) ) ) ).
% concat_conv_foldr
thf(fact_621_concat__conv__foldr,axiom,
( concat_o
= ( ^ [Xss3: list_list_o] : ( foldr_list_o_list_o @ append_o @ Xss3 @ nil_o ) ) ) ).
% concat_conv_foldr
thf(fact_622_prefixes_Osimps_I2_J,axiom,
! [X2: option_o,Xs: list_option_o] :
( ( prefixes_option_o @ ( cons_option_o @ X2 @ Xs ) )
= ( cons_list_option_o @ nil_option_o @ ( map_li7029392385944470980tion_o @ ( cons_option_o @ X2 ) @ ( prefixes_option_o @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_623_prefixes_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( prefixes_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_list_nat @ nil_list_nat @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X2 ) @ ( prefixes_list_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_624_prefixes_Osimps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( prefixes_list_o @ ( cons_list_o @ X2 @ Xs ) )
= ( cons_list_list_o @ nil_list_o @ ( map_li5464752787020143300list_o @ ( cons_list_o @ X2 ) @ ( prefixes_list_o @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_625_prefixes_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( prefixes_o @ ( cons_o @ X2 @ Xs ) )
= ( cons_list_o @ nil_o @ ( map_list_o_list_o @ ( cons_o @ X2 ) @ ( prefixes_o @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_626_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_627_prefix__snocD,axiom,
! [Xs: list_option_o,X2: option_o,Ys: list_option_o] :
( ( prefix_option_o @ ( append_option_o @ Xs @ ( cons_option_o @ X2 @ nil_option_o ) ) @ Ys )
=> ( strict4666211328587260336tion_o @ Xs @ Ys ) ) ).
% prefix_snocD
thf(fact_628_prefix__snocD,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ Ys )
=> ( strict6380099159608944142st_nat @ Xs @ Ys ) ) ).
% prefix_snocD
thf(fact_629_prefix__snocD,axiom,
! [Xs: list_list_o,X2: list_o,Ys: list_list_o] :
( ( prefix_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) ) @ Ys )
=> ( strict_prefix_list_o @ Xs @ Ys ) ) ).
% prefix_snocD
thf(fact_630_prefix__snocD,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ Ys )
=> ( strict_prefix_nat @ Xs @ Ys ) ) ).
% prefix_snocD
thf(fact_631_prefix__snocD,axiom,
! [Xs: list_o,X2: $o,Ys: list_o] :
( ( prefix_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ Ys )
=> ( strict_prefix_o @ Xs @ Ys ) ) ).
% prefix_snocD
thf(fact_632_map__tailrec__rev_Oelims,axiom,
! [X2: $o > $o,Xa: list_o,Xb: list_o,Y3: list_o] :
( ( ( map_tailrec_rev_o_o @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_o )
=> ( Y3 != Xb ) )
=> ~ ! [A4: $o,As: list_o] :
( ( Xa
= ( cons_o @ A4 @ As ) )
=> ( Y3
!= ( map_tailrec_rev_o_o @ X2 @ As @ ( cons_o @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_633_map__tailrec__rev_Oelims,axiom,
! [X2: $o > nat,Xa: list_o,Xb: list_nat,Y3: list_nat] :
( ( ( map_ta6980961046827203379_o_nat @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_o )
=> ( Y3 != Xb ) )
=> ~ ! [A4: $o,As: list_o] :
( ( Xa
= ( cons_o @ A4 @ As ) )
=> ( Y3
!= ( map_ta6980961046827203379_o_nat @ X2 @ As @ ( cons_nat @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_634_map__tailrec__rev_Oelims,axiom,
! [X2: nat > $o,Xa: list_nat,Xb: list_o,Y3: list_o] :
( ( ( map_ta860260371080612881_nat_o @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_nat )
=> ( Y3 != Xb ) )
=> ~ ! [A4: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A4 @ As ) )
=> ( Y3
!= ( map_ta860260371080612881_nat_o @ X2 @ As @ ( cons_o @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_635_map__tailrec__rev_Oelims,axiom,
! [X2: nat > nat,Xa: list_nat,Xb: list_nat,Y3: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_nat )
=> ( Y3 != Xb ) )
=> ~ ! [A4: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A4 @ As ) )
=> ( Y3
!= ( map_ta7164188454487880599at_nat @ X2 @ As @ ( cons_nat @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_636_map__tailrec__rev_Oelims,axiom,
! [X2: option_o > $o,Xa: list_option_o,Xb: list_o,Y3: list_o] :
( ( ( map_ta2677093932182474799on_o_o @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_option_o )
=> ( Y3 != Xb ) )
=> ~ ! [A4: option_o,As: list_option_o] :
( ( Xa
= ( cons_option_o @ A4 @ As ) )
=> ( Y3
!= ( map_ta2677093932182474799on_o_o @ X2 @ As @ ( cons_o @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_637_map__tailrec__rev_Oelims,axiom,
! [X2: list_nat > $o,Xa: list_list_nat,Xb: list_o,Y3: list_o] :
( ( ( map_ta2047065599740801025_nat_o @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_list_nat )
=> ( Y3 != Xb ) )
=> ~ ! [A4: list_nat,As: list_list_nat] :
( ( Xa
= ( cons_list_nat @ A4 @ As ) )
=> ( Y3
!= ( map_ta2047065599740801025_nat_o @ X2 @ As @ ( cons_o @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_638_map__tailrec__rev_Oelims,axiom,
! [X2: option_o > nat,Xa: list_option_o,Xb: list_nat,Y3: list_nat] :
( ( ( map_ta909136395571573497_o_nat @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_option_o )
=> ( Y3 != Xb ) )
=> ~ ! [A4: option_o,As: list_option_o] :
( ( Xa
= ( cons_option_o @ A4 @ As ) )
=> ( Y3
!= ( map_ta909136395571573497_o_nat @ X2 @ As @ ( cons_nat @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_639_map__tailrec__rev_Oelims,axiom,
! [X2: list_nat > nat,Xa: list_list_nat,Xb: list_nat,Y3: list_nat] :
( ( ( map_ta3159490279798172583at_nat @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_list_nat )
=> ( Y3 != Xb ) )
=> ~ ! [A4: list_nat,As: list_list_nat] :
( ( Xa
= ( cons_list_nat @ A4 @ As ) )
=> ( Y3
!= ( map_ta3159490279798172583at_nat @ X2 @ As @ ( cons_nat @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_640_map__tailrec__rev_Oelims,axiom,
! [X2: $o > list_o,Xa: list_o,Xb: list_list_o,Y3: list_list_o] :
( ( ( map_ta3905426649961562107list_o @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_o )
=> ( Y3 != Xb ) )
=> ~ ! [A4: $o,As: list_o] :
( ( Xa
= ( cons_o @ A4 @ As ) )
=> ( Y3
!= ( map_ta3905426649961562107list_o @ X2 @ As @ ( cons_list_o @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_641_map__tailrec__rev_Oelims,axiom,
! [X2: list_o > $o,Xa: list_list_o,Xb: list_o,Y3: list_o] :
( ( ( map_ta8539203598313750511st_o_o @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa = nil_list_o )
=> ( Y3 != Xb ) )
=> ~ ! [A4: list_o,As: list_list_o] :
( ( Xa
= ( cons_list_o @ A4 @ As ) )
=> ( Y3
!= ( map_ta8539203598313750511st_o_o @ X2 @ As @ ( cons_o @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_642_SuccI,axiom,
! [Kl: list_option_o,K: option_o,Kl2: set_list_option_o] :
( ( member_list_option_o @ ( append_option_o @ Kl @ ( cons_option_o @ K @ nil_option_o ) ) @ Kl2 )
=> ( member_option_o2 @ K @ ( bNF_Gr5056813354227606905tion_o @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_643_SuccI,axiom,
! [Kl: list_list_nat,K: list_nat,Kl2: set_list_list_nat] :
( ( member_list_list_nat @ ( append_list_nat @ Kl @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl2 )
=> ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_644_SuccI,axiom,
! [Kl: list_list_o,K: list_o,Kl2: set_list_list_o] :
( ( member_list_list_o @ ( append_list_o @ Kl @ ( cons_list_o @ K @ nil_list_o ) ) @ Kl2 )
=> ( member_list_o2 @ K @ ( bNF_Gr2388622420913600185list_o @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_645_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat2 @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_646_SuccI,axiom,
! [Kl: list_o,K: $o,Kl2: set_list_o] :
( ( member_list_o2 @ ( append_o @ Kl @ ( cons_o @ K @ nil_o ) ) @ Kl2 )
=> ( member_o2 @ K @ ( bNF_Greatest_Succ_o @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_647_list_Omap__disc__iff,axiom,
! [F: nat > $o,A: list_nat] :
( ( ( map_nat_o @ F @ A )
= nil_o )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_648_list_Omap__disc__iff,axiom,
! [F: $o > nat,A: list_o] :
( ( ( map_o_nat @ F @ A )
= nil_nat )
= ( A = nil_o ) ) ).
% list.map_disc_iff
thf(fact_649_list_Omap__disc__iff,axiom,
! [F: $o > $o,A: list_o] :
( ( ( map_o_o @ F @ A )
= nil_o )
= ( A = nil_o ) ) ).
% list.map_disc_iff
thf(fact_650_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_651_list_Omap__disc__iff,axiom,
! [F: option_o > $o,A: list_option_o] :
( ( ( map_option_o_o @ F @ A )
= nil_o )
= ( A = nil_option_o ) ) ).
% list.map_disc_iff
thf(fact_652_list_Omap__disc__iff,axiom,
! [F: list_o > $o,A: list_list_o] :
( ( ( map_list_o_o @ F @ A )
= nil_o )
= ( A = nil_list_o ) ) ).
% list.map_disc_iff
thf(fact_653_list_Omap__disc__iff,axiom,
! [F: list_nat > $o,A: list_list_nat] :
( ( ( map_list_nat_o @ F @ A )
= nil_o )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_654_list_Omap__disc__iff,axiom,
! [F: $o > option_o,A: list_o] :
( ( ( map_o_option_o @ F @ A )
= nil_option_o )
= ( A = nil_o ) ) ).
% list.map_disc_iff
thf(fact_655_list_Omap__disc__iff,axiom,
! [F: nat > option_o,A: list_nat] :
( ( ( map_nat_option_o @ F @ A )
= nil_option_o )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_656_list_Omap__disc__iff,axiom,
! [F: $o > list_o,A: list_o] :
( ( ( map_o_list_o @ F @ A )
= nil_list_o )
= ( A = nil_o ) ) ).
% list.map_disc_iff
thf(fact_657_Nil__is__map__conv,axiom,
! [F: nat > $o,Xs: list_nat] :
( ( nil_o
= ( map_nat_o @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_658_Nil__is__map__conv,axiom,
! [F: $o > nat,Xs: list_o] :
( ( nil_nat
= ( map_o_nat @ F @ Xs ) )
= ( Xs = nil_o ) ) ).
% Nil_is_map_conv
thf(fact_659_Nil__is__map__conv,axiom,
! [F: $o > $o,Xs: list_o] :
( ( nil_o
= ( map_o_o @ F @ Xs ) )
= ( Xs = nil_o ) ) ).
% Nil_is_map_conv
thf(fact_660_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_661_Nil__is__map__conv,axiom,
! [F: option_o > $o,Xs: list_option_o] :
( ( nil_o
= ( map_option_o_o @ F @ Xs ) )
= ( Xs = nil_option_o ) ) ).
% Nil_is_map_conv
thf(fact_662_Nil__is__map__conv,axiom,
! [F: list_o > $o,Xs: list_list_o] :
( ( nil_o
= ( map_list_o_o @ F @ Xs ) )
= ( Xs = nil_list_o ) ) ).
% Nil_is_map_conv
thf(fact_663_Nil__is__map__conv,axiom,
! [F: list_nat > $o,Xs: list_list_nat] :
( ( nil_o
= ( map_list_nat_o @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_664_Nil__is__map__conv,axiom,
! [F: $o > option_o,Xs: list_o] :
( ( nil_option_o
= ( map_o_option_o @ F @ Xs ) )
= ( Xs = nil_o ) ) ).
% Nil_is_map_conv
thf(fact_665_Nil__is__map__conv,axiom,
! [F: nat > option_o,Xs: list_nat] :
( ( nil_option_o
= ( map_nat_option_o @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_666_Nil__is__map__conv,axiom,
! [F: $o > list_o,Xs: list_o] :
( ( nil_list_o
= ( map_o_list_o @ F @ Xs ) )
= ( Xs = nil_o ) ) ).
% Nil_is_map_conv
thf(fact_667_map__is__Nil__conv,axiom,
! [F: nat > $o,Xs: list_nat] :
( ( ( map_nat_o @ F @ Xs )
= nil_o )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_668_map__is__Nil__conv,axiom,
! [F: $o > nat,Xs: list_o] :
( ( ( map_o_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_o ) ) ).
% map_is_Nil_conv
thf(fact_669_map__is__Nil__conv,axiom,
! [F: $o > $o,Xs: list_o] :
( ( ( map_o_o @ F @ Xs )
= nil_o )
= ( Xs = nil_o ) ) ).
% map_is_Nil_conv
thf(fact_670_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_671_map__is__Nil__conv,axiom,
! [F: option_o > $o,Xs: list_option_o] :
( ( ( map_option_o_o @ F @ Xs )
= nil_o )
= ( Xs = nil_option_o ) ) ).
% map_is_Nil_conv
thf(fact_672_map__is__Nil__conv,axiom,
! [F: list_o > $o,Xs: list_list_o] :
( ( ( map_list_o_o @ F @ Xs )
= nil_o )
= ( Xs = nil_list_o ) ) ).
% map_is_Nil_conv
thf(fact_673_map__is__Nil__conv,axiom,
! [F: list_nat > $o,Xs: list_list_nat] :
( ( ( map_list_nat_o @ F @ Xs )
= nil_o )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_674_map__is__Nil__conv,axiom,
! [F: $o > option_o,Xs: list_o] :
( ( ( map_o_option_o @ F @ Xs )
= nil_option_o )
= ( Xs = nil_o ) ) ).
% map_is_Nil_conv
thf(fact_675_map__is__Nil__conv,axiom,
! [F: nat > option_o,Xs: list_nat] :
( ( ( map_nat_option_o @ F @ Xs )
= nil_option_o )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_676_map__is__Nil__conv,axiom,
! [F: $o > list_o,Xs: list_o] :
( ( ( map_o_list_o @ F @ Xs )
= nil_list_o )
= ( Xs = nil_o ) ) ).
% map_is_Nil_conv
thf(fact_677_strict__prefix__simps_I3_J,axiom,
! [X2: list_o,Xs: list_list_o,Y3: list_o,Ys: list_list_o] :
( ( strict_prefix_list_o @ ( cons_list_o @ X2 @ Xs ) @ ( cons_list_o @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
& ( strict_prefix_list_o @ Xs @ Ys ) ) ) ).
% strict_prefix_simps(3)
thf(fact_678_strict__prefix__simps_I3_J,axiom,
! [X2: $o,Xs: list_o,Y3: $o,Ys: list_o] :
( ( strict_prefix_o @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
& ( strict_prefix_o @ Xs @ Ys ) ) ) ).
% strict_prefix_simps(3)
thf(fact_679_strict__prefix__simps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( strict_prefix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
& ( strict_prefix_nat @ Xs @ Ys ) ) ) ).
% strict_prefix_simps(3)
thf(fact_680_strict__prefix__simps_I1_J,axiom,
! [Xs: list_option_o] :
~ ( strict4666211328587260336tion_o @ Xs @ nil_option_o ) ).
% strict_prefix_simps(1)
thf(fact_681_strict__prefix__simps_I1_J,axiom,
! [Xs: list_list_o] :
~ ( strict_prefix_list_o @ Xs @ nil_list_o ) ).
% strict_prefix_simps(1)
thf(fact_682_strict__prefix__simps_I1_J,axiom,
! [Xs: list_list_nat] :
~ ( strict6380099159608944142st_nat @ Xs @ nil_list_nat ) ).
% strict_prefix_simps(1)
thf(fact_683_strict__prefix__simps_I1_J,axiom,
! [Xs: list_o] :
~ ( strict_prefix_o @ Xs @ nil_o ) ).
% strict_prefix_simps(1)
thf(fact_684_strict__prefix__simps_I1_J,axiom,
! [Xs: list_nat] :
~ ( strict_prefix_nat @ Xs @ nil_nat ) ).
% strict_prefix_simps(1)
thf(fact_685_map__append,axiom,
! [F: $o > list_o,Xs: list_o,Ys: list_o] :
( ( map_o_list_o @ F @ ( append_o @ Xs @ Ys ) )
= ( append_list_o @ ( map_o_list_o @ F @ Xs ) @ ( map_o_list_o @ F @ Ys ) ) ) ).
% map_append
thf(fact_686_map__append,axiom,
! [F: $o > nat,Xs: list_o,Ys: list_o] :
( ( map_o_nat @ F @ ( append_o @ Xs @ Ys ) )
= ( append_nat @ ( map_o_nat @ F @ Xs ) @ ( map_o_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_687_map__append,axiom,
! [F: list_o > $o,Xs: list_list_o,Ys: list_list_o] :
( ( map_list_o_o @ F @ ( append_list_o @ Xs @ Ys ) )
= ( append_o @ ( map_list_o_o @ F @ Xs ) @ ( map_list_o_o @ F @ Ys ) ) ) ).
% map_append
thf(fact_688_map__append,axiom,
! [F: list_o > list_o,Xs: list_list_o,Ys: list_list_o] :
( ( map_list_o_list_o @ F @ ( append_list_o @ Xs @ Ys ) )
= ( append_list_o @ ( map_list_o_list_o @ F @ Xs ) @ ( map_list_o_list_o @ F @ Ys ) ) ) ).
% map_append
thf(fact_689_map__append,axiom,
! [F: list_o > nat,Xs: list_list_o,Ys: list_list_o] :
( ( map_list_o_nat @ F @ ( append_list_o @ Xs @ Ys ) )
= ( append_nat @ ( map_list_o_nat @ F @ Xs ) @ ( map_list_o_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_690_map__append,axiom,
! [F: nat > $o,Xs: list_nat,Ys: list_nat] :
( ( map_nat_o @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_o @ ( map_nat_o @ F @ Xs ) @ ( map_nat_o @ F @ Ys ) ) ) ).
% map_append
thf(fact_691_map__append,axiom,
! [F: nat > list_o,Xs: list_nat,Ys: list_nat] :
( ( map_nat_list_o @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_list_o @ ( map_nat_list_o @ F @ Xs ) @ ( map_nat_list_o @ F @ Ys ) ) ) ).
% map_append
thf(fact_692_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_693_map__append,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( map_list_nat_nat @ F @ ( append_list_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_list_nat_nat @ F @ Xs ) @ ( map_list_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_694_map__append,axiom,
! [F: $o > $o,Xs: list_o,Ys: list_o] :
( ( map_o_o @ F @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( map_o_o @ F @ Xs ) @ ( map_o_o @ F @ Ys ) ) ) ).
% map_append
thf(fact_695_foldr__append,axiom,
! [F: list_o > list_o > list_o,Xs: list_list_o,Ys: list_list_o,A: list_o] :
( ( foldr_list_o_list_o @ F @ ( append_list_o @ Xs @ Ys ) @ A )
= ( foldr_list_o_list_o @ F @ Xs @ ( foldr_list_o_list_o @ F @ Ys @ A ) ) ) ).
% foldr_append
thf(fact_696_foldr__append,axiom,
! [F: nat > nat > nat,Xs: list_nat,Ys: list_nat,A: nat] :
( ( foldr_nat_nat @ F @ ( append_nat @ Xs @ Ys ) @ A )
= ( foldr_nat_nat @ F @ Xs @ ( foldr_nat_nat @ F @ Ys @ A ) ) ) ).
% foldr_append
thf(fact_697_rotate1__is__Nil__conv,axiom,
! [Xs: list_o] :
( ( ( rotate1_o @ Xs )
= nil_o )
= ( Xs = nil_o ) ) ).
% rotate1_is_Nil_conv
thf(fact_698_rotate1__is__Nil__conv,axiom,
! [Xs: list_option_o] :
( ( ( rotate1_option_o @ Xs )
= nil_option_o )
= ( Xs = nil_option_o ) ) ).
% rotate1_is_Nil_conv
thf(fact_699_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_o] :
( ( ( rotate1_list_o @ Xs )
= nil_list_o )
= ( Xs = nil_list_o ) ) ).
% rotate1_is_Nil_conv
thf(fact_700_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_701_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( ( rotate1_list_nat @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_702_strict__prefix__simps_I2_J,axiom,
! [X2: option_o,Xs: list_option_o] : ( strict4666211328587260336tion_o @ nil_option_o @ ( cons_option_o @ X2 @ Xs ) ) ).
% strict_prefix_simps(2)
thf(fact_703_strict__prefix__simps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] : ( strict6380099159608944142st_nat @ nil_list_nat @ ( cons_list_nat @ X2 @ Xs ) ) ).
% strict_prefix_simps(2)
thf(fact_704_strict__prefix__simps_I2_J,axiom,
! [X2: list_o,Xs: list_list_o] : ( strict_prefix_list_o @ nil_list_o @ ( cons_list_o @ X2 @ Xs ) ) ).
% strict_prefix_simps(2)
thf(fact_705_strict__prefix__simps_I2_J,axiom,
! [X2: $o,Xs: list_o] : ( strict_prefix_o @ nil_o @ ( cons_o @ X2 @ Xs ) ) ).
% strict_prefix_simps(2)
thf(fact_706_strict__prefix__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat] : ( strict_prefix_nat @ nil_nat @ ( cons_nat @ X2 @ Xs ) ) ).
% strict_prefix_simps(2)
thf(fact_707_map__concat,axiom,
! [F: $o > nat,Xs: list_list_o] :
( ( map_o_nat @ F @ ( concat_o @ Xs ) )
= ( concat_nat @ ( map_list_o_list_nat @ ( map_o_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_708_map__concat,axiom,
! [F: nat > $o,Xs: list_list_nat] :
( ( map_nat_o @ F @ ( concat_nat @ Xs ) )
= ( concat_o @ ( map_list_nat_list_o @ ( map_nat_o @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_709_map__concat,axiom,
! [F: $o > $o,Xs: list_list_o] :
( ( map_o_o @ F @ ( concat_o @ Xs ) )
= ( concat_o @ ( map_list_o_list_o @ ( map_o_o @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_710_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_711_map__concat,axiom,
! [F: list_nat > nat,Xs: list_list_list_nat] :
( ( map_list_nat_nat @ F @ ( concat_list_nat @ Xs ) )
= ( concat_nat @ ( map_li1587113026813354006st_nat @ ( map_list_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_712_prefix__order_Oasym,axiom,
! [A: list_o,B: list_o] :
( ( strict_prefix_o @ A @ B )
=> ~ ( strict_prefix_o @ B @ A ) ) ).
% prefix_order.asym
thf(fact_713_prefix__order_Oasym,axiom,
! [A: list_nat,B: list_nat] :
( ( strict_prefix_nat @ A @ B )
=> ~ ( strict_prefix_nat @ B @ A ) ) ).
% prefix_order.asym
thf(fact_714_prefix__order_Oless__asym,axiom,
! [X2: list_o,Y3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ~ ( strict_prefix_o @ Y3 @ X2 ) ) ).
% prefix_order.less_asym
thf(fact_715_prefix__order_Oless__asym,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ~ ( strict_prefix_nat @ Y3 @ X2 ) ) ).
% prefix_order.less_asym
thf(fact_716_prefix__order_Oless__asym_H,axiom,
! [A: list_o,B: list_o] :
( ( strict_prefix_o @ A @ B )
=> ~ ( strict_prefix_o @ B @ A ) ) ).
% prefix_order.less_asym'
thf(fact_717_prefix__order_Oless__asym_H,axiom,
! [A: list_nat,B: list_nat] :
( ( strict_prefix_nat @ A @ B )
=> ~ ( strict_prefix_nat @ B @ A ) ) ).
% prefix_order.less_asym'
thf(fact_718_prefix__order_Oless__trans,axiom,
! [X2: list_o,Y3: list_o,Z3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ( ( strict_prefix_o @ Y3 @ Z3 )
=> ( strict_prefix_o @ X2 @ Z3 ) ) ) ).
% prefix_order.less_trans
thf(fact_719_prefix__order_Oless__trans,axiom,
! [X2: list_nat,Y3: list_nat,Z3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ( ( strict_prefix_nat @ Y3 @ Z3 )
=> ( strict_prefix_nat @ X2 @ Z3 ) ) ) ).
% prefix_order.less_trans
thf(fact_720_prefix__order_Oless__irrefl,axiom,
! [X2: list_o] :
~ ( strict_prefix_o @ X2 @ X2 ) ).
% prefix_order.less_irrefl
thf(fact_721_prefix__order_Oless__irrefl,axiom,
! [X2: list_nat] :
~ ( strict_prefix_nat @ X2 @ X2 ) ).
% prefix_order.less_irrefl
thf(fact_722_prefix__order_Oless__imp__neq,axiom,
! [X2: list_o,Y3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% prefix_order.less_imp_neq
thf(fact_723_prefix__order_Oless__imp__neq,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% prefix_order.less_imp_neq
thf(fact_724_prefix__order_Oless__not__sym,axiom,
! [X2: list_o,Y3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ~ ( strict_prefix_o @ Y3 @ X2 ) ) ).
% prefix_order.less_not_sym
thf(fact_725_prefix__order_Oless__not__sym,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ~ ( strict_prefix_nat @ Y3 @ X2 ) ) ).
% prefix_order.less_not_sym
thf(fact_726_prefix__order_Ostrict__trans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( strict_prefix_o @ A @ B )
=> ( ( strict_prefix_o @ B @ C )
=> ( strict_prefix_o @ A @ C ) ) ) ).
% prefix_order.strict_trans
thf(fact_727_prefix__order_Ostrict__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( strict_prefix_nat @ A @ B )
=> ( ( strict_prefix_nat @ B @ C )
=> ( strict_prefix_nat @ A @ C ) ) ) ).
% prefix_order.strict_trans
thf(fact_728_prefix__order_Oless__imp__triv,axiom,
! [X2: list_o,Y3: list_o,P: $o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ( ( strict_prefix_o @ Y3 @ X2 )
=> P ) ) ).
% prefix_order.less_imp_triv
thf(fact_729_prefix__order_Oless__imp__triv,axiom,
! [X2: list_nat,Y3: list_nat,P: $o] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ( ( strict_prefix_nat @ Y3 @ X2 )
=> P ) ) ).
% prefix_order.less_imp_triv
thf(fact_730_prefix__order_Odual__order_Oasym,axiom,
! [B: list_o,A: list_o] :
( ( strict_prefix_o @ B @ A )
=> ~ ( strict_prefix_o @ A @ B ) ) ).
% prefix_order.dual_order.asym
thf(fact_731_prefix__order_Odual__order_Oasym,axiom,
! [B: list_nat,A: list_nat] :
( ( strict_prefix_nat @ B @ A )
=> ~ ( strict_prefix_nat @ A @ B ) ) ).
% prefix_order.dual_order.asym
thf(fact_732_prefix__order_Oless__imp__not__eq,axiom,
! [X2: list_o,Y3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% prefix_order.less_imp_not_eq
thf(fact_733_prefix__order_Oless__imp__not__eq,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% prefix_order.less_imp_not_eq
thf(fact_734_prefix__order_Oless__imp__not__eq2,axiom,
! [X2: list_o,Y3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% prefix_order.less_imp_not_eq2
thf(fact_735_prefix__order_Oless__imp__not__eq2,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% prefix_order.less_imp_not_eq2
thf(fact_736_prefix__order_Odual__order_Oirrefl,axiom,
! [A: list_o] :
~ ( strict_prefix_o @ A @ A ) ).
% prefix_order.dual_order.irrefl
thf(fact_737_prefix__order_Odual__order_Oirrefl,axiom,
! [A: list_nat] :
~ ( strict_prefix_nat @ A @ A ) ).
% prefix_order.dual_order.irrefl
thf(fact_738_prefix__order_Oless__imp__not__less,axiom,
! [X2: list_o,Y3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ~ ( strict_prefix_o @ Y3 @ X2 ) ) ).
% prefix_order.less_imp_not_less
thf(fact_739_prefix__order_Oless__imp__not__less,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ~ ( strict_prefix_nat @ Y3 @ X2 ) ) ).
% prefix_order.less_imp_not_less
thf(fact_740_prefix__order_Oord__eq__less__trans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( A = B )
=> ( ( strict_prefix_o @ B @ C )
=> ( strict_prefix_o @ A @ C ) ) ) ).
% prefix_order.ord_eq_less_trans
thf(fact_741_prefix__order_Oord__eq__less__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( strict_prefix_nat @ B @ C )
=> ( strict_prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_eq_less_trans
thf(fact_742_prefix__order_Oord__less__eq__trans,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( strict_prefix_o @ A @ B )
=> ( ( B = C )
=> ( strict_prefix_o @ A @ C ) ) ) ).
% prefix_order.ord_less_eq_trans
thf(fact_743_prefix__order_Oord__less__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( strict_prefix_nat @ A @ B )
=> ( ( B = C )
=> ( strict_prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_less_eq_trans
thf(fact_744_prefix__order_Ostrict__implies__not__eq,axiom,
! [A: list_o,B: list_o] :
( ( strict_prefix_o @ A @ B )
=> ( A != B ) ) ).
% prefix_order.strict_implies_not_eq
thf(fact_745_prefix__order_Ostrict__implies__not__eq,axiom,
! [A: list_nat,B: list_nat] :
( ( strict_prefix_nat @ A @ B )
=> ( A != B ) ) ).
% prefix_order.strict_implies_not_eq
thf(fact_746_prefix__order_Odual__order_Ostrict__trans,axiom,
! [B: list_o,A: list_o,C: list_o] :
( ( strict_prefix_o @ B @ A )
=> ( ( strict_prefix_o @ C @ B )
=> ( strict_prefix_o @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans
thf(fact_747_prefix__order_Odual__order_Ostrict__trans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( strict_prefix_nat @ B @ A )
=> ( ( strict_prefix_nat @ C @ B )
=> ( strict_prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans
thf(fact_748_prefix__order_Odual__order_Ostrict__implies__not__eq,axiom,
! [B: list_o,A: list_o] :
( ( strict_prefix_o @ B @ A )
=> ( A != B ) ) ).
% prefix_order.dual_order.strict_implies_not_eq
thf(fact_749_prefix__order_Odual__order_Ostrict__implies__not__eq,axiom,
! [B: list_nat,A: list_nat] :
( ( strict_prefix_nat @ B @ A )
=> ( A != B ) ) ).
% prefix_order.dual_order.strict_implies_not_eq
thf(fact_750_rotate1__map,axiom,
! [F: nat > $o,Xs: list_nat] :
( ( rotate1_o @ ( map_nat_o @ F @ Xs ) )
= ( map_nat_o @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_751_rotate1__map,axiom,
! [F: $o > nat,Xs: list_o] :
( ( rotate1_nat @ ( map_o_nat @ F @ Xs ) )
= ( map_o_nat @ F @ ( rotate1_o @ Xs ) ) ) ).
% rotate1_map
thf(fact_752_rotate1__map,axiom,
! [F: $o > $o,Xs: list_o] :
( ( rotate1_o @ ( map_o_o @ F @ Xs ) )
= ( map_o_o @ F @ ( rotate1_o @ Xs ) ) ) ).
% rotate1_map
thf(fact_753_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_754_rotate1__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( rotate1_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( map_list_nat_nat @ F @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_755_list_Osimps_I9_J,axiom,
! [F: list_nat > nat,X21: list_nat,X222: list_list_nat] :
( ( map_list_nat_nat @ F @ ( cons_list_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_list_nat_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_756_list_Osimps_I9_J,axiom,
! [F: $o > $o,X21: $o,X222: list_o] :
( ( map_o_o @ F @ ( cons_o @ X21 @ X222 ) )
= ( cons_o @ ( F @ X21 ) @ ( map_o_o @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_757_list_Osimps_I9_J,axiom,
! [F: $o > list_o,X21: $o,X222: list_o] :
( ( map_o_list_o @ F @ ( cons_o @ X21 @ X222 ) )
= ( cons_list_o @ ( F @ X21 ) @ ( map_o_list_o @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_758_list_Osimps_I9_J,axiom,
! [F: $o > nat,X21: $o,X222: list_o] :
( ( map_o_nat @ F @ ( cons_o @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_o_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_759_list_Osimps_I9_J,axiom,
! [F: list_o > $o,X21: list_o,X222: list_list_o] :
( ( map_list_o_o @ F @ ( cons_list_o @ X21 @ X222 ) )
= ( cons_o @ ( F @ X21 ) @ ( map_list_o_o @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_760_list_Osimps_I9_J,axiom,
! [F: list_o > list_o,X21: list_o,X222: list_list_o] :
( ( map_list_o_list_o @ F @ ( cons_list_o @ X21 @ X222 ) )
= ( cons_list_o @ ( F @ X21 ) @ ( map_list_o_list_o @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_761_list_Osimps_I9_J,axiom,
! [F: list_o > nat,X21: list_o,X222: list_list_o] :
( ( map_list_o_nat @ F @ ( cons_list_o @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_list_o_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_762_list_Osimps_I9_J,axiom,
! [F: nat > $o,X21: nat,X222: list_nat] :
( ( map_nat_o @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_o @ ( F @ X21 ) @ ( map_nat_o @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_763_list_Osimps_I9_J,axiom,
! [F: nat > list_o,X21: nat,X222: list_nat] :
( ( map_nat_list_o @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_list_o @ ( F @ X21 ) @ ( map_nat_list_o @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_764_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_765_Cons__eq__map__D,axiom,
! [X2: $o,Xs: list_o,F: $o > $o,Ys: list_o] :
( ( ( cons_o @ X2 @ Xs )
= ( map_o_o @ F @ Ys ) )
=> ? [Z5: $o,Zs2: list_o] :
( ( Ys
= ( cons_o @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_o_o @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_766_Cons__eq__map__D,axiom,
! [X2: $o,Xs: list_o,F: list_o > $o,Ys: list_list_o] :
( ( ( cons_o @ X2 @ Xs )
= ( map_list_o_o @ F @ Ys ) )
=> ? [Z5: list_o,Zs2: list_list_o] :
( ( Ys
= ( cons_list_o @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_list_o_o @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_767_Cons__eq__map__D,axiom,
! [X2: $o,Xs: list_o,F: nat > $o,Ys: list_nat] :
( ( ( cons_o @ X2 @ Xs )
= ( map_nat_o @ F @ Ys ) )
=> ? [Z5: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_nat_o @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_768_Cons__eq__map__D,axiom,
! [X2: list_o,Xs: list_list_o,F: $o > list_o,Ys: list_o] :
( ( ( cons_list_o @ X2 @ Xs )
= ( map_o_list_o @ F @ Ys ) )
=> ? [Z5: $o,Zs2: list_o] :
( ( Ys
= ( cons_o @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_o_list_o @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_769_Cons__eq__map__D,axiom,
! [X2: list_o,Xs: list_list_o,F: list_o > list_o,Ys: list_list_o] :
( ( ( cons_list_o @ X2 @ Xs )
= ( map_list_o_list_o @ F @ Ys ) )
=> ? [Z5: list_o,Zs2: list_list_o] :
( ( Ys
= ( cons_list_o @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_list_o_list_o @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_770_Cons__eq__map__D,axiom,
! [X2: list_o,Xs: list_list_o,F: nat > list_o,Ys: list_nat] :
( ( ( cons_list_o @ X2 @ Xs )
= ( map_nat_list_o @ F @ Ys ) )
=> ? [Z5: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_nat_list_o @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_771_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_list_nat_nat @ F @ Ys ) )
=> ? [Z5: list_nat,Zs2: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_list_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_772_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: $o > nat,Ys: list_o] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_o_nat @ F @ Ys ) )
=> ? [Z5: $o,Zs2: list_o] :
( ( Ys
= ( cons_o @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_o_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_773_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: list_o > nat,Ys: list_list_o] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_list_o_nat @ F @ Ys ) )
=> ? [Z5: list_o,Zs2: list_list_o] :
( ( Ys
= ( cons_list_o @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_list_o_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_774_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 ) )
=> ? [Z5: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs2 ) )
& ( X2
= ( F @ Z5 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_775_map__eq__Cons__D,axiom,
! [F: $o > $o,Xs: list_o,Y3: $o,Ys: list_o] :
( ( ( map_o_o @ F @ Xs )
= ( cons_o @ Y3 @ Ys ) )
=> ? [Z5: $o,Zs2: list_o] :
( ( Xs
= ( cons_o @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_o_o @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_776_map__eq__Cons__D,axiom,
! [F: list_o > $o,Xs: list_list_o,Y3: $o,Ys: list_o] :
( ( ( map_list_o_o @ F @ Xs )
= ( cons_o @ Y3 @ Ys ) )
=> ? [Z5: list_o,Zs2: list_list_o] :
( ( Xs
= ( cons_list_o @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_list_o_o @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_777_map__eq__Cons__D,axiom,
! [F: nat > $o,Xs: list_nat,Y3: $o,Ys: list_o] :
( ( ( map_nat_o @ F @ Xs )
= ( cons_o @ Y3 @ Ys ) )
=> ? [Z5: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_nat_o @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_778_map__eq__Cons__D,axiom,
! [F: $o > list_o,Xs: list_o,Y3: list_o,Ys: list_list_o] :
( ( ( map_o_list_o @ F @ Xs )
= ( cons_list_o @ Y3 @ Ys ) )
=> ? [Z5: $o,Zs2: list_o] :
( ( Xs
= ( cons_o @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_o_list_o @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_779_map__eq__Cons__D,axiom,
! [F: list_o > list_o,Xs: list_list_o,Y3: list_o,Ys: list_list_o] :
( ( ( map_list_o_list_o @ F @ Xs )
= ( cons_list_o @ Y3 @ Ys ) )
=> ? [Z5: list_o,Zs2: list_list_o] :
( ( Xs
= ( cons_list_o @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_list_o_list_o @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_780_map__eq__Cons__D,axiom,
! [F: nat > list_o,Xs: list_nat,Y3: list_o,Ys: list_list_o] :
( ( ( map_nat_list_o @ F @ Xs )
= ( cons_list_o @ Y3 @ Ys ) )
=> ? [Z5: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_nat_list_o @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_781_map__eq__Cons__D,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ? [Z5: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_list_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_782_map__eq__Cons__D,axiom,
! [F: $o > nat,Xs: list_o,Y3: nat,Ys: list_nat] :
( ( ( map_o_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ? [Z5: $o,Zs2: list_o] :
( ( Xs
= ( cons_o @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_o_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_783_map__eq__Cons__D,axiom,
! [F: list_o > nat,Xs: list_list_o,Y3: nat,Ys: list_nat] :
( ( ( map_list_o_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ? [Z5: list_o,Zs2: list_list_o] :
( ( Xs
= ( cons_list_o @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_list_o_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_784_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ? [Z5: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_785_Cons__eq__map__conv,axiom,
! [X2: $o,Xs: list_o,F: $o > $o,Ys: list_o] :
( ( ( cons_o @ X2 @ Xs )
= ( map_o_o @ F @ Ys ) )
= ( ? [Z4: $o,Zs3: list_o] :
( ( Ys
= ( cons_o @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_o_o @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_786_Cons__eq__map__conv,axiom,
! [X2: $o,Xs: list_o,F: list_o > $o,Ys: list_list_o] :
( ( ( cons_o @ X2 @ Xs )
= ( map_list_o_o @ F @ Ys ) )
= ( ? [Z4: list_o,Zs3: list_list_o] :
( ( Ys
= ( cons_list_o @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_list_o_o @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_787_Cons__eq__map__conv,axiom,
! [X2: $o,Xs: list_o,F: nat > $o,Ys: list_nat] :
( ( ( cons_o @ X2 @ Xs )
= ( map_nat_o @ F @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_o @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_788_Cons__eq__map__conv,axiom,
! [X2: list_o,Xs: list_list_o,F: $o > list_o,Ys: list_o] :
( ( ( cons_list_o @ X2 @ Xs )
= ( map_o_list_o @ F @ Ys ) )
= ( ? [Z4: $o,Zs3: list_o] :
( ( Ys
= ( cons_o @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_o_list_o @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_789_Cons__eq__map__conv,axiom,
! [X2: list_o,Xs: list_list_o,F: list_o > list_o,Ys: list_list_o] :
( ( ( cons_list_o @ X2 @ Xs )
= ( map_list_o_list_o @ F @ Ys ) )
= ( ? [Z4: list_o,Zs3: list_list_o] :
( ( Ys
= ( cons_list_o @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_list_o_list_o @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_790_Cons__eq__map__conv,axiom,
! [X2: list_o,Xs: list_list_o,F: nat > list_o,Ys: list_nat] :
( ( ( cons_list_o @ X2 @ Xs )
= ( map_nat_list_o @ F @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_list_o @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_791_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_list_nat_nat @ F @ Ys ) )
= ( ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_list_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_792_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: $o > nat,Ys: list_o] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_o_nat @ F @ Ys ) )
= ( ? [Z4: $o,Zs3: list_o] :
( ( Ys
= ( cons_o @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_o_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_793_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: list_o > nat,Ys: list_list_o] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_list_o_nat @ F @ Ys ) )
= ( ? [Z4: list_o,Zs3: list_list_o] :
( ( Ys
= ( cons_list_o @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_list_o_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_794_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 ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_795_map__eq__Cons__conv,axiom,
! [F: $o > $o,Xs: list_o,Y3: $o,Ys: list_o] :
( ( ( map_o_o @ F @ Xs )
= ( cons_o @ Y3 @ Ys ) )
= ( ? [Z4: $o,Zs3: list_o] :
( ( Xs
= ( cons_o @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_o_o @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_796_map__eq__Cons__conv,axiom,
! [F: list_o > $o,Xs: list_list_o,Y3: $o,Ys: list_o] :
( ( ( map_list_o_o @ F @ Xs )
= ( cons_o @ Y3 @ Ys ) )
= ( ? [Z4: list_o,Zs3: list_list_o] :
( ( Xs
= ( cons_list_o @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_list_o_o @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_797_map__eq__Cons__conv,axiom,
! [F: nat > $o,Xs: list_nat,Y3: $o,Ys: list_o] :
( ( ( map_nat_o @ F @ Xs )
= ( cons_o @ Y3 @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_nat_o @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_798_map__eq__Cons__conv,axiom,
! [F: $o > list_o,Xs: list_o,Y3: list_o,Ys: list_list_o] :
( ( ( map_o_list_o @ F @ Xs )
= ( cons_list_o @ Y3 @ Ys ) )
= ( ? [Z4: $o,Zs3: list_o] :
( ( Xs
= ( cons_o @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_o_list_o @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_799_map__eq__Cons__conv,axiom,
! [F: list_o > list_o,Xs: list_list_o,Y3: list_o,Ys: list_list_o] :
( ( ( map_list_o_list_o @ F @ Xs )
= ( cons_list_o @ Y3 @ Ys ) )
= ( ? [Z4: list_o,Zs3: list_list_o] :
( ( Xs
= ( cons_list_o @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_list_o_list_o @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_800_map__eq__Cons__conv,axiom,
! [F: nat > list_o,Xs: list_nat,Y3: list_o,Ys: list_list_o] :
( ( ( map_nat_list_o @ F @ Xs )
= ( cons_list_o @ Y3 @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_nat_list_o @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_801_map__eq__Cons__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
= ( ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_list_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_802_map__eq__Cons__conv,axiom,
! [F: $o > nat,Xs: list_o,Y3: nat,Ys: list_nat] :
( ( ( map_o_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
= ( ? [Z4: $o,Zs3: list_o] :
( ( Xs
= ( cons_o @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_o_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_803_map__eq__Cons__conv,axiom,
! [F: list_o > nat,Xs: list_list_o,Y3: nat,Ys: list_nat] :
( ( ( map_list_o_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
= ( ? [Z4: list_o,Zs3: list_list_o] :
( ( Xs
= ( cons_list_o @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_list_o_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_804_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y3 )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_805_list_Osimps_I8_J,axiom,
! [F: $o > nat] :
( ( map_o_nat @ F @ nil_o )
= nil_nat ) ).
% list.simps(8)
thf(fact_806_list_Osimps_I8_J,axiom,
! [F: nat > $o] :
( ( map_nat_o @ F @ nil_nat )
= nil_o ) ).
% list.simps(8)
thf(fact_807_list_Osimps_I8_J,axiom,
! [F: $o > $o] :
( ( map_o_o @ F @ nil_o )
= nil_o ) ).
% list.simps(8)
thf(fact_808_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_809_list_Osimps_I8_J,axiom,
! [F: $o > option_o] :
( ( map_o_option_o @ F @ nil_o )
= nil_option_o ) ).
% list.simps(8)
thf(fact_810_list_Osimps_I8_J,axiom,
! [F: $o > list_o] :
( ( map_o_list_o @ F @ nil_o )
= nil_list_o ) ).
% list.simps(8)
thf(fact_811_list_Osimps_I8_J,axiom,
! [F: $o > list_nat] :
( ( map_o_list_nat @ F @ nil_o )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_812_list_Osimps_I8_J,axiom,
! [F: option_o > $o] :
( ( map_option_o_o @ F @ nil_option_o )
= nil_o ) ).
% list.simps(8)
thf(fact_813_list_Osimps_I8_J,axiom,
! [F: option_o > nat] :
( ( map_option_o_nat @ F @ nil_option_o )
= nil_nat ) ).
% list.simps(8)
thf(fact_814_list_Osimps_I8_J,axiom,
! [F: list_o > $o] :
( ( map_list_o_o @ F @ nil_list_o )
= nil_o ) ).
% list.simps(8)
thf(fact_815_append__eq__map__conv,axiom,
! [Ys: list_o,Zs: list_o,F: list_o > $o,Xs: list_list_o] :
( ( ( append_o @ Ys @ Zs )
= ( map_list_o_o @ F @ Xs ) )
= ( ? [Us: list_list_o,Vs: list_list_o] :
( ( Xs
= ( append_list_o @ Us @ Vs ) )
& ( Ys
= ( map_list_o_o @ F @ Us ) )
& ( Zs
= ( map_list_o_o @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_816_append__eq__map__conv,axiom,
! [Ys: list_o,Zs: list_o,F: nat > $o,Xs: list_nat] :
( ( ( append_o @ Ys @ Zs )
= ( map_nat_o @ F @ Xs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_o @ F @ Us ) )
& ( Zs
= ( map_nat_o @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_817_append__eq__map__conv,axiom,
! [Ys: list_list_o,Zs: list_list_o,F: $o > list_o,Xs: list_o] :
( ( ( append_list_o @ Ys @ Zs )
= ( map_o_list_o @ F @ Xs ) )
= ( ? [Us: list_o,Vs: list_o] :
( ( Xs
= ( append_o @ Us @ Vs ) )
& ( Ys
= ( map_o_list_o @ F @ Us ) )
& ( Zs
= ( map_o_list_o @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_818_append__eq__map__conv,axiom,
! [Ys: list_list_o,Zs: list_list_o,F: list_o > list_o,Xs: list_list_o] :
( ( ( append_list_o @ Ys @ Zs )
= ( map_list_o_list_o @ F @ Xs ) )
= ( ? [Us: list_list_o,Vs: list_list_o] :
( ( Xs
= ( append_list_o @ Us @ Vs ) )
& ( Ys
= ( map_list_o_list_o @ F @ Us ) )
& ( Zs
= ( map_list_o_list_o @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_819_append__eq__map__conv,axiom,
! [Ys: list_list_o,Zs: list_list_o,F: nat > list_o,Xs: list_nat] :
( ( ( append_list_o @ Ys @ Zs )
= ( map_nat_list_o @ F @ Xs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_list_o @ F @ Us ) )
& ( Zs
= ( map_nat_list_o @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_820_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: $o > nat,Xs: list_o] :
( ( ( append_nat @ Ys @ Zs )
= ( map_o_nat @ F @ Xs ) )
= ( ? [Us: list_o,Vs: list_o] :
( ( Xs
= ( append_o @ Us @ Vs ) )
& ( Ys
= ( map_o_nat @ F @ Us ) )
& ( Zs
= ( map_o_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_821_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: list_o > nat,Xs: list_list_o] :
( ( ( append_nat @ Ys @ Zs )
= ( map_list_o_nat @ F @ Xs ) )
= ( ? [Us: list_list_o,Vs: list_list_o] :
( ( Xs
= ( append_list_o @ Us @ Vs ) )
& ( Ys
= ( map_list_o_nat @ F @ Us ) )
& ( Zs
= ( map_list_o_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_822_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 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_823_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: list_nat > nat,Xs: list_list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_list_nat_nat @ F @ Xs ) )
= ( ? [Us: list_list_nat,Vs: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us @ Vs ) )
& ( Ys
= ( map_list_nat_nat @ F @ Us ) )
& ( Zs
= ( map_list_nat_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_824_append__eq__map__conv,axiom,
! [Ys: list_o,Zs: list_o,F: $o > $o,Xs: list_o] :
( ( ( append_o @ Ys @ Zs )
= ( map_o_o @ F @ Xs ) )
= ( ? [Us: list_o,Vs: list_o] :
( ( Xs
= ( append_o @ Us @ Vs ) )
& ( Ys
= ( map_o_o @ F @ Us ) )
& ( Zs
= ( map_o_o @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_825_map__eq__append__conv,axiom,
! [F: list_o > $o,Xs: list_list_o,Ys: list_o,Zs: list_o] :
( ( ( map_list_o_o @ F @ Xs )
= ( append_o @ Ys @ Zs ) )
= ( ? [Us: list_list_o,Vs: list_list_o] :
( ( Xs
= ( append_list_o @ Us @ Vs ) )
& ( Ys
= ( map_list_o_o @ F @ Us ) )
& ( Zs
= ( map_list_o_o @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_826_map__eq__append__conv,axiom,
! [F: nat > $o,Xs: list_nat,Ys: list_o,Zs: list_o] :
( ( ( map_nat_o @ F @ Xs )
= ( append_o @ Ys @ Zs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_o @ F @ Us ) )
& ( Zs
= ( map_nat_o @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_827_map__eq__append__conv,axiom,
! [F: $o > list_o,Xs: list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( map_o_list_o @ F @ Xs )
= ( append_list_o @ Ys @ Zs ) )
= ( ? [Us: list_o,Vs: list_o] :
( ( Xs
= ( append_o @ Us @ Vs ) )
& ( Ys
= ( map_o_list_o @ F @ Us ) )
& ( Zs
= ( map_o_list_o @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_828_map__eq__append__conv,axiom,
! [F: list_o > list_o,Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( map_list_o_list_o @ F @ Xs )
= ( append_list_o @ Ys @ Zs ) )
= ( ? [Us: list_list_o,Vs: list_list_o] :
( ( Xs
= ( append_list_o @ Us @ Vs ) )
& ( Ys
= ( map_list_o_list_o @ F @ Us ) )
& ( Zs
= ( map_list_o_list_o @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_829_map__eq__append__conv,axiom,
! [F: nat > list_o,Xs: list_nat,Ys: list_list_o,Zs: list_list_o] :
( ( ( map_nat_list_o @ F @ Xs )
= ( append_list_o @ Ys @ Zs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_list_o @ F @ Us ) )
& ( Zs
= ( map_nat_list_o @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_830_map__eq__append__conv,axiom,
! [F: $o > nat,Xs: list_o,Ys: list_nat,Zs: list_nat] :
( ( ( map_o_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us: list_o,Vs: list_o] :
( ( Xs
= ( append_o @ Us @ Vs ) )
& ( Ys
= ( map_o_nat @ F @ Us ) )
& ( Zs
= ( map_o_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_831_map__eq__append__conv,axiom,
! [F: list_o > nat,Xs: list_list_o,Ys: list_nat,Zs: list_nat] :
( ( ( map_list_o_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us: list_list_o,Vs: list_list_o] :
( ( Xs
= ( append_list_o @ Us @ Vs ) )
& ( Ys
= ( map_list_o_nat @ F @ Us ) )
& ( Zs
= ( map_list_o_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_832_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 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_833_map__eq__append__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us: list_list_nat,Vs: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us @ Vs ) )
& ( Ys
= ( map_list_nat_nat @ F @ Us ) )
& ( Zs
= ( map_list_nat_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_834_map__eq__append__conv,axiom,
! [F: $o > $o,Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( map_o_o @ F @ Xs )
= ( append_o @ Ys @ Zs ) )
= ( ? [Us: list_o,Vs: list_o] :
( ( Xs
= ( append_o @ Us @ Vs ) )
& ( Ys
= ( map_o_o @ F @ Us ) )
& ( Zs
= ( map_o_o @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_835_map__mono__prefix,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > nat] :
( ( prefix_list_nat @ Xs @ Ys )
=> ( prefix_nat @ ( map_list_nat_nat @ F @ Xs ) @ ( map_list_nat_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_836_map__mono__prefix,axiom,
! [Xs: list_o,Ys: list_o,F: $o > nat] :
( ( prefix_o @ Xs @ Ys )
=> ( prefix_nat @ ( map_o_nat @ F @ Xs ) @ ( map_o_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_837_map__mono__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > $o] :
( ( prefix_nat @ Xs @ Ys )
=> ( prefix_o @ ( map_nat_o @ F @ Xs ) @ ( map_nat_o @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_838_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_839_map__mono__prefix,axiom,
! [Xs: list_o,Ys: list_o,F: $o > $o] :
( ( prefix_o @ Xs @ Ys )
=> ( prefix_o @ ( map_o_o @ F @ Xs ) @ ( map_o_o @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_840_prefix__map__rightE,axiom,
! [Xs: list_o,F: nat > $o,Ys: list_nat] :
( ( prefix_o @ Xs @ ( map_nat_o @ F @ Ys ) )
=> ? [Xs5: list_nat] :
( ( prefix_nat @ Xs5 @ Ys )
& ( Xs
= ( map_nat_o @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_841_prefix__map__rightE,axiom,
! [Xs: list_nat,F: list_nat > nat,Ys: list_list_nat] :
( ( prefix_nat @ Xs @ ( map_list_nat_nat @ F @ Ys ) )
=> ? [Xs5: list_list_nat] :
( ( prefix_list_nat @ Xs5 @ Ys )
& ( Xs
= ( map_list_nat_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_842_prefix__map__rightE,axiom,
! [Xs: list_nat,F: $o > nat,Ys: list_o] :
( ( prefix_nat @ Xs @ ( map_o_nat @ F @ Ys ) )
=> ? [Xs5: list_o] :
( ( prefix_o @ Xs5 @ Ys )
& ( Xs
= ( map_o_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_843_prefix__map__rightE,axiom,
! [Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( map_nat_nat @ F @ Ys ) )
=> ? [Xs5: list_nat] :
( ( prefix_nat @ Xs5 @ Ys )
& ( Xs
= ( map_nat_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_844_prefix__map__rightE,axiom,
! [Xs: list_o,F: $o > $o,Ys: list_o] :
( ( prefix_o @ Xs @ ( map_o_o @ F @ Ys ) )
=> ? [Xs5: list_o] :
( ( prefix_o @ Xs5 @ Ys )
& ( Xs
= ( map_o_o @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_845_prefix__bot_Oextremum__strict,axiom,
! [A: list_option_o] :
~ ( strict4666211328587260336tion_o @ A @ nil_option_o ) ).
% prefix_bot.extremum_strict
thf(fact_846_prefix__bot_Oextremum__strict,axiom,
! [A: list_list_o] :
~ ( strict_prefix_list_o @ A @ nil_list_o ) ).
% prefix_bot.extremum_strict
thf(fact_847_prefix__bot_Oextremum__strict,axiom,
! [A: list_list_nat] :
~ ( strict6380099159608944142st_nat @ A @ nil_list_nat ) ).
% prefix_bot.extremum_strict
thf(fact_848_prefix__bot_Oextremum__strict,axiom,
! [A: list_o] :
~ ( strict_prefix_o @ A @ nil_o ) ).
% prefix_bot.extremum_strict
thf(fact_849_prefix__bot_Oextremum__strict,axiom,
! [A: list_nat] :
~ ( strict_prefix_nat @ A @ nil_nat ) ).
% prefix_bot.extremum_strict
thf(fact_850_prefix__bot_Onot__eq__extremum,axiom,
! [A: list_option_o] :
( ( A != nil_option_o )
= ( strict4666211328587260336tion_o @ nil_option_o @ A ) ) ).
% prefix_bot.not_eq_extremum
thf(fact_851_prefix__bot_Onot__eq__extremum,axiom,
! [A: list_list_o] :
( ( A != nil_list_o )
= ( strict_prefix_list_o @ nil_list_o @ A ) ) ).
% prefix_bot.not_eq_extremum
thf(fact_852_prefix__bot_Onot__eq__extremum,axiom,
! [A: list_list_nat] :
( ( A != nil_list_nat )
= ( strict6380099159608944142st_nat @ nil_list_nat @ A ) ) ).
% prefix_bot.not_eq_extremum
thf(fact_853_prefix__bot_Onot__eq__extremum,axiom,
! [A: list_o] :
( ( A != nil_o )
= ( strict_prefix_o @ nil_o @ A ) ) ).
% prefix_bot.not_eq_extremum
thf(fact_854_prefix__bot_Onot__eq__extremum,axiom,
! [A: list_nat] :
( ( A != nil_nat )
= ( strict_prefix_nat @ nil_nat @ A ) ) ).
% prefix_bot.not_eq_extremum
thf(fact_855_prefix__order_OleD,axiom,
! [Y3: list_nat,X2: list_nat] :
( ( prefix_nat @ Y3 @ X2 )
=> ~ ( strict_prefix_nat @ X2 @ Y3 ) ) ).
% prefix_order.leD
thf(fact_856_prefix__order_OleD,axiom,
! [Y3: list_o,X2: list_o] :
( ( prefix_o @ Y3 @ X2 )
=> ~ ( strict_prefix_o @ X2 @ Y3 ) ) ).
% prefix_order.leD
thf(fact_857_prefix__order_Ole__less,axiom,
( prefix_nat
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( strict_prefix_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% prefix_order.le_less
thf(fact_858_prefix__order_Ole__less,axiom,
( prefix_o
= ( ^ [X3: list_o,Y4: list_o] :
( ( strict_prefix_o @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% prefix_order.le_less
thf(fact_859_prefix__order_Oless__le,axiom,
( strict_prefix_nat
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( prefix_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% prefix_order.less_le
thf(fact_860_prefix__order_Oless__le,axiom,
( strict_prefix_o
= ( ^ [X3: list_o,Y4: list_o] :
( ( prefix_o @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% prefix_order.less_le
thf(fact_861_prefix__order_Onless__le,axiom,
! [A: list_nat,B: list_nat] :
( ( ~ ( strict_prefix_nat @ A @ B ) )
= ( ~ ( prefix_nat @ A @ B )
| ( A = B ) ) ) ).
% prefix_order.nless_le
thf(fact_862_prefix__order_Onless__le,axiom,
! [A: list_o,B: list_o] :
( ( ~ ( strict_prefix_o @ A @ B ) )
= ( ~ ( prefix_o @ A @ B )
| ( A = B ) ) ) ).
% prefix_order.nless_le
thf(fact_863_prefix__order_Oless__imp__le,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ( prefix_nat @ X2 @ Y3 ) ) ).
% prefix_order.less_imp_le
thf(fact_864_prefix__order_Oless__imp__le,axiom,
! [X2: list_o,Y3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ( prefix_o @ X2 @ Y3 ) ) ).
% prefix_order.less_imp_le
thf(fact_865_prefix__order_Ole__neq__trans,axiom,
! [A: list_nat,B: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( A != B )
=> ( strict_prefix_nat @ A @ B ) ) ) ).
% prefix_order.le_neq_trans
thf(fact_866_prefix__order_Ole__neq__trans,axiom,
! [A: list_o,B: list_o] :
( ( prefix_o @ A @ B )
=> ( ( A != B )
=> ( strict_prefix_o @ A @ B ) ) ) ).
% prefix_order.le_neq_trans
thf(fact_867_prefix__order_Oantisym__conv1,axiom,
! [X2: list_nat,Y3: list_nat] :
( ~ ( strict_prefix_nat @ X2 @ Y3 )
=> ( ( prefix_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% prefix_order.antisym_conv1
thf(fact_868_prefix__order_Oantisym__conv1,axiom,
! [X2: list_o,Y3: list_o] :
( ~ ( strict_prefix_o @ X2 @ Y3 )
=> ( ( prefix_o @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% prefix_order.antisym_conv1
thf(fact_869_prefix__order_Oantisym__conv2,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( prefix_nat @ X2 @ Y3 )
=> ( ( ~ ( strict_prefix_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% prefix_order.antisym_conv2
thf(fact_870_prefix__order_Oantisym__conv2,axiom,
! [X2: list_o,Y3: list_o] :
( ( prefix_o @ X2 @ Y3 )
=> ( ( ~ ( strict_prefix_o @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% prefix_order.antisym_conv2
thf(fact_871_prefix__order_Ole__less__trans,axiom,
! [X2: list_nat,Y3: list_nat,Z3: list_nat] :
( ( prefix_nat @ X2 @ Y3 )
=> ( ( strict_prefix_nat @ Y3 @ Z3 )
=> ( strict_prefix_nat @ X2 @ Z3 ) ) ) ).
% prefix_order.le_less_trans
thf(fact_872_prefix__order_Ole__less__trans,axiom,
! [X2: list_o,Y3: list_o,Z3: list_o] :
( ( prefix_o @ X2 @ Y3 )
=> ( ( strict_prefix_o @ Y3 @ Z3 )
=> ( strict_prefix_o @ X2 @ Z3 ) ) ) ).
% prefix_order.le_less_trans
thf(fact_873_prefix__order_Oless__le__trans,axiom,
! [X2: list_nat,Y3: list_nat,Z3: list_nat] :
( ( strict_prefix_nat @ X2 @ Y3 )
=> ( ( prefix_nat @ Y3 @ Z3 )
=> ( strict_prefix_nat @ X2 @ Z3 ) ) ) ).
% prefix_order.less_le_trans
thf(fact_874_prefix__order_Oless__le__trans,axiom,
! [X2: list_o,Y3: list_o,Z3: list_o] :
( ( strict_prefix_o @ X2 @ Y3 )
=> ( ( prefix_o @ Y3 @ Z3 )
=> ( strict_prefix_o @ X2 @ Z3 ) ) ) ).
% prefix_order.less_le_trans
thf(fact_875_prefix__order_Ostrict__trans1,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( strict_prefix_nat @ B @ C )
=> ( strict_prefix_nat @ A @ C ) ) ) ).
% prefix_order.strict_trans1
thf(fact_876_prefix__order_Ostrict__trans1,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( prefix_o @ A @ B )
=> ( ( strict_prefix_o @ B @ C )
=> ( strict_prefix_o @ A @ C ) ) ) ).
% prefix_order.strict_trans1
thf(fact_877_prefix__order_Ostrict__trans2,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( strict_prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ C )
=> ( strict_prefix_nat @ A @ C ) ) ) ).
% prefix_order.strict_trans2
thf(fact_878_prefix__order_Ostrict__trans2,axiom,
! [A: list_o,B: list_o,C: list_o] :
( ( strict_prefix_o @ A @ B )
=> ( ( prefix_o @ B @ C )
=> ( strict_prefix_o @ A @ C ) ) ) ).
% prefix_order.strict_trans2
thf(fact_879_prefix__order_Oless__le__not__le,axiom,
( strict_prefix_nat
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( prefix_nat @ X3 @ Y4 )
& ~ ( prefix_nat @ Y4 @ X3 ) ) ) ) ).
% prefix_order.less_le_not_le
thf(fact_880_prefix__order_Oless__le__not__le,axiom,
( strict_prefix_o
= ( ^ [X3: list_o,Y4: list_o] :
( ( prefix_o @ X3 @ Y4 )
& ~ ( prefix_o @ Y4 @ X3 ) ) ) ) ).
% prefix_order.less_le_not_le
thf(fact_881_prefix__order_Ostrict__iff__not,axiom,
( strict_prefix_nat
= ( ^ [A2: list_nat,B2: list_nat] :
( ( prefix_nat @ A2 @ B2 )
& ~ ( prefix_nat @ B2 @ A2 ) ) ) ) ).
% prefix_order.strict_iff_not
thf(fact_882_prefix__order_Ostrict__iff__not,axiom,
( strict_prefix_o
= ( ^ [A2: list_o,B2: list_o] :
( ( prefix_o @ A2 @ B2 )
& ~ ( prefix_o @ B2 @ A2 ) ) ) ) ).
% prefix_order.strict_iff_not
thf(fact_883_prefix__order_Oorder__iff__strict,axiom,
( prefix_nat
= ( ^ [A2: list_nat,B2: list_nat] :
( ( strict_prefix_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% prefix_order.order_iff_strict
thf(fact_884_prefix__order_Oorder__iff__strict,axiom,
( prefix_o
= ( ^ [A2: list_o,B2: list_o] :
( ( strict_prefix_o @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% prefix_order.order_iff_strict
thf(fact_885_prefix__order_Ostrict__iff__order,axiom,
( strict_prefix_nat
= ( ^ [A2: list_nat,B2: list_nat] :
( ( prefix_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% prefix_order.strict_iff_order
thf(fact_886_prefix__order_Ostrict__iff__order,axiom,
( strict_prefix_o
= ( ^ [A2: list_o,B2: list_o] :
( ( prefix_o @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% prefix_order.strict_iff_order
thf(fact_887_prefix__order_Ole__imp__less__or__eq,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( prefix_nat @ X2 @ Y3 )
=> ( ( strict_prefix_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% prefix_order.le_imp_less_or_eq
thf(fact_888_prefix__order_Ole__imp__less__or__eq,axiom,
! [X2: list_o,Y3: list_o] :
( ( prefix_o @ X2 @ Y3 )
=> ( ( strict_prefix_o @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% prefix_order.le_imp_less_or_eq
thf(fact_889_prefix__order_Ostrict__implies__order,axiom,
! [A: list_nat,B: list_nat] :
( ( strict_prefix_nat @ A @ B )
=> ( prefix_nat @ A @ B ) ) ).
% prefix_order.strict_implies_order
thf(fact_890_prefix__order_Ostrict__implies__order,axiom,
! [A: list_o,B: list_o] :
( ( strict_prefix_o @ A @ B )
=> ( prefix_o @ A @ B ) ) ).
% prefix_order.strict_implies_order
thf(fact_891_prefix__order_Odual__order_Ostrict__trans1,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( strict_prefix_nat @ C @ B )
=> ( strict_prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans1
thf(fact_892_prefix__order_Odual__order_Ostrict__trans1,axiom,
! [B: list_o,A: list_o,C: list_o] :
( ( prefix_o @ B @ A )
=> ( ( strict_prefix_o @ C @ B )
=> ( strict_prefix_o @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans1
thf(fact_893_prefix__order_Odual__order_Ostrict__trans2,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( strict_prefix_nat @ B @ A )
=> ( ( prefix_nat @ C @ B )
=> ( strict_prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans2
thf(fact_894_prefix__order_Odual__order_Ostrict__trans2,axiom,
! [B: list_o,A: list_o,C: list_o] :
( ( strict_prefix_o @ B @ A )
=> ( ( prefix_o @ C @ B )
=> ( strict_prefix_o @ C @ A ) ) ) ).
% prefix_order.dual_order.strict_trans2
thf(fact_895_prefix__order_Odual__order_Ostrict__iff__not,axiom,
( strict_prefix_nat
= ( ^ [B2: list_nat,A2: list_nat] :
( ( prefix_nat @ B2 @ A2 )
& ~ ( prefix_nat @ A2 @ B2 ) ) ) ) ).
% prefix_order.dual_order.strict_iff_not
thf(fact_896_prefix__order_Odual__order_Ostrict__iff__not,axiom,
( strict_prefix_o
= ( ^ [B2: list_o,A2: list_o] :
( ( prefix_o @ B2 @ A2 )
& ~ ( prefix_o @ A2 @ B2 ) ) ) ) ).
% prefix_order.dual_order.strict_iff_not
thf(fact_897_prefix__order_Odual__order_Oorder__iff__strict,axiom,
( prefix_nat
= ( ^ [B2: list_nat,A2: list_nat] :
( ( strict_prefix_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% prefix_order.dual_order.order_iff_strict
thf(fact_898_prefix__order_Odual__order_Oorder__iff__strict,axiom,
( prefix_o
= ( ^ [B2: list_o,A2: list_o] :
( ( strict_prefix_o @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% prefix_order.dual_order.order_iff_strict
thf(fact_899_prefix__order_Odual__order_Ostrict__iff__order,axiom,
( strict_prefix_nat
= ( ^ [B2: list_nat,A2: list_nat] :
( ( prefix_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% prefix_order.dual_order.strict_iff_order
thf(fact_900_prefix__order_Odual__order_Ostrict__iff__order,axiom,
( strict_prefix_o
= ( ^ [B2: list_o,A2: list_o] :
( ( prefix_o @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% prefix_order.dual_order.strict_iff_order
thf(fact_901_prefix__order_Onot__eq__order__implies__strict,axiom,
! [A: list_nat,B: list_nat] :
( ( A != B )
=> ( ( prefix_nat @ A @ B )
=> ( strict_prefix_nat @ A @ B ) ) ) ).
% prefix_order.not_eq_order_implies_strict
thf(fact_902_prefix__order_Onot__eq__order__implies__strict,axiom,
! [A: list_o,B: list_o] :
( ( A != B )
=> ( ( prefix_o @ A @ B )
=> ( strict_prefix_o @ A @ B ) ) ) ).
% prefix_order.not_eq_order_implies_strict
thf(fact_903_prefix__order_Odual__order_Ostrict__implies__order,axiom,
! [B: list_nat,A: list_nat] :
( ( strict_prefix_nat @ B @ A )
=> ( prefix_nat @ B @ A ) ) ).
% prefix_order.dual_order.strict_implies_order
thf(fact_904_prefix__order_Odual__order_Ostrict__implies__order,axiom,
! [B: list_o,A: list_o] :
( ( strict_prefix_o @ B @ A )
=> ( prefix_o @ B @ A ) ) ).
% prefix_order.dual_order.strict_implies_order
thf(fact_905_prefix__order_Odual__order_Onot__eq__order__implies__strict,axiom,
! [A: list_nat,B: list_nat] :
( ( A != B )
=> ( ( prefix_nat @ B @ A )
=> ( strict_prefix_nat @ B @ A ) ) ) ).
% prefix_order.dual_order.not_eq_order_implies_strict
thf(fact_906_prefix__order_Odual__order_Onot__eq__order__implies__strict,axiom,
! [A: list_o,B: list_o] :
( ( A != B )
=> ( ( prefix_o @ B @ A )
=> ( strict_prefix_o @ B @ A ) ) ) ).
% prefix_order.dual_order.not_eq_order_implies_strict
thf(fact_907_strict__prefixE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( strict_prefix_nat @ Xs @ Ys )
=> ~ ( ( prefix_nat @ Xs @ Ys )
=> ( Xs = Ys ) ) ) ).
% strict_prefixE
thf(fact_908_strict__prefixE,axiom,
! [Xs: list_o,Ys: list_o] :
( ( strict_prefix_o @ Xs @ Ys )
=> ~ ( ( prefix_o @ Xs @ Ys )
=> ( Xs = Ys ) ) ) ).
% strict_prefixE
thf(fact_909_strict__prefixI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ( Xs != Ys )
=> ( strict_prefix_nat @ Xs @ Ys ) ) ) ).
% strict_prefixI
thf(fact_910_strict__prefixI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ( Xs != Ys )
=> ( strict_prefix_o @ Xs @ Ys ) ) ) ).
% strict_prefixI
thf(fact_911_strict__prefix__def,axiom,
( strict_prefix_nat
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
& ( Xs2 != Ys2 ) ) ) ) ).
% strict_prefix_def
thf(fact_912_strict__prefix__def,axiom,
( strict_prefix_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
( ( prefix_o @ Xs2 @ Ys2 )
& ( Xs2 != Ys2 ) ) ) ) ).
% strict_prefix_def
thf(fact_913_map__butlast,axiom,
! [F: $o > nat,Xs: list_o] :
( ( map_o_nat @ F @ ( butlast_o @ Xs ) )
= ( butlast_nat @ ( map_o_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_914_map__butlast,axiom,
! [F: nat > $o,Xs: list_nat] :
( ( map_nat_o @ F @ ( butlast_nat @ Xs ) )
= ( butlast_o @ ( map_nat_o @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_915_map__butlast,axiom,
! [F: $o > $o,Xs: list_o] :
( ( map_o_o @ F @ ( butlast_o @ Xs ) )
= ( butlast_o @ ( map_o_o @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_916_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_917_map__butlast,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( map_list_nat_nat @ F @ ( butlast_list_nat @ Xs ) )
= ( butlast_nat @ ( map_list_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_918_rotate1_Osimps_I1_J,axiom,
( ( rotate1_o @ nil_o )
= nil_o ) ).
% rotate1.simps(1)
thf(fact_919_rotate1_Osimps_I1_J,axiom,
( ( rotate1_option_o @ nil_option_o )
= nil_option_o ) ).
% rotate1.simps(1)
thf(fact_920_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_o @ nil_list_o )
= nil_list_o ) ).
% rotate1.simps(1)
thf(fact_921_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_922_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% rotate1.simps(1)
thf(fact_923_strict__prefixI_H,axiom,
! [Ys: list_o,Xs: list_o,Z3: $o,Zs: list_o] :
( ( Ys
= ( append_o @ Xs @ ( cons_o @ Z3 @ Zs ) ) )
=> ( strict_prefix_o @ Xs @ Ys ) ) ).
% strict_prefixI'
thf(fact_924_strict__prefixE_H,axiom,
! [Xs: list_o,Ys: list_o] :
( ( strict_prefix_o @ Xs @ Ys )
=> ~ ! [Z5: $o,Zs2: list_o] :
( Ys
!= ( append_o @ Xs @ ( cons_o @ Z5 @ Zs2 ) ) ) ) ).
% strict_prefixE'
thf(fact_925_SuccD,axiom,
! [K: $o,Kl2: set_list_o,Kl: list_o] :
( ( member_o2 @ K @ ( bNF_Greatest_Succ_o @ Kl2 @ Kl ) )
=> ( member_list_o2 @ ( append_o @ Kl @ ( cons_o @ K @ nil_o ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_926_suffix__snoc,axiom,
! [Xs: list_o,Ys: list_o,Y3: $o] :
( ( suffix_o @ Xs @ ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
= ( ( Xs = nil_o )
| ? [Zs3: list_o] :
( ( Xs
= ( append_o @ Zs3 @ ( cons_o @ Y3 @ nil_o ) ) )
& ( suffix_o @ Zs3 @ Ys ) ) ) ) ).
% suffix_snoc
thf(fact_927_snoc__suffix__snoc,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y3: $o] :
( ( suffix_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
= ( ( X2 = Y3 )
& ( suffix_o @ Xs @ Ys ) ) ) ).
% snoc_suffix_snoc
thf(fact_928_same__suffix__suffix,axiom,
! [Ys: list_o,Xs: list_o,Zs: list_o] :
( ( suffix_o @ ( append_o @ Ys @ Xs ) @ ( append_o @ Zs @ Xs ) )
= ( suffix_o @ Ys @ Zs ) ) ).
% same_suffix_suffix
thf(fact_929_same__suffix__nil,axiom,
! [Ys: list_o,Xs: list_o] :
( ( suffix_o @ ( append_o @ Ys @ Xs ) @ Xs )
= ( Ys = nil_o ) ) ).
% same_suffix_nil
thf(fact_930_suffix__appendI,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( suffix_o @ Xs @ Ys )
=> ( suffix_o @ Xs @ ( append_o @ Zs @ Ys ) ) ) ).
% suffix_appendI
thf(fact_931_suffix__appendD,axiom,
! [Zs: list_o,Xs: list_o,Ys: list_o] :
( ( suffix_o @ ( append_o @ Zs @ Xs ) @ Ys )
=> ( suffix_o @ Xs @ Ys ) ) ).
% suffix_appendD
thf(fact_932_suffix__append,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( suffix_o @ Xs @ ( append_o @ Ys @ Zs ) )
= ( ( suffix_o @ Xs @ Zs )
| ? [Xs4: list_o] :
( ( Xs
= ( append_o @ Xs4 @ Zs ) )
& ( suffix_o @ Xs4 @ Ys ) ) ) ) ).
% suffix_append
thf(fact_933_suffix__def,axiom,
( suffix_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
? [Zs3: list_o] :
( Ys2
= ( append_o @ Zs3 @ Xs2 ) ) ) ) ).
% suffix_def
thf(fact_934_suffixI,axiom,
! [Ys: list_o,Zs: list_o,Xs: list_o] :
( ( Ys
= ( append_o @ Zs @ Xs ) )
=> ( suffix_o @ Xs @ Ys ) ) ).
% suffixI
thf(fact_935_suffixE,axiom,
! [Xs: list_o,Ys: list_o] :
( ( suffix_o @ Xs @ Ys )
=> ~ ! [Zs2: list_o] :
( Ys
!= ( append_o @ Zs2 @ Xs ) ) ) ).
% suffixE
thf(fact_936_ord_Olexordp__eq__pref,axiom,
! [Less: $o > $o > $o,U: list_o,V: list_o] : ( lexordp_eq_o @ Less @ U @ ( append_o @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_937_Longest__common__prefix__prefix,axiom,
! [Xs: list_o,L: set_list_o] :
( ( member_list_o2 @ Xs @ L )
=> ( prefix_o @ ( longes2916941589570675010efix_o @ L ) @ Xs ) ) ).
% Longest_common_prefix_prefix
thf(fact_938_not__suffix__induct,axiom,
! [Ps: list_o,Ls: list_o,P: list_o > list_o > $o] :
( ~ ( suffix_o @ Ps @ Ls )
=> ( ! [X4: $o,Xs3: list_o] : ( P @ ( append_o @ Xs3 @ ( cons_o @ X4 @ nil_o ) ) @ nil_o )
=> ( ! [X4: $o,Xs3: list_o,Y5: $o,Ys3: list_o] :
( ( X4 != Y5 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X4 @ nil_o ) ) @ ( append_o @ Ys3 @ ( cons_o @ Y5 @ nil_o ) ) ) )
=> ( ! [X4: $o,Xs3: list_o,Y5: $o,Ys3: list_o] :
( ( X4 = Y5 )
=> ( ~ ( suffix_o @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X4 @ nil_o ) ) @ ( append_o @ Ys3 @ ( cons_o @ Y5 @ nil_o ) ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_suffix_induct
thf(fact_939_not__suffix__cases,axiom,
! [Ps: list_o,Ls: list_o] :
( ~ ( suffix_o @ Ps @ Ls )
=> ( ( ( Ps != nil_o )
=> ( Ls != nil_o ) )
=> ( ! [A4: $o,As: list_o] :
( ( Ps
= ( append_o @ As @ ( cons_o @ A4 @ nil_o ) ) )
=> ! [X4: $o,Xs3: list_o] :
( ( Ls
= ( append_o @ Xs3 @ ( cons_o @ X4 @ nil_o ) ) )
=> ( ( X4 = A4 )
=> ( suffix_o @ As @ Xs3 ) ) ) )
=> ~ ! [A4: $o] :
( ? [As: list_o] :
( Ps
= ( append_o @ As @ ( cons_o @ A4 @ nil_o ) ) )
=> ! [X4: $o] :
( ? [Xs3: list_o] :
( Ls
= ( append_o @ Xs3 @ ( cons_o @ X4 @ nil_o ) ) )
=> ( X4 = A4 ) ) ) ) ) ) ).
% not_suffix_cases
thf(fact_940_snoc__sublist__snoc,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y3: $o] :
( ( sublist_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
= ( ( ( X2 = Y3 )
& ( suffix_o @ Xs @ Ys ) )
| ( sublist_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) @ Ys ) ) ) ).
% snoc_sublist_snoc
thf(fact_941_sublist__snoc,axiom,
! [Xs: list_o,Ys: list_o,Y3: $o] :
( ( sublist_o @ Xs @ ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
= ( ( suffix_o @ Xs @ ( append_o @ Ys @ ( cons_o @ Y3 @ nil_o ) ) )
| ( sublist_o @ Xs @ Ys ) ) ) ).
% sublist_snoc
thf(fact_942_sublist__appendI,axiom,
! [Xs: list_o,Ps: list_o,Ss2: list_o] : ( sublist_o @ Xs @ ( append_o @ Ps @ ( append_o @ Xs @ Ss2 ) ) ) ).
% sublist_appendI
thf(fact_943_sublist__append__leftI,axiom,
! [Xs: list_o,Ps: list_o] : ( sublist_o @ Xs @ ( append_o @ Ps @ Xs ) ) ).
% sublist_append_leftI
thf(fact_944_sublist__append__rightI,axiom,
! [Xs: list_o,Ss2: list_o] : ( sublist_o @ Xs @ ( append_o @ Xs @ Ss2 ) ) ).
% sublist_append_rightI
thf(fact_945_prefix__imp__sublist,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( sublist_o @ Xs @ Ys ) ) ).
% prefix_imp_sublist
thf(fact_946_rev__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( rev_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( rev_o @ Ys ) @ ( rev_o @ Xs ) ) ) ).
% rev_append
thf(fact_947_rev__eq__Cons__iff,axiom,
! [Xs: list_o,Y3: $o,Ys: list_o] :
( ( ( rev_o @ Xs )
= ( cons_o @ Y3 @ Ys ) )
= ( Xs
= ( append_o @ ( rev_o @ Ys ) @ ( cons_o @ Y3 @ nil_o ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_948_sublist__def,axiom,
( sublist_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
? [Ps2: list_o,Ss3: list_o] :
( Ys2
= ( append_o @ Ps2 @ ( append_o @ Xs2 @ Ss3 ) ) ) ) ) ).
% sublist_def
thf(fact_949_suffix__to__prefix,axiom,
( suffix_o
= ( ^ [Xs2: list_o,Ys2: list_o] : ( prefix_o @ ( rev_o @ Xs2 ) @ ( rev_o @ Ys2 ) ) ) ) ).
% suffix_to_prefix
thf(fact_950_sublist__Cons__right,axiom,
! [Xs: list_o,Y3: $o,Ys: list_o] :
( ( sublist_o @ Xs @ ( cons_o @ Y3 @ Ys ) )
= ( ( prefix_o @ Xs @ ( cons_o @ Y3 @ Ys ) )
| ( sublist_o @ Xs @ Ys ) ) ) ).
% sublist_Cons_right
thf(fact_951_sublist__code_I3_J,axiom,
! [X2: $o,Xs: list_o,Y3: $o,Ys: list_o] :
( ( sublist_o @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y3 @ Ys ) )
= ( ( prefix_o @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y3 @ Ys ) )
| ( sublist_o @ ( cons_o @ X2 @ Xs ) @ Ys ) ) ) ).
% sublist_code(3)
thf(fact_952_sublist__altdef,axiom,
( sublist_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
? [Ys4: list_o] :
( ( prefix_o @ Ys4 @ Ys2 )
& ( suffix_o @ Xs2 @ Ys4 ) ) ) ) ).
% sublist_altdef
thf(fact_953_sublist__altdef_H,axiom,
( sublist_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
? [Ys4: list_o] :
( ( suffix_o @ Ys4 @ Ys2 )
& ( prefix_o @ Xs2 @ Ys4 ) ) ) ) ).
% sublist_altdef'
thf(fact_954_rev_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( rev_o @ ( cons_o @ X2 @ Xs ) )
= ( append_o @ ( rev_o @ Xs ) @ ( cons_o @ X2 @ nil_o ) ) ) ).
% rev.simps(2)
thf(fact_955_sublist__append,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( sublist_o @ Xs @ ( append_o @ Ys @ Zs ) )
= ( ( sublist_o @ Xs @ Ys )
| ( sublist_o @ Xs @ Zs )
| ? [Xs12: list_o,Xs22: list_o] :
( ( Xs
= ( append_o @ Xs12 @ Xs22 ) )
& ( suffix_o @ Xs12 @ Ys )
& ( prefix_o @ Xs22 @ Zs ) ) ) ) ).
% sublist_append
thf(fact_956_lexordp__eq__pref,axiom,
! [U: list_o,V: list_o] : ( ord_lexordp_eq_o @ U @ ( append_o @ U @ V ) ) ).
% lexordp_eq_pref
thf(fact_957_fold__append__concat__rev,axiom,
! [Xss2: list_list_o] :
( ( fold_list_o_list_o @ append_o @ Xss2 )
= ( append_o @ ( concat_o @ ( rev_list_o @ Xss2 ) ) ) ) ).
% fold_append_concat_rev
thf(fact_958_length__concat__rev,axiom,
! [Xs: list_list_nat] :
( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
= ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).
% length_concat_rev
thf(fact_959_lexordp__eq_Ocases,axiom,
! [A1: list_nat,A22: list_nat] :
( ( ord_lexordp_eq_nat @ A1 @ A22 )
=> ( ( A1 != nil_nat )
=> ( ! [X4: nat] :
( ? [Xs3: list_nat] :
( A1
= ( cons_nat @ X4 @ Xs3 ) )
=> ! [Y5: nat] :
( ? [Ys3: list_nat] :
( A22
= ( cons_nat @ Y5 @ Ys3 ) )
=> ~ ( ord_less_nat @ X4 @ Y5 ) ) )
=> ~ ! [X4: nat,Y5: nat,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X4 @ Xs3 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y5 @ Ys3 ) )
=> ( ~ ( ord_less_nat @ X4 @ Y5 )
=> ( ~ ( ord_less_nat @ Y5 @ X4 )
=> ~ ( ord_lexordp_eq_nat @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_960_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_nat
= ( ^ [A12: list_nat,A23: list_nat] :
( ? [Ys2: list_nat] :
( ( A12 = nil_nat )
& ( A23 = Ys2 ) )
| ? [X3: nat,Y4: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs2 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ord_less_nat @ X3 @ Y4 ) )
| ? [X3: nat,Y4: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs2 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys2 ) )
& ~ ( ord_less_nat @ X3 @ Y4 )
& ~ ( ord_less_nat @ Y4 @ X3 )
& ( ord_lexordp_eq_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_961_append__eq__append__conv,axiom,
! [Xs: list_o,Ys: list_o,Us2: list_o,Vs2: list_o] :
( ( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
| ( ( size_size_list_o @ Us2 )
= ( size_size_list_o @ Vs2 ) ) )
=> ( ( ( append_o @ Xs @ Us2 )
= ( append_o @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_962_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs2: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us2 )
= ( size_size_list_nat @ Vs2 ) ) )
=> ( ( ( append_nat @ Xs @ Us2 )
= ( append_nat @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_963_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_964_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_965_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_966_lexordp__eq__simps_I4_J,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ord_lexordp_eq_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ~ ( ord_less_nat @ Y3 @ X2 )
& ( ord_lexordp_eq_nat @ Xs @ Ys ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_967_suffix__length__less,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( strict_suffix_nat @ Xs @ Ys )
=> ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% suffix_length_less
thf(fact_968_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys6: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_969_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_970_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_971_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_972_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_973_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_974_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_975_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_976_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X4 )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_977_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_978_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_979_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_980_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_981_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_982_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_983_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_984_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_985_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_986_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_987_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_988_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_989_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_990_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_991_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_992_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_993_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_994_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_995_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_996_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_997_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_998_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_999_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1000_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_1001_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_1002_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_1003_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_1004_prefix__length__less,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( strict_prefix_nat @ Xs @ Ys )
=> ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% prefix_length_less
thf(fact_1005_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 )
=> ( ! [X4: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1006_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 )
=> ( ! [X4: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat,Z5: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1007_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > 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 ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat,Z5: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( 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 @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1008_lexordp__eq_OCons__eq,axiom,
! [X2: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ord_lexordp_eq_nat @ Xs @ Ys )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_1009_lexordp__eq_OCons,axiom,
! [X2: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ).
% lexordp_eq.Cons
thf(fact_1010_same__length__different,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != Ys )
=> ( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ? [Pre: list_o,X4: $o,Xs5: list_o,Y5: $o,Ys5: list_o] :
( ( X4 != Y5 )
& ( Xs
= ( append_o @ Pre @ ( append_o @ ( cons_o @ X4 @ nil_o ) @ Xs5 ) ) )
& ( Ys
= ( append_o @ Pre @ ( append_o @ ( cons_o @ Y5 @ nil_o ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1011_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,X4: nat,Xs5: list_nat,Y5: nat,Ys5: list_nat] :
( ( X4 != Y5 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y5 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1012_fold__Cons__rev,axiom,
! [Xs: list_o] :
( ( fold_o_list_o @ cons_o @ Xs )
= ( append_o @ ( rev_o @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_1013_Cons__less__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y3: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y3 ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X2 @ Y3 ) ) ) ) ).
% Cons_less_Cons
thf(fact_1014_less__list__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ( X2 = Y3 )
& ( ord_less_list_nat @ Xs @ Ys ) ) ) ) ).
% less_list_code(3)
thf(fact_1015_append__one__prefix,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) )
=> ( prefix_o @ ( append_o @ Xs @ ( cons_o @ ( nth_o @ Ys @ ( size_size_list_o @ Xs ) ) @ nil_o ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_1016_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_1017_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_1018_find__Some__iff2,axiom,
! [X2: list_o,P: list_o > $o,Xs: list_list_o] :
( ( ( some_list_o @ X2 )
= ( find_list_o @ P @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s2710708370519433104list_o @ Xs ) )
& ( P @ ( nth_list_o @ Xs @ I ) )
& ( X2
= ( nth_list_o @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_list_o @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1019_find__Some__iff2,axiom,
! [X2: nat,P: nat > $o,Xs: list_nat] :
( ( ( some_nat @ X2 )
= ( find_nat @ P @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I ) )
& ( X2
= ( nth_nat @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1020_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1021_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_1022_nth__append__length,axiom,
! [Xs: list_o,X2: $o,Ys: list_o] :
( ( nth_o @ ( append_o @ Xs @ ( cons_o @ X2 @ Ys ) ) @ ( size_size_list_o @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_1023_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_1024_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_1025_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1026_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X6: nat] : ( P @ I @ X6 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1027_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_1028_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1029_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_1030_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P3: nat > $o,Xs2: list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
& ( P3 @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_1031_find__Some__iff,axiom,
! [P: list_o > $o,Xs: list_list_o,X2: list_o] :
( ( ( find_list_o @ P @ Xs )
= ( some_list_o @ X2 ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s2710708370519433104list_o @ Xs ) )
& ( P @ ( nth_list_o @ Xs @ I ) )
& ( X2
= ( nth_list_o @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_list_o @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1032_find__Some__iff,axiom,
! [P: nat > $o,Xs: list_nat,X2: nat] :
( ( ( find_nat @ P @ Xs )
= ( some_nat @ X2 ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I ) )
& ( X2
= ( nth_nat @ Xs @ I ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ~ ( P @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1033_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_1034_nth__append,axiom,
! [Xs: list_o,Ys: list_o,N: nat] :
( ( nth_o @ ( append_o @ Xs @ Ys ) @ N )
= ( ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( nth_o @ Xs @ N ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( nth_o @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1035_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_1036_length__Suc__conv__rev,axiom,
! [Xs: list_o,N: nat] :
( ( ( size_size_list_o @ Xs )
= ( suc @ N ) )
= ( ? [Y4: $o,Ys2: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ Y4 @ nil_o ) ) )
& ( ( size_size_list_o @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1037_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1038_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_1039_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_1040_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_1041_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1042_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1043_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_1044_nths__all,axiom,
! [Xs: list_nat,I3: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ I2 @ I3 ) )
=> ( ( nths_nat @ Xs @ I3 )
= Xs ) ) ).
% nths_all
thf(fact_1045_length__append__singleton,axiom,
! [Xs: list_o,X2: $o] :
( ( size_size_list_o @ ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) ) )
= ( suc @ ( size_size_list_o @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1046_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_1047_option_Osize__gen_I1_J,axiom,
! [X2: list_o > nat] :
( ( size_option_list_o @ X2 @ none_list_o )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_1048_option_Osize_I3_J,axiom,
( ( size_s9086679804311813456list_o @ none_list_o )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1049_option_Osize_I4_J,axiom,
! [X22: list_o] :
( ( size_s9086679804311813456list_o @ ( some_list_o @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1050_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_1051_take__Suc__conv__app__nth,axiom,
! [I4: nat,Xs: list_o] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
=> ( ( take_o @ ( suc @ I4 ) @ Xs )
= ( append_o @ ( take_o @ I4 @ Xs ) @ ( cons_o @ ( nth_o @ Xs @ I4 ) @ nil_o ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1052_take__Suc__conv__app__nth,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I4 ) @ Xs )
= ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I4 ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1053_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_1054_nth__take,axiom,
! [I4: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I4 )
= ( nth_nat @ Xs @ I4 ) ) ) ).
% nth_take
thf(fact_1055_take__append,axiom,
! [N: nat,Xs: list_o,Ys: list_o] :
( ( take_o @ N @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( take_o @ N @ Xs ) @ ( take_o @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1056_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_1057_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_1058_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs2: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_conv_take
thf(fact_1059_take__is__prefix,axiom,
! [N: nat,Xs: list_o] : ( prefix_o @ ( take_o @ N @ Xs ) @ Xs ) ).
% take_is_prefix
thf(fact_1060_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_1061_suffix__take,axiom,
! [Xs: list_o,Ys: list_o] :
( ( suffix_o @ Xs @ Ys )
=> ( Ys
= ( append_o @ ( take_o @ ( minus_minus_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs ) ) @ Ys ) @ Xs ) ) ) ).
% suffix_take
thf(fact_1062_suffix__take,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( suffix_nat @ Xs @ Ys )
=> ( Ys
= ( append_nat @ ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs ) ) @ Ys ) @ Xs ) ) ) ).
% suffix_take
thf(fact_1063_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_1064_nth__non__equal__first__eq,axiom,
! [X2: nat,Y3: nat,Xs: list_nat,N: nat] :
( ( X2 != Y3 )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= Y3 )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1065_id__take__nth__drop,axiom,
! [I4: nat,Xs: list_o] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
=> ( Xs
= ( append_o @ ( take_o @ I4 @ Xs ) @ ( cons_o @ ( nth_o @ Xs @ I4 ) @ ( drop_o @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1066_id__take__nth__drop,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I4 ) @ ( drop_nat @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1067_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_1068_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_1069_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1070_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_1071_length__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( size_size_list_o @ ( append_o @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_append
thf(fact_1072_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_1073_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_1074_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_1075_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_1076_append__take__drop__id,axiom,
! [N: nat,Xs: list_o] :
( ( append_o @ ( take_o @ N @ Xs ) @ ( drop_o @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_1077_nth__append__length__plus,axiom,
! [Xs: list_o,Ys: list_o,N: nat] :
( ( nth_o @ ( append_o @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_o @ Xs ) @ N ) )
= ( nth_o @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1078_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_1079_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_1080_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_1081_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_1082_drop__append,axiom,
! [N: nat,Xs: list_o,Ys: list_o] :
( ( drop_o @ N @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( drop_o @ N @ Xs ) @ ( drop_o @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1083_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_1084_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_1085_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_1086_nth__drop,axiom,
! [N: nat,Xs: list_nat,I4: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I4 )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I4 ) ) ) ) ).
% nth_drop
thf(fact_1087_append__eq__append__conv__if,axiom,
! [Xs_1: list_o,Xs_2: list_o,Ys_1: list_o,Ys_2: list_o] :
( ( ( append_o @ Xs_1 @ Xs_2 )
= ( append_o @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs_1 ) @ ( size_size_list_o @ Ys_1 ) )
=> ( ( Xs_1
= ( take_o @ ( size_size_list_o @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_o @ ( drop_o @ ( size_size_list_o @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_o @ Xs_1 ) @ ( size_size_list_o @ Ys_1 ) )
=> ( ( ( take_o @ ( size_size_list_o @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_o @ ( drop_o @ ( size_size_list_o @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_1088_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_1089_take__add,axiom,
! [I4: nat,J2: nat,Xs: list_o] :
( ( take_o @ ( plus_plus_nat @ I4 @ J2 ) @ Xs )
= ( append_o @ ( take_o @ I4 @ Xs ) @ ( take_o @ J2 @ ( drop_o @ I4 @ Xs ) ) ) ) ).
% take_add
thf(fact_1090_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_1091_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_1092_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1093_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1094_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_1095_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_1096_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1097_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1098_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1099_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1100_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1101_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1102_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1103_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_1104_order__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_1105_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_1106_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_1107_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1108_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1109_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1110_le__cases3,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1111_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1112_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1113_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1114_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1115_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1116_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1117_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1118_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1119_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1120_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1121_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1122_order__less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1123_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_1124_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_1125_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1126_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1127_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1128_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1129_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1130_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1131_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1132_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1133_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1134_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1135_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1136_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1137_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1138_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1139_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_1140_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1141_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_1142_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_1143_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1144_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_1145_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_1146_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y3 ) ) ).
% nth_via_drop
thf(fact_1147_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_1148_prefix__order_Olift__Suc__antimono__le,axiom,
! [F: nat > list_o,N: nat,N3: nat] :
( ! [N4: nat] : ( prefix_o @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_o @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_1149_prefix__order_Olift__Suc__mono__le,axiom,
! [F: nat > list_o,N: nat,N3: nat] :
( ! [N4: nat] : ( prefix_o @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_o @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_1150_prefix__length__prefix,axiom,
! [Ps: list_o,Xs: list_o,Qs: list_o] :
( ( prefix_o @ Ps @ Xs )
=> ( ( prefix_o @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_o @ Ps ) @ ( size_size_list_o @ Qs ) )
=> ( prefix_o @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_1151_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs )
=> ( ( prefix_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_1152_prefix__length__le,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% prefix_length_le
thf(fact_1153_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_1154_suffix__length__le,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( suffix_nat @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% suffix_length_le
thf(fact_1155_suffix__length__suffix,axiom,
! [Ps: list_nat,Xs: list_nat,Qs: list_nat] :
( ( suffix_nat @ Ps @ Xs )
=> ( ( suffix_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( suffix_nat @ Ps @ Qs ) ) ) ) ).
% suffix_length_suffix
thf(fact_1156_sublist__length__le,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sublist_nat @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% sublist_length_le
thf(fact_1157_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_1158_append__eq__conv__conj,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( append_o @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_o @ ( size_size_list_o @ Xs ) @ Zs ) )
& ( Ys
= ( drop_o @ ( size_size_list_o @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1159_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_1160_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1161_drop__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_1162_rev__drop,axiom,
! [I4: nat,Xs: list_nat] :
( ( rev_nat @ ( drop_nat @ I4 @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I4 ) @ ( rev_nat @ Xs ) ) ) ).
% rev_drop
thf(fact_1163_rev__take,axiom,
! [I4: nat,Xs: list_nat] :
( ( rev_nat @ ( take_nat @ I4 @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I4 ) @ ( rev_nat @ Xs ) ) ) ).
% rev_take
thf(fact_1164_take__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_1165_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_1166_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1167_Cons__nth__drop__Suc,axiom,
! [I4: nat,Xs: list_nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I4 ) @ ( drop_nat @ ( suc @ I4 ) @ Xs ) )
= ( drop_nat @ I4 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1168_option_Osize__gen_I2_J,axiom,
! [X2: list_o > nat,X22: list_o] :
( ( size_option_list_o @ X2 @ ( some_list_o @ X22 ) )
= ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1169_take__hd__drop,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( append_o @ ( take_o @ N @ Xs ) @ ( cons_o @ ( hd_o @ ( drop_o @ N @ Xs ) ) @ nil_o ) )
= ( take_o @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1170_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_1171_upd__conv__take__nth__drop,axiom,
! [I4: nat,Xs: list_o,A: $o] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ Xs @ I4 @ A )
= ( append_o @ ( take_o @ I4 @ Xs ) @ ( cons_o @ A @ ( drop_o @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1172_upd__conv__take__nth__drop,axiom,
! [I4: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I4 @ A )
= ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1173_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1174_Cons__le__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y3 ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X2 @ Y3 ) ) ) ) ).
% Cons_le_Cons
thf(fact_1175_length__list__update,axiom,
! [Xs: list_nat,I4: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_1176_nth__list__update__neq,axiom,
! [I4: nat,J2: nat,Xs: list_nat,X2: nat] :
( ( I4 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_1177_list__update__id,axiom,
! [Xs: list_nat,I4: nat] :
( ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_1178_hd__append2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != nil_o )
=> ( ( hd_o @ ( append_o @ Xs @ Ys ) )
= ( hd_o @ Xs ) ) ) ).
% hd_append2
thf(fact_1179_list__update__beyond,axiom,
! [Xs: list_nat,I4: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I4 )
=> ( ( list_update_nat @ Xs @ I4 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_1180_list__update__length,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y3: $o] :
( ( list_update_o @ ( append_o @ Xs @ ( cons_o @ X2 @ Ys ) ) @ ( size_size_list_o @ Xs ) @ Y3 )
= ( append_o @ Xs @ ( cons_o @ Y3 @ Ys ) ) ) ).
% list_update_length
thf(fact_1181_list__update__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y3 )
= ( append_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) ) ).
% list_update_length
thf(fact_1182_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ I4 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_1183_sorted__take,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_take
thf(fact_1184_sorted__nths,axiom,
! [Xs: list_nat,I3: set_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( nths_nat @ Xs @ I3 ) ) ) ).
% sorted_nths
thf(fact_1185_sorted2,axiom,
! [X2: nat,Y3: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y3 @ Zs ) ) )
= ( ( ord_less_eq_nat @ X2 @ Y3 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y3 @ Zs ) ) ) ) ).
% sorted2
thf(fact_1186_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1187_sorted__wrt__drop,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_1188_sorted__drop,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_drop
thf(fact_1189_sorted__antimono__prefix,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( ( sorted_wrt_o @ ord_less_eq_o @ Ys )
=> ( sorted_wrt_o @ ord_less_eq_o @ Xs ) ) ) ).
% sorted_antimono_prefix
thf(fact_1190_sorted__antimono__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ) ).
% sorted_antimono_prefix
thf(fact_1191_sorted__antimono__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( suffix_nat @ Xs @ Ys )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ) ).
% sorted_antimono_suffix
thf(fact_1192_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_1193_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I4: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I4 @ ( nth_nat @ Ns @ I4 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1194_less__eq__list__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ( X2 = Y3 )
& ( ord_less_eq_list_nat @ Xs @ Ys ) ) ) ) ).
% less_eq_list_code(3)
thf(fact_1195_hd__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( hd_o @ ( append_o @ Xs @ Ys ) )
= ( ( ( Xs = nil_o )
=> ( hd_o @ Ys ) )
& ( ( Xs != nil_o )
=> ( hd_o @ Xs ) ) ) ) ).
% hd_append
thf(fact_1196_longest__common__prefix,axiom,
! [Xs: list_o,Ys: list_o] :
? [Ps3: list_o,Xs5: list_o,Ys5: list_o] :
( ( Xs
= ( append_o @ Ps3 @ Xs5 ) )
& ( Ys
= ( append_o @ Ps3 @ Ys5 ) )
& ( ( Xs5 = nil_o )
| ( Ys5 = nil_o )
| ( ( hd_o @ Xs5 )
!= ( hd_o @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_1197_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1198_sorted__wrt1,axiom,
! [P: nat > nat > $o,X2: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1199_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1200_sorted__wrt__take,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_1201_sorted1,axiom,
! [X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted1
thf(fact_1202_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_1203_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I4: nat,J2: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1204_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs2: list_nat] :
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P3 @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1205_sorted__butlast,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( butlast_nat @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1206_list__update__append1,axiom,
! [I4: nat,Xs: list_o,Ys: list_o,X2: $o] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ ( append_o @ Xs @ Ys ) @ I4 @ X2 )
= ( append_o @ ( list_update_o @ Xs @ I4 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1207_list__update__append1,axiom,
! [I4: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I4 @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1208_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1209_nth__list__update,axiom,
! [I4: nat,Xs: list_nat,J2: nat,X2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I4 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J2 )
= X2 ) )
& ( ( I4 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_1210_list__update__same__conv,axiom,
! [I4: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I4 @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I4 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1211_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_1212_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1213_list__update__append,axiom,
! [N: nat,Xs: list_o,Ys: list_o,X2: $o] :
( ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ ( append_o @ Xs @ Ys ) @ N @ X2 )
= ( append_o @ ( list_update_o @ Xs @ N @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( list_update_o @ ( append_o @ Xs @ Ys ) @ N @ X2 )
= ( append_o @ Xs @ ( list_update_o @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_1214_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_1215_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_1216_butlast__list__update,axiom,
! [K: nat,Xs: list_nat,X2: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_1217_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ ( suc @ I ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1218_sorted__nth__mono,axiom,
! [Xs: list_nat,I4: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1219_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1220_rev__update,axiom,
! [K: nat,Xs: list_nat,Y3: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y3 ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y3 ) ) ) ).
% rev_update
thf(fact_1221_last__list__update,axiom,
! [Xs: list_nat,K: nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1222_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I ) ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1223_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I4: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1224_foldr__max__sorted,axiom,
! [Xs: list_nat,Y3: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ( Xs = nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs @ Y3 )
= Y3 ) )
& ( ( Xs != nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs @ Y3 )
= ( ord_max_nat @ ( nth_nat @ Xs @ zero_zero_nat ) @ Y3 ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_1225_length__transpose__sorted,axiom,
! [Xs: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs ) ) )
=> ( ( ( Xs = nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= zero_zero_nat ) )
& ( ( Xs != nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= ( size_size_list_nat @ ( nth_list_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_1226_max__absorb2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_max_nat @ X2 @ Y3 )
= Y3 ) ) ).
% max_absorb2
thf(fact_1227_max__absorb1,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_max_nat @ X2 @ Y3 )
= X2 ) ) ).
% max_absorb1
thf(fact_1228_max__def,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_1229_sorted__transpose,axiom,
! [Xs: list_list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( transpose_nat @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_1230_distinct__adj__append__iff,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_adj_o @ ( append_o @ Xs @ Ys ) )
= ( ( distinct_adj_o @ Xs )
& ( distinct_adj_o @ Ys )
& ( ( Xs = nil_o )
| ( Ys = nil_o )
| ( ( last_o @ Xs )
!= ( hd_o @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1231_distinct__adj__appendD2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_adj_o @ ( append_o @ Xs @ Ys ) )
=> ( distinct_adj_o @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_1232_distinct__adj__appendD1,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_adj_o @ ( append_o @ Xs @ Ys ) )
=> ( distinct_adj_o @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_1233_rotate1__hd__tl,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( ( rotate1_o @ Xs )
= ( append_o @ ( tl_o @ Xs ) @ ( cons_o @ ( hd_o @ Xs ) @ nil_o ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1234_tl__append2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != nil_o )
=> ( ( tl_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( tl_o @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_1235_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_1236_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_1237_tl__append__if,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( Xs = nil_o )
=> ( ( tl_o @ ( append_o @ Xs @ Ys ) )
= ( tl_o @ Ys ) ) )
& ( ( Xs != nil_o )
=> ( ( tl_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ ( tl_o @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_1238_sorted__tl,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( tl_nat @ Xs ) ) ) ).
% sorted_tl
thf(fact_1239_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_1240_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs2: list_nat] : ( if_nat @ ( Xs2 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs2 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1241_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_1242_size__list__append,axiom,
! [F: $o > nat,Xs: list_o,Ys: list_o] :
( ( size_list_o @ F @ ( append_o @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_o @ F @ Xs ) @ ( size_list_o @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_1243_rotate__drop__take,axiom,
( rotate_o
= ( ^ [N2: nat,Xs2: list_o] : ( append_o @ ( drop_o @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Xs2 ) ) @ Xs2 ) @ ( take_o @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_1244_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) @ ( take_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_1245_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_1246_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_1247_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_1248_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_1249_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ).
% rotate_conv_mod
thf(fact_1250_rotate__append,axiom,
! [L2: list_o,Q: list_o] :
( ( rotate_o @ ( size_size_list_o @ L2 ) @ ( append_o @ L2 @ Q ) )
= ( append_o @ Q @ L2 ) ) ).
% rotate_append
thf(fact_1251_rotate__append,axiom,
! [L2: list_nat,Q: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L2 ) @ ( append_nat @ L2 @ Q ) )
= ( append_nat @ Q @ L2 ) ) ).
% rotate_append
thf(fact_1252_rotate__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1253_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1254_set__swap,axiom,
! [I4: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I4 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1255_comm__append__is__replicate,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != nil_o )
=> ( ( Ys != nil_o )
=> ( ( ( append_o @ Xs @ Ys )
= ( append_o @ Ys @ Xs ) )
=> ? [N4: nat,Zs2: list_o] :
( ( ord_less_nat @ one_one_nat @ N4 )
& ( ( concat_o @ ( replicate_list_o @ N4 @ Zs2 ) )
= ( append_o @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_1256_length__replicate,axiom,
! [N: nat,X2: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_1257_in__set__prefixes,axiom,
! [Xs: list_o,Ys: list_o] :
( ( member_list_o2 @ Xs @ ( set_list_o2 @ ( prefixes_o @ Ys ) ) )
= ( prefix_o @ Xs @ Ys ) ) ).
% in_set_prefixes
thf(fact_1258_nth__replicate,axiom,
! [I4: nat,N: nat,X2: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I4 )
= X2 ) ) ).
% nth_replicate
thf(fact_1259_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1260_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1261_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1262_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1263_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1264_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1265_replicate__append__same,axiom,
! [I4: nat,X2: $o] :
( ( append_o @ ( replicate_o @ I4 @ X2 ) @ ( cons_o @ X2 @ nil_o ) )
= ( cons_o @ X2 @ ( replicate_o @ I4 @ X2 ) ) ) ).
% replicate_append_same
thf(fact_1266_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1267_comm__append__are__replicate,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= ( append_o @ Ys @ Xs ) )
=> ? [M3: nat,N4: nat,Zs2: list_o] :
( ( ( concat_o @ ( replicate_list_o @ M3 @ Zs2 ) )
= Xs )
& ( ( concat_o @ ( replicate_list_o @ N4 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_1268_in__set__butlast__appendI,axiom,
! [X2: $o,Xs: list_o,Ys: list_o] :
( ( ( member_o2 @ X2 @ ( set_o2 @ ( butlast_o @ Xs ) ) )
| ( member_o2 @ X2 @ ( set_o2 @ ( butlast_o @ Ys ) ) ) )
=> ( member_o2 @ X2 @ ( set_o2 @ ( butlast_o @ ( append_o @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1269_find__None__iff,axiom,
! [P: list_o > $o,Xs: list_list_o] :
( ( ( find_list_o @ P @ Xs )
= none_list_o )
= ( ~ ? [X3: list_o] :
( ( member_list_o2 @ X3 @ ( set_list_o2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_1270_find__None__iff2,axiom,
! [P: list_o > $o,Xs: list_list_o] :
( ( none_list_o
= ( find_list_o @ P @ Xs ) )
= ( ~ ? [X3: list_o] :
( ( member_list_o2 @ X3 @ ( set_list_o2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_1271_replicate__app__Cons__same,axiom,
! [N: nat,X2: $o,Xs: list_o] :
( ( append_o @ ( replicate_o @ N @ X2 ) @ ( cons_o @ X2 @ Xs ) )
= ( cons_o @ X2 @ ( append_o @ ( replicate_o @ N @ X2 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X2: list_o,Y3: list_o] :
( ( if_list_o @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X2: list_o,Y3: list_o] :
( ( if_list_o @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( prefix_o @ ( append_o @ x2 @ y2 ) @ z2 )
| ( prefix_o @ z2 @ ( append_o @ x2 @ y2 ) ) ) ).
%------------------------------------------------------------------------------