TPTP Problem File: SLH0696^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc3/0007_Fair_Stream/prob_00034_000983__11895320_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1513 ( 631 unt; 237 typ; 0 def)
% Number of atoms : 3183 (1533 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10583 ( 338 ~; 67 |; 190 &;8638 @)
% ( 0 <=>;1350 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 578 ( 578 >; 0 *; 0 +; 0 <<)
% Number of symbols : 205 ( 202 usr; 15 con; 0-3 aty)
% Number of variables : 3362 ( 182 ^;3072 !; 108 ?;3362 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:27:19.102
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J_J,type,
stream2377611395989027862list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
stream3775083406817841430list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J_J,type,
stream1229541022664496662list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
stream2255243159586646806list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J_J,type,
set_li4183784480778324208list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_It__Nat__Onat_J_J_J,type,
stream6406556387131777664st_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J_J,type,
stream2307142169165677840ream_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
stream_list_list_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
stream_stream_list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__Stream__Ostream_Itf__a_J_J_J,type,
stream_list_stream_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
stream_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
list_stream_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
set_stream_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Stream__Ostream_Itf__a_J_J_J,type,
set_list_stream_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Nat__Onat_J_J,type,
stream_stream_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__Nat__Onat_J_J,type,
stream_list_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J,type,
stream_stream_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
stream_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Stream__Ostream_Itf__a_J_J,type,
list_stream_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_Itf__a_J_J,type,
set_stream_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Nat__Onat_J,type,
stream_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__Stream__Ostream_Itf__a_J,type,
stream_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (202)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Fair__Stream_Ofair_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
fair_f5205184157793188661list_a: stream_list_list_a > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__List__Olist_It__Nat__Onat_J,type,
fair_fair_list_nat: stream_list_nat > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__List__Olist_Itf__a_J,type,
fair_fair_list_a: stream_list_a > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Nat__Onat,type,
fair_fair_nat: stream_nat > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
fair_f1030357992759841077list_a: stream_stream_list_a > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Stream__Ostream_Itf__a_J,type,
fair_fair_stream_a: stream_stream_a > $o ).
thf(sy_c_Fair__Stream_Ofair_001tf__a,type,
fair_fair_a: stream_a > $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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
if_str8217234800680828380list_a: $o > stream2255243159586646806list_a > stream2255243159586646806list_a > stream2255243159586646806list_a ).
thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
if_str5104684111711982906st_nat: $o > stream_list_list_nat > stream_list_list_nat > stream_list_list_nat ).
thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
if_str7505741754068378070list_a: $o > stream_list_list_a > stream_list_list_a > stream_list_list_a ).
thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__Nat__Onat_J_J,type,
if_stream_list_nat: $o > stream_list_nat > stream_list_nat > stream_list_nat ).
thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
if_stream_list_a: $o > stream_list_a > stream_list_a > stream_list_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
nil_stream_list_a: list_stream_list_a ).
thf(sy_c_List_Olist_ONil_001t__Stream__Ostream_Itf__a_J,type,
nil_stream_a: list_stream_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
tl_list_list_a: list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__Nat__Onat_J,type,
tl_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
list_ex1_list_a: ( list_a > $o ) > list_list_a > $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_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_Itf__a_J,type,
list_ex_list_a: ( list_a > $o ) > list_list_a > $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_001tf__a,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
nth_stream_list_a: list_stream_list_a > nat > stream_list_a ).
thf(sy_c_List_Onth_001t__Stream__Ostream_Itf__a_J,type,
nth_stream_a: list_stream_a > nat > stream_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001t__List__Olist_Itf__a_J,type,
rev_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001tf__a,type,
rotate_a: nat > list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_Itf__a_J,type,
sorted_wrt_list_a: ( list_a > list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
size_s3694203809124329340list_a: list_stream_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Stream__Ostream_Itf__a_J_J,type,
size_s2142770077969500662ream_a: list_stream_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le8488217952732425610list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Stream_Ocycle_001t__List__Olist_Itf__a_J,type,
cycle_list_a: list_list_a > stream_list_a ).
thf(sy_c_Stream_Ocycle_001t__Nat__Onat,type,
cycle_nat: list_nat > stream_nat ).
thf(sy_c_Stream_Ocycle_001tf__a,type,
cycle_a: list_a > stream_a ).
thf(sy_c_Stream_Oflat_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
flat_list_list_a: stream2255243159586646806list_a > stream_list_list_a ).
thf(sy_c_Stream_Oflat_001t__List__Olist_It__Nat__Onat_J,type,
flat_list_nat: stream_list_list_nat > stream_list_nat ).
thf(sy_c_Stream_Oflat_001t__List__Olist_Itf__a_J,type,
flat_list_a: stream_list_list_a > stream_list_a ).
thf(sy_c_Stream_Oflat_001t__Nat__Onat,type,
flat_nat: stream_list_nat > stream_nat ).
thf(sy_c_Stream_Oflat_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
flat_stream_list_a: stream1229541022664496662list_a > stream_stream_list_a ).
thf(sy_c_Stream_Oflat_001t__Stream__Ostream_Itf__a_J,type,
flat_stream_a: stream_list_stream_a > stream_stream_a ).
thf(sy_c_Stream_Oflat_001tf__a,type,
flat_a: stream_list_a > stream_a ).
thf(sy_c_Stream_Osdrop_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sdrop_list_list_a: nat > stream_list_list_a > stream_list_list_a ).
thf(sy_c_Stream_Osdrop_001t__List__Olist_It__Nat__Onat_J,type,
sdrop_list_nat: nat > stream_list_nat > stream_list_nat ).
thf(sy_c_Stream_Osdrop_001t__List__Olist_Itf__a_J,type,
sdrop_list_a: nat > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osdrop_001t__Nat__Onat,type,
sdrop_nat: nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Osdrop_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
sdrop_stream_list_a: nat > stream_stream_list_a > stream_stream_list_a ).
thf(sy_c_Stream_Osdrop_001t__Stream__Ostream_Itf__a_J,type,
sdrop_stream_a: nat > stream_stream_a > stream_stream_a ).
thf(sy_c_Stream_Osdrop_001tf__a,type,
sdrop_a: nat > stream_a > stream_a ).
thf(sy_c_Stream_Oshift_001t__List__Olist_Itf__a_J,type,
shift_list_a: list_list_a > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Oshift_001t__Nat__Onat,type,
shift_nat: list_nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Oshift_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
shift_stream_list_a: list_stream_list_a > stream_stream_list_a > stream_stream_list_a ).
thf(sy_c_Stream_Oshift_001t__Stream__Ostream_Itf__a_J,type,
shift_stream_a: list_stream_a > stream_stream_a > stream_stream_a ).
thf(sy_c_Stream_Oshift_001tf__a,type,
shift_a: list_a > stream_a > stream_a ).
thf(sy_c_Stream_Osmember_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
smember_list_list_a: list_list_a > stream_list_list_a > $o ).
thf(sy_c_Stream_Osmember_001t__List__Olist_It__Nat__Onat_J,type,
smember_list_nat: list_nat > stream_list_nat > $o ).
thf(sy_c_Stream_Osmember_001t__List__Olist_Itf__a_J,type,
smember_list_a: list_a > stream_list_a > $o ).
thf(sy_c_Stream_Osmember_001tf__a,type,
smember_a: a > stream_a > $o ).
thf(sy_c_Stream_Osmerge_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
smerge_list_list_a: stream3775083406817841430list_a > stream_list_list_a ).
thf(sy_c_Stream_Osmerge_001t__List__Olist_It__Nat__Onat_J,type,
smerge_list_nat: stream6406556387131777664st_nat > stream_list_nat ).
thf(sy_c_Stream_Osmerge_001t__List__Olist_Itf__a_J,type,
smerge_list_a: stream_stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osmerge_001t__Nat__Onat,type,
smerge_nat: stream_stream_nat > stream_nat ).
thf(sy_c_Stream_Osmerge_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
smerge_stream_list_a: stream2377611395989027862list_a > stream_stream_list_a ).
thf(sy_c_Stream_Osmerge_001t__Stream__Ostream_Itf__a_J,type,
smerge_stream_a: stream2307142169165677840ream_a > stream_stream_a ).
thf(sy_c_Stream_Osmerge_001tf__a,type,
smerge_a: stream_stream_a > stream_a ).
thf(sy_c_Stream_Osnth_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
snth_list_list_a: stream_list_list_a > nat > list_list_a ).
thf(sy_c_Stream_Osnth_001t__List__Olist_It__Nat__Onat_J,type,
snth_list_nat: stream_list_nat > nat > list_nat ).
thf(sy_c_Stream_Osnth_001t__List__Olist_Itf__a_J,type,
snth_list_a: stream_list_a > nat > list_a ).
thf(sy_c_Stream_Osnth_001t__Nat__Onat,type,
snth_nat: stream_nat > nat > nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
snth_s3514276978948636852list_a: stream3775083406817841430list_a > nat > stream_list_list_a ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__List__Olist_It__Nat__Onat_J_J,type,
snth_stream_list_nat: stream6406556387131777664st_nat > nat > stream_list_nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
snth_stream_list_a: stream_stream_list_a > nat > stream_list_a ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Nat__Onat_J,type,
snth_stream_nat: stream_stream_nat > nat > stream_nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
snth_s4451090846259577524list_a: stream2377611395989027862list_a > nat > stream_stream_list_a ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J,type,
snth_stream_stream_a: stream2307142169165677840ream_a > nat > stream_stream_a ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_Itf__a_J,type,
snth_stream_a: stream_stream_a > nat > stream_a ).
thf(sy_c_Stream_Osnth_001tf__a,type,
snth_a: stream_a > nat > a ).
thf(sy_c_Stream_Ostake_001t__List__Olist_Itf__a_J,type,
stake_list_a: nat > stream_list_a > list_list_a ).
thf(sy_c_Stream_Ostake_001t__Nat__Onat,type,
stake_nat: nat > stream_nat > list_nat ).
thf(sy_c_Stream_Ostake_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
stake_stream_list_a: nat > stream_stream_list_a > list_stream_list_a ).
thf(sy_c_Stream_Ostake_001t__Stream__Ostream_Itf__a_J,type,
stake_stream_a: nat > stream_stream_a > list_stream_a ).
thf(sy_c_Stream_Ostake_001tf__a,type,
stake_a: nat > stream_a > list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
sCons_8165023923567507367list_a: list_list_list_a > stream2255243159586646806list_a > stream2255243159586646806list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
sCons_list_list_nat: list_list_nat > stream_list_list_nat > stream_list_list_nat ).
thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sCons_list_list_a: list_list_a > stream_list_list_a > stream_list_list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__Nat__Onat_J,type,
sCons_list_nat: list_nat > stream_list_nat > stream_list_nat ).
thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_Itf__a_J,type,
sCons_list_a: list_a > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Nat__Onat,type,
sCons_nat: nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Ostream_OSCons_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
sCons_stream_list_a: stream_list_a > stream_stream_list_a > stream_stream_list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Stream__Ostream_Itf__a_J,type,
sCons_stream_a: stream_a > stream_stream_a > stream_stream_a ).
thf(sy_c_Stream_Ostream_OSCons_001tf__a,type,
sCons_a: a > stream_a > stream_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
shd_list_list_list_a: stream2255243159586646806list_a > list_list_list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
shd_list_list_nat: stream_list_list_nat > list_list_nat ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
shd_list_list_a: stream_list_list_a > list_list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__Nat__Onat_J,type,
shd_list_nat: stream_list_nat > list_nat ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
shd_li3061069428927788988list_a: stream1229541022664496662list_a > list_stream_list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__Stream__Ostream_Itf__a_J_J,type,
shd_list_stream_a: stream_list_stream_a > list_stream_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_Itf__a_J,type,
shd_list_a: stream_list_a > list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Nat__Onat,type,
shd_nat: stream_nat > nat ).
thf(sy_c_Stream_Ostream_Oshd_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
shd_stream_list_a: stream_stream_list_a > stream_list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Stream__Ostream_Itf__a_J,type,
shd_stream_a: stream_stream_a > stream_a ).
thf(sy_c_Stream_Ostream_Oshd_001tf__a,type,
shd_a: stream_a > a ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sset_list_list_a: stream_list_list_a > set_list_list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__Nat__Onat_J,type,
sset_list_nat: stream_list_nat > set_list_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
sset_l2691409540454527268list_a: stream1229541022664496662list_a > set_li4183784480778324208list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__Stream__Ostream_Itf__a_J_J,type,
sset_list_stream_a: stream_list_stream_a > set_list_stream_a ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_Itf__a_J,type,
sset_list_a: stream_list_a > set_list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Nat__Onat,type,
sset_nat: stream_nat > set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
sset_stream_list_a: stream_stream_list_a > set_stream_list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_Itf__a_J,type,
sset_stream_a: stream_stream_a > set_stream_a ).
thf(sy_c_Stream_Ostream_Osset_001tf__a,type,
sset_a: stream_a > set_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
stl_list_list_list_a: stream2255243159586646806list_a > stream2255243159586646806list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
stl_list_list_nat: stream_list_list_nat > stream_list_list_nat ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
stl_list_list_a: stream_list_list_a > stream_list_list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__Nat__Onat_J,type,
stl_list_nat: stream_list_nat > stream_list_nat ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
stl_li3392269663658258880list_a: stream1229541022664496662list_a > stream1229541022664496662list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__Stream__Ostream_Itf__a_J_J,type,
stl_list_stream_a: stream_list_stream_a > stream_list_stream_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_Itf__a_J,type,
stl_list_a: stream_list_a > stream_list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Nat__Onat,type,
stl_nat: stream_nat > stream_nat ).
thf(sy_c_Stream_Ostream_Ostl_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
stl_stream_list_a: stream_stream_list_a > stream_stream_list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Stream__Ostream_Itf__a_J,type,
stl_stream_a: stream_stream_a > stream_stream_a ).
thf(sy_c_Stream_Ostream_Ostl_001tf__a,type,
stl_a: stream_a > stream_a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
member2241138852378865081list_a: list_stream_list_a > set_li4183784480778324208list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Stream__Ostream_Itf__a_J_J,type,
member_list_stream_a: list_stream_a > set_list_stream_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
member_stream_list_a: stream_list_a > set_stream_list_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_Itf__a_J,type,
member_stream_a: stream_a > set_stream_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_s,type,
s: stream_list_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1262)
thf(fact_0_assms_I3_J,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( sset_list_a @ s ) )
=> ( X != nil_a ) ) ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
ord_less_nat @ m @ ( size_size_list_a @ ( snth_list_a @ s @ n ) ) ).
% assms(2)
thf(fact_2_assms_I1_J,axiom,
( x
= ( nth_a @ ( snth_list_a @ s @ n ) @ m ) ) ).
% assms(1)
thf(fact_3_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X2: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ X2 ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_6_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% dual_order.refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_9_fair__def,axiom,
( fair_fair_nat
= ( ^ [S: stream_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ ( sset_nat @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_nat @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_10_fair__def,axiom,
( fair_fair_stream_a
= ( ^ [S: stream_stream_a] :
! [X3: stream_a] :
( ( member_stream_a @ X3 @ ( sset_stream_a @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_stream_a @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_11_fair__def,axiom,
( fair_f1030357992759841077list_a
= ( ^ [S: stream_stream_list_a] :
! [X3: stream_list_a] :
( ( member_stream_list_a @ X3 @ ( sset_stream_list_a @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_stream_list_a @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_12_fair__def,axiom,
( fair_fair_list_nat
= ( ^ [S: stream_list_nat] :
! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( sset_list_nat @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_list_nat @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_13_fair__def,axiom,
( fair_f5205184157793188661list_a
= ( ^ [S: stream_list_list_a] :
! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( sset_list_list_a @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_list_list_a @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_14_fair__def,axiom,
( fair_fair_a
= ( ^ [S: stream_a] :
! [X3: a] :
( ( member_a @ X3 @ ( sset_a @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_a @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_15_fair__def,axiom,
( fair_fair_list_a
= ( ^ [S: stream_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ ( sset_list_a @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_list_a @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_16_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_17_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_18_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_19_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_20_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_21_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_22_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M3: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_23_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_24_verit__comp__simplify1_I2_J,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_25_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_26_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_27_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_28_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_29_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_30_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_31_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_32_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X4 )
=> ( P @ Y2 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_33_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_34_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_35_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_36_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_37_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_38_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_39_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_40_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_41_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_42_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_43_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_44_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_45_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_46_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_47_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_48_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_49_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_50_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_51_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_52_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A3: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_56_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_57_size__neq__size__imp__neq,axiom,
! [X2: list_list_a,Y3: list_list_a] :
( ( ( size_s349497388124573686list_a @ X2 )
!= ( size_s349497388124573686list_a @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_58_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y3: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_59_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_60_size__neq__size__imp__neq,axiom,
! [X2: char,Y3: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_61_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_62_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_63_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_64_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_65_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_66_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_67_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_68_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,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_69_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,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_70_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_71_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_72_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_73_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_74_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_75_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_76_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_77_order__le__imp__less__or__eq,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ( ord_less_set_list_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_78_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_79_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_80_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_81_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_82_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,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_83_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_84_order__less__le__subst1,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( ord_less_set_list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_85_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_86_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_87_order__less__le__subst1,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( ord_less_set_list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_88_order__less__le__subst1,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_89_order__less__le__subst1,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_90_order__less__le__subst1,axiom,
! [A: set_list_a,F: set_list_a > set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_91_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,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_92_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_93_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_94_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_95_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_96_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_97_order__le__less__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_98_order__le__less__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_99_order__le__less__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_100_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,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_101_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_102_order__le__less__subst1,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_103_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,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_104_order__less__le__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_105_order__less__le__trans,axiom,
! [X2: set_list_a,Y3: set_list_a,Z: set_list_a] :
( ( ord_less_set_list_a @ X2 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ Y3 @ Z )
=> ( ord_less_set_list_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_106_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_107_order__le__less__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_108_order__le__less__trans,axiom,
! [X2: set_list_a,Y3: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ( ord_less_set_list_a @ Y3 @ Z )
=> ( ord_less_set_list_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_109_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_110_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_111_order__neq__le__trans,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A != B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_less_set_list_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_112_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_113_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_114_order__le__neq__trans,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_list_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_115_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_116_order__less__imp__le,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_117_order__less__imp__le,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ( ord_less_set_list_a @ X2 @ Y3 )
=> ( ord_le8861187494160871172list_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_118_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_119_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_120_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_121_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_122_order__less__le,axiom,
( ord_less_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_123_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_124_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_125_order__le__less,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_less_set_list_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_126_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_127_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_128_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_129_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_130_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_131_order_Ostrict__implies__order,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_132_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_133_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_134_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_list_a
= ( ^ [B3: set_list_a,A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A4 )
& ~ ( ord_le8861187494160871172list_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_135_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_136_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_137_dual__order_Ostrict__trans2,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_less_set_list_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_138_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_139_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_140_dual__order_Ostrict__trans1,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_less_set_list_a @ C @ B )
=> ( ord_less_set_list_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_141_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_142_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_143_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_list_a
= ( ^ [B3: set_list_a,A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_144_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_145_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_146_dual__order_Oorder__iff__strict,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B3: set_list_a,A4: set_list_a] :
( ( ord_less_set_list_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_147_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_148_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_149_order_Ostrict__iff__not,axiom,
( ord_less_set_list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B3 )
& ~ ( ord_le8861187494160871172list_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_150_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_151_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_152_order_Ostrict__trans2,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_153_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_154_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_155_order_Ostrict__trans1,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_156_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_157_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_158_order_Ostrict__iff__order,axiom,
( ord_less_set_list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_159_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_160_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_161_order_Oorder__iff__strict,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ( ord_less_set_list_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_162_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_163_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_164_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_165_less__le__not__le,axiom,
( ord_less_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ~ ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_166_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_167_antisym__conv2,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_168_antisym__conv2,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_list_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_169_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_170_antisym__conv1,axiom,
! [X2: set_a,Y3: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_171_antisym__conv1,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ~ ( ord_less_set_list_a @ X2 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_172_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_173_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_174_nless__le,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ~ ( ord_less_set_list_a @ A @ B ) )
= ( ~ ( ord_le8861187494160871172list_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_175_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_176_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_177_leD,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_178_leD,axiom,
! [Y3: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y3 @ X2 )
=> ~ ( ord_less_set_list_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_179_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_180_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_181_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_182_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_183_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_184_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_185_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_186_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_187_order__antisym__conv,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_188_order__antisym__conv,axiom,
! [Y3: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y3 @ X2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_189_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_190_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_191_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_192_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_193_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_194_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_195_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_196_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_197_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_198_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_199_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,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_200_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_201_ord__eq__le__subst,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_202_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_203_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_204_ord__eq__le__subst,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_205_ord__eq__le__subst,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_206_ord__eq__le__subst,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_207_ord__eq__le__subst,axiom,
! [A: set_list_a,F: set_list_a > set_list_a,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_208_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,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_209_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_210_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_211_order__eq__refl,axiom,
! [X2: set_a,Y3: set_a] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_212_order__eq__refl,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ( X2 = Y3 )
=> ( ord_le8861187494160871172list_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_213_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_214_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_215_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_216_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_217_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_218_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_219_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_220_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_221_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_222_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,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_223_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_224_order__subst1,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_225_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_226_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_227_order__subst1,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_228_order__subst1,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_229_order__subst1,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_230_order__subst1,axiom,
! [A: set_list_a,F: set_list_a > set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_231_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,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_232_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_233_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_a,Z2: set_list_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_234_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_235_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_236_antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_237_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_238_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_239_dual__order_Otrans,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_le8861187494160871172list_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_240_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_241_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_242_dual__order_Oantisym,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_243_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_244_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_245_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_list_a,Z2: set_list_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A4 )
& ( ord_le8861187494160871172list_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_246_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_247_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_248_order__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_249_order__trans,axiom,
! [X2: set_list_a,Y3: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ Y3 @ Z )
=> ( ord_le8861187494160871172list_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_250_order__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_251_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_252_order_Otrans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% order.trans
thf(fact_253_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_254_order__antisym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_255_order__antisym,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_256_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_257_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_258_ord__le__eq__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( B = C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_259_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_260_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_261_ord__eq__le__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( A = B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_262_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_263_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_264_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_a,Z2: set_list_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_265_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_266_le__cases3,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_267_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_268_nth__equalityI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I2 )
= ( nth_list_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_269_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_270_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_271_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: list_a] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_list_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_272_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: a] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_273_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: nat] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_274_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_list_a,Z2: list_list_a] : ( Y5 = Z2 ) )
= ( ^ [Xs2: list_list_a,Ys2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( ( nth_list_a @ Xs2 @ I3 )
= ( nth_list_a @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_275_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_a,Z2: list_a] : ( Y5 = Z2 ) )
= ( ^ [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_276_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z2: list_nat] : ( Y5 = Z2 ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_277_snth__sset,axiom,
! [S2: stream_nat,N2: nat] : ( member_nat @ ( snth_nat @ S2 @ N2 ) @ ( sset_nat @ S2 ) ) ).
% snth_sset
thf(fact_278_snth__sset,axiom,
! [S2: stream_stream_a,N2: nat] : ( member_stream_a @ ( snth_stream_a @ S2 @ N2 ) @ ( sset_stream_a @ S2 ) ) ).
% snth_sset
thf(fact_279_snth__sset,axiom,
! [S2: stream_stream_list_a,N2: nat] : ( member_stream_list_a @ ( snth_stream_list_a @ S2 @ N2 ) @ ( sset_stream_list_a @ S2 ) ) ).
% snth_sset
thf(fact_280_snth__sset,axiom,
! [S2: stream_list_nat,N2: nat] : ( member_list_nat @ ( snth_list_nat @ S2 @ N2 ) @ ( sset_list_nat @ S2 ) ) ).
% snth_sset
thf(fact_281_snth__sset,axiom,
! [S2: stream_list_list_a,N2: nat] : ( member_list_list_a @ ( snth_list_list_a @ S2 @ N2 ) @ ( sset_list_list_a @ S2 ) ) ).
% snth_sset
thf(fact_282_snth__sset,axiom,
! [S2: stream_list_a,N2: nat] : ( member_list_a @ ( snth_list_a @ S2 @ N2 ) @ ( sset_list_a @ S2 ) ) ).
% snth_sset
thf(fact_283_snth__sset,axiom,
! [S2: stream_a,N2: nat] : ( member_a @ ( snth_a @ S2 @ N2 ) @ ( sset_a @ S2 ) ) ).
% snth_sset
thf(fact_284_length__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ! [Xs3: list_list_a] :
( ! [Ys3: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys3 ) @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_285_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs3: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_286_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_287_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_288_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ C2 ) )
=> ( P @ X ) )
& ! [D: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_289_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ T ) ) ).
% pinf(6)
thf(fact_290_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ T @ X ) ) ).
% pinf(8)
thf(fact_291_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_292_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_293_shift__snth__less,axiom,
! [P4: nat,Xs: list_stream_a,S2: stream_stream_a] :
( ( ord_less_nat @ P4 @ ( size_s2142770077969500662ream_a @ Xs ) )
=> ( ( snth_stream_a @ ( shift_stream_a @ Xs @ S2 ) @ P4 )
= ( nth_stream_a @ Xs @ P4 ) ) ) ).
% shift_snth_less
thf(fact_294_shift__snth__less,axiom,
! [P4: nat,Xs: list_stream_list_a,S2: stream_stream_list_a] :
( ( ord_less_nat @ P4 @ ( size_s3694203809124329340list_a @ Xs ) )
=> ( ( snth_stream_list_a @ ( shift_stream_list_a @ Xs @ S2 ) @ P4 )
= ( nth_stream_list_a @ Xs @ P4 ) ) ) ).
% shift_snth_less
thf(fact_295_shift__snth__less,axiom,
! [P4: nat,Xs: list_list_a,S2: stream_list_a] :
( ( ord_less_nat @ P4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( snth_list_a @ ( shift_list_a @ Xs @ S2 ) @ P4 )
= ( nth_list_a @ Xs @ P4 ) ) ) ).
% shift_snth_less
thf(fact_296_shift__snth__less,axiom,
! [P4: nat,Xs: list_a,S2: stream_a] :
( ( ord_less_nat @ P4 @ ( size_size_list_a @ Xs ) )
=> ( ( snth_a @ ( shift_a @ Xs @ S2 ) @ P4 )
= ( nth_a @ Xs @ P4 ) ) ) ).
% shift_snth_less
thf(fact_297_shift__snth__less,axiom,
! [P4: nat,Xs: list_nat,S2: stream_nat] :
( ( ord_less_nat @ P4 @ ( size_size_list_nat @ Xs ) )
=> ( ( snth_nat @ ( shift_nat @ Xs @ S2 ) @ P4 )
= ( nth_nat @ Xs @ P4 ) ) ) ).
% shift_snth_less
thf(fact_298_shift__left__inj,axiom,
! [Xs: list_nat,S1: stream_nat,S22: stream_nat] :
( ( ( shift_nat @ Xs @ S1 )
= ( shift_nat @ Xs @ S22 ) )
= ( S1 = S22 ) ) ).
% shift_left_inj
thf(fact_299_shift__left__inj,axiom,
! [Xs: list_a,S1: stream_a,S22: stream_a] :
( ( ( shift_a @ Xs @ S1 )
= ( shift_a @ Xs @ S22 ) )
= ( S1 = S22 ) ) ).
% shift_left_inj
thf(fact_300_shift__left__inj,axiom,
! [Xs: list_list_a,S1: stream_list_a,S22: stream_list_a] :
( ( ( shift_list_a @ Xs @ S1 )
= ( shift_list_a @ Xs @ S22 ) )
= ( S1 = S22 ) ) ).
% shift_left_inj
thf(fact_301_shift_Osimps_I1_J,axiom,
! [S2: stream_nat] :
( ( shift_nat @ nil_nat @ S2 )
= S2 ) ).
% shift.simps(1)
thf(fact_302_shift_Osimps_I1_J,axiom,
! [S2: stream_list_a] :
( ( shift_list_a @ nil_list_a @ S2 )
= S2 ) ).
% shift.simps(1)
thf(fact_303_shift_Osimps_I1_J,axiom,
! [S2: stream_a] :
( ( shift_a @ nil_a @ S2 )
= S2 ) ).
% shift.simps(1)
thf(fact_304_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P5 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_305_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P5 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_306_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_307_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_308_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ~ ( ord_less_nat @ X @ T ) ) ).
% pinf(5)
thf(fact_309_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ord_less_nat @ T @ X ) ) ).
% pinf(7)
thf(fact_310_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P5 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_311_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P5 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_312_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( X != T ) ) ).
% minf(3)
thf(fact_313_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( X != T ) ) ).
% minf(4)
thf(fact_314_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ord_less_nat @ X @ T ) ) ).
% minf(5)
thf(fact_315_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ~ ( ord_less_nat @ T @ X ) ) ).
% minf(7)
thf(fact_316_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_317_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_318_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_319_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_list_a] :
( ( size_s349497388124573686list_a @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_320_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_321_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_322_snth__sset__smerge,axiom,
! [Ss: stream_stream_nat,N2: nat,M2: nat] : ( member_nat @ ( snth_nat @ ( snth_stream_nat @ Ss @ N2 ) @ M2 ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_323_snth__sset__smerge,axiom,
! [Ss: stream2307142169165677840ream_a,N2: nat,M2: nat] : ( member_stream_a @ ( snth_stream_a @ ( snth_stream_stream_a @ Ss @ N2 ) @ M2 ) @ ( sset_stream_a @ ( smerge_stream_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_324_snth__sset__smerge,axiom,
! [Ss: stream2377611395989027862list_a,N2: nat,M2: nat] : ( member_stream_list_a @ ( snth_stream_list_a @ ( snth_s4451090846259577524list_a @ Ss @ N2 ) @ M2 ) @ ( sset_stream_list_a @ ( smerge_stream_list_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_325_snth__sset__smerge,axiom,
! [Ss: stream6406556387131777664st_nat,N2: nat,M2: nat] : ( member_list_nat @ ( snth_list_nat @ ( snth_stream_list_nat @ Ss @ N2 ) @ M2 ) @ ( sset_list_nat @ ( smerge_list_nat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_326_snth__sset__smerge,axiom,
! [Ss: stream3775083406817841430list_a,N2: nat,M2: nat] : ( member_list_list_a @ ( snth_list_list_a @ ( snth_s3514276978948636852list_a @ Ss @ N2 ) @ M2 ) @ ( sset_list_list_a @ ( smerge_list_list_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_327_snth__sset__smerge,axiom,
! [Ss: stream_stream_list_a,N2: nat,M2: nat] : ( member_list_a @ ( snth_list_a @ ( snth_stream_list_a @ Ss @ N2 ) @ M2 ) @ ( sset_list_a @ ( smerge_list_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_328_snth__sset__smerge,axiom,
! [Ss: stream_stream_a,N2: nat,M2: nat] : ( member_a @ ( snth_a @ ( snth_stream_a @ Ss @ N2 ) @ M2 ) @ ( sset_a @ ( smerge_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_329_shift__snth,axiom,
! [N2: nat,Xs: list_stream_a,S2: stream_stream_a] :
( ( ( ord_less_nat @ N2 @ ( size_s2142770077969500662ream_a @ Xs ) )
=> ( ( snth_stream_a @ ( shift_stream_a @ Xs @ S2 ) @ N2 )
= ( nth_stream_a @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s2142770077969500662ream_a @ Xs ) )
=> ( ( snth_stream_a @ ( shift_stream_a @ Xs @ S2 ) @ N2 )
= ( snth_stream_a @ S2 @ ( minus_minus_nat @ N2 @ ( size_s2142770077969500662ream_a @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_330_shift__snth,axiom,
! [N2: nat,Xs: list_stream_list_a,S2: stream_stream_list_a] :
( ( ( ord_less_nat @ N2 @ ( size_s3694203809124329340list_a @ Xs ) )
=> ( ( snth_stream_list_a @ ( shift_stream_list_a @ Xs @ S2 ) @ N2 )
= ( nth_stream_list_a @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s3694203809124329340list_a @ Xs ) )
=> ( ( snth_stream_list_a @ ( shift_stream_list_a @ Xs @ S2 ) @ N2 )
= ( snth_stream_list_a @ S2 @ ( minus_minus_nat @ N2 @ ( size_s3694203809124329340list_a @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_331_shift__snth,axiom,
! [N2: nat,Xs: list_list_a,S2: stream_list_a] :
( ( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( snth_list_a @ ( shift_list_a @ Xs @ S2 ) @ N2 )
= ( nth_list_a @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( snth_list_a @ ( shift_list_a @ Xs @ S2 ) @ N2 )
= ( snth_list_a @ S2 @ ( minus_minus_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_332_shift__snth,axiom,
! [N2: nat,Xs: list_a,S2: stream_a] :
( ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( snth_a @ ( shift_a @ Xs @ S2 ) @ N2 )
= ( nth_a @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( snth_a @ ( shift_a @ Xs @ S2 ) @ N2 )
= ( snth_a @ S2 @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_333_shift__snth,axiom,
! [N2: nat,Xs: list_nat,S2: stream_nat] :
( ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( snth_nat @ ( shift_nat @ Xs @ S2 ) @ N2 )
= ( nth_nat @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( snth_nat @ ( shift_nat @ Xs @ S2 ) @ N2 )
= ( snth_nat @ S2 @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_334_shift__snth__ge,axiom,
! [Xs: list_stream_a,P4: nat,S2: stream_stream_a] :
( ( ord_less_eq_nat @ ( size_s2142770077969500662ream_a @ Xs ) @ P4 )
=> ( ( snth_stream_a @ ( shift_stream_a @ Xs @ S2 ) @ P4 )
= ( snth_stream_a @ S2 @ ( minus_minus_nat @ P4 @ ( size_s2142770077969500662ream_a @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_335_shift__snth__ge,axiom,
! [Xs: list_stream_list_a,P4: nat,S2: stream_stream_list_a] :
( ( ord_less_eq_nat @ ( size_s3694203809124329340list_a @ Xs ) @ P4 )
=> ( ( snth_stream_list_a @ ( shift_stream_list_a @ Xs @ S2 ) @ P4 )
= ( snth_stream_list_a @ S2 @ ( minus_minus_nat @ P4 @ ( size_s3694203809124329340list_a @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_336_shift__snth__ge,axiom,
! [Xs: list_list_a,P4: nat,S2: stream_list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ P4 )
=> ( ( snth_list_a @ ( shift_list_a @ Xs @ S2 ) @ P4 )
= ( snth_list_a @ S2 @ ( minus_minus_nat @ P4 @ ( size_s349497388124573686list_a @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_337_shift__snth__ge,axiom,
! [Xs: list_a,P4: nat,S2: stream_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ P4 )
=> ( ( snth_a @ ( shift_a @ Xs @ S2 ) @ P4 )
= ( snth_a @ S2 @ ( minus_minus_nat @ P4 @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_338_shift__snth__ge,axiom,
! [Xs: list_nat,P4: nat,S2: stream_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ P4 )
=> ( ( snth_nat @ ( shift_nat @ Xs @ S2 ) @ P4 )
= ( snth_nat @ S2 @ ( minus_minus_nat @ P4 @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_339_stake__nth,axiom,
! [M2: nat,N2: nat,S2: stream_stream_a] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( nth_stream_a @ ( stake_stream_a @ N2 @ S2 ) @ M2 )
= ( snth_stream_a @ S2 @ M2 ) ) ) ).
% stake_nth
thf(fact_340_stake__nth,axiom,
! [M2: nat,N2: nat,S2: stream_stream_list_a] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( nth_stream_list_a @ ( stake_stream_list_a @ N2 @ S2 ) @ M2 )
= ( snth_stream_list_a @ S2 @ M2 ) ) ) ).
% stake_nth
thf(fact_341_stake__nth,axiom,
! [M2: nat,N2: nat,S2: stream_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( nth_nat @ ( stake_nat @ N2 @ S2 ) @ M2 )
= ( snth_nat @ S2 @ M2 ) ) ) ).
% stake_nth
thf(fact_342_stake__nth,axiom,
! [M2: nat,N2: nat,S2: stream_a] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( nth_a @ ( stake_a @ N2 @ S2 ) @ M2 )
= ( snth_a @ S2 @ M2 ) ) ) ).
% stake_nth
thf(fact_343_stake__nth,axiom,
! [M2: nat,N2: nat,S2: stream_list_a] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( nth_list_a @ ( stake_list_a @ N2 @ S2 ) @ M2 )
= ( snth_list_a @ S2 @ M2 ) ) ) ).
% stake_nth
thf(fact_344_list__ex__length,axiom,
( list_ex_list_a
= ( ^ [P3: list_a > $o,Xs2: list_list_a] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs2 ) )
& ( P3 @ ( nth_list_a @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_345_list__ex__length,axiom,
( list_ex_a
= ( ^ [P3: a > $o,Xs2: list_a] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
& ( P3 @ ( nth_a @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_346_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P3: nat > $o,Xs2: list_nat] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
& ( P3 @ ( nth_nat @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_347_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_348_list__ex1__simps_I1_J,axiom,
! [P: list_a > $o] :
~ ( list_ex1_list_a @ P @ nil_list_a ) ).
% list_ex1_simps(1)
thf(fact_349_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_350_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( 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_nth_mono
thf(fact_351_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_352_Stream_Osmember__def,axiom,
( smember_list_nat
= ( ^ [X3: list_nat,S: stream_list_nat] : ( member_list_nat @ X3 @ ( sset_list_nat @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_353_Stream_Osmember__def,axiom,
( smember_list_list_a
= ( ^ [X3: list_list_a,S: stream_list_list_a] : ( member_list_list_a @ X3 @ ( sset_list_list_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_354_Stream_Osmember__def,axiom,
( smember_list_a
= ( ^ [X3: list_a,S: stream_list_a] : ( member_list_a @ X3 @ ( sset_list_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_355_Stream_Osmember__def,axiom,
( smember_a
= ( ^ [X3: a,S: stream_a] : ( member_a @ X3 @ ( sset_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_356_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_357_length__stake,axiom,
! [N2: nat,S2: stream_list_a] :
( ( size_s349497388124573686list_a @ ( stake_list_a @ N2 @ S2 ) )
= N2 ) ).
% length_stake
thf(fact_358_length__stake,axiom,
! [N2: nat,S2: stream_a] :
( ( size_size_list_a @ ( stake_a @ N2 @ S2 ) )
= N2 ) ).
% length_stake
thf(fact_359_length__stake,axiom,
! [N2: nat,S2: stream_nat] :
( ( size_size_list_nat @ ( stake_nat @ N2 @ S2 ) )
= N2 ) ).
% length_stake
thf(fact_360_list__ex__simps_I2_J,axiom,
! [P: nat > $o] :
~ ( list_ex_nat @ P @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_361_list__ex__simps_I2_J,axiom,
! [P: list_a > $o] :
~ ( list_ex_list_a @ P @ nil_list_a ) ).
% list_ex_simps(2)
thf(fact_362_list__ex__simps_I2_J,axiom,
! [P: a > $o] :
~ ( list_ex_a @ P @ nil_a ) ).
% list_ex_simps(2)
thf(fact_363_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_364_sorted__wrt_Osimps_I1_J,axiom,
! [P: list_a > list_a > $o] : ( sorted_wrt_list_a @ P @ nil_list_a ) ).
% sorted_wrt.simps(1)
thf(fact_365_sorted__wrt_Osimps_I1_J,axiom,
! [P: a > a > $o] : ( sorted_wrt_a @ P @ nil_a ) ).
% sorted_wrt.simps(1)
thf(fact_366_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_367_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_368_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_369_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_370_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_371_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_372_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_373_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_374_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_375_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_376_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_377_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_378_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_379_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_380_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_381_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_382_sorted__wrt__nth__less,axiom,
! [P: list_a > list_a > $o,Xs: list_list_a,I: nat,J: nat] :
( ( sorted_wrt_list_a @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P @ ( nth_list_a @ Xs @ I ) @ ( nth_list_a @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_383_sorted__wrt__nth__less,axiom,
! [P: a > a > $o,Xs: list_a,I: nat,J: nat] :
( ( sorted_wrt_a @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I ) @ ( nth_a @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_384_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_385_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_list_a
= ( ^ [P3: list_a > list_a > $o,Xs2: list_list_a] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P3 @ ( nth_list_a @ Xs2 @ I3 ) @ ( nth_list_a @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_386_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_a
= ( ^ [P3: a > a > $o,Xs2: list_a] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs2 ) )
=> ( P3 @ ( nth_a @ Xs2 @ I3 ) @ ( nth_a @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_387_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs2: list_nat] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P3 @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_388_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_389_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_390_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( 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_nth_mono
thf(fact_391_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_392_stake__cycle__eq,axiom,
! [U: list_list_a] :
( ( U != nil_list_a )
=> ( ( stake_list_a @ ( size_s349497388124573686list_a @ U ) @ ( cycle_list_a @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_393_stake__cycle__eq,axiom,
! [U: list_a] :
( ( U != nil_a )
=> ( ( stake_a @ ( size_size_list_a @ U ) @ ( cycle_a @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_394_stake__cycle__eq,axiom,
! [U: list_nat] :
( ( U != nil_nat )
=> ( ( stake_nat @ ( size_size_list_nat @ U ) @ ( cycle_nat @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_395_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_396_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_397_flat__snth,axiom,
! [S2: stream_list_stream_a,N2: nat] :
( ! [X4: list_stream_a] :
( ( member_list_stream_a @ X4 @ ( sset_list_stream_a @ S2 ) )
=> ( X4 != nil_stream_a ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_s2142770077969500662ream_a @ ( shd_list_stream_a @ S2 ) ) )
=> ( ( snth_stream_a @ ( flat_stream_a @ S2 ) @ N2 )
= ( nth_stream_a @ ( shd_list_stream_a @ S2 ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s2142770077969500662ream_a @ ( shd_list_stream_a @ S2 ) ) )
=> ( ( snth_stream_a @ ( flat_stream_a @ S2 ) @ N2 )
= ( snth_stream_a @ ( flat_stream_a @ ( stl_list_stream_a @ S2 ) ) @ ( minus_minus_nat @ N2 @ ( size_s2142770077969500662ream_a @ ( shd_list_stream_a @ S2 ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_398_flat__snth,axiom,
! [S2: stream1229541022664496662list_a,N2: nat] :
( ! [X4: list_stream_list_a] :
( ( member2241138852378865081list_a @ X4 @ ( sset_l2691409540454527268list_a @ S2 ) )
=> ( X4 != nil_stream_list_a ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_s3694203809124329340list_a @ ( shd_li3061069428927788988list_a @ S2 ) ) )
=> ( ( snth_stream_list_a @ ( flat_stream_list_a @ S2 ) @ N2 )
= ( nth_stream_list_a @ ( shd_li3061069428927788988list_a @ S2 ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s3694203809124329340list_a @ ( shd_li3061069428927788988list_a @ S2 ) ) )
=> ( ( snth_stream_list_a @ ( flat_stream_list_a @ S2 ) @ N2 )
= ( snth_stream_list_a @ ( flat_stream_list_a @ ( stl_li3392269663658258880list_a @ S2 ) ) @ ( minus_minus_nat @ N2 @ ( size_s3694203809124329340list_a @ ( shd_li3061069428927788988list_a @ S2 ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_399_flat__snth,axiom,
! [S2: stream_list_list_a,N2: nat] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( sset_list_list_a @ S2 ) )
=> ( X4 != nil_list_a ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ ( shd_list_list_a @ S2 ) ) )
=> ( ( snth_list_a @ ( flat_list_a @ S2 ) @ N2 )
= ( nth_list_a @ ( shd_list_list_a @ S2 ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ ( shd_list_list_a @ S2 ) ) )
=> ( ( snth_list_a @ ( flat_list_a @ S2 ) @ N2 )
= ( snth_list_a @ ( flat_list_a @ ( stl_list_list_a @ S2 ) ) @ ( minus_minus_nat @ N2 @ ( size_s349497388124573686list_a @ ( shd_list_list_a @ S2 ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_400_flat__snth,axiom,
! [S2: stream_list_a,N2: nat] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( sset_list_a @ S2 ) )
=> ( X4 != nil_a ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_size_list_a @ ( shd_list_a @ S2 ) ) )
=> ( ( snth_a @ ( flat_a @ S2 ) @ N2 )
= ( nth_a @ ( shd_list_a @ S2 ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_a @ ( shd_list_a @ S2 ) ) )
=> ( ( snth_a @ ( flat_a @ S2 ) @ N2 )
= ( snth_a @ ( flat_a @ ( stl_list_a @ S2 ) ) @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ ( shd_list_a @ S2 ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_401_flat__snth,axiom,
! [S2: stream_list_nat,N2: nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( sset_list_nat @ S2 ) )
=> ( X4 != nil_nat ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ ( shd_list_nat @ S2 ) ) )
=> ( ( snth_nat @ ( flat_nat @ S2 ) @ N2 )
= ( nth_nat @ ( shd_list_nat @ S2 ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ ( shd_list_nat @ S2 ) ) )
=> ( ( snth_nat @ ( flat_nat @ S2 ) @ N2 )
= ( snth_nat @ ( flat_nat @ ( stl_list_nat @ S2 ) ) @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ ( shd_list_nat @ S2 ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_402_flat__Stream,axiom,
! [Xs: list_nat,Ws: stream_list_nat] :
( ( Xs != nil_nat )
=> ( ( flat_nat @ ( sCons_list_nat @ Xs @ Ws ) )
= ( shift_nat @ Xs @ ( flat_nat @ Ws ) ) ) ) ).
% flat_Stream
thf(fact_403_flat__Stream,axiom,
! [Xs: list_list_a,Ws: stream_list_list_a] :
( ( Xs != nil_list_a )
=> ( ( flat_list_a @ ( sCons_list_list_a @ Xs @ Ws ) )
= ( shift_list_a @ Xs @ ( flat_list_a @ Ws ) ) ) ) ).
% flat_Stream
thf(fact_404_flat__Stream,axiom,
! [Xs: list_a,Ws: stream_list_a] :
( ( Xs != nil_a )
=> ( ( flat_a @ ( sCons_list_a @ Xs @ Ws ) )
= ( shift_a @ Xs @ ( flat_a @ Ws ) ) ) ) ).
% flat_Stream
thf(fact_405_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_406_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_407_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_408_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_409_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_410_rev__is__rev__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( rev_nat @ Xs )
= ( rev_nat @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_411_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_412_rev__rev__ident,axiom,
! [Xs: list_nat] :
( ( rev_nat @ ( rev_nat @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_413_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_414_stream_Oinject,axiom,
! [X1: list_a,X22: stream_list_a,Y1: list_a,Y22: stream_list_a] :
( ( ( sCons_list_a @ X1 @ X22 )
= ( sCons_list_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_415_stream_Oinject,axiom,
! [X1: a,X22: stream_a,Y1: a,Y22: stream_a] :
( ( ( sCons_a @ X1 @ X22 )
= ( sCons_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_416_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_417_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_418_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_419_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_420_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_421_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_422_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_423_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_424_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_425_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_426_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_427_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_428_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_429_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_430_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_431_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_432_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_433_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_434_Nil__is__rev__conv,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_435_Nil__is__rev__conv,axiom,
! [Xs: list_list_a] :
( ( nil_list_a
= ( rev_list_a @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_rev_conv
thf(fact_436_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_437_rev__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rev_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_438_rev__is__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( ( rev_list_a @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% rev_is_Nil_conv
thf(fact_439_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_440_length__rev,axiom,
! [Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( rev_list_a @ Xs ) )
= ( size_s349497388124573686list_a @ Xs ) ) ).
% length_rev
thf(fact_441_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_442_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_443_list__ex__rev,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( list_ex_nat @ P @ ( rev_nat @ Xs ) )
= ( list_ex_nat @ P @ Xs ) ) ).
% list_ex_rev
thf(fact_444_list__ex__rev,axiom,
! [P: a > $o,Xs: list_a] :
( ( list_ex_a @ P @ ( rev_a @ Xs ) )
= ( list_ex_a @ P @ Xs ) ) ).
% list_ex_rev
thf(fact_445_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_446_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_447_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_448_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_449_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_450_smember__code,axiom,
! [X2: list_a,Y3: list_a,S2: stream_list_a] :
( ( smember_list_a @ X2 @ ( sCons_list_a @ Y3 @ S2 ) )
= ( ( X2 != Y3 )
=> ( smember_list_a @ X2 @ S2 ) ) ) ).
% smember_code
thf(fact_451_smember__code,axiom,
! [X2: a,Y3: a,S2: stream_a] :
( ( smember_a @ X2 @ ( sCons_a @ Y3 @ S2 ) )
= ( ( X2 != Y3 )
=> ( smember_a @ X2 @ S2 ) ) ) ).
% smember_code
thf(fact_452_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_453_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_454_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_455_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_456_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_457_stake__invert__Nil,axiom,
! [N2: nat,S2: stream_nat] :
( ( ( stake_nat @ N2 @ S2 )
= nil_nat )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_458_stake__invert__Nil,axiom,
! [N2: nat,S2: stream_list_a] :
( ( ( stake_list_a @ N2 @ S2 )
= nil_list_a )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_459_stake__invert__Nil,axiom,
! [N2: nat,S2: stream_a] :
( ( ( stake_a @ N2 @ S2 )
= nil_a )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_460_stream_Ocollapse,axiom,
! [Stream: stream_list_nat] :
( ( sCons_list_nat @ ( shd_list_nat @ Stream ) @ ( stl_list_nat @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_461_stream_Ocollapse,axiom,
! [Stream: stream_list_list_a] :
( ( sCons_list_list_a @ ( shd_list_list_a @ Stream ) @ ( stl_list_list_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_462_stream_Ocollapse,axiom,
! [Stream: stream_list_a] :
( ( sCons_list_a @ ( shd_list_a @ Stream ) @ ( stl_list_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_463_stream_Ocollapse,axiom,
! [Stream: stream_a] :
( ( sCons_a @ ( shd_a @ Stream ) @ ( stl_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_464_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_465_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_466_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_467_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_468_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_469_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_470_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_471_sset__induct,axiom,
! [Y3: list_nat,S2: stream_list_nat,P: list_nat > stream_list_nat > $o] :
( ( member_list_nat @ Y3 @ ( sset_list_nat @ S2 ) )
=> ( ! [S3: stream_list_nat] : ( P @ ( shd_list_nat @ S3 ) @ S3 )
=> ( ! [S3: stream_list_nat,Y: list_nat] :
( ( member_list_nat @ Y @ ( sset_list_nat @ ( stl_list_nat @ S3 ) ) )
=> ( ( P @ Y @ ( stl_list_nat @ S3 ) )
=> ( P @ Y @ S3 ) ) )
=> ( P @ Y3 @ S2 ) ) ) ) ).
% sset_induct
thf(fact_472_sset__induct,axiom,
! [Y3: list_list_a,S2: stream_list_list_a,P: list_list_a > stream_list_list_a > $o] :
( ( member_list_list_a @ Y3 @ ( sset_list_list_a @ S2 ) )
=> ( ! [S3: stream_list_list_a] : ( P @ ( shd_list_list_a @ S3 ) @ S3 )
=> ( ! [S3: stream_list_list_a,Y: list_list_a] :
( ( member_list_list_a @ Y @ ( sset_list_list_a @ ( stl_list_list_a @ S3 ) ) )
=> ( ( P @ Y @ ( stl_list_list_a @ S3 ) )
=> ( P @ Y @ S3 ) ) )
=> ( P @ Y3 @ S2 ) ) ) ) ).
% sset_induct
thf(fact_473_sset__induct,axiom,
! [Y3: list_a,S2: stream_list_a,P: list_a > stream_list_a > $o] :
( ( member_list_a @ Y3 @ ( sset_list_a @ S2 ) )
=> ( ! [S3: stream_list_a] : ( P @ ( shd_list_a @ S3 ) @ S3 )
=> ( ! [S3: stream_list_a,Y: list_a] :
( ( member_list_a @ Y @ ( sset_list_a @ ( stl_list_a @ S3 ) ) )
=> ( ( P @ Y @ ( stl_list_a @ S3 ) )
=> ( P @ Y @ S3 ) ) )
=> ( P @ Y3 @ S2 ) ) ) ) ).
% sset_induct
thf(fact_474_sset__induct,axiom,
! [Y3: a,S2: stream_a,P: a > stream_a > $o] :
( ( member_a @ Y3 @ ( sset_a @ S2 ) )
=> ( ! [S3: stream_a] : ( P @ ( shd_a @ S3 ) @ S3 )
=> ( ! [S3: stream_a,Y: a] :
( ( member_a @ Y @ ( sset_a @ ( stl_a @ S3 ) ) )
=> ( ( P @ Y @ ( stl_a @ S3 ) )
=> ( P @ Y @ S3 ) ) )
=> ( P @ Y3 @ S2 ) ) ) ) ).
% sset_induct
thf(fact_475_rev__swap,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( rev_nat @ Xs )
= Ys )
= ( Xs
= ( rev_nat @ Ys ) ) ) ).
% rev_swap
thf(fact_476_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_477_stream_Ocoinduct__strong,axiom,
! [R: stream_list_nat > stream_list_nat > $o,Stream: stream_list_nat,Stream2: stream_list_nat] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_nat,Stream4: stream_list_nat] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_nat @ Stream3 )
= ( shd_list_nat @ Stream4 ) )
& ( ( R @ ( stl_list_nat @ Stream3 ) @ ( stl_list_nat @ Stream4 ) )
| ( ( stl_list_nat @ Stream3 )
= ( stl_list_nat @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_478_stream_Ocoinduct__strong,axiom,
! [R: stream_list_a > stream_list_a > $o,Stream: stream_list_a,Stream2: stream_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_a,Stream4: stream_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_a @ Stream3 )
= ( shd_list_a @ Stream4 ) )
& ( ( R @ ( stl_list_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) )
| ( ( stl_list_a @ Stream3 )
= ( stl_list_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_479_stream_Ocoinduct__strong,axiom,
! [R: stream_list_list_a > stream_list_list_a > $o,Stream: stream_list_list_a,Stream2: stream_list_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_list_a,Stream4: stream_list_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_list_a @ Stream3 )
= ( shd_list_list_a @ Stream4 ) )
& ( ( R @ ( stl_list_list_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) )
| ( ( stl_list_list_a @ Stream3 )
= ( stl_list_list_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_480_stream_Ocoinduct__strong,axiom,
! [R: stream_a > stream_a > $o,Stream: stream_a,Stream2: stream_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_a,Stream4: stream_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_a @ Stream3 )
= ( shd_a @ Stream4 ) )
& ( ( R @ ( stl_a @ Stream3 ) @ ( stl_a @ Stream4 ) )
| ( ( stl_a @ Stream3 )
= ( stl_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_481_stream_Oexhaust__sel,axiom,
! [Stream: stream_list_nat] :
( Stream
= ( sCons_list_nat @ ( shd_list_nat @ Stream ) @ ( stl_list_nat @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_482_stream_Oexhaust__sel,axiom,
! [Stream: stream_list_list_a] :
( Stream
= ( sCons_list_list_a @ ( shd_list_list_a @ Stream ) @ ( stl_list_list_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_483_stream_Oexhaust__sel,axiom,
! [Stream: stream_list_a] :
( Stream
= ( sCons_list_a @ ( shd_list_a @ Stream ) @ ( stl_list_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_484_stream_Oexhaust__sel,axiom,
! [Stream: stream_a] :
( Stream
= ( sCons_a @ ( shd_a @ Stream ) @ ( stl_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_485_stream_Ocoinduct,axiom,
! [R: stream_list_nat > stream_list_nat > $o,Stream: stream_list_nat,Stream2: stream_list_nat] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_nat,Stream4: stream_list_nat] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_nat @ Stream3 )
= ( shd_list_nat @ Stream4 ) )
& ( R @ ( stl_list_nat @ Stream3 ) @ ( stl_list_nat @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_486_stream_Ocoinduct,axiom,
! [R: stream_list_a > stream_list_a > $o,Stream: stream_list_a,Stream2: stream_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_a,Stream4: stream_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_a @ Stream3 )
= ( shd_list_a @ Stream4 ) )
& ( R @ ( stl_list_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_487_stream_Ocoinduct,axiom,
! [R: stream_list_list_a > stream_list_list_a > $o,Stream: stream_list_list_a,Stream2: stream_list_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_list_a,Stream4: stream_list_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_list_a @ Stream3 )
= ( shd_list_list_a @ Stream4 ) )
& ( R @ ( stl_list_list_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_488_stream_Ocoinduct,axiom,
! [R: stream_a > stream_a > $o,Stream: stream_a,Stream2: stream_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_a,Stream4: stream_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_a @ Stream3 )
= ( shd_a @ Stream4 ) )
& ( R @ ( stl_a @ Stream3 ) @ ( stl_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_489_stream_Oexhaust,axiom,
! [Y3: stream_list_a] :
~ ! [X12: list_a,X23: stream_list_a] :
( Y3
!= ( sCons_list_a @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_490_stream_Oexhaust,axiom,
! [Y3: stream_a] :
~ ! [X12: a,X23: stream_a] :
( Y3
!= ( sCons_a @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_491_stream_Oexpand,axiom,
! [Stream: stream_list_nat,Stream2: stream_list_nat] :
( ( ( ( shd_list_nat @ Stream )
= ( shd_list_nat @ Stream2 ) )
& ( ( stl_list_nat @ Stream )
= ( stl_list_nat @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_492_stream_Oexpand,axiom,
! [Stream: stream_list_a,Stream2: stream_list_a] :
( ( ( ( shd_list_a @ Stream )
= ( shd_list_a @ Stream2 ) )
& ( ( stl_list_a @ Stream )
= ( stl_list_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_493_stream_Oexpand,axiom,
! [Stream: stream_list_list_a,Stream2: stream_list_list_a] :
( ( ( ( shd_list_list_a @ Stream )
= ( shd_list_list_a @ Stream2 ) )
& ( ( stl_list_list_a @ Stream )
= ( stl_list_list_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_494_stream_Oexpand,axiom,
! [Stream: stream_a,Stream2: stream_a] :
( ( ( ( shd_a @ Stream )
= ( shd_a @ Stream2 ) )
& ( ( stl_a @ Stream )
= ( stl_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_495_stream_Osel_I1_J,axiom,
! [X1: list_nat,X22: stream_list_nat] :
( ( shd_list_nat @ ( sCons_list_nat @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_496_stream_Osel_I1_J,axiom,
! [X1: list_list_a,X22: stream_list_list_a] :
( ( shd_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_497_stream_Osel_I1_J,axiom,
! [X1: list_a,X22: stream_list_a] :
( ( shd_list_a @ ( sCons_list_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_498_stream_Osel_I1_J,axiom,
! [X1: a,X22: stream_a] :
( ( shd_a @ ( sCons_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_499_stream_Osel_I2_J,axiom,
! [X1: list_nat,X22: stream_list_nat] :
( ( stl_list_nat @ ( sCons_list_nat @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_500_stream_Osel_I2_J,axiom,
! [X1: list_list_a,X22: stream_list_list_a] :
( ( stl_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_501_stream_Osel_I2_J,axiom,
! [X1: list_a,X22: stream_list_a] :
( ( stl_list_a @ ( sCons_list_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_502_stream_Osel_I2_J,axiom,
! [X1: a,X22: stream_a] :
( ( stl_a @ ( sCons_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_503_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_504_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_505_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_506_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_507_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_508_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_509_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X4: nat,Y: nat] :
( ( P @ X4 @ Y )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_510_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_511_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_512_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_513_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_514_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_515_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_516_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_517_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_518_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_519_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_520_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_521_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_522_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_523_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_524_snth__Stream,axiom,
! [X2: nat,S2: stream_nat,I: nat] :
( ( snth_nat @ ( sCons_nat @ X2 @ S2 ) @ ( suc @ I ) )
= ( snth_nat @ S2 @ I ) ) ).
% snth_Stream
thf(fact_525_snth__Stream,axiom,
! [X2: stream_a,S2: stream_stream_a,I: nat] :
( ( snth_stream_a @ ( sCons_stream_a @ X2 @ S2 ) @ ( suc @ I ) )
= ( snth_stream_a @ S2 @ I ) ) ).
% snth_Stream
thf(fact_526_snth__Stream,axiom,
! [X2: stream_list_a,S2: stream_stream_list_a,I: nat] :
( ( snth_stream_list_a @ ( sCons_stream_list_a @ X2 @ S2 ) @ ( suc @ I ) )
= ( snth_stream_list_a @ S2 @ I ) ) ).
% snth_Stream
thf(fact_527_snth__Stream,axiom,
! [X2: a,S2: stream_a,I: nat] :
( ( snth_a @ ( sCons_a @ X2 @ S2 ) @ ( suc @ I ) )
= ( snth_a @ S2 @ I ) ) ).
% snth_Stream
thf(fact_528_snth__Stream,axiom,
! [X2: list_a,S2: stream_list_a,I: nat] :
( ( snth_list_a @ ( sCons_list_a @ X2 @ S2 ) @ ( suc @ I ) )
= ( snth_list_a @ S2 @ I ) ) ).
% snth_Stream
thf(fact_529_snth_Osimps_I1_J,axiom,
! [S2: stream_nat] :
( ( snth_nat @ S2 @ zero_zero_nat )
= ( shd_nat @ S2 ) ) ).
% snth.simps(1)
thf(fact_530_snth_Osimps_I1_J,axiom,
! [S2: stream_stream_a] :
( ( snth_stream_a @ S2 @ zero_zero_nat )
= ( shd_stream_a @ S2 ) ) ).
% snth.simps(1)
thf(fact_531_snth_Osimps_I1_J,axiom,
! [S2: stream_stream_list_a] :
( ( snth_stream_list_a @ S2 @ zero_zero_nat )
= ( shd_stream_list_a @ S2 ) ) ).
% snth.simps(1)
thf(fact_532_snth_Osimps_I1_J,axiom,
! [S2: stream_list_nat] :
( ( snth_list_nat @ S2 @ zero_zero_nat )
= ( shd_list_nat @ S2 ) ) ).
% snth.simps(1)
thf(fact_533_snth_Osimps_I1_J,axiom,
! [S2: stream_list_list_a] :
( ( snth_list_list_a @ S2 @ zero_zero_nat )
= ( shd_list_list_a @ S2 ) ) ).
% snth.simps(1)
thf(fact_534_snth_Osimps_I1_J,axiom,
! [S2: stream_a] :
( ( snth_a @ S2 @ zero_zero_nat )
= ( shd_a @ S2 ) ) ).
% snth.simps(1)
thf(fact_535_snth_Osimps_I1_J,axiom,
! [S2: stream_list_a] :
( ( snth_list_a @ S2 @ zero_zero_nat )
= ( shd_list_a @ S2 ) ) ).
% snth.simps(1)
thf(fact_536_snth_Osimps_I2_J,axiom,
! [S2: stream_nat,N2: nat] :
( ( snth_nat @ S2 @ ( suc @ N2 ) )
= ( snth_nat @ ( stl_nat @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_537_snth_Osimps_I2_J,axiom,
! [S2: stream_stream_a,N2: nat] :
( ( snth_stream_a @ S2 @ ( suc @ N2 ) )
= ( snth_stream_a @ ( stl_stream_a @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_538_snth_Osimps_I2_J,axiom,
! [S2: stream_stream_list_a,N2: nat] :
( ( snth_stream_list_a @ S2 @ ( suc @ N2 ) )
= ( snth_stream_list_a @ ( stl_stream_list_a @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_539_snth_Osimps_I2_J,axiom,
! [S2: stream_list_nat,N2: nat] :
( ( snth_list_nat @ S2 @ ( suc @ N2 ) )
= ( snth_list_nat @ ( stl_list_nat @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_540_snth_Osimps_I2_J,axiom,
! [S2: stream_list_list_a,N2: nat] :
( ( snth_list_list_a @ S2 @ ( suc @ N2 ) )
= ( snth_list_list_a @ ( stl_list_list_a @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_541_snth_Osimps_I2_J,axiom,
! [S2: stream_a,N2: nat] :
( ( snth_a @ S2 @ ( suc @ N2 ) )
= ( snth_a @ ( stl_a @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_542_snth_Osimps_I2_J,axiom,
! [S2: stream_list_a,N2: nat] :
( ( snth_list_a @ S2 @ ( suc @ N2 ) )
= ( snth_list_a @ ( stl_list_a @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_543_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_544_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_545_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_546_rev_Osimps_I1_J,axiom,
( ( rev_list_a @ nil_list_a )
= nil_list_a ) ).
% rev.simps(1)
thf(fact_547_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_548_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_549_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_550_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_551_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_552_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_553_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_554_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_555_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_556_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_557_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_558_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_559_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_560_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_561_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_562_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_563_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_564_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_565_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y: nat,Z3: nat] :
( ( R @ X4 @ Y )
=> ( ( R @ Y @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_566_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_567_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_568_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_569_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_570_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_571_Suc__le__D,axiom,
! [N2: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_572_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_573_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_574_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_575_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_576_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_577_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_578_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_579_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_580_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_581_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_582_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_583_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_584_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_585_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_586_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_587_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_588_stream_Oset__intros_I2_J,axiom,
! [Y3: list_nat,X22: stream_list_nat,X1: list_nat] :
( ( member_list_nat @ Y3 @ ( sset_list_nat @ X22 ) )
=> ( member_list_nat @ Y3 @ ( sset_list_nat @ ( sCons_list_nat @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_589_stream_Oset__intros_I2_J,axiom,
! [Y3: list_list_a,X22: stream_list_list_a,X1: list_list_a] :
( ( member_list_list_a @ Y3 @ ( sset_list_list_a @ X22 ) )
=> ( member_list_list_a @ Y3 @ ( sset_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_590_stream_Oset__intros_I2_J,axiom,
! [Y3: list_a,X22: stream_list_a,X1: list_a] :
( ( member_list_a @ Y3 @ ( sset_list_a @ X22 ) )
=> ( member_list_a @ Y3 @ ( sset_list_a @ ( sCons_list_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_591_stream_Oset__intros_I2_J,axiom,
! [Y3: a,X22: stream_a,X1: a] :
( ( member_a @ Y3 @ ( sset_a @ X22 ) )
=> ( member_a @ Y3 @ ( sset_a @ ( sCons_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_592_stream_Oset__intros_I1_J,axiom,
! [X1: list_nat,X22: stream_list_nat] : ( member_list_nat @ X1 @ ( sset_list_nat @ ( sCons_list_nat @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_593_stream_Oset__intros_I1_J,axiom,
! [X1: list_list_a,X22: stream_list_list_a] : ( member_list_list_a @ X1 @ ( sset_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_594_stream_Oset__intros_I1_J,axiom,
! [X1: list_a,X22: stream_list_a] : ( member_list_a @ X1 @ ( sset_list_a @ ( sCons_list_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_595_stream_Oset__intros_I1_J,axiom,
! [X1: a,X22: stream_a] : ( member_a @ X1 @ ( sset_a @ ( sCons_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_596_stream_Oset__cases,axiom,
! [E: list_nat,A: stream_list_nat] :
( ( member_list_nat @ E @ ( sset_list_nat @ A ) )
=> ( ! [Z22: stream_list_nat] :
( A
!= ( sCons_list_nat @ E @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: stream_list_nat] :
( ( A
= ( sCons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E @ ( sset_list_nat @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_597_stream_Oset__cases,axiom,
! [E: list_list_a,A: stream_list_list_a] :
( ( member_list_list_a @ E @ ( sset_list_list_a @ A ) )
=> ( ! [Z22: stream_list_list_a] :
( A
!= ( sCons_list_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_list_a,Z22: stream_list_list_a] :
( ( A
= ( sCons_list_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_list_a @ E @ ( sset_list_list_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_598_stream_Oset__cases,axiom,
! [E: list_a,A: stream_list_a] :
( ( member_list_a @ E @ ( sset_list_a @ A ) )
=> ( ! [Z22: stream_list_a] :
( A
!= ( sCons_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: stream_list_a] :
( ( A
= ( sCons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E @ ( sset_list_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_599_stream_Oset__cases,axiom,
! [E: a,A: stream_a] :
( ( member_a @ E @ ( sset_a @ A ) )
=> ( ! [Z22: stream_a] :
( A
!= ( sCons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: stream_a] :
( ( A
= ( sCons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( sset_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_600_stream_Oset__induct,axiom,
! [X2: list_nat,A: stream_list_nat,P: list_nat > stream_list_nat > $o] :
( ( member_list_nat @ X2 @ ( sset_list_nat @ A ) )
=> ( ! [Z1: list_nat,Z22: stream_list_nat] : ( P @ Z1 @ ( sCons_list_nat @ Z1 @ Z22 ) )
=> ( ! [Z1: list_nat,Z22: stream_list_nat,Xa: list_nat] :
( ( member_list_nat @ Xa @ ( sset_list_nat @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( sCons_list_nat @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A ) ) ) ) ).
% stream.set_induct
thf(fact_601_stream_Oset__induct,axiom,
! [X2: list_list_a,A: stream_list_list_a,P: list_list_a > stream_list_list_a > $o] :
( ( member_list_list_a @ X2 @ ( sset_list_list_a @ A ) )
=> ( ! [Z1: list_list_a,Z22: stream_list_list_a] : ( P @ Z1 @ ( sCons_list_list_a @ Z1 @ Z22 ) )
=> ( ! [Z1: list_list_a,Z22: stream_list_list_a,Xa: list_list_a] :
( ( member_list_list_a @ Xa @ ( sset_list_list_a @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( sCons_list_list_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A ) ) ) ) ).
% stream.set_induct
thf(fact_602_stream_Oset__induct,axiom,
! [X2: list_a,A: stream_list_a,P: list_a > stream_list_a > $o] :
( ( member_list_a @ X2 @ ( sset_list_a @ A ) )
=> ( ! [Z1: list_a,Z22: stream_list_a] : ( P @ Z1 @ ( sCons_list_a @ Z1 @ Z22 ) )
=> ( ! [Z1: list_a,Z22: stream_list_a,Xa: list_a] :
( ( member_list_a @ Xa @ ( sset_list_a @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( sCons_list_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A ) ) ) ) ).
% stream.set_induct
thf(fact_603_stream_Oset__induct,axiom,
! [X2: a,A: stream_a,P: a > stream_a > $o] :
( ( member_a @ X2 @ ( sset_a @ A ) )
=> ( ! [Z1: a,Z22: stream_a] : ( P @ Z1 @ ( sCons_a @ Z1 @ Z22 ) )
=> ( ! [Z1: a,Z22: stream_a,Xa: a] :
( ( member_a @ Xa @ ( sset_a @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( sCons_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A ) ) ) ) ).
% stream.set_induct
thf(fact_604_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_605_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_606_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_607_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_608_stl__sset,axiom,
! [X2: list_nat,A: stream_list_nat] :
( ( member_list_nat @ X2 @ ( sset_list_nat @ ( stl_list_nat @ A ) ) )
=> ( member_list_nat @ X2 @ ( sset_list_nat @ A ) ) ) ).
% stl_sset
thf(fact_609_stl__sset,axiom,
! [X2: list_list_a,A: stream_list_list_a] :
( ( member_list_list_a @ X2 @ ( sset_list_list_a @ ( stl_list_list_a @ A ) ) )
=> ( member_list_list_a @ X2 @ ( sset_list_list_a @ A ) ) ) ).
% stl_sset
thf(fact_610_stl__sset,axiom,
! [X2: list_a,A: stream_list_a] :
( ( member_list_a @ X2 @ ( sset_list_a @ ( stl_list_a @ A ) ) )
=> ( member_list_a @ X2 @ ( sset_list_a @ A ) ) ) ).
% stl_sset
thf(fact_611_stl__sset,axiom,
! [X2: a,A: stream_a] :
( ( member_a @ X2 @ ( sset_a @ ( stl_a @ A ) ) )
=> ( member_a @ X2 @ ( sset_a @ A ) ) ) ).
% stl_sset
thf(fact_612_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_613_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_614_shd__sset,axiom,
! [A: stream_list_nat] : ( member_list_nat @ ( shd_list_nat @ A ) @ ( sset_list_nat @ A ) ) ).
% shd_sset
thf(fact_615_shd__sset,axiom,
! [A: stream_list_list_a] : ( member_list_list_a @ ( shd_list_list_a @ A ) @ ( sset_list_list_a @ A ) ) ).
% shd_sset
thf(fact_616_shd__sset,axiom,
! [A: stream_list_a] : ( member_list_a @ ( shd_list_a @ A ) @ ( sset_list_a @ A ) ) ).
% shd_sset
thf(fact_617_shd__sset,axiom,
! [A: stream_a] : ( member_a @ ( shd_a @ A ) @ ( sset_a @ A ) ) ).
% shd_sset
thf(fact_618_flat__unfold,axiom,
! [Ws: stream_list_nat] :
( ( ( shd_list_nat @ Ws )
!= nil_nat )
=> ( ( flat_nat @ Ws )
= ( shift_nat @ ( shd_list_nat @ Ws ) @ ( flat_nat @ ( stl_list_nat @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_619_flat__unfold,axiom,
! [Ws: stream_list_list_a] :
( ( ( shd_list_list_a @ Ws )
!= nil_list_a )
=> ( ( flat_list_a @ Ws )
= ( shift_list_a @ ( shd_list_list_a @ Ws ) @ ( flat_list_a @ ( stl_list_list_a @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_620_flat__unfold,axiom,
! [Ws: stream_list_a] :
( ( ( shd_list_a @ Ws )
!= nil_a )
=> ( ( flat_a @ Ws )
= ( shift_a @ ( shd_list_a @ Ws ) @ ( flat_a @ ( stl_list_a @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_621_rev__nth,axiom,
! [N2: nat,Xs: list_list_a] :
( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( rev_list_a @ Xs ) @ N2 )
= ( nth_list_a @ Xs @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_622_rev__nth,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rev_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_623_rev__nth,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_624_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_625_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8861187494160871172list_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_le8861187494160871172list_a @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_626_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_627_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_628_lift__Suc__mono__le,axiom,
! [F: nat > set_list_a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8861187494160871172list_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_le8861187494160871172list_a @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_629_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_630_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_631_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_632_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_633_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_634_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_635_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_636_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_637_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_638_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_639_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_640_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_641_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_642_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_643_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_644_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_645_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_646_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_647_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_648_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_649_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I3 ) ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_650_sset__flat__stl,axiom,
! [S2: stream_list_nat] : ( ord_less_eq_set_nat @ ( sset_nat @ ( flat_nat @ ( stl_list_nat @ S2 ) ) ) @ ( sset_nat @ ( flat_nat @ S2 ) ) ) ).
% sset_flat_stl
thf(fact_651_sset__flat__stl,axiom,
! [S2: stream_list_list_nat] : ( ord_le6045566169113846134st_nat @ ( sset_list_nat @ ( flat_list_nat @ ( stl_list_list_nat @ S2 ) ) ) @ ( sset_list_nat @ ( flat_list_nat @ S2 ) ) ) ).
% sset_flat_stl
thf(fact_652_sset__flat__stl,axiom,
! [S2: stream2255243159586646806list_a] : ( ord_le8488217952732425610list_a @ ( sset_list_list_a @ ( flat_list_list_a @ ( stl_list_list_list_a @ S2 ) ) ) @ ( sset_list_list_a @ ( flat_list_list_a @ S2 ) ) ) ).
% sset_flat_stl
thf(fact_653_sset__flat__stl,axiom,
! [S2: stream_list_list_a] : ( ord_le8861187494160871172list_a @ ( sset_list_a @ ( flat_list_a @ ( stl_list_list_a @ S2 ) ) ) @ ( sset_list_a @ ( flat_list_a @ S2 ) ) ) ).
% sset_flat_stl
thf(fact_654_sset__flat__stl,axiom,
! [S2: stream_list_a] : ( ord_less_eq_set_a @ ( sset_a @ ( flat_a @ ( stl_list_a @ S2 ) ) ) @ ( sset_a @ ( flat_a @ S2 ) ) ) ).
% sset_flat_stl
thf(fact_655_stake_Osimps_I1_J,axiom,
! [S2: stream_nat] :
( ( stake_nat @ zero_zero_nat @ S2 )
= nil_nat ) ).
% stake.simps(1)
thf(fact_656_stake_Osimps_I1_J,axiom,
! [S2: stream_list_a] :
( ( stake_list_a @ zero_zero_nat @ S2 )
= nil_list_a ) ).
% stake.simps(1)
thf(fact_657_stake_Osimps_I1_J,axiom,
! [S2: stream_a] :
( ( stake_a @ zero_zero_nat @ S2 )
= nil_a ) ).
% stake.simps(1)
thf(fact_658_cycle__decomp,axiom,
! [U: list_nat] :
( ( U != nil_nat )
=> ( ( cycle_nat @ U )
= ( shift_nat @ U @ ( cycle_nat @ U ) ) ) ) ).
% cycle_decomp
thf(fact_659_cycle__decomp,axiom,
! [U: list_list_a] :
( ( U != nil_list_a )
=> ( ( cycle_list_a @ U )
= ( shift_list_a @ U @ ( cycle_list_a @ U ) ) ) ) ).
% cycle_decomp
thf(fact_660_cycle__decomp,axiom,
! [U: list_a] :
( ( U != nil_a )
=> ( ( cycle_a @ U )
= ( shift_a @ U @ ( cycle_a @ U ) ) ) ) ).
% cycle_decomp
thf(fact_661_sorted__wrt01,axiom,
! [Xs: list_list_a,P: list_a > list_a > $o] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ one_one_nat )
=> ( sorted_wrt_list_a @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_662_sorted__wrt01,axiom,
! [Xs: list_a,P: a > a > $o] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( sorted_wrt_a @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_663_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_664_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_665_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_666_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_667_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_668_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_669_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_670_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_671_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_672_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_673_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_674_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_675_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_676_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_677_flat_Osimps_I2_J,axiom,
! [Ws: stream_list_nat] :
( ( stl_nat @ ( flat_nat @ Ws ) )
= ( flat_nat
@ ( if_stream_list_nat
@ ( ( tl_nat @ ( shd_list_nat @ Ws ) )
= nil_nat )
@ ( stl_list_nat @ Ws )
@ ( sCons_list_nat @ ( tl_nat @ ( shd_list_nat @ Ws ) ) @ ( stl_list_nat @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_678_flat_Osimps_I2_J,axiom,
! [Ws: stream_list_list_nat] :
( ( stl_list_nat @ ( flat_list_nat @ Ws ) )
= ( flat_list_nat
@ ( if_str5104684111711982906st_nat
@ ( ( tl_list_nat @ ( shd_list_list_nat @ Ws ) )
= nil_list_nat )
@ ( stl_list_list_nat @ Ws )
@ ( sCons_list_list_nat @ ( tl_list_nat @ ( shd_list_list_nat @ Ws ) ) @ ( stl_list_list_nat @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_679_flat_Osimps_I2_J,axiom,
! [Ws: stream_list_list_a] :
( ( stl_list_a @ ( flat_list_a @ Ws ) )
= ( flat_list_a
@ ( if_str7505741754068378070list_a
@ ( ( tl_list_a @ ( shd_list_list_a @ Ws ) )
= nil_list_a )
@ ( stl_list_list_a @ Ws )
@ ( sCons_list_list_a @ ( tl_list_a @ ( shd_list_list_a @ Ws ) ) @ ( stl_list_list_a @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_680_flat_Osimps_I2_J,axiom,
! [Ws: stream2255243159586646806list_a] :
( ( stl_list_list_a @ ( flat_list_list_a @ Ws ) )
= ( flat_list_list_a
@ ( if_str8217234800680828380list_a
@ ( ( tl_list_list_a @ ( shd_list_list_list_a @ Ws ) )
= nil_list_list_a )
@ ( stl_list_list_list_a @ Ws )
@ ( sCons_8165023923567507367list_a @ ( tl_list_list_a @ ( shd_list_list_list_a @ Ws ) ) @ ( stl_list_list_list_a @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_681_flat_Osimps_I2_J,axiom,
! [Ws: stream_list_a] :
( ( stl_a @ ( flat_a @ Ws ) )
= ( flat_a
@ ( if_stream_list_a
@ ( ( tl_a @ ( shd_list_a @ Ws ) )
= nil_a )
@ ( stl_list_a @ Ws )
@ ( sCons_list_a @ ( tl_a @ ( shd_list_a @ Ws ) ) @ ( stl_list_a @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_682_id__stake__snth__sdrop,axiom,
! [S2: stream_stream_a,I: nat] :
( S2
= ( shift_stream_a @ ( stake_stream_a @ I @ S2 ) @ ( sCons_stream_a @ ( snth_stream_a @ S2 @ I ) @ ( sdrop_stream_a @ ( suc @ I ) @ S2 ) ) ) ) ).
% id_stake_snth_sdrop
thf(fact_683_id__stake__snth__sdrop,axiom,
! [S2: stream_stream_list_a,I: nat] :
( S2
= ( shift_stream_list_a @ ( stake_stream_list_a @ I @ S2 ) @ ( sCons_stream_list_a @ ( snth_stream_list_a @ S2 @ I ) @ ( sdrop_stream_list_a @ ( suc @ I ) @ S2 ) ) ) ) ).
% id_stake_snth_sdrop
thf(fact_684_id__stake__snth__sdrop,axiom,
! [S2: stream_nat,I: nat] :
( S2
= ( shift_nat @ ( stake_nat @ I @ S2 ) @ ( sCons_nat @ ( snth_nat @ S2 @ I ) @ ( sdrop_nat @ ( suc @ I ) @ S2 ) ) ) ) ).
% id_stake_snth_sdrop
thf(fact_685_id__stake__snth__sdrop,axiom,
! [S2: stream_a,I: nat] :
( S2
= ( shift_a @ ( stake_a @ I @ S2 ) @ ( sCons_a @ ( snth_a @ S2 @ I ) @ ( sdrop_a @ ( suc @ I ) @ S2 ) ) ) ) ).
% id_stake_snth_sdrop
thf(fact_686_id__stake__snth__sdrop,axiom,
! [S2: stream_list_a,I: nat] :
( S2
= ( shift_list_a @ ( stake_list_a @ I @ S2 ) @ ( sCons_list_a @ ( snth_list_a @ S2 @ I ) @ ( sdrop_list_a @ ( suc @ I ) @ S2 ) ) ) ) ).
% id_stake_snth_sdrop
thf(fact_687_nth__Cons__pos,axiom,
! [N2: nat,X2: list_a,Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_list_a @ ( cons_list_a @ X2 @ Xs ) @ N2 )
= ( nth_list_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_688_nth__Cons__pos,axiom,
! [N2: nat,X2: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_689_nth__Cons__pos,axiom,
! [N2: nat,X2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_690_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_691_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y222: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_692_list_Oinject,axiom,
! [X21: list_a,X222: list_list_a,Y21: list_a,Y222: list_list_a] :
( ( ( cons_list_a @ X21 @ X222 )
= ( cons_list_a @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_693_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_694_list__ex__simps_I1_J,axiom,
! [P: a > $o,X2: a,Xs: list_a] :
( ( list_ex_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_a @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_695_list__ex__simps_I1_J,axiom,
! [P: list_a > $o,X2: list_a,Xs: list_list_a] :
( ( list_ex_list_a @ P @ ( cons_list_a @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_list_a @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_696_nth__Cons__Suc,axiom,
! [X2: list_a,Xs: list_list_a,N2: nat] :
( ( nth_list_a @ ( cons_list_a @ X2 @ Xs ) @ ( suc @ N2 ) )
= ( nth_list_a @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_697_nth__Cons__Suc,axiom,
! [X2: a,Xs: list_a,N2: nat] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ ( suc @ N2 ) )
= ( nth_a @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_698_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N2: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N2 ) )
= ( nth_nat @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_699_nth__Cons__0,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( nth_list_a @ ( cons_list_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_700_nth__Cons__0,axiom,
! [X2: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_701_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_702_singleton__rev__conv,axiom,
! [X2: nat,Xs: list_nat] :
( ( ( cons_nat @ X2 @ nil_nat )
= ( rev_nat @ Xs ) )
= ( ( cons_nat @ X2 @ nil_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_703_singleton__rev__conv,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( ( cons_list_a @ X2 @ nil_list_a )
= ( rev_list_a @ Xs ) )
= ( ( cons_list_a @ X2 @ nil_list_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_704_singleton__rev__conv,axiom,
! [X2: a,Xs: list_a] :
( ( ( cons_a @ X2 @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X2 @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_705_rev__singleton__conv,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ X2 @ nil_nat ) )
= ( Xs
= ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_706_rev__singleton__conv,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( ( rev_list_a @ Xs )
= ( cons_list_a @ X2 @ nil_list_a ) )
= ( Xs
= ( cons_list_a @ X2 @ nil_list_a ) ) ) ).
% rev_singleton_conv
thf(fact_707_rev__singleton__conv,axiom,
! [Xs: list_a,X2: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X2 @ nil_a ) )
= ( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_708_sdrop__simps_I2_J,axiom,
! [N2: nat,S2: stream_nat] :
( ( stl_nat @ ( sdrop_nat @ N2 @ S2 ) )
= ( sdrop_nat @ ( suc @ N2 ) @ S2 ) ) ).
% sdrop_simps(2)
thf(fact_709_sdrop__simps_I2_J,axiom,
! [N2: nat,S2: stream_list_nat] :
( ( stl_list_nat @ ( sdrop_list_nat @ N2 @ S2 ) )
= ( sdrop_list_nat @ ( suc @ N2 ) @ S2 ) ) ).
% sdrop_simps(2)
thf(fact_710_sdrop__simps_I2_J,axiom,
! [N2: nat,S2: stream_list_a] :
( ( stl_list_a @ ( sdrop_list_a @ N2 @ S2 ) )
= ( sdrop_list_a @ ( suc @ N2 ) @ S2 ) ) ).
% sdrop_simps(2)
thf(fact_711_sdrop__simps_I2_J,axiom,
! [N2: nat,S2: stream_list_list_a] :
( ( stl_list_list_a @ ( sdrop_list_list_a @ N2 @ S2 ) )
= ( sdrop_list_list_a @ ( suc @ N2 ) @ S2 ) ) ).
% sdrop_simps(2)
thf(fact_712_sdrop__simps_I2_J,axiom,
! [N2: nat,S2: stream_a] :
( ( stl_a @ ( sdrop_a @ N2 @ S2 ) )
= ( sdrop_a @ ( suc @ N2 ) @ S2 ) ) ).
% sdrop_simps(2)
thf(fact_713_sdrop__simps_I1_J,axiom,
! [N2: nat,S2: stream_stream_a] :
( ( shd_stream_a @ ( sdrop_stream_a @ N2 @ S2 ) )
= ( snth_stream_a @ S2 @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_714_sdrop__simps_I1_J,axiom,
! [N2: nat,S2: stream_stream_list_a] :
( ( shd_stream_list_a @ ( sdrop_stream_list_a @ N2 @ S2 ) )
= ( snth_stream_list_a @ S2 @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_715_sdrop__simps_I1_J,axiom,
! [N2: nat,S2: stream_nat] :
( ( shd_nat @ ( sdrop_nat @ N2 @ S2 ) )
= ( snth_nat @ S2 @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_716_sdrop__simps_I1_J,axiom,
! [N2: nat,S2: stream_list_nat] :
( ( shd_list_nat @ ( sdrop_list_nat @ N2 @ S2 ) )
= ( snth_list_nat @ S2 @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_717_sdrop__simps_I1_J,axiom,
! [N2: nat,S2: stream_list_list_a] :
( ( shd_list_list_a @ ( sdrop_list_list_a @ N2 @ S2 ) )
= ( snth_list_list_a @ S2 @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_718_sdrop__simps_I1_J,axiom,
! [N2: nat,S2: stream_a] :
( ( shd_a @ ( sdrop_a @ N2 @ S2 ) )
= ( snth_a @ S2 @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_719_sdrop__simps_I1_J,axiom,
! [N2: nat,S2: stream_list_a] :
( ( shd_list_a @ ( sdrop_list_a @ N2 @ S2 ) )
= ( snth_list_a @ S2 @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_720_length__tl,axiom,
! [Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( tl_list_a @ Xs ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_721_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_722_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_723_shift__simps_I2_J,axiom,
! [Xs: list_a,S2: stream_a] :
( ( ( Xs = nil_a )
=> ( ( stl_a @ ( shift_a @ Xs @ S2 ) )
= ( stl_a @ S2 ) ) )
& ( ( Xs != nil_a )
=> ( ( stl_a @ ( shift_a @ Xs @ S2 ) )
= ( shift_a @ ( tl_a @ Xs ) @ S2 ) ) ) ) ).
% shift_simps(2)
thf(fact_724_sdrop__cycle__eq,axiom,
! [U: list_a] :
( ( U != nil_a )
=> ( ( sdrop_a @ ( size_size_list_a @ U ) @ ( cycle_a @ U ) )
= ( cycle_a @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_725_sdrop__cycle__eq,axiom,
! [U: list_nat] :
( ( U != nil_nat )
=> ( ( sdrop_nat @ ( size_size_list_nat @ U ) @ ( cycle_nat @ U ) )
= ( cycle_nat @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_726_flat__Cons,axiom,
! [X2: a,Xs: list_a,Ws: stream_list_a] :
( ( flat_a @ ( sCons_list_a @ ( cons_a @ X2 @ Xs ) @ Ws ) )
= ( sCons_a @ X2 @ ( flat_a @ ( if_stream_list_a @ ( Xs = nil_a ) @ Ws @ ( sCons_list_a @ Xs @ Ws ) ) ) ) ) ).
% flat_Cons
thf(fact_727_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_728_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_729_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a] : ( P @ ( cons_a @ X4 @ Xs3 ) @ nil_a )
=> ( ! [Y: a,Ys4: list_a] : ( P @ nil_a @ ( cons_a @ Y @ Ys4 ) )
=> ( ! [X4: a,Xs3: list_a,Y: a,Ys4: list_a] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_730_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_731_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X4: a] :
( X2
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y: a,Xs3: list_a] :
( X2
!= ( cons_a @ X4 @ ( cons_a @ Y @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_732_list_Oexhaust,axiom,
! [Y3: list_a] :
( ( Y3 != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y3
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_733_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_734_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_735_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_736_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_737_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs3: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_738_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_739_length__Suc__conv,axiom,
! [Xs: list_a,N2: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_740_length__Suc__conv,axiom,
! [Xs: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_741_Suc__length__conv,axiom,
! [N2: nat,Xs: list_a] :
( ( ( suc @ N2 )
= ( size_size_list_a @ Xs ) )
= ( ? [Y4: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_742_Suc__length__conv,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( suc @ N2 )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_743_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_744_list__induct2,axiom,
! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X4: a,Xs3: list_a,Y: nat,Ys4: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_745_list__induct2,axiom,
! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X4: nat,Xs3: list_nat,Y: a,Ys4: list_a] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_a @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_746_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,Y: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_747_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y: a,Ys4: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_748_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,P: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat )
=> ( ! [X4: a,Xs3: list_a,Y: a,Ys4: list_a,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_749_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,P: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y: nat,Ys4: list_nat,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_750_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,P: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X4: a,Xs3: list_a,Y: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_751_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,P: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a )
=> ( ! [X4: nat,Xs3: list_nat,Y: a,Ys4: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_a @ Y @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_752_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_nat,P: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y: a,Ys4: list_a,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_a @ Y @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_753_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_a,P: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X4: nat,Xs3: list_nat,Y: nat,Ys4: list_nat,Z3: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_754_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,Y: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_755_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_756_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_757_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_758_sorted__wrt1,axiom,
! [P: a > a > $o,X2: a] : ( sorted_wrt_a @ P @ ( cons_a @ X2 @ nil_a ) ) ).
% sorted_wrt1
thf(fact_759_sorted__wrt1,axiom,
! [P: nat > nat > $o,X2: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_760_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_a @ Xs ) )
= ( ? [X3: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N2 @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_761_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_762_sorted1,axiom,
! [X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted1
thf(fact_763_nth__tl,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ ( tl_a @ Xs ) ) )
=> ( ( nth_a @ ( tl_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( suc @ N2 ) ) ) ) ).
% nth_tl
thf(fact_764_nth__tl,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( suc @ N2 ) ) ) ) ).
% nth_tl
thf(fact_765_nth__Cons_H,axiom,
! [N2: nat,X2: a,Xs: list_a] :
( ( ( N2 = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N2 )
= X2 ) )
& ( ( N2 != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_766_nth__Cons_H,axiom,
! [N2: nat,X2: nat,Xs: list_nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
= X2 ) )
& ( ( N2 != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_767_nth__non__equal__first__eq,axiom,
! [X2: a,Y3: a,Xs: list_a,N2: nat] :
( ( X2 != Y3 )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N2 )
= Y3 )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_768_nth__non__equal__first__eq,axiom,
! [X2: nat,Y3: nat,Xs: list_nat,N2: nat] :
( ( X2 != Y3 )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
= Y3 )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= Y3 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_769_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs2: list_a] : ( if_nat @ ( Xs2 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs2 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_770_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_771_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_772_length__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X2 @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_773_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_774_flat_Ocode,axiom,
( flat_a
= ( ^ [Ws2: stream_list_a] :
( sCons_a @ ( hd_a @ ( shd_list_a @ Ws2 ) )
@ ( flat_a
@ ( if_stream_list_a
@ ( ( tl_a @ ( shd_list_a @ Ws2 ) )
= nil_a )
@ ( stl_list_a @ Ws2 )
@ ( sCons_list_a @ ( tl_a @ ( shd_list_a @ Ws2 ) ) @ ( stl_list_a @ Ws2 ) ) ) ) ) ) ) ).
% flat.code
thf(fact_775_sdrop__cycle__eq__mod__0,axiom,
! [U: list_a,N2: nat] :
( ( U != nil_a )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ U ) )
= zero_zero_nat )
=> ( ( sdrop_a @ N2 @ ( cycle_a @ U ) )
= ( cycle_a @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_776_sdrop__cycle__eq__mod__0,axiom,
! [U: list_nat,N2: nat] :
( ( U != nil_nat )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ U ) )
= zero_zero_nat )
=> ( ( sdrop_nat @ N2 @ ( cycle_nat @ U ) )
= ( cycle_nat @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_777_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_778_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_779_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_780_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_781_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_782_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_783_shift__simps_I1_J,axiom,
! [Xs: list_a,S2: stream_a] :
( ( ( Xs = nil_a )
=> ( ( shd_a @ ( shift_a @ Xs @ S2 ) )
= ( shd_a @ S2 ) ) )
& ( ( Xs != nil_a )
=> ( ( shd_a @ ( shift_a @ Xs @ S2 ) )
= ( hd_a @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_784_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_785_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_786_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_787_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_788_flat_Osimps_I1_J,axiom,
! [Ws: stream_list_a] :
( ( shd_a @ ( flat_a @ Ws ) )
= ( hd_a @ ( shd_list_a @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_789_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_790_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_791_mod__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( modulo_modulo_nat @ M2 @ N2 )
= M2 ) ) ).
% mod_less
thf(fact_792_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_793_mod__Suc__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N2 ) ) ).
% mod_Suc_Suc_eq
thf(fact_794_mod__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).
% mod_Suc_eq
thf(fact_795_mod__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_796_mod__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
!= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ) ).
% mod_Suc
thf(fact_797_mod__induct,axiom,
! [P: nat > $o,N2: nat,P4: nat,M2: nat] :
( ( P @ N2 )
=> ( ( ord_less_nat @ N2 @ P4 )
=> ( ( ord_less_nat @ M2 @ P4 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P4 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_798_mod__less__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_799_mod__Suc__le__divisor,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_800_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M @ N ) @ M @ ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ).
% mod_if
thf(fact_801_le__mod__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( modulo_modulo_nat @ M2 @ N2 )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_802_mod__le__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_803_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_a @ N2 @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_a @ N2 @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_804_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
=> ( P @ M4 @ N3 ) ) )
=> ( P @ M2 @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_805_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_806_nth__rotate1,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate1_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_807_nth__rotate1,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_808_sdrop__cycle,axiom,
! [U: list_a,N2: nat] :
( ( U != nil_a )
=> ( ( sdrop_a @ N2 @ ( cycle_a @ U ) )
= ( cycle_a @ ( rotate_a @ ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_809_sdrop__cycle,axiom,
! [U: list_nat,N2: nat] :
( ( U != nil_nat )
=> ( ( sdrop_nat @ N2 @ ( cycle_nat @ U ) )
= ( cycle_nat @ ( rotate_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_810_cycle__rotated,axiom,
! [V: list_a,U: list_a,S2: stream_a] :
( ( V != nil_a )
=> ( ( ( cycle_a @ U )
= ( shift_a @ V @ S2 ) )
=> ( ( cycle_a @ ( append_a @ ( tl_a @ U ) @ ( cons_a @ ( hd_a @ U ) @ nil_a ) ) )
= ( shift_a @ ( tl_a @ V ) @ S2 ) ) ) ) ).
% cycle_rotated
thf(fact_811_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_812_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_813_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_814_self__append__conv2,axiom,
! [Y3: list_a,Xs: list_a] :
( ( Y3
= ( append_a @ Xs @ Y3 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_815_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_816_self__append__conv,axiom,
! [Y3: list_a,Ys: list_a] :
( ( Y3
= ( append_a @ Y3 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_817_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_818_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_819_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_820_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_821_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_822_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_a] :
( ( ( rotate_a @ N2 @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate_is_Nil_conv
thf(fact_823_length__rotate,axiom,
! [N2: nat,Xs: list_a] :
( ( size_size_list_a @ ( rotate_a @ N2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate
thf(fact_824_length__rotate,axiom,
! [N2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_825_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_826_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_827_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_828_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y3: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y3 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_829_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_830_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_831_nth__append__length,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_832_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_833_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y3: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y3 @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_834_rotate__length01,axiom,
! [Xs: list_a,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate_a @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_835_rotate__length01,axiom,
! [Xs: list_nat,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_836_rotate__id,axiom,
! [N2: nat,Xs: list_a] :
( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_a @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_837_rotate__id,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_838_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_839_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_840_rotate1_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_841_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_842_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_843_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_844_rotate__append,axiom,
! [L: list_a,Q3: list_a] :
( ( rotate_a @ ( size_size_list_a @ L ) @ ( append_a @ L @ Q3 ) )
= ( append_a @ Q3 @ L ) ) ).
% rotate_append
thf(fact_845_rotate__append,axiom,
! [L: list_nat,Q3: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q3 ) )
= ( append_nat @ Q3 @ L ) ) ).
% rotate_append
thf(fact_846_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_847_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_848_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_849_Cons__eq__append__conv,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_850_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys4: list_a,Y: a] :
( Xs
!= ( append_a @ Ys4 @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_851_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X4: a,Xs3: list_a] :
( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_852_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_a @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_853_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_854_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_855_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_856_rotate__conv__mod,axiom,
( rotate_a
= ( ^ [N: nat,Xs2: list_a] : ( rotate_a @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Xs2 ) ) ) ).
% rotate_conv_mod
thf(fact_857_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N: nat,Xs2: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ).
% rotate_conv_mod
thf(fact_858_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X4: a,Xs4: list_a,Y: a,Ys6: list_a] :
( ( X4 != Y )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X4 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_859_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,Xs4: list_nat,Y: nat,Ys6: list_nat] :
( ( X4 != Y )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Xs4 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_860_rev_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_861_length__Suc__conv__rev,axiom,
! [Xs: list_a,N2: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: a,Ys2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ Y4 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_862_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_863_length__append__singleton,axiom,
! [Xs: list_a,X2: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_864_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_865_nth__append,axiom,
! [N2: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N2 )
= ( nth_a @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N2 )
= ( nth_a @ Ys @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_866_nth__append,axiom,
! [N2: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N2 )
= ( nth_nat @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N2 )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_867_cycle__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( cycle_a @ ( cons_a @ X2 @ Xs ) )
= ( sCons_a @ X2 @ ( cycle_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ) ) ).
% cycle_Cons
thf(fact_868_stake__Suc,axiom,
! [N2: nat,S2: stream_list_a] :
( ( stake_list_a @ ( suc @ N2 ) @ S2 )
= ( append_list_a @ ( stake_list_a @ N2 @ S2 ) @ ( cons_list_a @ ( snth_list_a @ S2 @ N2 ) @ nil_list_a ) ) ) ).
% stake_Suc
thf(fact_869_stake__Suc,axiom,
! [N2: nat,S2: stream_a] :
( ( stake_a @ ( suc @ N2 ) @ S2 )
= ( append_a @ ( stake_a @ N2 @ S2 ) @ ( cons_a @ ( snth_a @ S2 @ N2 ) @ nil_a ) ) ) ).
% stake_Suc
thf(fact_870_rotate__rev,axiom,
! [N2: nat,Xs: list_a] :
( ( rotate_a @ N2 @ ( rev_a @ Xs ) )
= ( rev_a @ ( rotate_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_871_rotate__rev,axiom,
! [N2: nat,Xs: list_nat] :
( ( rotate_nat @ N2 @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_872_cycle_Ocode,axiom,
( cycle_a
= ( ^ [Xs2: list_a] : ( sCons_a @ ( hd_a @ Xs2 ) @ ( cycle_a @ ( append_a @ ( tl_a @ Xs2 ) @ ( cons_a @ ( hd_a @ Xs2 ) @ nil_a ) ) ) ) ) ) ).
% cycle.code
thf(fact_873_cycle_Osimps_I2_J,axiom,
! [Xs: list_a] :
( ( stl_a @ ( cycle_a @ Xs ) )
= ( cycle_a @ ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% cycle.simps(2)
thf(fact_874_hd__rotate__conv__nth,axiom,
! [Xs: list_a,N2: nat] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( rotate_a @ N2 @ Xs ) )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_875_hd__rotate__conv__nth,axiom,
! [Xs: list_nat,N2: nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( rotate_nat @ N2 @ Xs ) )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_876_SuccD,axiom,
! [K: a,Kl: set_list_a,Kl2: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
=> ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_877_SuccI,axiom,
! [Kl2: list_a,K: a,Kl: set_list_a] :
( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_878_empty__Shift,axiom,
! [Kl: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_879_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_880_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_881_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_882_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_a] :
( ( nil_a
= ( take_a @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_883_take__eq__Nil,axiom,
! [N2: nat,Xs: list_a] :
( ( ( take_a @ N2 @ Xs )
= nil_a )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_884_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs2: list_a] : nil_a ) ) ).
% take0
thf(fact_885_take__all__iff,axiom,
! [N2: nat,Xs: list_a] :
( ( ( take_a @ N2 @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N2 ) ) ).
% take_all_iff
thf(fact_886_take__all__iff,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( take_nat @ N2 @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% take_all_iff
thf(fact_887_take__all,axiom,
! [Xs: list_a,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N2 )
=> ( ( take_a @ N2 @ Xs )
= Xs ) ) ).
% take_all
thf(fact_888_take__all,axiom,
! [Xs: list_nat,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 )
=> ( ( take_nat @ N2 @ Xs )
= Xs ) ) ).
% take_all
thf(fact_889_nth__take,axiom,
! [I: nat,N2: nat,Xs: list_a] :
( ( ord_less_nat @ I @ N2 )
=> ( ( nth_a @ ( take_a @ N2 @ Xs ) @ I )
= ( nth_a @ Xs @ I ) ) ) ).
% nth_take
thf(fact_890_nth__take,axiom,
! [I: nat,N2: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ( nth_nat @ ( take_nat @ N2 @ Xs ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_891_take__append,axiom,
! [N2: nat,Xs: list_a,Ys: list_a] :
( ( take_a @ N2 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( take_a @ N2 @ Xs ) @ ( take_a @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_892_take__append,axiom,
! [N2: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N2 @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_893_stake__cycle__le,axiom,
! [U: list_a,N2: nat] :
( ( U != nil_a )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_a @ U ) )
=> ( ( stake_a @ N2 @ ( cycle_a @ U ) )
= ( take_a @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_894_stake__cycle__le,axiom,
! [U: list_nat,N2: nat] :
( ( U != nil_nat )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ U ) )
=> ( ( stake_nat @ N2 @ ( cycle_nat @ U ) )
= ( take_nat @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_895_take__Nil,axiom,
! [N2: nat] :
( ( take_a @ N2 @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_896_sorted__wrt__take,axiom,
! [F: nat > nat > $o,Xs: list_nat,N2: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( take_nat @ N2 @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_897_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_898_sorted__take,axiom,
! [Xs: list_nat,N2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N2 @ Xs ) ) ) ).
% sorted_take
thf(fact_899_nth__take__lemma,axiom,
! [K: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( ( take_a @ K @ Xs )
= ( take_a @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_900_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_901_take__Cons_H,axiom,
! [N2: nat,X2: a,Xs: list_a] :
( ( ( N2 = zero_zero_nat )
=> ( ( take_a @ N2 @ ( cons_a @ X2 @ Xs ) )
= nil_a ) )
& ( ( N2 != zero_zero_nat )
=> ( ( take_a @ N2 @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( take_a @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_902_take__Suc,axiom,
! [Xs: list_a,N2: nat] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ N2 ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ ( take_a @ N2 @ ( tl_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_903_stake__shift,axiom,
! [I: nat,W: list_a,S2: stream_a] :
( ( stake_a @ I @ ( shift_a @ W @ S2 ) )
= ( append_a @ ( take_a @ I @ W ) @ ( stake_a @ ( minus_minus_nat @ I @ ( size_size_list_a @ W ) ) @ S2 ) ) ) ).
% stake_shift
thf(fact_904_stake__shift,axiom,
! [I: nat,W: list_nat,S2: stream_nat] :
( ( stake_nat @ I @ ( shift_nat @ W @ S2 ) )
= ( append_nat @ ( take_nat @ I @ W ) @ ( stake_nat @ ( minus_minus_nat @ I @ ( size_size_list_nat @ W ) ) @ S2 ) ) ) ).
% stake_shift
thf(fact_905_take__hd__drop,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( append_a @ ( take_a @ N2 @ Xs ) @ ( cons_a @ ( hd_a @ ( drop_a @ N2 @ Xs ) ) @ nil_a ) )
= ( take_a @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_906_take__hd__drop,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N2 @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N2 @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_907_id__take__nth__drop,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_908_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_909_length__drop,axiom,
! [N2: nat,Xs: list_a] :
( ( size_size_list_a @ ( drop_a @ N2 @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N2 ) ) ).
% length_drop
thf(fact_910_length__drop,axiom,
! [N2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N2 @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% length_drop
thf(fact_911_drop__all,axiom,
! [Xs: list_a,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N2 )
=> ( ( drop_a @ N2 @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_912_drop__all,axiom,
! [Xs: list_nat,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 )
=> ( ( drop_nat @ N2 @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_913_drop__eq__Nil,axiom,
! [N2: nat,Xs: list_a] :
( ( ( drop_a @ N2 @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N2 ) ) ).
% drop_eq_Nil
thf(fact_914_drop__eq__Nil,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( drop_nat @ N2 @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% drop_eq_Nil
thf(fact_915_drop__eq__Nil2,axiom,
! [N2: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N2 @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N2 ) ) ).
% drop_eq_Nil2
thf(fact_916_drop__eq__Nil2,axiom,
! [N2: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N2 @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% drop_eq_Nil2
thf(fact_917_drop__append,axiom,
! [N2: nat,Xs: list_a,Ys: list_a] :
( ( drop_a @ N2 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( drop_a @ N2 @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_918_drop__append,axiom,
! [N2: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N2 @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_919_sorted__wrt__drop,axiom,
! [F: nat > nat > $o,Xs: list_nat,N2: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( drop_nat @ N2 @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_920_drop__Nil,axiom,
! [N2: nat] :
( ( drop_a @ N2 @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_921_sorted__drop,axiom,
! [Xs: list_nat,N2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N2 @ Xs ) ) ) ).
% sorted_drop
thf(fact_922_nth__via__drop,axiom,
! [N2: nat,Xs: list_a,Y3: a,Ys: list_a] :
( ( ( drop_a @ N2 @ Xs )
= ( cons_a @ Y3 @ Ys ) )
=> ( ( nth_a @ Xs @ N2 )
= Y3 ) ) ).
% nth_via_drop
thf(fact_923_nth__via__drop,axiom,
! [N2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
( ( ( drop_nat @ N2 @ Xs )
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( nth_nat @ Xs @ N2 )
= Y3 ) ) ).
% nth_via_drop
thf(fact_924_append__eq__conv__conj,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_a @ ( size_size_list_a @ Xs ) @ Zs ) )
& ( Ys
= ( drop_a @ ( size_size_list_a @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_925_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_926_append__eq__append__conv__if,axiom,
! [Xs_1: list_a,Xs_2: list_a,Ys_1: list_a,Ys_2: list_a] :
( ( ( append_a @ Xs_1 @ Xs_2 )
= ( append_a @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( Xs_1
= ( take_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_a @ ( drop_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( ( take_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_a @ ( drop_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_927_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_928_hd__drop__conv__nth,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( hd_a @ ( drop_a @ N2 @ Xs ) )
= ( nth_a @ Xs @ N2 ) ) ) ).
% hd_drop_conv_nth
thf(fact_929_hd__drop__conv__nth,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N2 @ Xs ) )
= ( nth_nat @ Xs @ N2 ) ) ) ).
% hd_drop_conv_nth
thf(fact_930_drop__rev,axiom,
! [N2: nat,Xs: list_a] :
( ( drop_a @ N2 @ ( rev_a @ Xs ) )
= ( rev_a @ ( take_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N2 ) @ Xs ) ) ) ).
% drop_rev
thf(fact_931_drop__rev,axiom,
! [N2: nat,Xs: list_nat] :
( ( drop_nat @ N2 @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N2 ) @ Xs ) ) ) ).
% drop_rev
thf(fact_932_rev__drop,axiom,
! [I: nat,Xs: list_a] :
( ( rev_a @ ( drop_a @ I @ Xs ) )
= ( take_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ I ) @ ( rev_a @ Xs ) ) ) ).
% rev_drop
thf(fact_933_rev__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( rev_nat @ ( drop_nat @ I @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I ) @ ( rev_nat @ Xs ) ) ) ).
% rev_drop
thf(fact_934_rev__take,axiom,
! [I: nat,Xs: list_a] :
( ( rev_a @ ( take_a @ I @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ I ) @ ( rev_a @ Xs ) ) ) ).
% rev_take
thf(fact_935_rev__take,axiom,
! [I: nat,Xs: list_nat] :
( ( rev_nat @ ( take_nat @ I @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I ) @ ( rev_nat @ Xs ) ) ) ).
% rev_take
thf(fact_936_take__rev,axiom,
! [N2: nat,Xs: list_a] :
( ( take_a @ N2 @ ( rev_a @ Xs ) )
= ( rev_a @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N2 ) @ Xs ) ) ) ).
% take_rev
thf(fact_937_take__rev,axiom,
! [N2: nat,Xs: list_nat] :
( ( take_nat @ N2 @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N2 ) @ Xs ) ) ) ).
% take_rev
thf(fact_938_sdrop__shift,axiom,
! [I: nat,W: list_a,S2: stream_a] :
( ( sdrop_a @ I @ ( shift_a @ W @ S2 ) )
= ( shift_a @ ( drop_a @ I @ W ) @ ( sdrop_a @ ( minus_minus_nat @ I @ ( size_size_list_a @ W ) ) @ S2 ) ) ) ).
% sdrop_shift
thf(fact_939_sdrop__shift,axiom,
! [I: nat,W: list_nat,S2: stream_nat] :
( ( sdrop_nat @ I @ ( shift_nat @ W @ S2 ) )
= ( shift_nat @ ( drop_nat @ I @ W ) @ ( sdrop_nat @ ( minus_minus_nat @ I @ ( size_size_list_nat @ W ) ) @ S2 ) ) ) ).
% sdrop_shift
thf(fact_940_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) )
= ( drop_a @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_941_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
= ( drop_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_942_rotate__drop__take,axiom,
( rotate_a
= ( ^ [N: nat,Xs2: list_a] : ( append_a @ ( drop_a @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Xs2 ) @ ( take_a @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_943_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N: nat,Xs2: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) @ ( take_nat @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ) ).
% rotate_drop_take
thf(fact_944_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_a,A: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I @ A )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_945_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_946_stake__append,axiom,
! [N2: nat,U: list_a,S2: stream_a] :
( ( stake_a @ N2 @ ( shift_a @ U @ S2 ) )
= ( append_a @ ( take_a @ ( ord_min_nat @ ( size_size_list_a @ U ) @ N2 ) @ U ) @ ( stake_a @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ U ) ) @ S2 ) ) ) ).
% stake_append
thf(fact_947_stake__append,axiom,
! [N2: nat,U: list_nat,S2: stream_nat] :
( ( stake_nat @ N2 @ ( shift_nat @ U @ S2 ) )
= ( append_nat @ ( take_nat @ ( ord_min_nat @ ( size_size_list_nat @ U ) @ N2 ) @ U ) @ ( stake_nat @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ U ) ) @ S2 ) ) ) ).
% stake_append
thf(fact_948_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_949_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_950_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_951_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_952_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_953_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_954_min__less__iff__conj,axiom,
! [Z: nat,X2: nat,Y3: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Z @ X2 )
& ( ord_less_nat @ Z @ Y3 ) ) ) ).
% min_less_iff_conj
thf(fact_955_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_956_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_957_list__update__nonempty,axiom,
! [Xs: list_a,K: nat,X2: a] :
( ( ( list_update_a @ Xs @ K @ X2 )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_958_length__list__update,axiom,
! [Xs: list_a,I: nat,X2: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X2 ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_959_length__list__update,axiom,
! [Xs: list_nat,I: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_960_min__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min_nat @ M2 @ N2 ) ) ) ).
% min_Suc_Suc
thf(fact_961_min__0L,axiom,
! [N2: nat] :
( ( ord_min_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% min_0L
thf(fact_962_min__0R,axiom,
! [N2: nat] :
( ( ord_min_nat @ N2 @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_963_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_a,X2: a] :
( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X2 ) @ J )
= ( nth_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_964_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X2: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_965_list__update__id,axiom,
! [Xs: list_a,I: nat] :
( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_966_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_967_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_968_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_969_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_970_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_971_length__take,axiom,
! [N2: nat,Xs: list_a] :
( ( size_size_list_a @ ( take_a @ N2 @ Xs ) )
= ( ord_min_nat @ ( size_size_list_a @ Xs ) @ N2 ) ) ).
% length_take
thf(fact_972_length__take,axiom,
! [N2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( take_nat @ N2 @ Xs ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% length_take
thf(fact_973_list__update__length,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y3: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y3 )
= ( append_a @ Xs @ ( cons_a @ Y3 @ Ys ) ) ) ).
% list_update_length
thf(fact_974_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_975_nth__list__update__eq,axiom,
! [I: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_976_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_977_list__update__code_I1_J,axiom,
! [I: nat,Y3: a] :
( ( list_update_a @ nil_a @ I @ Y3 )
= nil_a ) ).
% list_update_code(1)
thf(fact_978_list__update_Osimps_I1_J,axiom,
! [I: nat,V: a] :
( ( list_update_a @ nil_a @ I @ V )
= nil_a ) ).
% list_update.simps(1)
thf(fact_979_min__le__iff__disj,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
| ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_980_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_981_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_982_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_min_nat @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_983_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_min_nat @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_984_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_985_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_986_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( ord_min_nat @ A4 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_987_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_988_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_989_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_990_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_991_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D2 ) ) ) ) ).
% min.mono
thf(fact_992_min__def,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_993_min__absorb1,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_min_nat @ X2 @ Y3 )
= X2 ) ) ).
% min_absorb1
thf(fact_994_min__absorb2,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_min_nat @ X2 @ Y3 )
= Y3 ) ) ).
% min_absorb2
thf(fact_995_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_996_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_997_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( ord_min_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_998_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_999_min__less__iff__disj,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_nat @ X2 @ Z )
| ( ord_less_nat @ Y3 @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_1000_min__diff,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N2 @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M2 @ N2 ) @ I ) ) ).
% min_diff
thf(fact_1001_min_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.left_commute
thf(fact_1002_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( ord_min_nat @ B3 @ A4 ) ) ) ).
% min.commute
thf(fact_1003_min_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.assoc
thf(fact_1004_list__update__append1,axiom,
! [I: nat,Xs: list_a,Ys: list_a,X2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I @ X2 )
= ( append_a @ ( list_update_a @ Xs @ I @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1005_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1006_nth__list__update,axiom,
! [I: nat,Xs: list_a,J: nat,X2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X2 ) @ J )
= ( nth_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1007_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1008_list__update__same__conv,axiom,
! [I: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I @ X2 )
= Xs )
= ( ( nth_a @ Xs @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1009_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1010_list__update__append,axiom,
! [N2: nat,Xs: list_a,Ys: list_a,X2: a] :
( ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N2 @ X2 )
= ( append_a @ ( list_update_a @ Xs @ N2 @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N2 @ X2 )
= ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_1011_list__update__append,axiom,
! [N2: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N2 @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ N2 @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N2 @ X2 )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_1012_rev__update,axiom,
! [K: nat,Xs: list_a,Y3: a] :
( ( ord_less_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( rev_a @ ( list_update_a @ Xs @ K @ Y3 ) )
= ( list_update_a @ ( rev_a @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ K ) @ one_one_nat ) @ Y3 ) ) ) ).
% rev_update
thf(fact_1013_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_1014_butlast__take,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( butlast_a @ ( take_a @ N2 @ Xs ) )
= ( take_a @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1015_butlast__take,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( butlast_nat @ ( take_nat @ N2 @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1016_last__list__update,axiom,
! [Xs: list_a,K: nat,X2: a] :
( ( Xs != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X2 ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1017_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_1018_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_1019_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_1020_last__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_1021_butlast__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1022_length__butlast,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( butlast_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1023_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_1024_last__drop,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( last_a @ ( drop_a @ N2 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_drop
thf(fact_1025_last__drop,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( last_nat @ ( drop_nat @ N2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_drop
thf(fact_1026_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_1027_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_1028_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1029_last_Osimps,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_1030_last__ConsL,axiom,
! [Xs: list_a,X2: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_1031_last__ConsR,axiom,
! [Xs: list_a,X2: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_1032_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss2: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Xs4 @ Ss2 ) )
& ( Ys
= ( append_a @ Ys6 @ Ss2 ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_1033_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_1034_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1035_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_1036_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_1037_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1038_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_1039_nth__butlast,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ ( butlast_a @ Xs ) ) )
=> ( ( nth_a @ ( butlast_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ N2 ) ) ) ).
% nth_butlast
thf(fact_1040_nth__butlast,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ N2 ) ) ) ).
% nth_butlast
thf(fact_1041_take__butlast,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ N2 @ ( butlast_a @ Xs ) )
= ( take_a @ N2 @ Xs ) ) ) ).
% take_butlast
thf(fact_1042_take__butlast,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N2 @ ( butlast_nat @ Xs ) )
= ( take_nat @ N2 @ Xs ) ) ) ).
% take_butlast
thf(fact_1043_butlast__conv__take,axiom,
( butlast_a
= ( ^ [Xs2: list_a] : ( take_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_conv_take
thf(fact_1044_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_1045_butlast__list__update,axiom,
! [K: nat,Xs: list_a,X2: a] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
= ( butlast_a @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
= ( list_update_a @ ( butlast_a @ Xs ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_1046_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_1047_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1048_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_1049_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_1050_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1051_nth__rotate,axiom,
! [N2: nat,Xs: list_a,M2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate_a @ M2 @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1052_nth__rotate,axiom,
! [N2: nat,Xs: list_nat,M2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M2 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1053_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1054_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1055_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1056_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1057_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1058_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1059_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1060_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1061_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1062_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1063_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1064_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1065_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1066_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1067_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1068_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1069_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1070_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1071_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1072_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1073_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1074_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1075_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1076_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1077_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1078_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1079_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1080_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1081_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1082_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1083_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1084_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1085_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1086_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1087_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1088_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1089_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1090_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1091_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1092_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_1093_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_1094_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1095_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1096_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N2: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N2 ) )
= ( nth_a @ Ys @ N2 ) ) ).
% nth_append_length_plus
thf(fact_1097_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N2: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N2 ) )
= ( nth_nat @ Ys @ N2 ) ) ).
% nth_append_length_plus
thf(fact_1098_nth__drop,axiom,
! [N2: nat,Xs: list_a,I: nat] :
( ( ord_less_eq_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( drop_a @ N2 @ Xs ) @ I )
= ( nth_a @ Xs @ ( plus_plus_nat @ N2 @ I ) ) ) ) ).
% nth_drop
thf(fact_1099_nth__drop,axiom,
! [N2: nat,Xs: list_nat,I: nat] :
( ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N2 @ Xs ) @ I )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N2 @ I ) ) ) ) ).
% nth_drop
thf(fact_1100_min__add__distrib__right,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( plus_plus_nat @ X2 @ ( ord_min_nat @ Y3 @ Z ) )
= ( ord_min_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).
% min_add_distrib_right
thf(fact_1101_min__add__distrib__left,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X2 @ Y3 ) @ Z )
= ( ord_min_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% min_add_distrib_left
thf(fact_1102_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1103_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1104_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1105_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1106_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1107_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1108_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1109_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1110_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1111_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1112_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1113_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1114_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1115_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1116_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1117_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1118_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1119_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1120_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1121_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1122_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1123_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1124_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1125_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1126_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1127_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1128_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1129_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1130_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1131_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1132_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1133_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_1134_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
= ( P @ B2 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1135_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1136_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1137_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1138_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1139_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1140_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1141_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1142_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1143_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1144_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1145_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_1146_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1147_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1148_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1149_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1150_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1151_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1152_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1153_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1154_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1155_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1156_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1157_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1158_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1159_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1160_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1161_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1162_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1163_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1164_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1165_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_1166_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_1167_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1168_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1169_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1170_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1171_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1172_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1173_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1174_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1175_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1176_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1177_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1178_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1179_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1180_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1181_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1182_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1183_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1184_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1185_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1186_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1187_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1188_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1189_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1190_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1191_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1192_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1193_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1194_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1195_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1196_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1197_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q4: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1198_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1199_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1200_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1201_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1202_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1203_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1204_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1205_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1206_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1207_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1208_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1209_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1210_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1211_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1212_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1213_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1214_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1215_sdrop__snth,axiom,
! [N2: nat,S2: stream_a,M2: nat] :
( ( snth_a @ ( sdrop_a @ N2 @ S2 ) @ M2 )
= ( snth_a @ S2 @ ( plus_plus_nat @ N2 @ M2 ) ) ) ).
% sdrop_snth
thf(fact_1216_sdrop__snth,axiom,
! [N2: nat,S2: stream_list_a,M2: nat] :
( ( snth_list_a @ ( sdrop_list_a @ N2 @ S2 ) @ M2 )
= ( snth_list_a @ S2 @ ( plus_plus_nat @ N2 @ M2 ) ) ) ).
% sdrop_snth
thf(fact_1217_distinct__adj__singleton,axiom,
! [X2: a] : ( distinct_adj_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_1218_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1219_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1220_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1221_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1222_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1223_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1224_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1225_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1226_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1227_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1228_distinct__adj__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X2
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_1229_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1230_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_1231_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1232_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F2: a > nat,Xs2: list_a] : ( if_nat @ ( Xs2 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_a @ Xs2 ) ) @ ( size_list_a @ F2 @ ( tl_a @ Xs2 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1233_list_Osize__gen_I1_J,axiom,
! [X2: a > nat] :
( ( size_list_a @ X2 @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1234_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_1235_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1236_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_1237_verit__le__mono__div,axiom,
! [A3: nat,B5: nat,N2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N2 )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N2 )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1238_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1239_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1240_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1241_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1242_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1243_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1244_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1245_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_1246_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_1247_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1248_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1249_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_1250_div__le__mono,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1251_Suc__div__le__mono,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1252_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1253_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1254_div__less__mono,axiom,
! [A3: nat,B5: nat,N2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( modulo_modulo_nat @ A3 @ N2 )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N2 )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A3 @ N2 ) @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1255_div__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( divide_divide_nat @ M2 @ N2 ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M2 ) @ N2 )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% div_Suc
thf(fact_1256_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1257_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1258_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1259_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1260_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1261_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
% Helper facts (13)
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_2_1_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: stream_list_a,Y3: stream_list_a] :
( ( if_stream_list_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: stream_list_a,Y3: stream_list_a] :
( ( if_stream_list_a @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X2: stream_list_nat,Y3: stream_list_nat] :
( ( if_stream_list_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X2: stream_list_nat,Y3: stream_list_nat] :
( ( if_stream_list_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream_list_list_a,Y3: stream_list_list_a] :
( ( if_str7505741754068378070list_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream_list_list_a,Y3: stream_list_list_a] :
( ( if_str7505741754068378070list_a @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
! [X2: stream_list_list_nat,Y3: stream_list_list_nat] :
( ( if_str5104684111711982906st_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
! [X2: stream_list_list_nat,Y3: stream_list_list_nat] :
( ( if_str5104684111711982906st_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
! [X2: stream2255243159586646806list_a,Y3: stream2255243159586646806list_a] :
( ( if_str8217234800680828380list_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
! [X2: stream2255243159586646806list_a,Y3: stream2255243159586646806list_a] :
( ( if_str8217234800680828380list_a @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [N5: nat] :
( ( ord_less_eq_nat @ n @ N5 )
& ( x
= ( snth_a @ ( flat_a @ s ) @ N5 ) ) ) ).
%------------------------------------------------------------------------------