TPTP Problem File: SLH0600^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc2/0003_SeCaV/prob_00168_005068__12673988_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1472 ( 769 unt; 193 typ; 0 def)
% Number of atoms : 3336 (1879 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 10811 ( 442 ~; 93 |; 313 &;8741 @)
% ( 0 <=>;1222 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 692 ( 692 >; 0 *; 0 +; 0 <<)
% Number of symbols : 183 ( 180 usr; 13 con; 0-3 aty)
% Number of variables : 3482 ( 93 ^;3153 !; 236 ?;3482 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:38:41.807
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
list_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__SeCaV__Otm,type,
tm: $tType ).
thf(ty_n_t__SeCaV__Ofm,type,
fm: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (180)
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Int__Oint_J,type,
append_list_int: list_list_int > list_list_int > list_list_int ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Ofm_J,type,
append_list_fm: list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Otm_J,type,
append_list_tm: list_list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__SeCaV__Ofm,type,
append_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Oappend_001t__SeCaV__Otm,type,
append_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Obind_001t__Int__Oint_001t__Int__Oint,type,
bind_int_int: list_int > ( int > list_int ) > list_int ).
thf(sy_c_List_Obind_001t__Int__Oint_001t__SeCaV__Ofm,type,
bind_int_fm: list_int > ( int > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__Int__Oint_001t__SeCaV__Otm,type,
bind_int_tm: list_int > ( int > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__Int__Oint,type,
bind_fm_int: list_fm > ( fm > list_int ) > list_int ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
bind_fm_fm: list_fm > ( fm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
bind_fm_tm: list_fm > ( fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__Int__Oint,type,
bind_tm_int: list_tm > ( tm > list_int ) > list_int ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
bind_tm_fm: list_tm > ( tm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
bind_tm_tm: list_tm > ( tm > list_tm ) > list_tm ).
thf(sy_c_List_Oconcat_001t__Int__Oint,type,
concat_int: list_list_int > list_int ).
thf(sy_c_List_Oconcat_001t__SeCaV__Ofm,type,
concat_fm: list_list_fm > list_fm ).
thf(sy_c_List_Oconcat_001t__SeCaV__Otm,type,
concat_tm: list_list_tm > list_tm ).
thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
gen_length_int: nat > list_int > nat ).
thf(sy_c_List_Ogen__length_001t__SeCaV__Ofm,type,
gen_length_fm: nat > list_fm > nat ).
thf(sy_c_List_Ogen__length_001t__SeCaV__Otm,type,
gen_length_tm: nat > list_tm > nat ).
thf(sy_c_List_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Oinsert_001t__SeCaV__Ofm,type,
insert_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Otm,type,
insert_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olast_001t__Int__Oint,type,
last_int: list_int > int ).
thf(sy_c_List_Olast_001t__SeCaV__Ofm,type,
last_fm: list_fm > fm ).
thf(sy_c_List_Olast_001t__SeCaV__Otm,type,
last_tm: list_tm > tm ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Ofm_J,type,
cons_list_fm: list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Otm_J,type,
cons_list_tm: list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
cons_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
cons_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Ofm_J,type,
nil_list_fm: list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Otm_J,type,
nil_list_tm: list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Ofm,type,
nil_fm: list_fm ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Otm,type,
nil_tm: list_tm ).
thf(sy_c_List_Olist__ex1_001t__Int__Oint,type,
list_ex1_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Ofm,type,
list_ex1_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Otm,type,
list_ex1_tm: ( tm > $o ) > list_tm > $o ).
thf(sy_c_List_Olistrelp_001t__Int__Oint_001t__Int__Oint,type,
listrelp_int_int: ( int > int > $o ) > list_int > list_int > $o ).
thf(sy_c_List_Olistrelp_001t__Int__Oint_001t__SeCaV__Ofm,type,
listrelp_int_fm: ( int > fm > $o ) > list_int > list_fm > $o ).
thf(sy_c_List_Olistrelp_001t__Int__Oint_001t__SeCaV__Otm,type,
listrelp_int_tm: ( int > tm > $o ) > list_int > list_tm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Ofm_001t__Int__Oint,type,
listrelp_fm_int: ( fm > int > $o ) > list_fm > list_int > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
listrelp_fm_fm: ( fm > fm > $o ) > list_fm > list_fm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
listrelp_fm_tm: ( fm > tm > $o ) > list_fm > list_tm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Otm_001t__Int__Oint,type,
listrelp_tm_int: ( tm > int > $o ) > list_tm > list_int > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
listrelp_tm_fm: ( tm > fm > $o ) > list_tm > list_fm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
listrelp_tm_tm: ( tm > tm > $o ) > list_tm > list_tm > $o ).
thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__Int__Oint,type,
map_ta8161051695879884879nt_int: ( int > int ) > list_int > list_int > list_int ).
thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__SeCaV__Ofm,type,
map_ta6489050020625333520int_fm: ( int > fm ) > list_int > list_fm > list_fm ).
thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__SeCaV__Otm,type,
map_ta6489050020626251906int_tm: ( int > tm ) > list_int > list_tm > list_tm ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Ofm_001t__Int__Oint,type,
map_ta6140500833912806250fm_int: ( fm > int ) > list_fm > list_int > list_int ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
map_ta229225445362710709_fm_fm: ( fm > fm ) > list_fm > list_fm > list_fm ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
map_ta229225445363629095_fm_tm: ( fm > tm ) > list_fm > list_tm > list_tm ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Otm_001t__Int__Oint,type,
map_ta7804880090983306972tm_int: ( tm > int ) > list_tm > list_int > list_int ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
map_ta4789309763158333891_tm_fm: ( tm > fm ) > list_tm > list_fm > list_fm ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
map_ta4789309763159252277_tm_tm: ( tm > tm ) > list_tm > list_tm > list_tm ).
thf(sy_c_List_Omaps_001t__Int__Oint_001t__Int__Oint,type,
maps_int_int: ( int > list_int ) > list_int > list_int ).
thf(sy_c_List_Omaps_001t__Int__Oint_001t__SeCaV__Ofm,type,
maps_int_fm: ( int > list_fm ) > list_int > list_fm ).
thf(sy_c_List_Omaps_001t__Int__Oint_001t__SeCaV__Otm,type,
maps_int_tm: ( int > list_tm ) > list_int > list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__Int__Oint,type,
maps_fm_int: ( fm > list_int ) > list_fm > list_int ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
maps_fm_fm: ( fm > list_fm ) > list_fm > list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
maps_fm_tm: ( fm > list_tm ) > list_fm > list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__Int__Oint,type,
maps_tm_int: ( tm > list_int ) > list_tm > list_int ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
maps_tm_fm: ( tm > list_fm ) > list_tm > list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
maps_tm_tm: ( tm > list_tm ) > list_tm > list_tm ).
thf(sy_c_List_Omember_001t__Int__Oint,type,
member_int: list_int > int > $o ).
thf(sy_c_List_Omember_001t__SeCaV__Ofm,type,
member_fm: list_fm > fm > $o ).
thf(sy_c_List_Omember_001t__SeCaV__Otm,type,
member_tm: list_tm > tm > $o ).
thf(sy_c_List_On__lists_001t__Int__Oint,type,
n_lists_int: nat > list_int > list_list_int ).
thf(sy_c_List_On__lists_001t__SeCaV__Ofm,type,
n_lists_fm: nat > list_fm > list_list_fm ).
thf(sy_c_List_On__lists_001t__SeCaV__Otm,type,
n_lists_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__SeCaV__Ofm,type,
nth_fm: list_fm > nat > fm ).
thf(sy_c_List_Onth_001t__SeCaV__Otm,type,
nth_tm: list_tm > nat > tm ).
thf(sy_c_List_Onths_001t__Int__Oint,type,
nths_int: list_int > set_nat > list_int ).
thf(sy_c_List_Onths_001t__SeCaV__Ofm,type,
nths_fm: list_fm > set_nat > list_fm ).
thf(sy_c_List_Onths_001t__SeCaV__Otm,type,
nths_tm: list_tm > set_nat > list_tm ).
thf(sy_c_List_Oord_Olexordp__eq_001t__Int__Oint,type,
lexordp_eq_int: ( int > int > $o ) > list_int > list_int > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__SeCaV__Ofm,type,
lexordp_eq_fm: ( fm > fm > $o ) > list_fm > list_fm > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__SeCaV__Otm,type,
lexordp_eq_tm: ( tm > tm > $o ) > list_tm > list_tm > $o ).
thf(sy_c_List_Oord__class_Olexordp_001t__Int__Oint,type,
ord_lexordp_int: list_int > list_int > $o ).
thf(sy_c_List_Oord__class_Olexordp_001t__Nat__Onat,type,
ord_lexordp_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Int__Oint,type,
ord_lexordp_eq_int: list_int > list_int > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Nat__Onat,type,
ord_lexordp_eq_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
product_lists_int: list_list_int > list_list_int ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Ofm,type,
product_lists_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Otm,type,
product_lists_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
replicate_int: nat > int > list_int ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Int__Oint_J,type,
replicate_list_int: nat > list_int > list_list_int ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__SeCaV__Ofm_J,type,
replicate_list_fm: nat > list_fm > list_list_fm ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__SeCaV__Otm_J,type,
replicate_list_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Oreplicate_001t__SeCaV__Ofm,type,
replicate_fm: nat > fm > list_fm ).
thf(sy_c_List_Oreplicate_001t__SeCaV__Otm,type,
replicate_tm: nat > tm > list_tm ).
thf(sy_c_List_Orotate1_001t__Int__Oint,type,
rotate1_int: list_int > list_int ).
thf(sy_c_List_Orotate1_001t__SeCaV__Ofm,type,
rotate1_fm: list_fm > list_fm ).
thf(sy_c_List_Orotate1_001t__SeCaV__Otm,type,
rotate1_tm: list_tm > list_tm ).
thf(sy_c_List_Osplice_001t__Int__Oint,type,
splice_int: list_int > list_int > list_int ).
thf(sy_c_List_Osplice_001t__SeCaV__Ofm,type,
splice_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Osplice_001t__SeCaV__Otm,type,
splice_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
subseqs_int: list_int > list_list_int ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Ofm,type,
subseqs_fm: list_fm > list_list_fm ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Otm,type,
subseqs_tm: list_tm > list_list_tm ).
thf(sy_c_List_Oupto,type,
upto: int > int > list_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Ofm_J,type,
size_size_list_fm: list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Otm_J,type,
size_size_list_tm: list_tm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Ofm,type,
size_size_fm: fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Otm,type,
size_size_tm: tm > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_SeCaV_Oext_001t__Int__Oint,type,
ext_int: list_int > list_int > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Ofm,type,
ext_fm: list_fm > list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Otm,type,
ext_tm: list_tm > list_tm > $o ).
thf(sy_c_SeCaV_Ofm_OCon,type,
con: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ODis,type,
dis: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_OExi,type,
exi: fm > fm ).
thf(sy_c_SeCaV_Ofm_OImp,type,
imp: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ONeg,type,
neg: fm > fm ).
thf(sy_c_SeCaV_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_SeCaV_Ofm_OUni,type,
uni: fm > fm ).
thf(sy_c_SeCaV_Ofm_Osize__fm,type,
size_fm: fm > nat ).
thf(sy_c_SeCaV_Oinc__list,type,
inc_list: list_tm > list_tm ).
thf(sy_c_SeCaV_Oinc__term,type,
inc_term: tm > tm ).
thf(sy_c_SeCaV_Omember_001t__Int__Oint,type,
member_int2: int > list_int > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Ofm,type,
member_fm2: fm > list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Otm,type,
member_tm2: tm > list_tm > $o ).
thf(sy_c_SeCaV_Onew,type,
new: nat > fm > $o ).
thf(sy_c_SeCaV_Onew__list,type,
new_list: nat > list_tm > $o ).
thf(sy_c_SeCaV_Onew__term,type,
new_term: nat > tm > $o ).
thf(sy_c_SeCaV_Onews,type,
news: nat > list_fm > $o ).
thf(sy_c_SeCaV_Osemantics__list_001t__Int__Oint,type,
semantics_list_int: ( nat > int ) > ( nat > list_int > int ) > list_tm > list_int ).
thf(sy_c_SeCaV_Osemantics__list_001t__SeCaV__Ofm,type,
semantics_list_fm: ( nat > fm ) > ( nat > list_fm > fm ) > list_tm > list_fm ).
thf(sy_c_SeCaV_Osemantics__list_001t__SeCaV__Otm,type,
semantics_list_tm: ( nat > tm ) > ( nat > list_tm > tm ) > list_tm > list_tm ).
thf(sy_c_SeCaV_Osemantics__term_001t__Int__Oint,type,
semantics_term_int: ( nat > int ) > ( nat > list_int > int ) > tm > int ).
thf(sy_c_SeCaV_Osemantics__term_001t__SeCaV__Ofm,type,
semantics_term_fm: ( nat > fm ) > ( nat > list_fm > fm ) > tm > fm ).
thf(sy_c_SeCaV_Osemantics__term_001t__SeCaV__Otm,type,
semantics_term_tm: ( nat > tm ) > ( nat > list_tm > tm ) > tm > tm ).
thf(sy_c_SeCaV_Osequent__calculus,type,
sequent_calculus: list_fm > $o ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__list,type,
sub_list: nat > tm > list_tm > list_tm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_SeCaV_Otm_OVar,type,
var: nat > tm ).
thf(sy_c_SeCaV_Otm_Osize__tm,type,
size_tm: tm > nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
% Relevant facts (1271)
thf(fact_0_Ext,axiom,
! [Z: list_fm,Y: list_fm] :
( ( sequent_calculus @ Z )
=> ( ( ext_fm @ Y @ Z )
=> ( sequent_calculus @ Y ) ) ) ).
% Ext
thf(fact_1__092_060open_062_092_060tturnstile_062_A_091Neg_A_IPre_A0_A_091_093_J_M_APre_A0_A_091_093_093_A_092_060Longrightarrow_062_A_092_060tturnstile_062_A_091Imp_A_IPre_A0_A_091_093_J_A_IPre_A0_A_091_093_J_093_092_060close_062,axiom,
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( pre @ zero_zero_nat @ nil_tm ) ) @ ( cons_fm @ ( pre @ zero_zero_nat @ nil_tm ) @ nil_fm ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( imp @ ( pre @ zero_zero_nat @ nil_tm ) @ ( pre @ zero_zero_nat @ nil_tm ) ) @ nil_fm ) ) ) ).
% \<open>\<tturnstile> [Neg (Pre 0 []), Pre 0 []] \<Longrightarrow> \<tturnstile> [Imp (Pre 0 []) (Pre 0 [])]\<close>
thf(fact_2_that,axiom,
sequent_calculus @ ( cons_fm @ ( pre @ zero_zero_nat @ nil_tm ) @ ( cons_fm @ ( neg @ ( pre @ zero_zero_nat @ nil_tm ) ) @ nil_fm ) ) ).
% that
thf(fact_3_fm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( pre @ X11 @ X12 )
= ( pre @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fm.inject(1)
thf(fact_4_fm_Oinject_I2_J,axiom,
! [X21: fm,X22: fm,Y21: fm,Y22: fm] :
( ( ( imp @ X21 @ X22 )
= ( imp @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% fm.inject(2)
thf(fact_5_fm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X21: fm,X22: fm] :
( ( pre @ X11 @ X12 )
!= ( imp @ X21 @ X22 ) ) ).
% fm.distinct(1)
thf(fact_6_BetaImp,axiom,
! [P: fm,Z: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( imp @ P @ Q ) ) @ Z ) ) ) ) ).
% BetaImp
thf(fact_7_AlphaImp,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ Q @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( imp @ P @ Q ) @ Z ) ) ) ).
% AlphaImp
thf(fact_8_ext_Osimps_I1_J,axiom,
! [Y: list_tm] : ( ext_tm @ Y @ nil_tm ) ).
% ext.simps(1)
thf(fact_9_ext_Osimps_I1_J,axiom,
! [Y: list_fm] : ( ext_fm @ Y @ nil_fm ) ).
% ext.simps(1)
thf(fact_10_ext_Osimps_I1_J,axiom,
! [Y: list_int] : ( ext_int @ Y @ nil_int ) ).
% ext.simps(1)
thf(fact_11_Neg,axiom,
! [P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( neg @ P ) ) @ Z ) ) ) ).
% Neg
thf(fact_12_list_Oinject,axiom,
! [X21: fm,X22: list_fm,Y21: fm,Y22: list_fm] :
( ( ( cons_fm @ X21 @ X22 )
= ( cons_fm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_13_list_Oinject,axiom,
! [X21: tm,X22: list_tm,Y21: tm,Y22: list_tm] :
( ( ( cons_tm @ X21 @ X22 )
= ( cons_tm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_14_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_15_fm_Odistinct_I21_J,axiom,
! [X21: fm,X22: fm,X7: fm] :
( ( imp @ X21 @ X22 )
!= ( neg @ X7 ) ) ).
% fm.distinct(21)
thf(fact_16_fm_Oinject_I7_J,axiom,
! [X7: fm,Y7: fm] :
( ( ( neg @ X7 )
= ( neg @ Y7 ) )
= ( X7 = Y7 ) ) ).
% fm.inject(7)
thf(fact_17_fm_Odistinct_I11_J,axiom,
! [X11: nat,X12: list_tm,X7: fm] :
( ( pre @ X11 @ X12 )
!= ( neg @ X7 ) ) ).
% fm.distinct(11)
thf(fact_18_list_Odistinct_I1_J,axiom,
! [X21: fm,X22: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_19_list_Odistinct_I1_J,axiom,
! [X21: tm,X22: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_20_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_21_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X22: list_fm] :
( ( List
= ( cons_fm @ X21 @ X22 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_22_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X22: list_tm] :
( ( List
= ( cons_tm @ X21 @ X22 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_23_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_24_list_Oexhaust,axiom,
! [Y: list_fm] :
( ( Y != nil_fm )
=> ~ ! [X212: fm,X222: list_fm] :
( Y
!= ( cons_fm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_25_list_Oexhaust,axiom,
! [Y: list_tm] :
( ( Y != nil_tm )
=> ~ ! [X212: tm,X222: list_tm] :
( Y
!= ( cons_tm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_26_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_27_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_28_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_29_not__Cons__self2,axiom,
! [X: fm,Xs: list_fm] :
( ( cons_fm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_30_not__Cons__self2,axiom,
! [X: tm,Xs: list_tm] :
( ( cons_tm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_31_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_32_list__nonempty__induct,axiom,
! [Xs: list_fm,P2: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X2: fm] : ( P2 @ ( cons_fm @ X2 @ nil_fm ) )
=> ( ! [X2: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_33_list__nonempty__induct,axiom,
! [Xs: list_tm,P2: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X2: tm] : ( P2 @ ( cons_tm @ X2 @ nil_tm ) )
=> ( ! [X2: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_34_list__nonempty__induct,axiom,
! [Xs: list_int,P2: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X2: int] : ( P2 @ ( cons_int @ X2 @ nil_int ) )
=> ( ! [X2: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_35_list__induct2_H,axiom,
! [P2: list_fm > list_fm > $o,Xs: list_fm,Ys: list_fm] :
( ( P2 @ nil_fm @ nil_fm )
=> ( ! [X2: fm,Xs2: list_fm] : ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ nil_fm )
=> ( ! [Y2: fm,Ys2: list_fm] : ( P2 @ nil_fm @ ( cons_fm @ Y2 @ Ys2 ) )
=> ( ! [X2: fm,Xs2: list_fm,Y2: fm,Ys2: list_fm] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_36_list__induct2_H,axiom,
! [P2: list_fm > list_tm > $o,Xs: list_fm,Ys: list_tm] :
( ( P2 @ nil_fm @ nil_tm )
=> ( ! [X2: fm,Xs2: list_fm] : ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ nil_tm )
=> ( ! [Y2: tm,Ys2: list_tm] : ( P2 @ nil_fm @ ( cons_tm @ Y2 @ Ys2 ) )
=> ( ! [X2: fm,Xs2: list_fm,Y2: tm,Ys2: list_tm] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_37_list__induct2_H,axiom,
! [P2: list_fm > list_int > $o,Xs: list_fm,Ys: list_int] :
( ( P2 @ nil_fm @ nil_int )
=> ( ! [X2: fm,Xs2: list_fm] : ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P2 @ nil_fm @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X2: fm,Xs2: list_fm,Y2: int,Ys2: list_int] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_38_list__induct2_H,axiom,
! [P2: list_tm > list_fm > $o,Xs: list_tm,Ys: list_fm] :
( ( P2 @ nil_tm @ nil_fm )
=> ( ! [X2: tm,Xs2: list_tm] : ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ nil_fm )
=> ( ! [Y2: fm,Ys2: list_fm] : ( P2 @ nil_tm @ ( cons_fm @ Y2 @ Ys2 ) )
=> ( ! [X2: tm,Xs2: list_tm,Y2: fm,Ys2: list_fm] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_39_list__induct2_H,axiom,
! [P2: list_tm > list_tm > $o,Xs: list_tm,Ys: list_tm] :
( ( P2 @ nil_tm @ nil_tm )
=> ( ! [X2: tm,Xs2: list_tm] : ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ nil_tm )
=> ( ! [Y2: tm,Ys2: list_tm] : ( P2 @ nil_tm @ ( cons_tm @ Y2 @ Ys2 ) )
=> ( ! [X2: tm,Xs2: list_tm,Y2: tm,Ys2: list_tm] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_40_list__induct2_H,axiom,
! [P2: list_tm > list_int > $o,Xs: list_tm,Ys: list_int] :
( ( P2 @ nil_tm @ nil_int )
=> ( ! [X2: tm,Xs2: list_tm] : ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P2 @ nil_tm @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X2: tm,Xs2: list_tm,Y2: int,Ys2: list_int] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_41_list__induct2_H,axiom,
! [P2: list_int > list_fm > $o,Xs: list_int,Ys: list_fm] :
( ( P2 @ nil_int @ nil_fm )
=> ( ! [X2: int,Xs2: list_int] : ( P2 @ ( cons_int @ X2 @ Xs2 ) @ nil_fm )
=> ( ! [Y2: fm,Ys2: list_fm] : ( P2 @ nil_int @ ( cons_fm @ Y2 @ Ys2 ) )
=> ( ! [X2: int,Xs2: list_int,Y2: fm,Ys2: list_fm] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_42_list__induct2_H,axiom,
! [P2: list_int > list_tm > $o,Xs: list_int,Ys: list_tm] :
( ( P2 @ nil_int @ nil_tm )
=> ( ! [X2: int,Xs2: list_int] : ( P2 @ ( cons_int @ X2 @ Xs2 ) @ nil_tm )
=> ( ! [Y2: tm,Ys2: list_tm] : ( P2 @ nil_int @ ( cons_tm @ Y2 @ Ys2 ) )
=> ( ! [X2: int,Xs2: list_int,Y2: tm,Ys2: list_tm] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_43_list__induct2_H,axiom,
! [P2: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P2 @ nil_int @ nil_int )
=> ( ! [X2: int,Xs2: list_int] : ( P2 @ ( cons_int @ X2 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P2 @ nil_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X2: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_44_neq__Nil__conv,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
= ( ? [Y3: fm,Ys3: list_fm] :
( Xs
= ( cons_fm @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_45_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y3: tm,Ys3: list_tm] :
( Xs
= ( cons_tm @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_46_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y3: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_47_remdups__adj_Ocases,axiom,
! [X: list_fm] :
( ( X != nil_fm )
=> ( ! [X2: fm] :
( X
!= ( cons_fm @ X2 @ nil_fm ) )
=> ~ ! [X2: fm,Y2: fm,Xs2: list_fm] :
( X
!= ( cons_fm @ X2 @ ( cons_fm @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_48_remdups__adj_Ocases,axiom,
! [X: list_tm] :
( ( X != nil_tm )
=> ( ! [X2: tm] :
( X
!= ( cons_tm @ X2 @ nil_tm ) )
=> ~ ! [X2: tm,Y2: tm,Xs2: list_tm] :
( X
!= ( cons_tm @ X2 @ ( cons_tm @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_49_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X2: int] :
( X
!= ( cons_int @ X2 @ nil_int ) )
=> ~ ! [X2: int,Y2: int,Xs2: list_int] :
( X
!= ( cons_int @ X2 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_50_transpose_Ocases,axiom,
! [X: list_list_fm] :
( ( X != nil_list_fm )
=> ( ! [Xss: list_list_fm] :
( X
!= ( cons_list_fm @ nil_fm @ Xss ) )
=> ~ ! [X2: fm,Xs2: list_fm,Xss: list_list_fm] :
( X
!= ( cons_list_fm @ ( cons_fm @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_51_transpose_Ocases,axiom,
! [X: list_list_tm] :
( ( X != nil_list_tm )
=> ( ! [Xss: list_list_tm] :
( X
!= ( cons_list_tm @ nil_tm @ Xss ) )
=> ~ ! [X2: tm,Xs2: list_tm,Xss: list_list_tm] :
( X
!= ( cons_list_tm @ ( cons_tm @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_52_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X2: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_53_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X2: int,Xs2: list_int] :
( X
!= ( cons_int @ X2 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_54_Basic,axiom,
! [P: fm,Z: list_fm] :
( ( member_fm2 @ ( neg @ P ) @ Z )
=> ( sequent_calculus @ ( cons_fm @ P @ Z ) ) ) ).
% Basic
thf(fact_55_ext_Osimps_I2_J,axiom,
! [Y: list_fm,P: fm,Z: list_fm] :
( ( ext_fm @ Y @ ( cons_fm @ P @ Z ) )
= ( ( ( member_fm2 @ P @ Y )
=> ( ext_fm @ Y @ Z ) )
& ( member_fm2 @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_56_ext_Osimps_I2_J,axiom,
! [Y: list_tm,P: tm,Z: list_tm] :
( ( ext_tm @ Y @ ( cons_tm @ P @ Z ) )
= ( ( ( member_tm2 @ P @ Y )
=> ( ext_tm @ Y @ Z ) )
& ( member_tm2 @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_57_ext_Osimps_I2_J,axiom,
! [Y: list_int,P: int,Z: list_int] :
( ( ext_int @ Y @ ( cons_int @ P @ Z ) )
= ( ( ( member_int2 @ P @ Y )
=> ( ext_int @ Y @ Z ) )
& ( member_int2 @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_58_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= nil_list_tm ) ) ) ).
% n_lists_Nil
thf(fact_59_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_fm @ N @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_fm @ N @ nil_fm )
= nil_list_fm ) ) ) ).
% n_lists_Nil
thf(fact_60_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= nil_list_int ) ) ) ).
% n_lists_Nil
thf(fact_61_semantics__list_Osimps_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm] :
( ( semantics_list_tm @ E @ F @ nil_tm )
= nil_tm ) ).
% semantics_list.simps(1)
thf(fact_62_semantics__list_Osimps_I1_J,axiom,
! [E: nat > fm,F: nat > list_fm > fm] :
( ( semantics_list_fm @ E @ F @ nil_tm )
= nil_fm ) ).
% semantics_list.simps(1)
thf(fact_63_semantics__list_Osimps_I1_J,axiom,
! [E: nat > int,F: nat > list_int > int] :
( ( semantics_list_int @ E @ F @ nil_tm )
= nil_int ) ).
% semantics_list.simps(1)
thf(fact_64_AlphaCon,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ ( neg @ Q ) @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( con @ P @ Q ) ) @ Z ) ) ) ).
% AlphaCon
thf(fact_65_BetaDis,axiom,
! [P: fm,Z: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( dis @ P @ Q ) ) @ Z ) ) ) ) ).
% BetaDis
thf(fact_66_insert__Nil,axiom,
! [X: fm] :
( ( insert_fm @ X @ nil_fm )
= ( cons_fm @ X @ nil_fm ) ) ).
% insert_Nil
thf(fact_67_insert__Nil,axiom,
! [X: tm] :
( ( insert_tm @ X @ nil_tm )
= ( cons_tm @ X @ nil_tm ) ) ).
% insert_Nil
thf(fact_68_insert__Nil,axiom,
! [X: int] :
( ( insert_int @ X @ nil_int )
= ( cons_int @ X @ nil_int ) ) ).
% insert_Nil
thf(fact_69_nths__singleton,axiom,
! [A: set_nat,X: fm] :
( ( ( member_nat @ zero_zero_nat @ A )
=> ( ( nths_fm @ ( cons_fm @ X @ nil_fm ) @ A )
= ( cons_fm @ X @ nil_fm ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ( nths_fm @ ( cons_fm @ X @ nil_fm ) @ A )
= nil_fm ) ) ) ).
% nths_singleton
thf(fact_70_nths__singleton,axiom,
! [A: set_nat,X: tm] :
( ( ( member_nat @ zero_zero_nat @ A )
=> ( ( nths_tm @ ( cons_tm @ X @ nil_tm ) @ A )
= ( cons_tm @ X @ nil_tm ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ( nths_tm @ ( cons_tm @ X @ nil_tm ) @ A )
= nil_tm ) ) ) ).
% nths_singleton
thf(fact_71_nths__singleton,axiom,
! [A: set_nat,X: int] :
( ( ( member_nat @ zero_zero_nat @ A )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A )
= ( cons_int @ X @ nil_int ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A )
= nil_int ) ) ) ).
% nths_singleton
thf(fact_72_news_Osimps_I1_J,axiom,
! [C: nat] : ( news @ C @ nil_fm ) ).
% news.simps(1)
thf(fact_73_list__ex1__simps_I1_J,axiom,
! [P2: tm > $o] :
~ ( list_ex1_tm @ P2 @ nil_tm ) ).
% list_ex1_simps(1)
thf(fact_74_list__ex1__simps_I1_J,axiom,
! [P2: fm > $o] :
~ ( list_ex1_fm @ P2 @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_75_list__ex1__simps_I1_J,axiom,
! [P2: int > $o] :
~ ( list_ex1_int @ P2 @ nil_int ) ).
% list_ex1_simps(1)
thf(fact_76_n__lists_Osimps_I1_J,axiom,
! [Xs: list_tm] :
( ( n_lists_tm @ zero_zero_nat @ Xs )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% n_lists.simps(1)
thf(fact_77_n__lists_Osimps_I1_J,axiom,
! [Xs: list_fm] :
( ( n_lists_fm @ zero_zero_nat @ Xs )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% n_lists.simps(1)
thf(fact_78_n__lists_Osimps_I1_J,axiom,
! [Xs: list_int] :
( ( n_lists_int @ zero_zero_nat @ Xs )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% n_lists.simps(1)
thf(fact_79_inc__list_Osimps_I1_J,axiom,
( ( inc_list @ nil_tm )
= nil_tm ) ).
% inc_list.simps(1)
thf(fact_80_fm_Oinject_I4_J,axiom,
! [X41: fm,X42: fm,Y41: fm,Y42: fm] :
( ( ( con @ X41 @ X42 )
= ( con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(4)
thf(fact_81_fm_Oinject_I3_J,axiom,
! [X31: fm,X32: fm,Y31: fm,Y32: fm] :
( ( ( dis @ X31 @ X32 )
= ( dis @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(3)
thf(fact_82_nths__nil,axiom,
! [A: set_nat] :
( ( nths_tm @ nil_tm @ A )
= nil_tm ) ).
% nths_nil
thf(fact_83_nths__nil,axiom,
! [A: set_nat] :
( ( nths_fm @ nil_fm @ A )
= nil_fm ) ).
% nths_nil
thf(fact_84_nths__nil,axiom,
! [A: set_nat] :
( ( nths_int @ nil_int @ A )
= nil_int ) ).
% nths_nil
thf(fact_85_fm_Odistinct_I23_J,axiom,
! [X31: fm,X32: fm,X41: fm,X42: fm] :
( ( dis @ X31 @ X32 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(23)
thf(fact_86_fm_Odistinct_I29_J,axiom,
! [X31: fm,X32: fm,X7: fm] :
( ( dis @ X31 @ X32 )
!= ( neg @ X7 ) ) ).
% fm.distinct(29)
thf(fact_87_fm_Odistinct_I35_J,axiom,
! [X41: fm,X42: fm,X7: fm] :
( ( con @ X41 @ X42 )
!= ( neg @ X7 ) ) ).
% fm.distinct(35)
thf(fact_88_fm_Odistinct_I5_J,axiom,
! [X11: nat,X12: list_tm,X41: fm,X42: fm] :
( ( pre @ X11 @ X12 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_89_fm_Odistinct_I3_J,axiom,
! [X11: nat,X12: list_tm,X31: fm,X32: fm] :
( ( pre @ X11 @ X12 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_90_fm_Odistinct_I15_J,axiom,
! [X21: fm,X22: fm,X41: fm,X42: fm] :
( ( imp @ X21 @ X22 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(15)
thf(fact_91_fm_Odistinct_I13_J,axiom,
! [X21: fm,X22: fm,X31: fm,X32: fm] :
( ( imp @ X21 @ X22 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(13)
thf(fact_92_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( member_fm2 @ P @ ( cons_fm @ Q @ Z ) )
= ( ( P != Q )
=> ( member_fm2 @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_93_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: tm,Q: tm,Z: list_tm] :
( ( member_tm2 @ P @ ( cons_tm @ Q @ Z ) )
= ( ( P != Q )
=> ( member_tm2 @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_94_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: int,Q: int,Z: list_int] :
( ( member_int2 @ P @ ( cons_int @ Q @ Z ) )
= ( ( P != Q )
=> ( member_int2 @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_95_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: tm] :
~ ( member_tm2 @ P @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_96_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: fm] :
~ ( member_fm2 @ P @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_97_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: int] :
~ ( member_int2 @ P @ nil_int ) ).
% SeCaV.member.simps(1)
thf(fact_98_BetaCon,axiom,
! [P: fm,Z: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ Q @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( con @ P @ Q ) @ Z ) ) ) ) ).
% BetaCon
thf(fact_99_AlphaDis,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ ( cons_fm @ Q @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( dis @ P @ Q ) @ Z ) ) ) ).
% AlphaDis
thf(fact_100_semantics__list_Osimps_I2_J,axiom,
! [E: nat > fm,F: nat > list_fm > fm,T: tm,L: list_tm] :
( ( semantics_list_fm @ E @ F @ ( cons_tm @ T @ L ) )
= ( cons_fm @ ( semantics_term_fm @ E @ F @ T ) @ ( semantics_list_fm @ E @ F @ L ) ) ) ).
% semantics_list.simps(2)
thf(fact_101_semantics__list_Osimps_I2_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,T: tm,L: list_tm] :
( ( semantics_list_tm @ E @ F @ ( cons_tm @ T @ L ) )
= ( cons_tm @ ( semantics_term_tm @ E @ F @ T ) @ ( semantics_list_tm @ E @ F @ L ) ) ) ).
% semantics_list.simps(2)
thf(fact_102_semantics__list_Osimps_I2_J,axiom,
! [E: nat > int,F: nat > list_int > int,T: tm,L: list_tm] :
( ( semantics_list_int @ E @ F @ ( cons_tm @ T @ L ) )
= ( cons_int @ ( semantics_term_int @ E @ F @ T ) @ ( semantics_list_int @ E @ F @ L ) ) ) ).
% semantics_list.simps(2)
thf(fact_103_product__lists_Osimps_I1_J,axiom,
( ( product_lists_tm @ nil_list_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% product_lists.simps(1)
thf(fact_104_product__lists_Osimps_I1_J,axiom,
( ( product_lists_fm @ nil_list_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_105_product__lists_Osimps_I1_J,axiom,
( ( product_lists_int @ nil_list_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% product_lists.simps(1)
thf(fact_106_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_107_subseqs_Osimps_I1_J,axiom,
( ( subseqs_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_108_subseqs_Osimps_I1_J,axiom,
( ( subseqs_int @ nil_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% subseqs.simps(1)
thf(fact_109_inc__list_Osimps_I2_J,axiom,
! [T: tm,L: list_tm] :
( ( inc_list @ ( cons_tm @ T @ L ) )
= ( cons_tm @ ( inc_term @ T ) @ ( inc_list @ L ) ) ) ).
% inc_list.simps(2)
thf(fact_110_news_Osimps_I2_J,axiom,
! [C: nat,P: fm,Z: list_fm] :
( ( news @ C @ ( cons_fm @ P @ Z ) )
= ( ( ( new @ C @ P )
=> ( news @ C @ Z ) )
& ( new @ C @ P ) ) ) ).
% news.simps(2)
thf(fact_111_fm_Oexhaust,axiom,
! [Y: fm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( pre @ X112 @ X122 ) )
=> ( ! [X212: fm,X222: fm] :
( Y
!= ( imp @ X212 @ X222 ) )
=> ( ! [X312: fm,X322: fm] :
( Y
!= ( dis @ X312 @ X322 ) )
=> ( ! [X412: fm,X422: fm] :
( Y
!= ( con @ X412 @ X422 ) )
=> ( ! [X5: fm] :
( Y
!= ( exi @ X5 ) )
=> ( ! [X6: fm] :
( Y
!= ( uni @ X6 ) )
=> ~ ! [X72: fm] :
( Y
!= ( neg @ X72 ) ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_112_map__tailrec__rev_Oelims,axiom,
! [X: fm > fm,Xa: list_fm,Xb: list_fm,Y: list_fm] :
( ( ( map_ta229225445362710709_fm_fm @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_fm )
=> ( Y != Xb ) )
=> ~ ! [A2: fm,As: list_fm] :
( ( Xa
= ( cons_fm @ A2 @ As ) )
=> ( Y
!= ( map_ta229225445362710709_fm_fm @ X @ As @ ( cons_fm @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_113_map__tailrec__rev_Oelims,axiom,
! [X: fm > tm,Xa: list_fm,Xb: list_tm,Y: list_tm] :
( ( ( map_ta229225445363629095_fm_tm @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_fm )
=> ( Y != Xb ) )
=> ~ ! [A2: fm,As: list_fm] :
( ( Xa
= ( cons_fm @ A2 @ As ) )
=> ( Y
!= ( map_ta229225445363629095_fm_tm @ X @ As @ ( cons_tm @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_114_map__tailrec__rev_Oelims,axiom,
! [X: fm > int,Xa: list_fm,Xb: list_int,Y: list_int] :
( ( ( map_ta6140500833912806250fm_int @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_fm )
=> ( Y != Xb ) )
=> ~ ! [A2: fm,As: list_fm] :
( ( Xa
= ( cons_fm @ A2 @ As ) )
=> ( Y
!= ( map_ta6140500833912806250fm_int @ X @ As @ ( cons_int @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_115_map__tailrec__rev_Oelims,axiom,
! [X: tm > fm,Xa: list_tm,Xb: list_fm,Y: list_fm] :
( ( ( map_ta4789309763158333891_tm_fm @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_tm )
=> ( Y != Xb ) )
=> ~ ! [A2: tm,As: list_tm] :
( ( Xa
= ( cons_tm @ A2 @ As ) )
=> ( Y
!= ( map_ta4789309763158333891_tm_fm @ X @ As @ ( cons_fm @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_116_map__tailrec__rev_Oelims,axiom,
! [X: tm > tm,Xa: list_tm,Xb: list_tm,Y: list_tm] :
( ( ( map_ta4789309763159252277_tm_tm @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_tm )
=> ( Y != Xb ) )
=> ~ ! [A2: tm,As: list_tm] :
( ( Xa
= ( cons_tm @ A2 @ As ) )
=> ( Y
!= ( map_ta4789309763159252277_tm_tm @ X @ As @ ( cons_tm @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_117_map__tailrec__rev_Oelims,axiom,
! [X: tm > int,Xa: list_tm,Xb: list_int,Y: list_int] :
( ( ( map_ta7804880090983306972tm_int @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_tm )
=> ( Y != Xb ) )
=> ~ ! [A2: tm,As: list_tm] :
( ( Xa
= ( cons_tm @ A2 @ As ) )
=> ( Y
!= ( map_ta7804880090983306972tm_int @ X @ As @ ( cons_int @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_118_map__tailrec__rev_Oelims,axiom,
! [X: int > fm,Xa: list_int,Xb: list_fm,Y: list_fm] :
( ( ( map_ta6489050020625333520int_fm @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_int )
=> ( Y != Xb ) )
=> ~ ! [A2: int,As: list_int] :
( ( Xa
= ( cons_int @ A2 @ As ) )
=> ( Y
!= ( map_ta6489050020625333520int_fm @ X @ As @ ( cons_fm @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_119_map__tailrec__rev_Oelims,axiom,
! [X: int > tm,Xa: list_int,Xb: list_tm,Y: list_tm] :
( ( ( map_ta6489050020626251906int_tm @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_int )
=> ( Y != Xb ) )
=> ~ ! [A2: int,As: list_int] :
( ( Xa
= ( cons_int @ A2 @ As ) )
=> ( Y
!= ( map_ta6489050020626251906int_tm @ X @ As @ ( cons_tm @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_120_map__tailrec__rev_Oelims,axiom,
! [X: int > int,Xa: list_int,Xb: list_int,Y: list_int] :
( ( ( map_ta8161051695879884879nt_int @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_int )
=> ( Y != Xb ) )
=> ~ ! [A2: int,As: list_int] :
( ( Xa
= ( cons_int @ A2 @ As ) )
=> ( Y
!= ( map_ta8161051695879884879nt_int @ X @ As @ ( cons_int @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_121_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: fm > fm > $o,X: fm,Xs: list_fm] :
~ ( lexordp_eq_fm @ Less @ ( cons_fm @ X @ Xs ) @ nil_fm ) ).
% ord.lexordp_eq_simps(3)
thf(fact_122_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: tm > tm > $o,X: tm,Xs: list_tm] :
~ ( lexordp_eq_tm @ Less @ ( cons_tm @ X @ Xs ) @ nil_tm ) ).
% ord.lexordp_eq_simps(3)
thf(fact_123_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: int > int > $o,X: int,Xs: list_int] :
~ ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ nil_int ) ).
% ord.lexordp_eq_simps(3)
thf(fact_124_fm_Oinject_I5_J,axiom,
! [X52: fm,Y5: fm] :
( ( ( exi @ X52 )
= ( exi @ Y5 ) )
= ( X52 = Y5 ) ) ).
% fm.inject(5)
thf(fact_125_fm_Oinject_I6_J,axiom,
! [X62: fm,Y6: fm] :
( ( ( uni @ X62 )
= ( uni @ Y6 ) )
= ( X62 = Y6 ) ) ).
% fm.inject(6)
thf(fact_126_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: fm > fm > $o,X: fm,Xs: list_fm,Y: fm,Ys: list_fm] :
( ( lexordp_eq_fm @ Less @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_fm @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_127_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: tm > tm > $o,X: tm,Xs: list_tm,Y: tm,Ys: list_tm] :
( ( lexordp_eq_tm @ Less @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_tm @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_128_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: int > int > $o,X: int,Xs: list_int,Y: int,Ys: list_int] :
( ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_int @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_129_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: tm > tm > $o,Ys: list_tm] : ( lexordp_eq_tm @ Less @ nil_tm @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_130_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: fm > fm > $o,Ys: list_fm] : ( lexordp_eq_fm @ Less @ nil_fm @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_131_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: int > int > $o,Ys: list_int] : ( lexordp_eq_int @ Less @ nil_int @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_132_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: tm > tm > $o,Xs: list_tm] :
( ( lexordp_eq_tm @ Less @ Xs @ nil_tm )
= ( Xs = nil_tm ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_133_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: fm > fm > $o,Xs: list_fm] :
( ( lexordp_eq_fm @ Less @ Xs @ nil_fm )
= ( Xs = nil_fm ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_134_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: int > int > $o,Xs: list_int] :
( ( lexordp_eq_int @ Less @ Xs @ nil_int )
= ( Xs = nil_int ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_135_new_Osimps_I6_J,axiom,
! [C: nat,P: fm] :
( ( new @ C @ ( uni @ P ) )
= ( new @ C @ P ) ) ).
% new.simps(6)
thf(fact_136_new_Osimps_I5_J,axiom,
! [C: nat,P: fm] :
( ( new @ C @ ( exi @ P ) )
= ( new @ C @ P ) ) ).
% new.simps(5)
thf(fact_137_fm_Odistinct_I37_J,axiom,
! [X52: fm,X62: fm] :
( ( exi @ X52 )
!= ( uni @ X62 ) ) ).
% fm.distinct(37)
thf(fact_138_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: fm > fm > $o,X: fm,Y: fm,Xs: list_fm,Ys: list_fm] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_fm @ Less @ Xs @ Ys )
=> ( lexordp_eq_fm @ Less @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_139_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: tm > tm > $o,X: tm,Y: tm,Xs: list_tm,Ys: list_tm] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_tm @ Less @ Xs @ Ys )
=> ( lexordp_eq_tm @ Less @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_140_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: int > int > $o,X: int,Y: int,Xs: list_int,Ys: list_int] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_int @ Less @ Xs @ Ys )
=> ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_141_ord_Olexordp__eq_OCons,axiom,
! [Less: fm > fm > $o,X: fm,Y: fm,Xs: list_fm,Ys: list_fm] :
( ( Less @ X @ Y )
=> ( lexordp_eq_fm @ Less @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_142_ord_Olexordp__eq_OCons,axiom,
! [Less: tm > tm > $o,X: tm,Y: tm,Xs: list_tm,Ys: list_tm] :
( ( Less @ X @ Y )
=> ( lexordp_eq_tm @ Less @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_143_ord_Olexordp__eq_OCons,axiom,
! [Less: int > int > $o,X: int,Y: int,Xs: list_int,Ys: list_int] :
( ( Less @ X @ Y )
=> ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_144_ord_Olexordp__eq_ONil,axiom,
! [Less: tm > tm > $o,Ys: list_tm] : ( lexordp_eq_tm @ Less @ nil_tm @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_145_ord_Olexordp__eq_ONil,axiom,
! [Less: fm > fm > $o,Ys: list_fm] : ( lexordp_eq_fm @ Less @ nil_fm @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_146_ord_Olexordp__eq_ONil,axiom,
! [Less: int > int > $o,Ys: list_int] : ( lexordp_eq_int @ Less @ nil_int @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_147_fm_Odistinct_I41_J,axiom,
! [X62: fm,X7: fm] :
( ( uni @ X62 )
!= ( neg @ X7 ) ) ).
% fm.distinct(41)
thf(fact_148_fm_Odistinct_I39_J,axiom,
! [X52: fm,X7: fm] :
( ( exi @ X52 )
!= ( neg @ X7 ) ) ).
% fm.distinct(39)
thf(fact_149_fm_Odistinct_I7_J,axiom,
! [X11: nat,X12: list_tm,X52: fm] :
( ( pre @ X11 @ X12 )
!= ( exi @ X52 ) ) ).
% fm.distinct(7)
thf(fact_150_fm_Odistinct_I9_J,axiom,
! [X11: nat,X12: list_tm,X62: fm] :
( ( pre @ X11 @ X12 )
!= ( uni @ X62 ) ) ).
% fm.distinct(9)
thf(fact_151_fm_Odistinct_I17_J,axiom,
! [X21: fm,X22: fm,X52: fm] :
( ( imp @ X21 @ X22 )
!= ( exi @ X52 ) ) ).
% fm.distinct(17)
thf(fact_152_fm_Odistinct_I19_J,axiom,
! [X21: fm,X22: fm,X62: fm] :
( ( imp @ X21 @ X22 )
!= ( uni @ X62 ) ) ).
% fm.distinct(19)
thf(fact_153_fm_Odistinct_I33_J,axiom,
! [X41: fm,X42: fm,X62: fm] :
( ( con @ X41 @ X42 )
!= ( uni @ X62 ) ) ).
% fm.distinct(33)
thf(fact_154_fm_Odistinct_I31_J,axiom,
! [X41: fm,X42: fm,X52: fm] :
( ( con @ X41 @ X42 )
!= ( exi @ X52 ) ) ).
% fm.distinct(31)
thf(fact_155_fm_Odistinct_I27_J,axiom,
! [X31: fm,X32: fm,X62: fm] :
( ( dis @ X31 @ X32 )
!= ( uni @ X62 ) ) ).
% fm.distinct(27)
thf(fact_156_fm_Odistinct_I25_J,axiom,
! [X31: fm,X32: fm,X52: fm] :
( ( dis @ X31 @ X32 )
!= ( exi @ X52 ) ) ).
% fm.distinct(25)
thf(fact_157_new_Osimps_I7_J,axiom,
! [C: nat,P: fm] :
( ( new @ C @ ( neg @ P ) )
= ( new @ C @ P ) ) ).
% new.simps(7)
thf(fact_158_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: fm > fm,A3: fm,As2: list_fm,Bs: list_fm] :
( ( map_ta229225445362710709_fm_fm @ F @ ( cons_fm @ A3 @ As2 ) @ Bs )
= ( map_ta229225445362710709_fm_fm @ F @ As2 @ ( cons_fm @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_159_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: fm > tm,A3: fm,As2: list_fm,Bs: list_tm] :
( ( map_ta229225445363629095_fm_tm @ F @ ( cons_fm @ A3 @ As2 ) @ Bs )
= ( map_ta229225445363629095_fm_tm @ F @ As2 @ ( cons_tm @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_160_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: fm > int,A3: fm,As2: list_fm,Bs: list_int] :
( ( map_ta6140500833912806250fm_int @ F @ ( cons_fm @ A3 @ As2 ) @ Bs )
= ( map_ta6140500833912806250fm_int @ F @ As2 @ ( cons_int @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_161_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: tm > fm,A3: tm,As2: list_tm,Bs: list_fm] :
( ( map_ta4789309763158333891_tm_fm @ F @ ( cons_tm @ A3 @ As2 ) @ Bs )
= ( map_ta4789309763158333891_tm_fm @ F @ As2 @ ( cons_fm @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_162_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: tm > tm,A3: tm,As2: list_tm,Bs: list_tm] :
( ( map_ta4789309763159252277_tm_tm @ F @ ( cons_tm @ A3 @ As2 ) @ Bs )
= ( map_ta4789309763159252277_tm_tm @ F @ As2 @ ( cons_tm @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_163_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: tm > int,A3: tm,As2: list_tm,Bs: list_int] :
( ( map_ta7804880090983306972tm_int @ F @ ( cons_tm @ A3 @ As2 ) @ Bs )
= ( map_ta7804880090983306972tm_int @ F @ As2 @ ( cons_int @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_164_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: int > fm,A3: int,As2: list_int,Bs: list_fm] :
( ( map_ta6489050020625333520int_fm @ F @ ( cons_int @ A3 @ As2 ) @ Bs )
= ( map_ta6489050020625333520int_fm @ F @ As2 @ ( cons_fm @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_165_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: int > tm,A3: int,As2: list_int,Bs: list_tm] :
( ( map_ta6489050020626251906int_tm @ F @ ( cons_int @ A3 @ As2 ) @ Bs )
= ( map_ta6489050020626251906int_tm @ F @ As2 @ ( cons_tm @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_166_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: int > int,A3: int,As2: list_int,Bs: list_int] :
( ( map_ta8161051695879884879nt_int @ F @ ( cons_int @ A3 @ As2 ) @ Bs )
= ( map_ta8161051695879884879nt_int @ F @ As2 @ ( cons_int @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_167_new_Osimps_I2_J,axiom,
! [C: nat,P: fm,Q: fm] :
( ( new @ C @ ( imp @ P @ Q ) )
= ( ( ( new @ C @ P )
=> ( new @ C @ Q ) )
& ( new @ C @ P ) ) ) ).
% new.simps(2)
thf(fact_168_new_Osimps_I4_J,axiom,
! [C: nat,P: fm,Q: fm] :
( ( new @ C @ ( con @ P @ Q ) )
= ( ( ( new @ C @ P )
=> ( new @ C @ Q ) )
& ( new @ C @ P ) ) ) ).
% new.simps(4)
thf(fact_169_new_Osimps_I3_J,axiom,
! [C: nat,P: fm,Q: fm] :
( ( new @ C @ ( dis @ P @ Q ) )
= ( ( ( new @ C @ P )
=> ( new @ C @ Q ) )
& ( new @ C @ P ) ) ) ).
% new.simps(3)
thf(fact_170_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_fm
= ( ^ [Less2: fm > fm > $o,A1: list_fm,A22: list_fm] :
( ? [Ys3: list_fm] :
( ( A1 = nil_fm )
& ( A22 = Ys3 ) )
| ? [X3: fm,Y3: fm,Xs3: list_fm,Ys3: list_fm] :
( ( A1
= ( cons_fm @ X3 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: fm,Y3: fm,Xs3: list_fm,Ys3: list_fm] :
( ( A1
= ( cons_fm @ X3 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq_fm @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_171_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_tm
= ( ^ [Less2: tm > tm > $o,A1: list_tm,A22: list_tm] :
( ? [Ys3: list_tm] :
( ( A1 = nil_tm )
& ( A22 = Ys3 ) )
| ? [X3: tm,Y3: tm,Xs3: list_tm,Ys3: list_tm] :
( ( A1
= ( cons_tm @ X3 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: tm,Y3: tm,Xs3: list_tm,Ys3: list_tm] :
( ( A1
= ( cons_tm @ X3 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq_tm @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_172_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_int
= ( ^ [Less2: int > int > $o,A1: list_int,A22: list_int] :
( ? [Ys3: list_int] :
( ( A1 = nil_int )
& ( A22 = Ys3 ) )
| ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq_int @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_173_ord_Olexordp__eq_Ocases,axiom,
! [Less: fm > fm > $o,A12: list_fm,A23: list_fm] :
( ( lexordp_eq_fm @ Less @ A12 @ A23 )
=> ( ( A12 != nil_fm )
=> ( ! [X2: fm] :
( ? [Xs2: list_fm] :
( A12
= ( cons_fm @ X2 @ Xs2 ) )
=> ! [Y2: fm] :
( ? [Ys2: list_fm] :
( A23
= ( cons_fm @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: fm,Y2: fm,Xs2: list_fm] :
( ( A12
= ( cons_fm @ X2 @ Xs2 ) )
=> ! [Ys2: list_fm] :
( ( A23
= ( cons_fm @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_eq_fm @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_174_ord_Olexordp__eq_Ocases,axiom,
! [Less: tm > tm > $o,A12: list_tm,A23: list_tm] :
( ( lexordp_eq_tm @ Less @ A12 @ A23 )
=> ( ( A12 != nil_tm )
=> ( ! [X2: tm] :
( ? [Xs2: list_tm] :
( A12
= ( cons_tm @ X2 @ Xs2 ) )
=> ! [Y2: tm] :
( ? [Ys2: list_tm] :
( A23
= ( cons_tm @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: tm,Y2: tm,Xs2: list_tm] :
( ( A12
= ( cons_tm @ X2 @ Xs2 ) )
=> ! [Ys2: list_tm] :
( ( A23
= ( cons_tm @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_eq_tm @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_175_ord_Olexordp__eq_Ocases,axiom,
! [Less: int > int > $o,A12: list_int,A23: list_int] :
( ( lexordp_eq_int @ Less @ A12 @ A23 )
=> ( ( A12 != nil_int )
=> ( ! [X2: int] :
( ? [Xs2: list_int] :
( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Y2: int] :
( ? [Ys2: list_int] :
( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: int,Y2: int,Xs2: list_int] :
( ( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_eq_int @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_176_GammaUni,axiom,
! [T: tm,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T @ P ) ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( uni @ P ) ) @ Z ) ) ) ).
% GammaUni
thf(fact_177_GammaExi,axiom,
! [T: tm,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T @ P ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( exi @ P ) @ Z ) ) ) ).
% GammaExi
thf(fact_178_new_Osimps_I1_J,axiom,
! [C: nat,I: nat,L: list_tm] :
( ( new @ C @ ( pre @ I @ L ) )
= ( new_list @ C @ L ) ) ).
% new.simps(1)
thf(fact_179_DeltaExi,axiom,
! [I: nat,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P ) ) @ Z ) )
=> ( ( news @ I @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( exi @ P ) ) @ Z ) ) ) ) ).
% DeltaExi
thf(fact_180_DeltaUni,axiom,
! [I: nat,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P ) @ Z ) )
=> ( ( news @ I @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( uni @ P ) @ Z ) ) ) ) ).
% DeltaUni
thf(fact_181_inc__term_Osimps_I2_J,axiom,
! [I: nat,L: list_tm] :
( ( inc_term @ ( fun @ I @ L ) )
= ( fun @ I @ ( inc_list @ L ) ) ) ).
% inc_term.simps(2)
thf(fact_182_sub__list_Osimps_I1_J,axiom,
! [V: nat,S: tm] :
( ( sub_list @ V @ S @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_183_tm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( fun @ X11 @ X12 )
= ( fun @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tm.inject(1)
thf(fact_184_sub_Osimps_I1_J,axiom,
! [V: nat,S: tm,I: nat,L: list_tm] :
( ( sub @ V @ S @ ( pre @ I @ L ) )
= ( pre @ I @ ( sub_list @ V @ S @ L ) ) ) ).
% sub.simps(1)
thf(fact_185_sub_Osimps_I7_J,axiom,
! [V: nat,S: tm,P: fm] :
( ( sub @ V @ S @ ( neg @ P ) )
= ( neg @ ( sub @ V @ S @ P ) ) ) ).
% sub.simps(7)
thf(fact_186_sub_Osimps_I2_J,axiom,
! [V: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V @ S @ ( imp @ P @ Q ) )
= ( imp @ ( sub @ V @ S @ P ) @ ( sub @ V @ S @ Q ) ) ) ).
% sub.simps(2)
thf(fact_187_sub_Osimps_I3_J,axiom,
! [V: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V @ S @ ( dis @ P @ Q ) )
= ( dis @ ( sub @ V @ S @ P ) @ ( sub @ V @ S @ Q ) ) ) ).
% sub.simps(3)
thf(fact_188_sub_Osimps_I4_J,axiom,
! [V: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V @ S @ ( con @ P @ Q ) )
= ( con @ ( sub @ V @ S @ P ) @ ( sub @ V @ S @ Q ) ) ) ).
% sub.simps(4)
thf(fact_189_new__list_Osimps_I1_J,axiom,
! [C: nat] : ( new_list @ C @ nil_tm ) ).
% new_list.simps(1)
thf(fact_190_sub__list_Osimps_I2_J,axiom,
! [V: nat,S: tm,T: tm,L: list_tm] :
( ( sub_list @ V @ S @ ( cons_tm @ T @ L ) )
= ( cons_tm @ ( sub_term @ V @ S @ T ) @ ( sub_list @ V @ S @ L ) ) ) ).
% sub_list.simps(2)
thf(fact_191_new__list_Osimps_I2_J,axiom,
! [C: nat,T: tm,L: list_tm] :
( ( new_list @ C @ ( cons_tm @ T @ L ) )
= ( ( ( new_term @ C @ T )
=> ( new_list @ C @ L ) )
& ( new_term @ C @ T ) ) ) ).
% new_list.simps(2)
thf(fact_192_sub__term_Osimps_I2_J,axiom,
! [V: nat,S: tm,I: nat,L: list_tm] :
( ( sub_term @ V @ S @ ( fun @ I @ L ) )
= ( fun @ I @ ( sub_list @ V @ S @ L ) ) ) ).
% sub_term.simps(2)
thf(fact_193_new__term_Osimps_I2_J,axiom,
! [C: nat,I: nat,L: list_tm] :
( ( new_term @ C @ ( fun @ I @ L ) )
= ( ( I != C )
& ( ( I != C )
=> ( new_list @ C @ L ) ) ) ) ).
% new_term.simps(2)
thf(fact_194_bind__simps_I1_J,axiom,
! [F: tm > list_tm] :
( ( bind_tm_tm @ nil_tm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_195_bind__simps_I1_J,axiom,
! [F: tm > list_fm] :
( ( bind_tm_fm @ nil_tm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_196_bind__simps_I1_J,axiom,
! [F: tm > list_int] :
( ( bind_tm_int @ nil_tm @ F )
= nil_int ) ).
% bind_simps(1)
thf(fact_197_bind__simps_I1_J,axiom,
! [F: fm > list_tm] :
( ( bind_fm_tm @ nil_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_198_bind__simps_I1_J,axiom,
! [F: fm > list_fm] :
( ( bind_fm_fm @ nil_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_199_bind__simps_I1_J,axiom,
! [F: fm > list_int] :
( ( bind_fm_int @ nil_fm @ F )
= nil_int ) ).
% bind_simps(1)
thf(fact_200_bind__simps_I1_J,axiom,
! [F: int > list_tm] :
( ( bind_int_tm @ nil_int @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_201_bind__simps_I1_J,axiom,
! [F: int > list_fm] :
( ( bind_int_fm @ nil_int @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_202_bind__simps_I1_J,axiom,
! [F: int > list_int] :
( ( bind_int_int @ nil_int @ F )
= nil_int ) ).
% bind_simps(1)
thf(fact_203_fm_Osize__gen_I1_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size_gen(1)
thf(fact_204_tm_Oinject_I2_J,axiom,
! [X23: nat,Y23: nat] :
( ( ( var @ X23 )
= ( var @ Y23 ) )
= ( X23 = Y23 ) ) ).
% tm.inject(2)
thf(fact_205_new__term_Osimps_I1_J,axiom,
! [C: nat,N: nat] : ( new_term @ C @ ( var @ N ) ) ).
% new_term.simps(1)
thf(fact_206_tm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X23: nat] :
( ( fun @ X11 @ X12 )
!= ( var @ X23 ) ) ).
% tm.distinct(1)
thf(fact_207_tm_Oexhaust,axiom,
! [Y: tm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( fun @ X112 @ X122 ) )
=> ~ ! [X24: nat] :
( Y
!= ( var @ X24 ) ) ) ).
% tm.exhaust
thf(fact_208_tm_Osize__gen_I2_J,axiom,
! [X23: nat] :
( ( size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size_gen(2)
thf(fact_209_tm_Osize_I4_J,axiom,
! [X23: nat] :
( ( size_size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size(4)
thf(fact_210_listrelp_Osimps,axiom,
( listrelp_fm_fm
= ( ^ [R: fm > fm > $o,A1: list_fm,A22: list_fm] :
( ( ( A1 = nil_fm )
& ( A22 = nil_fm ) )
| ? [X3: fm,Y3: fm,Xs3: list_fm,Ys3: list_fm] :
( ( A1
= ( cons_fm @ X3 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_fm_fm @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_211_listrelp_Osimps,axiom,
( listrelp_fm_tm
= ( ^ [R: fm > tm > $o,A1: list_fm,A22: list_tm] :
( ( ( A1 = nil_fm )
& ( A22 = nil_tm ) )
| ? [X3: fm,Y3: tm,Xs3: list_fm,Ys3: list_tm] :
( ( A1
= ( cons_fm @ X3 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_fm_tm @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_212_listrelp_Osimps,axiom,
( listrelp_fm_int
= ( ^ [R: fm > int > $o,A1: list_fm,A22: list_int] :
( ( ( A1 = nil_fm )
& ( A22 = nil_int ) )
| ? [X3: fm,Y3: int,Xs3: list_fm,Ys3: list_int] :
( ( A1
= ( cons_fm @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_fm_int @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_213_listrelp_Osimps,axiom,
( listrelp_tm_fm
= ( ^ [R: tm > fm > $o,A1: list_tm,A22: list_fm] :
( ( ( A1 = nil_tm )
& ( A22 = nil_fm ) )
| ? [X3: tm,Y3: fm,Xs3: list_tm,Ys3: list_fm] :
( ( A1
= ( cons_tm @ X3 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_tm_fm @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_214_listrelp_Osimps,axiom,
( listrelp_tm_tm
= ( ^ [R: tm > tm > $o,A1: list_tm,A22: list_tm] :
( ( ( A1 = nil_tm )
& ( A22 = nil_tm ) )
| ? [X3: tm,Y3: tm,Xs3: list_tm,Ys3: list_tm] :
( ( A1
= ( cons_tm @ X3 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_tm_tm @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_215_listrelp_Osimps,axiom,
( listrelp_tm_int
= ( ^ [R: tm > int > $o,A1: list_tm,A22: list_int] :
( ( ( A1 = nil_tm )
& ( A22 = nil_int ) )
| ? [X3: tm,Y3: int,Xs3: list_tm,Ys3: list_int] :
( ( A1
= ( cons_tm @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_tm_int @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_216_listrelp_Osimps,axiom,
( listrelp_int_fm
= ( ^ [R: int > fm > $o,A1: list_int,A22: list_fm] :
( ( ( A1 = nil_int )
& ( A22 = nil_fm ) )
| ? [X3: int,Y3: fm,Xs3: list_int,Ys3: list_fm] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_int_fm @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_217_listrelp_Osimps,axiom,
( listrelp_int_tm
= ( ^ [R: int > tm > $o,A1: list_int,A22: list_tm] :
( ( ( A1 = nil_int )
& ( A22 = nil_tm ) )
| ? [X3: int,Y3: tm,Xs3: list_int,Ys3: list_tm] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_int_tm @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_218_listrelp_Osimps,axiom,
( listrelp_int_int
= ( ^ [R: int > int > $o,A1: list_int,A22: list_int] :
( ( ( A1 = nil_int )
& ( A22 = nil_int ) )
| ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ( R @ X3 @ Y3 )
& ( listrelp_int_int @ R @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_219_listrelp_Ocases,axiom,
! [R2: fm > fm > $o,A12: list_fm,A23: list_fm] :
( ( listrelp_fm_fm @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_fm )
=> ( A23 != nil_fm ) )
=> ~ ! [X2: fm,Y2: fm,Xs2: list_fm] :
( ( A12
= ( cons_fm @ X2 @ Xs2 ) )
=> ! [Ys2: list_fm] :
( ( A23
= ( cons_fm @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_fm_fm @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_220_listrelp_Ocases,axiom,
! [R2: fm > tm > $o,A12: list_fm,A23: list_tm] :
( ( listrelp_fm_tm @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_fm )
=> ( A23 != nil_tm ) )
=> ~ ! [X2: fm,Y2: tm,Xs2: list_fm] :
( ( A12
= ( cons_fm @ X2 @ Xs2 ) )
=> ! [Ys2: list_tm] :
( ( A23
= ( cons_tm @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_fm_tm @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_221_listrelp_Ocases,axiom,
! [R2: fm > int > $o,A12: list_fm,A23: list_int] :
( ( listrelp_fm_int @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_fm )
=> ( A23 != nil_int ) )
=> ~ ! [X2: fm,Y2: int,Xs2: list_fm] :
( ( A12
= ( cons_fm @ X2 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_fm_int @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_222_listrelp_Ocases,axiom,
! [R2: tm > fm > $o,A12: list_tm,A23: list_fm] :
( ( listrelp_tm_fm @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_tm )
=> ( A23 != nil_fm ) )
=> ~ ! [X2: tm,Y2: fm,Xs2: list_tm] :
( ( A12
= ( cons_tm @ X2 @ Xs2 ) )
=> ! [Ys2: list_fm] :
( ( A23
= ( cons_fm @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_tm_fm @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_223_listrelp_Ocases,axiom,
! [R2: tm > tm > $o,A12: list_tm,A23: list_tm] :
( ( listrelp_tm_tm @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_tm )
=> ( A23 != nil_tm ) )
=> ~ ! [X2: tm,Y2: tm,Xs2: list_tm] :
( ( A12
= ( cons_tm @ X2 @ Xs2 ) )
=> ! [Ys2: list_tm] :
( ( A23
= ( cons_tm @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_tm_tm @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_224_listrelp_Ocases,axiom,
! [R2: tm > int > $o,A12: list_tm,A23: list_int] :
( ( listrelp_tm_int @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_tm )
=> ( A23 != nil_int ) )
=> ~ ! [X2: tm,Y2: int,Xs2: list_tm] :
( ( A12
= ( cons_tm @ X2 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_tm_int @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_225_listrelp_Ocases,axiom,
! [R2: int > fm > $o,A12: list_int,A23: list_fm] :
( ( listrelp_int_fm @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_int )
=> ( A23 != nil_fm ) )
=> ~ ! [X2: int,Y2: fm,Xs2: list_int] :
( ( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Ys2: list_fm] :
( ( A23
= ( cons_fm @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_int_fm @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_226_listrelp_Ocases,axiom,
! [R2: int > tm > $o,A12: list_int,A23: list_tm] :
( ( listrelp_int_tm @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_int )
=> ( A23 != nil_tm ) )
=> ~ ! [X2: int,Y2: tm,Xs2: list_int] :
( ( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Ys2: list_tm] :
( ( A23
= ( cons_tm @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_int_tm @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_227_listrelp_Ocases,axiom,
! [R2: int > int > $o,A12: list_int,A23: list_int] :
( ( listrelp_int_int @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_int )
=> ( A23 != nil_int ) )
=> ~ ! [X2: int,Y2: int,Xs2: list_int] :
( ( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ( R2 @ X2 @ Y2 )
=> ~ ( listrelp_int_int @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_228_lexordp__eq__simps_I3_J,axiom,
! [X: int,Xs: list_int] :
~ ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ nil_int ) ).
% lexordp_eq_simps(3)
thf(fact_229_fm_Osize_I8_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size(8)
thf(fact_230_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_231_lexordp__eq__simps_I2_J,axiom,
! [Xs: list_int] :
( ( ord_lexordp_eq_int @ Xs @ nil_int )
= ( Xs = nil_int ) ) ).
% lexordp_eq_simps(2)
thf(fact_232_lexordp__eq__simps_I1_J,axiom,
! [Ys: list_int] : ( ord_lexordp_eq_int @ nil_int @ Ys ) ).
% lexordp_eq_simps(1)
thf(fact_233_lexordp__eq_ONil,axiom,
! [Ys: list_int] : ( ord_lexordp_eq_int @ nil_int @ Ys ) ).
% lexordp_eq.Nil
thf(fact_234_listrelp_OCons,axiom,
! [R2: fm > fm > $o,X: fm,Y: fm,Xs: list_fm,Ys: list_fm] :
( ( R2 @ X @ Y )
=> ( ( listrelp_fm_fm @ R2 @ Xs @ Ys )
=> ( listrelp_fm_fm @ R2 @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_235_listrelp_OCons,axiom,
! [R2: fm > tm > $o,X: fm,Y: tm,Xs: list_fm,Ys: list_tm] :
( ( R2 @ X @ Y )
=> ( ( listrelp_fm_tm @ R2 @ Xs @ Ys )
=> ( listrelp_fm_tm @ R2 @ ( cons_fm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_236_listrelp_OCons,axiom,
! [R2: fm > int > $o,X: fm,Y: int,Xs: list_fm,Ys: list_int] :
( ( R2 @ X @ Y )
=> ( ( listrelp_fm_int @ R2 @ Xs @ Ys )
=> ( listrelp_fm_int @ R2 @ ( cons_fm @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_237_listrelp_OCons,axiom,
! [R2: tm > fm > $o,X: tm,Y: fm,Xs: list_tm,Ys: list_fm] :
( ( R2 @ X @ Y )
=> ( ( listrelp_tm_fm @ R2 @ Xs @ Ys )
=> ( listrelp_tm_fm @ R2 @ ( cons_tm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_238_listrelp_OCons,axiom,
! [R2: tm > tm > $o,X: tm,Y: tm,Xs: list_tm,Ys: list_tm] :
( ( R2 @ X @ Y )
=> ( ( listrelp_tm_tm @ R2 @ Xs @ Ys )
=> ( listrelp_tm_tm @ R2 @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_239_listrelp_OCons,axiom,
! [R2: tm > int > $o,X: tm,Y: int,Xs: list_tm,Ys: list_int] :
( ( R2 @ X @ Y )
=> ( ( listrelp_tm_int @ R2 @ Xs @ Ys )
=> ( listrelp_tm_int @ R2 @ ( cons_tm @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_240_listrelp_OCons,axiom,
! [R2: int > fm > $o,X: int,Y: fm,Xs: list_int,Ys: list_fm] :
( ( R2 @ X @ Y )
=> ( ( listrelp_int_fm @ R2 @ Xs @ Ys )
=> ( listrelp_int_fm @ R2 @ ( cons_int @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_241_listrelp_OCons,axiom,
! [R2: int > tm > $o,X: int,Y: tm,Xs: list_int,Ys: list_tm] :
( ( R2 @ X @ Y )
=> ( ( listrelp_int_tm @ R2 @ Xs @ Ys )
=> ( listrelp_int_tm @ R2 @ ( cons_int @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_242_listrelp_OCons,axiom,
! [R2: int > int > $o,X: int,Y: int,Xs: list_int,Ys: list_int] :
( ( R2 @ X @ Y )
=> ( ( listrelp_int_int @ R2 @ Xs @ Ys )
=> ( listrelp_int_int @ R2 @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_243_listrelp_ONil,axiom,
! [R2: tm > tm > $o] : ( listrelp_tm_tm @ R2 @ nil_tm @ nil_tm ) ).
% listrelp.Nil
thf(fact_244_listrelp_ONil,axiom,
! [R2: tm > fm > $o] : ( listrelp_tm_fm @ R2 @ nil_tm @ nil_fm ) ).
% listrelp.Nil
thf(fact_245_listrelp_ONil,axiom,
! [R2: tm > int > $o] : ( listrelp_tm_int @ R2 @ nil_tm @ nil_int ) ).
% listrelp.Nil
thf(fact_246_listrelp_ONil,axiom,
! [R2: fm > tm > $o] : ( listrelp_fm_tm @ R2 @ nil_fm @ nil_tm ) ).
% listrelp.Nil
thf(fact_247_listrelp_ONil,axiom,
! [R2: fm > fm > $o] : ( listrelp_fm_fm @ R2 @ nil_fm @ nil_fm ) ).
% listrelp.Nil
thf(fact_248_listrelp_ONil,axiom,
! [R2: fm > int > $o] : ( listrelp_fm_int @ R2 @ nil_fm @ nil_int ) ).
% listrelp.Nil
thf(fact_249_listrelp_ONil,axiom,
! [R2: int > tm > $o] : ( listrelp_int_tm @ R2 @ nil_int @ nil_tm ) ).
% listrelp.Nil
thf(fact_250_listrelp_ONil,axiom,
! [R2: int > fm > $o] : ( listrelp_int_fm @ R2 @ nil_int @ nil_fm ) ).
% listrelp.Nil
thf(fact_251_listrelp_ONil,axiom,
! [R2: int > int > $o] : ( listrelp_int_int @ R2 @ nil_int @ nil_int ) ).
% listrelp.Nil
thf(fact_252_lexordp__eq_Ocases,axiom,
! [A12: list_nat,A23: list_nat] :
( ( ord_lexordp_eq_nat @ A12 @ A23 )
=> ( ( A12 != nil_nat )
=> ( ! [X2: nat] :
( ? [Xs2: list_nat] :
( A12
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Y2: nat] :
( ? [Ys2: list_nat] :
( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) )
=> ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ~ ( ord_less_nat @ Y2 @ X2 )
=> ~ ( ord_lexordp_eq_nat @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_253_lexordp__eq_Ocases,axiom,
! [A12: list_int,A23: list_int] :
( ( ord_lexordp_eq_int @ A12 @ A23 )
=> ( ( A12 != nil_int )
=> ( ! [X2: int] :
( ? [Xs2: list_int] :
( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Y2: int] :
( ? [Ys2: list_int] :
( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) )
=> ~ ! [X2: int,Y2: int,Xs2: list_int] :
( ( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ~ ( ord_less_int @ Y2 @ X2 )
=> ~ ( ord_lexordp_eq_int @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_254_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_nat
= ( ^ [A1: list_nat,A22: list_nat] :
( ? [Ys3: list_nat] :
( ( A1 = nil_nat )
& ( A22 = Ys3 ) )
| ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ord_less_nat @ X3 @ Y3 ) )
| ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y3 @ Ys3 ) )
& ~ ( ord_less_nat @ X3 @ Y3 )
& ~ ( ord_less_nat @ Y3 @ X3 )
& ( ord_lexordp_eq_nat @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_255_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_int
= ( ^ [A1: list_int,A22: list_int] :
( ? [Ys3: list_int] :
( ( A1 = nil_int )
& ( A22 = Ys3 ) )
| ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ( ord_less_int @ X3 @ Y3 ) )
| ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ~ ( ord_less_int @ X3 @ Y3 )
& ~ ( ord_less_int @ Y3 @ X3 )
& ( ord_lexordp_eq_int @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_256_member__rec_I2_J,axiom,
! [Y: tm] :
~ ( member_tm @ nil_tm @ Y ) ).
% member_rec(2)
thf(fact_257_member__rec_I2_J,axiom,
! [Y: fm] :
~ ( member_fm @ nil_fm @ Y ) ).
% member_rec(2)
thf(fact_258_member__rec_I2_J,axiom,
! [Y: int] :
~ ( member_int @ nil_int @ Y ) ).
% member_rec(2)
thf(fact_259_member__rec_I1_J,axiom,
! [X: fm,Xs: list_fm,Y: fm] :
( ( member_fm @ ( cons_fm @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_fm @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_260_member__rec_I1_J,axiom,
! [X: tm,Xs: list_tm,Y: tm] :
( ( member_tm @ ( cons_tm @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_tm @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_261_member__rec_I1_J,axiom,
! [X: int,Xs: list_int,Y: int] :
( ( member_int @ ( cons_int @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_int @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_262_splice_Oelims,axiom,
! [X: list_fm,Xa: list_fm,Y: list_fm] :
( ( ( splice_fm @ X @ Xa )
= Y )
=> ( ( ( X = nil_fm )
=> ( Y != Xa ) )
=> ~ ! [X2: fm,Xs2: list_fm] :
( ( X
= ( cons_fm @ X2 @ Xs2 ) )
=> ( Y
!= ( cons_fm @ X2 @ ( splice_fm @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_263_splice_Oelims,axiom,
! [X: list_tm,Xa: list_tm,Y: list_tm] :
( ( ( splice_tm @ X @ Xa )
= Y )
=> ( ( ( X = nil_tm )
=> ( Y != Xa ) )
=> ~ ! [X2: tm,Xs2: list_tm] :
( ( X
= ( cons_tm @ X2 @ Xs2 ) )
=> ( Y
!= ( cons_tm @ X2 @ ( splice_tm @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_264_splice_Oelims,axiom,
! [X: list_int,Xa: list_int,Y: list_int] :
( ( ( splice_int @ X @ Xa )
= Y )
=> ( ( ( X = nil_int )
=> ( Y != Xa ) )
=> ~ ! [X2: int,Xs2: list_int] :
( ( X
= ( cons_int @ X2 @ Xs2 ) )
=> ( Y
!= ( cons_int @ X2 @ ( splice_int @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_265_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_266_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_267_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_268_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_269_split__Nil__iff,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( splice_tm @ Xs @ Ys )
= nil_tm )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% split_Nil_iff
thf(fact_270_split__Nil__iff,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( splice_fm @ Xs @ Ys )
= nil_fm )
= ( ( Xs = nil_fm )
& ( Ys = nil_fm ) ) ) ).
% split_Nil_iff
thf(fact_271_split__Nil__iff,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( splice_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% split_Nil_iff
thf(fact_272_splice__Nil2,axiom,
! [Xs: list_tm] :
( ( splice_tm @ Xs @ nil_tm )
= Xs ) ).
% splice_Nil2
thf(fact_273_splice__Nil2,axiom,
! [Xs: list_fm] :
( ( splice_fm @ Xs @ nil_fm )
= Xs ) ).
% splice_Nil2
thf(fact_274_splice__Nil2,axiom,
! [Xs: list_int] :
( ( splice_int @ Xs @ nil_int )
= Xs ) ).
% splice_Nil2
thf(fact_275_lexordp__eq__simps_I4_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ~ ( ord_less_nat @ Y @ X )
& ( ord_lexordp_eq_nat @ Xs @ Ys ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_276_lexordp__eq__simps_I4_J,axiom,
! [X: int,Xs: list_int,Y: int,Ys: list_int] :
( ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
= ( ( ord_less_int @ X @ Y )
| ( ~ ( ord_less_int @ Y @ X )
& ( ord_lexordp_eq_int @ Xs @ Ys ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_277_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_278_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_279_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_280_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_281_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_282_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_283_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_284_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_285_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_286_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_287_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_288_splice_Osimps_I2_J,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm] :
( ( splice_fm @ ( cons_fm @ X @ Xs ) @ Ys )
= ( cons_fm @ X @ ( splice_fm @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_289_splice_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm] :
( ( splice_tm @ ( cons_tm @ X @ Xs ) @ Ys )
= ( cons_tm @ X @ ( splice_tm @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_290_splice_Osimps_I2_J,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( splice_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( splice_int @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_291_splice_Osimps_I1_J,axiom,
! [Ys: list_tm] :
( ( splice_tm @ nil_tm @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_292_splice_Osimps_I1_J,axiom,
! [Ys: list_fm] :
( ( splice_fm @ nil_fm @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_293_splice_Osimps_I1_J,axiom,
! [Ys: list_int] :
( ( splice_int @ nil_int @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_294_lexordp__eq_OCons__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ~ ( ord_less_nat @ Y @ X )
=> ( ( ord_lexordp_eq_nat @ Xs @ Ys )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_295_lexordp__eq_OCons__eq,axiom,
! [X: int,Y: int,Xs: list_int,Ys: list_int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ~ ( ord_less_int @ Y @ X )
=> ( ( ord_lexordp_eq_int @ Xs @ Ys )
=> ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_296_lexordp__eq_OCons,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).
% lexordp_eq.Cons
thf(fact_297_lexordp__eq_OCons,axiom,
! [X: int,Y: int,Xs: list_int,Ys: list_int] :
( ( ord_less_int @ X @ Y )
=> ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ).
% lexordp_eq.Cons
thf(fact_298_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_299_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_300_lexordp__induct,axiom,
! [Xs: list_nat,Ys: list_nat,P2: list_nat > list_nat > $o] :
( ( ord_lexordp_nat @ Xs @ Ys )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( P2 @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( ! [X2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_lexordp_nat @ Xs2 @ Ys2 )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ X2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% lexordp_induct
thf(fact_301_lexordp__induct,axiom,
! [Xs: list_int,Ys: list_int,P2: list_int > list_int > $o] :
( ( ord_lexordp_int @ Xs @ Ys )
=> ( ! [Y2: int,Ys2: list_int] : ( P2 @ nil_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X2: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( ! [X2: int,Xs2: list_int,Ys2: list_int] :
( ( ord_lexordp_int @ Xs2 @ Ys2 )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_int @ X2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% lexordp_induct
thf(fact_302_lexordp__cases,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ord_lexordp_nat @ Xs @ Ys )
=> ( ( ( Xs = nil_nat )
=> ! [Y2: nat,Ys4: list_nat] :
( Ys
!= ( cons_nat @ Y2 @ Ys4 ) ) )
=> ( ! [X2: nat] :
( ? [Xs4: list_nat] :
( Xs
= ( cons_nat @ X2 @ Xs4 ) )
=> ! [Y2: nat] :
( ? [Ys4: list_nat] :
( Ys
= ( cons_nat @ Y2 @ Ys4 ) )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) )
=> ~ ! [X2: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X2 @ Xs4 ) )
=> ! [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys4 ) )
=> ~ ( ord_lexordp_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_303_lexordp__cases,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ord_lexordp_int @ Xs @ Ys )
=> ( ( ( Xs = nil_int )
=> ! [Y2: int,Ys4: list_int] :
( Ys
!= ( cons_int @ Y2 @ Ys4 ) ) )
=> ( ! [X2: int] :
( ? [Xs4: list_int] :
( Xs
= ( cons_int @ X2 @ Xs4 ) )
=> ! [Y2: int] :
( ? [Ys4: list_int] :
( Ys
= ( cons_int @ Y2 @ Ys4 ) )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) )
=> ~ ! [X2: int,Xs4: list_int] :
( ( Xs
= ( cons_int @ X2 @ Xs4 ) )
=> ! [Ys4: list_int] :
( ( Ys
= ( cons_int @ X2 @ Ys4 ) )
=> ~ ( ord_lexordp_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_304_lexordp_Osimps,axiom,
( ord_lexordp_nat
= ( ^ [A1: list_nat,A22: list_nat] :
( ? [Y3: nat,Ys3: list_nat] :
( ( A1 = nil_nat )
& ( A22
= ( cons_nat @ Y3 @ Ys3 ) ) )
| ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ord_less_nat @ X3 @ Y3 ) )
| ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y3 @ Ys3 ) )
& ~ ( ord_less_nat @ X3 @ Y3 )
& ~ ( ord_less_nat @ Y3 @ X3 )
& ( ord_lexordp_nat @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.simps
thf(fact_305_lexordp_Osimps,axiom,
( ord_lexordp_int
= ( ^ [A1: list_int,A22: list_int] :
( ? [Y3: int,Ys3: list_int] :
( ( A1 = nil_int )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) ) )
| ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ( ord_less_int @ X3 @ Y3 ) )
| ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys3 ) )
& ~ ( ord_less_int @ X3 @ Y3 )
& ~ ( ord_less_int @ Y3 @ X3 )
& ( ord_lexordp_int @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.simps
thf(fact_306_lexordp_Ocases,axiom,
! [A12: list_nat,A23: list_nat] :
( ( ord_lexordp_nat @ A12 @ A23 )
=> ( ( ( A12 = nil_nat )
=> ! [Y2: nat,Ys2: list_nat] :
( A23
!= ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( ! [X2: nat] :
( ? [Xs2: list_nat] :
( A12
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Y2: nat] :
( ? [Ys2: list_nat] :
( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) )
=> ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ~ ( ord_less_nat @ Y2 @ X2 )
=> ~ ( ord_lexordp_nat @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_307_lexordp_Ocases,axiom,
! [A12: list_int,A23: list_int] :
( ( ord_lexordp_int @ A12 @ A23 )
=> ( ( ( A12 = nil_int )
=> ! [Y2: int,Ys2: list_int] :
( A23
!= ( cons_int @ Y2 @ Ys2 ) ) )
=> ( ! [X2: int] :
( ? [Xs2: list_int] :
( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Y2: int] :
( ? [Ys2: list_int] :
( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) )
=> ~ ! [X2: int,Y2: int,Xs2: list_int] :
( ( A12
= ( cons_int @ X2 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A23
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ~ ( ord_less_int @ Y2 @ X2 )
=> ~ ( ord_lexordp_int @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_308_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_309_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_310_lexordp__simps_I2_J,axiom,
! [Xs: list_int] :
~ ( ord_lexordp_int @ Xs @ nil_int ) ).
% lexordp_simps(2)
thf(fact_311_lexordp__simps_I1_J,axiom,
! [Ys: list_int] :
( ( ord_lexordp_int @ nil_int @ Ys )
= ( Ys != nil_int ) ) ).
% lexordp_simps(1)
thf(fact_312_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_313_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_314_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_315_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_316_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_317_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_318_lexordp__simps_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_lexordp_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ~ ( ord_less_nat @ Y @ X )
& ( ord_lexordp_nat @ Xs @ Ys ) ) ) ) ).
% lexordp_simps(3)
thf(fact_319_lexordp__simps_I3_J,axiom,
! [X: int,Xs: list_int,Y: int,Ys: list_int] :
( ( ord_lexordp_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
= ( ( ord_less_int @ X @ Y )
| ( ~ ( ord_less_int @ Y @ X )
& ( ord_lexordp_int @ Xs @ Ys ) ) ) ) ).
% lexordp_simps(3)
thf(fact_320_lexordp__irreflexive,axiom,
! [Xs: list_nat] :
( ! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 )
=> ~ ( ord_lexordp_nat @ Xs @ Xs ) ) ).
% lexordp_irreflexive
thf(fact_321_lexordp__irreflexive,axiom,
! [Xs: list_int] :
( ! [X2: int] :
~ ( ord_less_int @ X2 @ X2 )
=> ~ ( ord_lexordp_int @ Xs @ Xs ) ) ).
% lexordp_irreflexive
thf(fact_322_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_323_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_324_lexordp_OCons,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_lexordp_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).
% lexordp.Cons
thf(fact_325_lexordp_OCons,axiom,
! [X: int,Y: int,Xs: list_int,Ys: list_int] :
( ( ord_less_int @ X @ Y )
=> ( ord_lexordp_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ).
% lexordp.Cons
thf(fact_326_lexordp_OCons__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ~ ( ord_less_nat @ Y @ X )
=> ( ( ord_lexordp_nat @ Xs @ Ys )
=> ( ord_lexordp_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_327_lexordp_OCons__eq,axiom,
! [X: int,Y: int,Xs: list_int,Ys: list_int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ~ ( ord_less_int @ Y @ X )
=> ( ( ord_lexordp_int @ Xs @ Ys )
=> ( ord_lexordp_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_328_lexordp_ONil,axiom,
! [Y: int,Ys: list_int] : ( ord_lexordp_int @ nil_int @ ( cons_int @ Y @ Ys ) ) ).
% lexordp.Nil
thf(fact_329_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_330_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_331_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_332_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_333_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_334_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_335_length__greater__0__conv,axiom,
! [Xs: list_tm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) )
= ( Xs != nil_tm ) ) ).
% length_greater_0_conv
thf(fact_336_length__greater__0__conv,axiom,
! [Xs: list_fm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) )
= ( Xs != nil_fm ) ) ).
% length_greater_0_conv
thf(fact_337_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_338_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_339_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_340_lexordp__iff,axiom,
( ord_lexordp_nat
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ? [X3: nat,Vs: list_nat] :
( Ys3
= ( append_nat @ Xs3 @ ( cons_nat @ X3 @ Vs ) ) )
| ? [Us: list_nat,A4: nat,B: nat,Vs: list_nat,Ws: list_nat] :
( ( ord_less_nat @ A4 @ B )
& ( Xs3
= ( append_nat @ Us @ ( cons_nat @ A4 @ Vs ) ) )
& ( Ys3
= ( append_nat @ Us @ ( cons_nat @ B @ Ws ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_341_lexordp__iff,axiom,
( ord_lexordp_int
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ? [X3: int,Vs: list_int] :
( Ys3
= ( append_int @ Xs3 @ ( cons_int @ X3 @ Vs ) ) )
| ? [Us: list_int,A4: int,B: int,Vs: list_int,Ws: list_int] :
( ( ord_less_int @ A4 @ B )
& ( Xs3
= ( append_int @ Us @ ( cons_int @ A4 @ Vs ) ) )
& ( Ys3
= ( append_int @ Us @ ( cons_int @ B @ Ws ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_342_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_343_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_344_same__append__eq,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_345_append__same__eq,axiom,
! [Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( append_int @ Ys @ Xs )
= ( append_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_346_append__assoc,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs )
= ( append_int @ Xs @ ( append_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_347_append_Oassoc,axiom,
! [A3: list_int,B2: list_int,C: list_int] :
( ( append_int @ ( append_int @ A3 @ B2 ) @ C )
= ( append_int @ A3 @ ( append_int @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_348_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_349_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_350_append__is__Nil__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= nil_tm )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_351_append__is__Nil__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= nil_fm )
= ( ( Xs = nil_fm )
& ( Ys = nil_fm ) ) ) ).
% append_is_Nil_conv
thf(fact_352_append__is__Nil__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_353_Nil__is__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( nil_tm
= ( append_tm @ Xs @ Ys ) )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_354_Nil__is__append__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( nil_fm
= ( append_fm @ Xs @ Ys ) )
= ( ( Xs = nil_fm )
& ( Ys = nil_fm ) ) ) ).
% Nil_is_append_conv
thf(fact_355_Nil__is__append__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( nil_int
= ( append_int @ Xs @ Ys ) )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_356_self__append__conv2,axiom,
! [Y: list_tm,Xs: list_tm] :
( ( Y
= ( append_tm @ Xs @ Y ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_357_self__append__conv2,axiom,
! [Y: list_fm,Xs: list_fm] :
( ( Y
= ( append_fm @ Xs @ Y ) )
= ( Xs = nil_fm ) ) ).
% self_append_conv2
thf(fact_358_self__append__conv2,axiom,
! [Y: list_int,Xs: list_int] :
( ( Y
= ( append_int @ Xs @ Y ) )
= ( Xs = nil_int ) ) ).
% self_append_conv2
thf(fact_359_append__self__conv2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_360_append__self__conv2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Ys )
= ( Xs = nil_fm ) ) ).
% append_self_conv2
thf(fact_361_append__self__conv2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_int ) ) ).
% append_self_conv2
thf(fact_362_self__append__conv,axiom,
! [Y: list_tm,Ys: list_tm] :
( ( Y
= ( append_tm @ Y @ Ys ) )
= ( Ys = nil_tm ) ) ).
% self_append_conv
thf(fact_363_self__append__conv,axiom,
! [Y: list_fm,Ys: list_fm] :
( ( Y
= ( append_fm @ Y @ Ys ) )
= ( Ys = nil_fm ) ) ).
% self_append_conv
thf(fact_364_self__append__conv,axiom,
! [Y: list_int,Ys: list_int] :
( ( Y
= ( append_int @ Y @ Ys ) )
= ( Ys = nil_int ) ) ).
% self_append_conv
thf(fact_365_append__self__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_tm ) ) ).
% append_self_conv
thf(fact_366_append__self__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Xs )
= ( Ys = nil_fm ) ) ).
% append_self_conv
thf(fact_367_append__self__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_int ) ) ).
% append_self_conv
thf(fact_368_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_369_append__Nil2,axiom,
! [Xs: list_fm] :
( ( append_fm @ Xs @ nil_fm )
= Xs ) ).
% append_Nil2
thf(fact_370_append__Nil2,axiom,
! [Xs: list_int] :
( ( append_int @ Xs @ nil_int )
= Xs ) ).
% append_Nil2
thf(fact_371_append_Oright__neutral,axiom,
! [A3: list_tm] :
( ( append_tm @ A3 @ nil_tm )
= A3 ) ).
% append.right_neutral
thf(fact_372_append_Oright__neutral,axiom,
! [A3: list_fm] :
( ( append_fm @ A3 @ nil_fm )
= A3 ) ).
% append.right_neutral
thf(fact_373_append_Oright__neutral,axiom,
! [A3: list_int] :
( ( append_int @ A3 @ nil_int )
= A3 ) ).
% append.right_neutral
thf(fact_374_append__eq__append__conv,axiom,
! [Xs: list_int,Ys: list_int,Us2: list_int,Vs2: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
| ( ( size_size_list_int @ Us2 )
= ( size_size_list_int @ Vs2 ) ) )
=> ( ( ( append_int @ Xs @ Us2 )
= ( append_int @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_375_length__0__conv,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ Xs )
= zero_zero_nat )
= ( Xs = nil_tm ) ) ).
% length_0_conv
thf(fact_376_length__0__conv,axiom,
! [Xs: list_fm] :
( ( ( size_size_list_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_fm ) ) ).
% length_0_conv
thf(fact_377_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_378_append1__eq__conv,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm,Y: fm] :
( ( ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) )
= ( append_fm @ Ys @ ( cons_fm @ Y @ nil_fm ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_379_append1__eq__conv,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm,Y: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) )
= ( append_tm @ Ys @ ( cons_tm @ Y @ nil_tm ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_380_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_381_bind__simps_I2_J,axiom,
! [X: fm,Xs: list_fm,F: fm > list_int] :
( ( bind_fm_int @ ( cons_fm @ X @ Xs ) @ F )
= ( append_int @ ( F @ X ) @ ( bind_fm_int @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_382_bind__simps_I2_J,axiom,
! [X: tm,Xs: list_tm,F: tm > list_int] :
( ( bind_tm_int @ ( cons_tm @ X @ Xs ) @ F )
= ( append_int @ ( F @ X ) @ ( bind_tm_int @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_383_bind__simps_I2_J,axiom,
! [X: int,Xs: list_int,F: int > list_int] :
( ( bind_int_int @ ( cons_int @ X @ Xs ) @ F )
= ( append_int @ ( F @ X ) @ ( bind_int_int @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_384_power__eq__0__iff,axiom,
! [A3: int,N: nat] :
( ( ( power_power_int @ A3 @ N )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_385_power__eq__0__iff,axiom,
! [A3: nat,N: nat] :
( ( ( power_power_nat @ A3 @ N )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_386_power__mono__iff,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_387_power__mono__iff,axiom,
! [A3: int,B2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_388_power__less__imp__less__base,axiom,
! [A3: nat,N: nat,B2: nat] :
( ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_389_power__less__imp__less__base,axiom,
! [A3: int,N: nat,B2: int] :
( ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ A3 @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_390_impossible__Cons,axiom,
! [Xs: list_fm,Ys: list_fm,X: fm] :
( ( ord_less_eq_nat @ ( size_size_list_fm @ Xs ) @ ( size_size_list_fm @ Ys ) )
=> ( Xs
!= ( cons_fm @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_391_impossible__Cons,axiom,
! [Xs: list_tm,Ys: list_tm,X: tm] :
( ( ord_less_eq_nat @ ( size_size_list_tm @ Xs ) @ ( size_size_list_tm @ Ys ) )
=> ( Xs
!= ( cons_tm @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_392_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_393_power__not__zero,axiom,
! [A3: int,N: nat] :
( ( A3 != zero_zero_int )
=> ( ( power_power_int @ A3 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_394_power__not__zero,axiom,
! [A3: nat,N: nat] :
( ( A3 != zero_zero_nat )
=> ( ( power_power_nat @ A3 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_395_append__eq__append__conv2,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ts: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Ts ) )
= ( ? [Us: list_int] :
( ( ( Xs
= ( append_int @ Zs @ Us ) )
& ( ( append_int @ Us @ Ys )
= Ts ) )
| ( ( ( append_int @ Xs @ Us )
= Zs )
& ( Ys
= ( append_int @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_396_append__eq__appendI,axiom,
! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys: list_int,Us2: list_int] :
( ( ( append_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_int @ Xs1 @ Us2 ) )
=> ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_397_zero__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_398_zero__le__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_399_power__mono,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_400_power__mono,axiom,
! [A3: int,B2: int,N: nat] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_401_same__length__different,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs != Ys )
=> ( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ? [Pre: list_fm,X2: fm,Xs4: list_fm,Y2: fm,Ys4: list_fm] :
( ( X2 != Y2 )
& ( Xs
= ( append_fm @ Pre @ ( append_fm @ ( cons_fm @ X2 @ nil_fm ) @ Xs4 ) ) )
& ( Ys
= ( append_fm @ Pre @ ( append_fm @ ( cons_fm @ Y2 @ nil_fm ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_402_same__length__different,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != Ys )
=> ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ? [Pre: list_tm,X2: tm,Xs4: list_tm,Y2: tm,Ys4: list_tm] :
( ( X2 != Y2 )
& ( Xs
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ X2 @ nil_tm ) @ Xs4 ) ) )
& ( Ys
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ Y2 @ nil_tm ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_403_same__length__different,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != Ys )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ? [Pre: list_int,X2: int,Xs4: list_int,Y2: int,Ys4: list_int] :
( ( X2 != Y2 )
& ( Xs
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X2 @ nil_int ) @ Xs4 ) ) )
& ( Ys
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y2 @ nil_int ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_404_power__eq__imp__eq__base,axiom,
! [A3: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A3 @ N )
= ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_405_power__eq__imp__eq__base,axiom,
! [A3: int,N: nat,B2: int] :
( ( ( power_power_int @ A3 @ N )
= ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_406_power__eq__iff__eq__base,axiom,
! [N: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ( power_power_nat @ A3 @ N )
= ( power_power_nat @ B2 @ N ) )
= ( A3 = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_407_power__eq__iff__eq__base,axiom,
! [N: nat,A3: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ( power_power_int @ A3 @ N )
= ( power_power_int @ B2 @ N ) )
= ( A3 = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_408_Cons__eq__appendI,axiom,
! [X: fm,Xs1: list_fm,Ys: list_fm,Xs: list_fm,Zs: list_fm] :
( ( ( cons_fm @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_fm @ Xs1 @ Zs ) )
=> ( ( cons_fm @ X @ Xs )
= ( append_fm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_409_Cons__eq__appendI,axiom,
! [X: tm,Xs1: list_tm,Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_tm @ Xs1 @ Zs ) )
=> ( ( cons_tm @ X @ Xs )
= ( append_tm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_410_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_411_append__Cons,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm] :
( ( append_fm @ ( cons_fm @ X @ Xs ) @ Ys )
= ( cons_fm @ X @ ( append_fm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_412_append__Cons,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm] :
( ( append_tm @ ( cons_tm @ X @ Xs ) @ Ys )
= ( cons_tm @ X @ ( append_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_413_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_414_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_tm @ nil_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_415_eq__Nil__appendI,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs = Ys )
=> ( Xs
= ( append_fm @ nil_fm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_416_eq__Nil__appendI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_int @ nil_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_417_append_Oleft__neutral,axiom,
! [A3: list_tm] :
( ( append_tm @ nil_tm @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_418_append_Oleft__neutral,axiom,
! [A3: list_fm] :
( ( append_fm @ nil_fm @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_419_append_Oleft__neutral,axiom,
! [A3: list_int] :
( ( append_int @ nil_int @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_420_append__Nil,axiom,
! [Ys: list_tm] :
( ( append_tm @ nil_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_421_append__Nil,axiom,
! [Ys: list_fm] :
( ( append_fm @ nil_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_422_append__Nil,axiom,
! [Ys: list_int] :
( ( append_int @ nil_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_423_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_424_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_425_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_426_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_427_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_428_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_429_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_430_zero__less__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_431_zero__less__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_432_lexordp__append__leftI,axiom,
! [Us2: list_int,Vs2: list_int,Xs: list_int] :
( ( ord_lexordp_int @ Us2 @ Vs2 )
=> ( ord_lexordp_int @ ( append_int @ Xs @ Us2 ) @ ( append_int @ Xs @ Vs2 ) ) ) ).
% lexordp_append_leftI
thf(fact_433_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_434_lexordp__eq__pref,axiom,
! [U: list_int,V: list_int] : ( ord_lexordp_eq_int @ U @ ( append_int @ U @ V ) ) ).
% lexordp_eq_pref
thf(fact_435_ord_Olexordp__eq__pref,axiom,
! [Less: int > int > $o,U: list_int,V: list_int] : ( lexordp_eq_int @ Less @ U @ ( append_int @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_436_power__strict__mono,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_437_power__strict__mono,axiom,
! [A3: int,B2: int,N: nat] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_438_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_439_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_440_rev__nonempty__induct,axiom,
! [Xs: list_fm,P2: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X2: fm] : ( P2 @ ( cons_fm @ X2 @ nil_fm ) )
=> ( ! [X2: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_fm @ Xs2 @ ( cons_fm @ X2 @ nil_fm ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_441_rev__nonempty__induct,axiom,
! [Xs: list_tm,P2: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X2: tm] : ( P2 @ ( cons_tm @ X2 @ nil_tm ) )
=> ( ! [X2: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_tm @ Xs2 @ ( cons_tm @ X2 @ nil_tm ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_442_rev__nonempty__induct,axiom,
! [Xs: list_int,P2: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X2: int] : ( P2 @ ( cons_int @ X2 @ nil_int ) )
=> ( ! [X2: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_int @ Xs2 @ ( cons_int @ X2 @ nil_int ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_443_append__eq__Cons__conv,axiom,
! [Ys: list_fm,Zs: list_fm,X: fm,Xs: list_fm] :
( ( ( append_fm @ Ys @ Zs )
= ( cons_fm @ X @ Xs ) )
= ( ( ( Ys = nil_fm )
& ( Zs
= ( cons_fm @ X @ Xs ) ) )
| ? [Ys5: list_fm] :
( ( Ys
= ( cons_fm @ X @ Ys5 ) )
& ( ( append_fm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_444_append__eq__Cons__conv,axiom,
! [Ys: list_tm,Zs: list_tm,X: tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( cons_tm @ X @ Xs ) )
= ( ( ( Ys = nil_tm )
& ( Zs
= ( cons_tm @ X @ Xs ) ) )
| ? [Ys5: list_tm] :
( ( Ys
= ( cons_tm @ X @ Ys5 ) )
& ( ( append_tm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_445_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs: list_int,X: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( cons_int @ X @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs
= ( cons_int @ X @ Xs ) ) )
| ? [Ys5: list_int] :
( ( Ys
= ( cons_int @ X @ Ys5 ) )
& ( ( append_int @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_446_Cons__eq__append__conv,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( cons_fm @ X @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ( ( Ys = nil_fm )
& ( ( cons_fm @ X @ Xs )
= Zs ) )
| ? [Ys5: list_fm] :
( ( ( cons_fm @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_fm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_447_Cons__eq__append__conv,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Ys = nil_tm )
& ( ( cons_tm @ X @ Xs )
= Zs ) )
| ? [Ys5: list_tm] :
( ( ( cons_tm @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_tm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_448_Cons__eq__append__conv,axiom,
! [X: int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X @ Xs )
= Zs ) )
| ? [Ys5: list_int] :
( ( ( cons_int @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_int @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_449_rev__exhaust,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ~ ! [Ys2: list_fm,Y2: fm] :
( Xs
!= ( append_fm @ Ys2 @ ( cons_fm @ Y2 @ nil_fm ) ) ) ) ).
% rev_exhaust
thf(fact_450_rev__exhaust,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ~ ! [Ys2: list_tm,Y2: tm] :
( Xs
!= ( append_tm @ Ys2 @ ( cons_tm @ Y2 @ nil_tm ) ) ) ) ).
% rev_exhaust
thf(fact_451_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys2: list_int,Y2: int] :
( Xs
!= ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_452_rev__induct,axiom,
! [P2: list_fm > $o,Xs: list_fm] :
( ( P2 @ nil_fm )
=> ( ! [X2: fm,Xs2: list_fm] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_fm @ Xs2 @ ( cons_fm @ X2 @ nil_fm ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_453_rev__induct,axiom,
! [P2: list_tm > $o,Xs: list_tm] :
( ( P2 @ nil_tm )
=> ( ! [X2: tm,Xs2: list_tm] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_tm @ Xs2 @ ( cons_tm @ X2 @ nil_tm ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_454_rev__induct,axiom,
! [P2: list_int > $o,Xs: list_int] :
( ( P2 @ nil_int )
=> ( ! [X2: int,Xs2: list_int] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_int @ Xs2 @ ( cons_int @ X2 @ nil_int ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_455_list_Osize_I3_J,axiom,
( ( size_size_list_tm @ nil_tm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_456_list_Osize_I3_J,axiom,
( ( size_size_list_fm @ nil_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_457_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_458_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,P2: list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: fm,Ys2: list_fm,Z2: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) @ ( cons_fm @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_459_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,P2: list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: fm,Ys2: list_fm,Z2: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) @ ( cons_tm @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_460_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_int,P2: list_fm > list_fm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_fm @ nil_int )
=> ( ! [X2: fm,Xs2: list_fm,Y2: fm,Ys2: list_fm,Z2: int,Zs2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) @ ( cons_int @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_461_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,P2: list_fm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_tm @ nil_fm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: tm,Ys2: list_tm,Z2: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( ( size_size_list_tm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) @ ( cons_fm @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_462_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_tm,P2: list_fm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_tm @ nil_tm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: tm,Ys2: list_tm,Z2: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( ( size_size_list_tm @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) @ ( cons_tm @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_463_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_int,P2: list_fm > list_tm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_tm @ nil_int )
=> ( ! [X2: fm,Xs2: list_fm,Y2: tm,Ys2: list_tm,Z2: int,Zs2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( ( size_size_list_tm @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) @ ( cons_int @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_464_list__induct3,axiom,
! [Xs: list_fm,Ys: list_int,Zs: list_fm,P2: list_fm > list_int > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_int @ nil_fm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: int,Ys2: list_int,Z2: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_fm @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_465_list__induct3,axiom,
! [Xs: list_fm,Ys: list_int,Zs: list_tm,P2: list_fm > list_int > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_int @ nil_tm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: int,Ys2: list_int,Z2: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_tm @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_466_list__induct3,axiom,
! [Xs: list_fm,Ys: list_int,Zs: list_int,P2: list_fm > list_int > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_fm @ nil_int @ nil_int )
=> ( ! [X2: fm,Xs2: list_fm,Y2: int,Ys2: list_int,Z2: int,Zs2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_int @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_467_list__induct3,axiom,
! [Xs: list_tm,Ys: list_fm,Zs: list_fm,P2: list_tm > list_fm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P2 @ nil_tm @ nil_fm @ nil_fm )
=> ( ! [X2: tm,Xs2: list_tm,Y2: fm,Ys2: list_fm,Z2: fm,Zs2: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) @ ( cons_fm @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_468_list__induct2,axiom,
! [Xs: list_fm,Ys: list_fm,P2: list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P2 @ nil_fm @ nil_fm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: fm,Ys2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_469_list__induct2,axiom,
! [Xs: list_fm,Ys: list_tm,P2: list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P2 @ nil_fm @ nil_tm )
=> ( ! [X2: fm,Xs2: list_fm,Y2: tm,Ys2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_470_list__induct2,axiom,
! [Xs: list_fm,Ys: list_int,P2: list_fm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_fm @ nil_int )
=> ( ! [X2: fm,Xs2: list_fm,Y2: int,Ys2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_fm @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_471_list__induct2,axiom,
! [Xs: list_tm,Ys: list_fm,P2: list_tm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P2 @ nil_tm @ nil_fm )
=> ( ! [X2: tm,Xs2: list_tm,Y2: fm,Ys2: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_472_list__induct2,axiom,
! [Xs: list_tm,Ys: list_tm,P2: list_tm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P2 @ nil_tm @ nil_tm )
=> ( ! [X2: tm,Xs2: list_tm,Y2: tm,Ys2: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_473_list__induct2,axiom,
! [Xs: list_tm,Ys: list_int,P2: list_tm > list_int > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_tm @ nil_int )
=> ( ! [X2: tm,Xs2: list_tm,Y2: int,Ys2: list_int] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_tm @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_474_list__induct2,axiom,
! [Xs: list_int,Ys: list_fm,P2: list_int > list_fm > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P2 @ nil_int @ nil_fm )
=> ( ! [X2: int,Xs2: list_int,Y2: fm,Ys2: list_fm] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_fm @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_475_list__induct2,axiom,
! [Xs: list_int,Ys: list_tm,P2: list_int > list_tm > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P2 @ nil_int @ nil_tm )
=> ( ! [X2: int,Xs2: list_int,Y2: tm,Ys2: list_tm] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_tm @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_476_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P2: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_int @ nil_int )
=> ( ! [X2: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_int @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_477_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_478_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_479_lexordp__append__leftD,axiom,
! [Xs: list_nat,Us2: list_nat,Vs2: list_nat] :
( ( ord_lexordp_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Xs @ Vs2 ) )
=> ( ! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 )
=> ( ord_lexordp_nat @ Us2 @ Vs2 ) ) ) ).
% lexordp_append_leftD
thf(fact_480_lexordp__append__leftD,axiom,
! [Xs: list_int,Us2: list_int,Vs2: list_int] :
( ( ord_lexordp_int @ ( append_int @ Xs @ Us2 ) @ ( append_int @ Xs @ Vs2 ) )
=> ( ! [A2: int] :
~ ( ord_less_int @ A2 @ A2 )
=> ( ord_lexordp_int @ Us2 @ Vs2 ) ) ) ).
% lexordp_append_leftD
thf(fact_481_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_482_lexordp__append__rightI,axiom,
! [Ys: list_int,Xs: list_int] :
( ( Ys != nil_int )
=> ( ord_lexordp_int @ Xs @ ( append_int @ Xs @ Ys ) ) ) ).
% lexordp_append_rightI
thf(fact_483_lexordp__append__left__rightI,axiom,
! [X: nat,Y: nat,Us2: list_nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_lexordp_nat @ ( append_nat @ Us2 @ ( cons_nat @ X @ Xs ) ) @ ( append_nat @ Us2 @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_484_lexordp__append__left__rightI,axiom,
! [X: int,Y: int,Us2: list_int,Xs: list_int,Ys: list_int] :
( ( ord_less_int @ X @ Y )
=> ( ord_lexordp_int @ ( append_int @ Us2 @ ( cons_int @ X @ Xs ) ) @ ( append_int @ Us2 @ ( cons_int @ Y @ Ys ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_485_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_486_power__strict__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_487_power__strict__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_488_power__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_489_power__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_490_maps__simps_I1_J,axiom,
! [F: fm > list_int,X: fm,Xs: list_fm] :
( ( maps_fm_int @ F @ ( cons_fm @ X @ Xs ) )
= ( append_int @ ( F @ X ) @ ( maps_fm_int @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_491_maps__simps_I1_J,axiom,
! [F: tm > list_int,X: tm,Xs: list_tm] :
( ( maps_tm_int @ F @ ( cons_tm @ X @ Xs ) )
= ( append_int @ ( F @ X ) @ ( maps_tm_int @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_492_maps__simps_I1_J,axiom,
! [F: int > list_int,X: int,Xs: list_int] :
( ( maps_int_int @ F @ ( cons_int @ X @ Xs ) )
= ( append_int @ ( F @ X ) @ ( maps_int_int @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_493_concat__eq__append__conv,axiom,
! [Xss2: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss2 )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_tm )
=> ( ( Ys = nil_tm )
& ( Zs = nil_tm ) ) )
& ( ( Xss2 != nil_list_tm )
=> ? [Xss1: list_list_tm,Xs3: list_tm,Xs5: list_tm,Xss22: list_list_tm] :
( ( Xss2
= ( append_list_tm @ Xss1 @ ( cons_list_tm @ ( append_tm @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_tm @ Xs5 @ ( concat_tm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_494_concat__eq__append__conv,axiom,
! [Xss2: list_list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( concat_fm @ Xss2 )
= ( append_fm @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_fm )
=> ( ( Ys = nil_fm )
& ( Zs = nil_fm ) ) )
& ( ( Xss2 != nil_list_fm )
=> ? [Xss1: list_list_fm,Xs3: list_fm,Xs5: list_fm,Xss22: list_list_fm] :
( ( Xss2
= ( append_list_fm @ Xss1 @ ( cons_list_fm @ ( append_fm @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_fm @ ( concat_fm @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_fm @ Xs5 @ ( concat_fm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_495_concat__eq__append__conv,axiom,
! [Xss2: list_list_int,Ys: list_int,Zs: list_int] :
( ( ( concat_int @ Xss2 )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_int )
=> ( ( Ys = nil_int )
& ( Zs = nil_int ) ) )
& ( ( Xss2 != nil_list_int )
=> ? [Xss1: list_list_int,Xs3: list_int,Xs5: list_int,Xss22: list_list_int] :
( ( Xss2
= ( append_list_int @ Xss1 @ ( cons_list_int @ ( append_int @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_int @ ( concat_int @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_int @ Xs5 @ ( concat_int @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_496_add_Oinverse__inverse,axiom,
! [A3: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_497_neg__equal__iff__equal,axiom,
! [A3: int,B2: int] :
( ( ( uminus_uminus_int @ A3 )
= ( uminus_uminus_int @ B2 ) )
= ( A3 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_498_neg__le__iff__le,axiom,
! [B2: int,A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_499_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_500_neg__0__equal__iff__equal,axiom,
! [A3: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A3 ) )
= ( zero_zero_int = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_501_neg__equal__0__iff__equal,axiom,
! [A3: int] :
( ( ( uminus_uminus_int @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_502_equal__neg__zero,axiom,
! [A3: int] :
( ( A3
= ( uminus_uminus_int @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_503_neg__equal__zero,axiom,
! [A3: int] :
( ( ( uminus_uminus_int @ A3 )
= A3 )
= ( A3 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_504_neg__less__iff__less,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_505_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_506_neg__0__le__iff__le,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_507_neg__le__0__iff__le,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_508_less__eq__neg__nonpos,axiom,
! [A3: int] :
( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_509_neg__less__eq__nonneg,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_510_less__neg__neg,axiom,
! [A3: int] :
( ( ord_less_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_511_neg__less__pos,axiom,
! [A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% neg_less_pos
thf(fact_512_neg__0__less__iff__less,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_513_neg__less__0__iff__less,axiom,
! [A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_514_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_515_concat__append,axiom,
! [Xs: list_list_int,Ys: list_list_int] :
( ( concat_int @ ( append_list_int @ Xs @ Ys ) )
= ( append_int @ ( concat_int @ Xs ) @ ( concat_int @ Ys ) ) ) ).
% concat_append
thf(fact_516_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_517_equation__minus__iff,axiom,
! [A3: int,B2: int] :
( ( A3
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_518_minus__equation__iff,axiom,
! [A3: int,B2: int] :
( ( ( uminus_uminus_int @ A3 )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_519_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_520_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_521_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_522_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_523_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_524_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_525_le__minus__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_minus_iff
thf(fact_526_minus__le__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A3 ) ) ).
% minus_le_iff
thf(fact_527_le__imp__neg__le,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_528_less__minus__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A3 ) ) ) ).
% less_minus_iff
thf(fact_529_minus__less__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_530_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_531_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_532_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_533_power__0,axiom,
! [A3: int] :
( ( power_power_int @ A3 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_534_power__0,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_535_concat_Osimps_I1_J,axiom,
( ( concat_tm @ nil_list_tm )
= nil_tm ) ).
% concat.simps(1)
thf(fact_536_concat_Osimps_I1_J,axiom,
( ( concat_fm @ nil_list_fm )
= nil_fm ) ).
% concat.simps(1)
thf(fact_537_concat_Osimps_I1_J,axiom,
( ( concat_int @ nil_list_int )
= nil_int ) ).
% concat.simps(1)
thf(fact_538_concat_Osimps_I2_J,axiom,
! [X: list_int,Xs: list_list_int] :
( ( concat_int @ ( cons_list_int @ X @ Xs ) )
= ( append_int @ X @ ( concat_int @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_539_maps__simps_I2_J,axiom,
! [F: tm > list_tm] :
( ( maps_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_540_maps__simps_I2_J,axiom,
! [F: tm > list_fm] :
( ( maps_tm_fm @ F @ nil_tm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_541_maps__simps_I2_J,axiom,
! [F: tm > list_int] :
( ( maps_tm_int @ F @ nil_tm )
= nil_int ) ).
% maps_simps(2)
thf(fact_542_maps__simps_I2_J,axiom,
! [F: fm > list_tm] :
( ( maps_fm_tm @ F @ nil_fm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_543_maps__simps_I2_J,axiom,
! [F: fm > list_fm] :
( ( maps_fm_fm @ F @ nil_fm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_544_maps__simps_I2_J,axiom,
! [F: fm > list_int] :
( ( maps_fm_int @ F @ nil_fm )
= nil_int ) ).
% maps_simps(2)
thf(fact_545_maps__simps_I2_J,axiom,
! [F: int > list_tm] :
( ( maps_int_tm @ F @ nil_int )
= nil_tm ) ).
% maps_simps(2)
thf(fact_546_maps__simps_I2_J,axiom,
! [F: int > list_fm] :
( ( maps_int_fm @ F @ nil_int )
= nil_fm ) ).
% maps_simps(2)
thf(fact_547_maps__simps_I2_J,axiom,
! [F: int > list_int] :
( ( maps_int_int @ F @ nil_int )
= nil_int ) ).
% maps_simps(2)
thf(fact_548_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_tm @ N @ nil_tm )
= N ) ).
% gen_length_code(1)
thf(fact_549_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_fm @ N @ nil_fm )
= N ) ).
% gen_length_code(1)
thf(fact_550_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_int @ N @ nil_int )
= N ) ).
% gen_length_code(1)
thf(fact_551_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_552_power__le__one,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_553_power__le__one,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_554_power__decreasing,axiom,
! [N: nat,N3: nat,A3: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N3 ) @ ( power_power_nat @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_555_power__decreasing,axiom,
! [N: nat,N3: nat,A3: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ N3 ) @ ( power_power_int @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_556_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_557_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_558_concat__eq__appendD,axiom,
! [Xss2: list_list_int,Ys: list_int,Zs: list_int] :
( ( ( concat_int @ Xss2 )
= ( append_int @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_int )
=> ? [Xss12: list_list_int,Xs2: list_int,Xs4: list_int,Xss23: list_list_int] :
( ( Xss2
= ( append_list_int @ Xss12 @ ( cons_list_int @ ( append_int @ Xs2 @ Xs4 ) @ Xss23 ) ) )
& ( Ys
= ( append_int @ ( concat_int @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_int @ Xs4 @ ( concat_int @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_559_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_560_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A3: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N3 ) @ ( power_power_nat @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_561_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A3: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A3 @ N3 ) @ ( power_power_int @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_562_self__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_563_self__le__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_564_one__less__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_565_one__less__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_566_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_567_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_568_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_569_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_570_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_571_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_572_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_573_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_574_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_575_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_576_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_577_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_578_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_579_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_580_sub_Osimps_I6_J,axiom,
! [V: nat,S: tm,P: fm] :
( ( sub @ V @ S @ ( uni @ P ) )
= ( uni @ ( sub @ ( plus_plus_nat @ V @ one_one_nat ) @ ( inc_term @ S ) @ P ) ) ) ).
% sub.simps(6)
thf(fact_581_sub_Osimps_I5_J,axiom,
! [V: nat,S: tm,P: fm] :
( ( sub @ V @ S @ ( exi @ P ) )
= ( exi @ ( sub @ ( plus_plus_nat @ V @ one_one_nat ) @ ( inc_term @ S ) @ P ) ) ) ).
% sub.simps(5)
thf(fact_582_sub__term_Osimps_I1_J,axiom,
! [N: nat,V: nat,S: tm] :
( ( ( ord_less_nat @ N @ V )
=> ( ( sub_term @ V @ S @ ( var @ N ) )
= ( var @ N ) ) )
& ( ~ ( ord_less_nat @ N @ V )
=> ( ( ( N = V )
=> ( ( sub_term @ V @ S @ ( var @ N ) )
= S ) )
& ( ( N != V )
=> ( ( sub_term @ V @ S @ ( var @ N ) )
= ( var @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% sub_term.simps(1)
thf(fact_583_comm__append__is__replicate,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != nil_tm )
=> ( ( Ys != nil_tm )
=> ( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Ys @ Xs ) )
=> ? [N2: nat,Zs2: list_tm] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_tm @ ( replicate_list_tm @ N2 @ Zs2 ) )
= ( append_tm @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_584_comm__append__is__replicate,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs != nil_fm )
=> ( ( Ys != nil_fm )
=> ( ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Ys @ Xs ) )
=> ? [N2: nat,Zs2: list_fm] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_fm @ ( replicate_list_fm @ N2 @ Zs2 ) )
= ( append_fm @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_585_comm__append__is__replicate,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != nil_int )
=> ( ( Ys != nil_int )
=> ( ( ( append_int @ Xs @ Ys )
= ( append_int @ Ys @ Xs ) )
=> ? [N2: nat,Zs2: list_int] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_int @ ( replicate_list_int @ N2 @ Zs2 ) )
= ( append_int @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_586_add__left__cancel,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_587_add__left__cancel,axiom,
! [A3: int,B2: int,C: int] :
( ( ( plus_plus_int @ A3 @ B2 )
= ( plus_plus_int @ A3 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_588_add__right__cancel,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_589_add__right__cancel,axiom,
! [B2: int,A3: int,C: int] :
( ( ( plus_plus_int @ B2 @ A3 )
= ( plus_plus_int @ C @ A3 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_590_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_591_add__le__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A3 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_592_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_593_add__le__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A3 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_594_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_595_add_Oright__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.right_neutral
thf(fact_596_double__zero__sym,axiom,
! [A3: int] :
( ( zero_zero_int
= ( plus_plus_int @ A3 @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_597_add__cancel__left__left,axiom,
! [B2: nat,A3: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= A3 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_598_add__cancel__left__left,axiom,
! [B2: int,A3: int] :
( ( ( plus_plus_int @ B2 @ A3 )
= A3 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_599_add__cancel__left__right,axiom,
! [A3: nat,B2: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= A3 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_600_add__cancel__left__right,axiom,
! [A3: int,B2: int] :
( ( ( plus_plus_int @ A3 @ B2 )
= A3 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_601_add__cancel__right__left,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ A3 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_602_add__cancel__right__left,axiom,
! [A3: int,B2: int] :
( ( A3
= ( plus_plus_int @ B2 @ A3 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_603_add__cancel__right__right,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_604_add__cancel__right__right,axiom,
! [A3: int,B2: int] :
( ( A3
= ( plus_plus_int @ A3 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_605_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_606_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_607_add__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add_0
thf(fact_608_add__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add_0
thf(fact_609_double__eq__0__iff,axiom,
! [A3: int] :
( ( ( plus_plus_int @ A3 @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_610_add__less__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_611_add__less__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_612_add__less__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_613_add__less__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_614_diff__self,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% diff_self
thf(fact_615_diff__0__right,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_0_right
thf(fact_616_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_617_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_618_diff__zero,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_zero
thf(fact_619_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_620_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_621_add__diff__cancel,axiom,
! [A3: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel
thf(fact_622_diff__add__cancel,axiom,
! [A3: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A3 @ B2 ) @ B2 )
= A3 ) ).
% diff_add_cancel
thf(fact_623_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_624_add__diff__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A3 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_625_add__diff__cancel__left_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ A3 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_626_add__diff__cancel__left_H,axiom,
! [A3: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B2 ) @ A3 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_627_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_628_add__diff__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A3 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_629_add__diff__cancel__right_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_630_add__diff__cancel__right_H,axiom,
! [A3: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_631_add__minus__cancel,axiom,
! [A3: int,B2: int] :
( ( plus_plus_int @ A3 @ ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_632_minus__add__cancel,axiom,
! [A3: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ ( plus_plus_int @ A3 @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_633_minus__add__distrib,axiom,
! [A3: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_634_minus__diff__eq,axiom,
! [A3: int,B2: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A3 @ B2 ) )
= ( minus_minus_int @ B2 @ A3 ) ) ).
% minus_diff_eq
thf(fact_635_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_636_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_637_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_638_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_639_rotate1__is__Nil__conv,axiom,
! [Xs: list_tm] :
( ( ( rotate1_tm @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% rotate1_is_Nil_conv
thf(fact_640_rotate1__is__Nil__conv,axiom,
! [Xs: list_fm] :
( ( ( rotate1_fm @ Xs )
= nil_fm )
= ( Xs = nil_fm ) ) ).
% rotate1_is_Nil_conv
thf(fact_641_rotate1__is__Nil__conv,axiom,
! [Xs: list_int] :
( ( ( rotate1_int @ Xs )
= nil_int )
= ( Xs = nil_int ) ) ).
% rotate1_is_Nil_conv
thf(fact_642_add__le__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_643_add__le__same__cancel1,axiom,
! [B2: int,A3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A3 ) @ B2 )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_644_add__le__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_645_add__le__same__cancel2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_646_le__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_647_le__add__same__cancel1,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ A3 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_648_le__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_649_le__add__same__cancel2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ B2 @ A3 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_650_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_651_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_652_diff__ge__0__iff__ge,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_653_add__less__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_654_add__less__same__cancel1,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A3 ) @ B2 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_655_add__less__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_656_add__less__same__cancel2,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ B2 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_657_less__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_658_less__add__same__cancel1,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_659_less__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_660_less__add__same__cancel2,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ B2 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_661_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_662_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_663_diff__gt__0__iff__gt,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B2 ) )
= ( ord_less_int @ B2 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_664_diff__add__zero,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_665_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_666_ab__left__minus,axiom,
! [A3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_667_add_Oright__inverse,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ ( uminus_uminus_int @ A3 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_668_diff__0,axiom,
! [A3: int] :
( ( minus_minus_int @ zero_zero_int @ A3 )
= ( uminus_uminus_int @ A3 ) ) ).
% diff_0
thf(fact_669_verit__minus__simplify_I3_J,axiom,
! [B2: int] :
( ( minus_minus_int @ zero_zero_int @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_670_diff__minus__eq__add,axiom,
! [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ ( uminus_uminus_int @ B2 ) )
= ( plus_plus_int @ A3 @ B2 ) ) ).
% diff_minus_eq_add
thf(fact_671_uminus__add__conv__diff,axiom,
! [A3: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ B2 )
= ( minus_minus_int @ B2 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_672_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_673_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_674_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_675_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_676_length__append,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int @ ( append_int @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_append
thf(fact_677_empty__replicate,axiom,
! [N: nat,X: tm] :
( ( nil_tm
= ( replicate_tm @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_678_empty__replicate,axiom,
! [N: nat,X: fm] :
( ( nil_fm
= ( replicate_fm @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_679_empty__replicate,axiom,
! [N: nat,X: int] :
( ( nil_int
= ( replicate_int @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_680_replicate__empty,axiom,
! [N: nat,X: tm] :
( ( ( replicate_tm @ N @ X )
= nil_tm )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_681_replicate__empty,axiom,
! [N: nat,X: fm] :
( ( ( replicate_fm @ N @ X )
= nil_fm )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_682_replicate__empty,axiom,
! [N: nat,X: int] :
( ( ( replicate_int @ N @ X )
= nil_int )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_683_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_tm @ ( replicate_list_tm @ I @ nil_tm ) )
= nil_tm ) ).
% concat_replicate_trivial
thf(fact_684_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_fm @ ( replicate_list_fm @ I @ nil_fm ) )
= nil_fm ) ).
% concat_replicate_trivial
thf(fact_685_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_int @ ( replicate_list_int @ I @ nil_int ) )
= nil_int ) ).
% concat_replicate_trivial
thf(fact_686_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_687_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_688_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_689_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A3 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_690_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B2 @ A3 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_691_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_692_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_693_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_694_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_695_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_696_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_697_le__add__diff,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_698_diff__add,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ A3 )
= B2 ) ) ).
% diff_add
thf(fact_699_le__diff__eq,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ A3 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_700_diff__le__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ C )
= ( ord_less_eq_int @ A3 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_701_group__cancel_Osub2,axiom,
! [B3: int,K: int,B2: int,A3: int] :
( ( B3
= ( plus_plus_int @ K @ B2 ) )
=> ( ( minus_minus_int @ A3 @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A3 @ B2 ) ) ) ) ).
% group_cancel.sub2
thf(fact_702_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_703_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_704_nat__diff__split,axiom,
! [P2: nat > $o,A3: nat,B2: nat] :
( ( P2 @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ( ( ord_less_nat @ A3 @ B2 )
=> ( P2 @ zero_zero_nat ) )
& ! [D: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D ) )
=> ( P2 @ D ) ) ) ) ).
% nat_diff_split
thf(fact_705_nat__diff__split__asm,axiom,
! [P2: nat > $o,A3: nat,B2: nat] :
( ( P2 @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B2 )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D ) )
& ~ ( P2 @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_706_less__diff__eq,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ A3 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ C ) ) ).
% less_diff_eq
thf(fact_707_diff__less__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ C )
= ( ord_less_int @ A3 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_less_eq
thf(fact_708_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_709_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_710_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B2 ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_711_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_712_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_713_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_714_group__cancel_Oadd1,axiom,
! [A: int,K: int,A3: int,B2: int] :
( ( A
= ( plus_plus_int @ K @ A3 ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_715_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B2: nat,A3: nat] :
( ( B3
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_716_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B2: int,A3: int] :
( ( B3
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A3 @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_717_group__cancel_Osub1,axiom,
! [A: int,K: int,A3: int,B2: int] :
( ( A
= ( plus_plus_int @ K @ A3 ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( minus_minus_int @ A3 @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_718_diff__eq__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( ( minus_minus_int @ A3 @ B2 )
= C )
= ( A3
= ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_719_eq__diff__eq,axiom,
! [A3: int,C: int,B2: int] :
( ( A3
= ( minus_minus_int @ C @ B2 ) )
= ( ( plus_plus_int @ A3 @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_720_add__diff__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( plus_plus_int @ A3 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A3 @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_721_diff__diff__eq2,axiom,
! [A3: int,B2: int,C: int] :
( ( minus_minus_int @ A3 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A3 @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_722_add_Oassoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_723_add_Oassoc,axiom,
! [A3: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B2 ) @ C )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_724_diff__add__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A3 @ B2 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A3 @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_725_add_Oleft__cancel,axiom,
! [A3: int,B2: int,C: int] :
( ( ( plus_plus_int @ A3 @ B2 )
= ( plus_plus_int @ A3 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_726_diff__eq__diff__eq,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B2 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A3 = B2 )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_727_add_Oright__cancel,axiom,
! [B2: int,A3: int,C: int] :
( ( ( plus_plus_int @ B2 @ A3 )
= ( plus_plus_int @ C @ A3 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_728_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B: nat] : ( plus_plus_nat @ B @ A4 ) ) ) ).
% add.commute
thf(fact_729_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B: int] : ( plus_plus_int @ B @ A4 ) ) ) ).
% add.commute
thf(fact_730_add_Oleft__commute,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_731_add_Oleft__commute,axiom,
! [B2: int,A3: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A3 @ C ) )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_732_add__left__imp__eq,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_733_add__left__imp__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( ( plus_plus_int @ A3 @ B2 )
= ( plus_plus_int @ A3 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_734_diff__add__eq__diff__diff__swap,axiom,
! [A3: int,B2: int,C: int] :
( ( minus_minus_int @ A3 @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A3 @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_735_add__right__imp__eq,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_736_add__right__imp__eq,axiom,
! [B2: int,A3: int,C: int] :
( ( ( plus_plus_int @ B2 @ A3 )
= ( plus_plus_int @ C @ A3 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_737_add__implies__diff,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A3 )
=> ( C
= ( minus_minus_nat @ A3 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_738_add__implies__diff,axiom,
! [C: int,B2: int,A3: int] :
( ( ( plus_plus_int @ C @ B2 )
= A3 )
=> ( C
= ( minus_minus_int @ A3 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_739_diff__right__commute,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_740_diff__right__commute,axiom,
! [A3: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A3 @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A3 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_741_diff__diff__eq,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_742_diff__diff__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A3 @ B2 ) @ C )
= ( minus_minus_int @ A3 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_743_replicate__add,axiom,
! [N: nat,M: nat,X: int] :
( ( replicate_int @ ( plus_plus_nat @ N @ M ) @ X )
= ( append_int @ ( replicate_int @ N @ X ) @ ( replicate_int @ M @ X ) ) ) ).
% replicate_add
thf(fact_744_diff__mono,axiom,
! [A3: int,B2: int,D2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_745_diff__left__mono,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_eq_int @ B2 @ A3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_746_diff__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_747_diff__eq__diff__less__eq,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B2 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A3 @ B2 )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_748_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B: int] :
( ( minus_minus_int @ A4 @ B )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_749_diff__strict__mono,axiom,
! [A3: int,B2: int,D2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_750_diff__eq__diff__less,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B2 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A3 @ B2 )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_751_diff__strict__left__mono,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_752_diff__strict__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_753_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_754_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_755_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_756_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_757_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_758_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_759_add__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_760_add__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_761_add__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_762_add__left__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_763_less__eqE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A3 @ C2 ) ) ) ).
% less_eqE
thf(fact_764_add__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_765_add__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_766_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B: nat] :
? [C3: nat] :
( B
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_767_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_768_add__le__imp__le__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_769_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_770_add__le__imp__le__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_771_minus__diff__commute,axiom,
! [B2: int,A3: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A3 )
= ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_772_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_773_comm__monoid__add__class_Oadd__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_774_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_775_add_Ocomm__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.comm_neutral
thf(fact_776_add_Ogroup__left__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_777_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_778_verit__sum__simplify,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% verit_sum_simplify
thf(fact_779_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_780_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_781_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_782_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_783_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_784_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_785_add__strict__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_786_add__strict__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_787_add__strict__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_788_add__strict__left__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_789_add__strict__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_790_add__strict__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_791_add__less__imp__less__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_792_add__less__imp__less__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_793_add__less__imp__less__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_794_add__less__imp__less__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_795_group__cancel_Oneg1,axiom,
! [A: int,K: int,A3: int] :
( ( A
= ( plus_plus_int @ K @ A3 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_796_add_Oinverse__distrib__swap,axiom,
! [A3: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_797_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_798_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_799_append__replicate__commute,axiom,
! [N: nat,X: int,K: nat] :
( ( append_int @ ( replicate_int @ N @ X ) @ ( replicate_int @ K @ X ) )
= ( append_int @ ( replicate_int @ K @ X ) @ ( replicate_int @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_800_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_801_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_802_rotate1_Osimps_I1_J,axiom,
( ( rotate1_tm @ nil_tm )
= nil_tm ) ).
% rotate1.simps(1)
thf(fact_803_rotate1_Osimps_I1_J,axiom,
( ( rotate1_fm @ nil_fm )
= nil_fm ) ).
% rotate1.simps(1)
thf(fact_804_rotate1_Osimps_I1_J,axiom,
( ( rotate1_int @ nil_int )
= nil_int ) ).
% rotate1.simps(1)
thf(fact_805_Cons__replicate__eq,axiom,
! [X: fm,Xs: list_fm,N: nat,Y: fm] :
( ( ( cons_fm @ X @ Xs )
= ( replicate_fm @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_fm @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_806_Cons__replicate__eq,axiom,
! [X: tm,Xs: list_tm,N: nat,Y: tm] :
( ( ( cons_tm @ X @ Xs )
= ( replicate_tm @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_tm @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_807_Cons__replicate__eq,axiom,
! [X: int,Xs: list_int,N: nat,Y: int] :
( ( ( cons_int @ X @ Xs )
= ( replicate_int @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_int @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_808_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_809_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_810_add__decreasing,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_811_add__decreasing,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_812_add__increasing,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_813_add__increasing,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_814_add__decreasing2,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_815_add__decreasing2,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A3 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_816_add__increasing2,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_817_add__increasing2,axiom,
! [C: int,B2: int,A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A3 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_818_add__nonneg__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_819_add__nonneg__nonneg,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_820_add__nonpos__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_821_add__nonpos__nonpos,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_822_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_823_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_824_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_825_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_826_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_827_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_828_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_829_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_830_add__le__less__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_831_add__le__less__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_832_add__less__le__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_833_add__less__le__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_834_add__neg__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_835_add__neg__neg,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_836_add__pos__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_837_add__pos__pos,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_838_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A3 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_839_pos__add__strict,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_840_pos__add__strict,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_841_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_842_replicate__0,axiom,
! [X: tm] :
( ( replicate_tm @ zero_zero_nat @ X )
= nil_tm ) ).
% replicate_0
thf(fact_843_replicate__0,axiom,
! [X: fm] :
( ( replicate_fm @ zero_zero_nat @ X )
= nil_fm ) ).
% replicate_0
thf(fact_844_replicate__0,axiom,
! [X: int] :
( ( replicate_int @ zero_zero_nat @ X )
= nil_int ) ).
% replicate_0
thf(fact_845_neg__eq__iff__add__eq__0,axiom,
! [A3: int,B2: int] :
( ( ( uminus_uminus_int @ A3 )
= B2 )
= ( ( plus_plus_int @ A3 @ B2 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_846_eq__neg__iff__add__eq__0,axiom,
! [A3: int,B2: int] :
( ( A3
= ( uminus_uminus_int @ B2 ) )
= ( ( plus_plus_int @ A3 @ B2 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_847_add_Oinverse__unique,axiom,
! [A3: int,B2: int] :
( ( ( plus_plus_int @ A3 @ B2 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A3 )
= B2 ) ) ).
% add.inverse_unique
thf(fact_848_ab__group__add__class_Oab__left__minus,axiom,
! [A3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ A3 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_849_add__eq__0__iff,axiom,
! [A3: int,B2: int] :
( ( ( plus_plus_int @ A3 @ B2 )
= zero_zero_int )
= ( B2
= ( uminus_uminus_int @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_850_replicate__app__Cons__same,axiom,
! [N: nat,X: fm,Xs: list_fm] :
( ( append_fm @ ( replicate_fm @ N @ X ) @ ( cons_fm @ X @ Xs ) )
= ( cons_fm @ X @ ( append_fm @ ( replicate_fm @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_851_replicate__app__Cons__same,axiom,
! [N: nat,X: tm,Xs: list_tm] :
( ( append_tm @ ( replicate_tm @ N @ X ) @ ( cons_tm @ X @ Xs ) )
= ( cons_tm @ X @ ( append_tm @ ( replicate_tm @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_852_replicate__app__Cons__same,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( append_int @ ( replicate_int @ N @ X ) @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( append_int @ ( replicate_int @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_853_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_854_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_855_comm__append__are__replicate,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Ys @ Xs ) )
=> ? [M3: nat,N2: nat,Zs2: list_int] :
( ( ( concat_int @ ( replicate_list_int @ M3 @ Zs2 ) )
= Xs )
& ( ( concat_int @ ( replicate_list_int @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_856_add__neg__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_857_add__neg__nonpos,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_858_add__nonneg__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_859_add__nonneg__pos,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_860_add__nonpos__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_861_add__nonpos__neg,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_862_add__pos__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_863_add__pos__nonneg,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_864_add__strict__increasing,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_865_add__strict__increasing,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_866_add__strict__increasing2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_867_add__strict__increasing2,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_868_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_869_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_870_replicate__append__same,axiom,
! [I: nat,X: fm] :
( ( append_fm @ ( replicate_fm @ I @ X ) @ ( cons_fm @ X @ nil_fm ) )
= ( cons_fm @ X @ ( replicate_fm @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_871_replicate__append__same,axiom,
! [I: nat,X: tm] :
( ( append_tm @ ( replicate_tm @ I @ X ) @ ( cons_tm @ X @ nil_tm ) )
= ( cons_tm @ X @ ( replicate_tm @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_872_replicate__append__same,axiom,
! [I: nat,X: int] :
( ( append_int @ ( replicate_int @ I @ X ) @ ( cons_int @ X @ nil_int ) )
= ( cons_int @ X @ ( replicate_int @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_873_inc__term_Osimps_I1_J,axiom,
! [N: nat] :
( ( inc_term @ ( var @ N ) )
= ( var @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% inc_term.simps(1)
thf(fact_874_rotate1_Osimps_I2_J,axiom,
! [X: fm,Xs: list_fm] :
( ( rotate1_fm @ ( cons_fm @ X @ Xs ) )
= ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) ) ) ).
% rotate1.simps(2)
thf(fact_875_rotate1_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( rotate1_tm @ ( cons_tm @ X @ Xs ) )
= ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) ) ).
% rotate1.simps(2)
thf(fact_876_rotate1_Osimps_I2_J,axiom,
! [X: int,Xs: list_int] :
( ( rotate1_int @ ( cons_int @ X @ Xs ) )
= ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) ) ).
% rotate1.simps(2)
thf(fact_877_Euclid__induct,axiom,
! [P2: nat > nat > $o,A3: nat,B2: nat] :
( ! [A2: nat,B4: nat] :
( ( P2 @ A2 @ B4 )
= ( P2 @ B4 @ A2 ) )
=> ( ! [A2: nat] : ( P2 @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B4: nat] :
( ( P2 @ A2 @ B4 )
=> ( P2 @ A2 @ ( plus_plus_nat @ A2 @ B4 ) ) )
=> ( P2 @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_878_add__0__iff,axiom,
! [B2: nat,A3: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_879_add__0__iff,axiom,
! [B2: int,A3: int] :
( ( B2
= ( plus_plus_int @ B2 @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_880_nth__Cons__pos,axiom,
! [N: nat,X: fm,Xs: list_fm] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_fm @ ( cons_fm @ X @ Xs ) @ N )
= ( nth_fm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_881_nth__Cons__pos,axiom,
! [N: nat,X: tm,Xs: list_tm] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_tm @ ( cons_tm @ X @ Xs ) @ N )
= ( nth_tm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_882_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_883_nth__Cons__0,axiom,
! [X: fm,Xs: list_fm] :
( ( nth_fm @ ( cons_fm @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_884_nth__Cons__0,axiom,
! [X: tm,Xs: list_tm] :
( ( nth_tm @ ( cons_tm @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_885_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_886_nth__replicate,axiom,
! [I: nat,N: nat,X: int] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_887_nth__append__length,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm] :
( ( nth_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ Ys ) ) @ ( size_size_list_fm @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_888_nth__append__length,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm] :
( ( nth_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ Ys ) ) @ ( size_size_list_tm @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_889_nth__append__length,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_890_nth__append__length__plus,axiom,
! [Xs: list_int,Ys: list_int,N: nat] :
( ( nth_int @ ( append_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
= ( nth_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_891_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I3 )
= ( nth_int @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_892_Skolem__list__nth,axiom,
! [K: nat,P2: nat > int > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: int] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_int @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_893_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z3: list_int] : ( Y4 = Z3 ) )
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs3 ) )
=> ( ( nth_int @ Xs3 @ I4 )
= ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_894_nth__Cons_H,axiom,
! [N: nat,X: fm,Xs: list_fm] :
( ( ( N = zero_zero_nat )
=> ( ( nth_fm @ ( cons_fm @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_fm @ ( cons_fm @ X @ Xs ) @ N )
= ( nth_fm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_895_nth__Cons_H,axiom,
! [N: nat,X: tm,Xs: list_tm] :
( ( ( N = zero_zero_nat )
=> ( ( nth_tm @ ( cons_tm @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_tm @ ( cons_tm @ X @ Xs ) @ N )
= ( nth_tm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_896_nth__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_897_nth__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_898_nth__non__equal__first__eq,axiom,
! [X: fm,Y: fm,Xs: list_fm,N: nat] :
( ( X != Y )
=> ( ( ( nth_fm @ ( cons_fm @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_fm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_899_nth__non__equal__first__eq,axiom,
! [X: tm,Y: tm,Xs: list_tm,N: nat] :
( ( X != Y )
=> ( ( ( nth_tm @ ( cons_tm @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_tm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_900_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs: list_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_901_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_902_last__conv__nth,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( ( last_tm @ Xs )
= ( nth_tm @ Xs @ ( minus_minus_nat @ ( size_size_list_tm @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_903_last__conv__nth,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( ( last_fm @ Xs )
= ( nth_fm @ Xs @ ( minus_minus_nat @ ( size_size_list_fm @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_904_last__conv__nth,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( last_int @ Xs )
= ( nth_int @ Xs @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_905_nth__Cons__numeral,axiom,
! [X: fm,Xs: list_fm,V: num] :
( ( nth_fm @ ( cons_fm @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_fm @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_906_nth__Cons__numeral,axiom,
! [X: tm,Xs: list_tm,V: num] :
( ( nth_tm @ ( cons_tm @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_tm @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_907_nth__Cons__numeral,axiom,
! [X: int,Xs: list_int,V: num] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_int @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_908_power__minus__mult,axiom,
! [N: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_nat @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_909_power__minus__mult,axiom,
! [N: nat,A3: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_int @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_910_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_911_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_912_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_913_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_914_abs__idempotent,axiom,
! [A3: int] :
( ( abs_abs_int @ ( abs_abs_int @ A3 ) )
= ( abs_abs_int @ A3 ) ) ).
% abs_idempotent
thf(fact_915_mult__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A3 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_916_mult__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A3 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_917_mult__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A3 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_918_mult__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A3 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_919_mult__eq__0__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ B2 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_920_mult__eq__0__iff,axiom,
! [A3: int,B2: int] :
( ( ( times_times_int @ A3 @ B2 )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_921_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_922_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_923_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_924_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_925_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_926_mult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% mult_1
thf(fact_927_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_928_mult_Oright__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.right_neutral
thf(fact_929_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_930_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_931_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_932_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_933_abs__eq__0,axiom,
! [A3: int] :
( ( ( abs_abs_int @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_934_abs__0__eq,axiom,
! [A3: int] :
( ( zero_zero_int
= ( abs_abs_int @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_935_abs__add__abs,axiom,
! [A3: int,B2: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_add_abs
thf(fact_936_abs__minus__cancel,axiom,
! [A3: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A3 ) )
= ( abs_abs_int @ A3 ) ) ).
% abs_minus_cancel
thf(fact_937_mult__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ( times_times_int @ A3 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_938_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_939_mult__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ( times_times_int @ C @ A3 )
= C )
= ( ( C = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_940_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_941_abs__of__nonneg,axiom,
! [A3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( abs_abs_int @ A3 )
= A3 ) ) ).
% abs_of_nonneg
thf(fact_942_abs__le__self__iff,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ A3 )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).
% abs_le_self_iff
thf(fact_943_abs__le__zero__iff,axiom,
! [A3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_944_zero__less__abs__iff,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A3 ) )
= ( A3 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_945_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_946_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_947_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_948_last__appendL,axiom,
! [Ys: list_tm,Xs: list_tm] :
( ( Ys = nil_tm )
=> ( ( last_tm @ ( append_tm @ Xs @ Ys ) )
= ( last_tm @ Xs ) ) ) ).
% last_appendL
thf(fact_949_last__appendL,axiom,
! [Ys: list_fm,Xs: list_fm] :
( ( Ys = nil_fm )
=> ( ( last_fm @ ( append_fm @ Xs @ Ys ) )
= ( last_fm @ Xs ) ) ) ).
% last_appendL
thf(fact_950_last__appendL,axiom,
! [Ys: list_int,Xs: list_int] :
( ( Ys = nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Xs ) ) ) ).
% last_appendL
thf(fact_951_last__appendR,axiom,
! [Ys: list_tm,Xs: list_tm] :
( ( Ys != nil_tm )
=> ( ( last_tm @ ( append_tm @ Xs @ Ys ) )
= ( last_tm @ Ys ) ) ) ).
% last_appendR
thf(fact_952_last__appendR,axiom,
! [Ys: list_fm,Xs: list_fm] :
( ( Ys != nil_fm )
=> ( ( last_fm @ ( append_fm @ Xs @ Ys ) )
= ( last_fm @ Ys ) ) ) ).
% last_appendR
thf(fact_953_last__appendR,axiom,
! [Ys: list_int,Xs: list_int] :
( ( Ys != nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Ys ) ) ) ).
% last_appendR
thf(fact_954_abs__of__nonpos,axiom,
! [A3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( abs_abs_int @ A3 )
= ( uminus_uminus_int @ A3 ) ) ) ).
% abs_of_nonpos
thf(fact_955_last__snoc,axiom,
! [Xs: list_fm,X: fm] :
( ( last_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) ) )
= X ) ).
% last_snoc
thf(fact_956_last__snoc,axiom,
! [Xs: list_tm,X: tm] :
( ( last_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) )
= X ) ).
% last_snoc
thf(fact_957_last__snoc,axiom,
! [Xs: list_int,X: int] :
( ( last_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% last_snoc
thf(fact_958_zero__less__power__abs__iff,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A3 ) @ N ) )
= ( ( A3 != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_959_abs__minus__commute,axiom,
! [A3: int,B2: int] :
( ( abs_abs_int @ ( minus_minus_int @ A3 @ B2 ) )
= ( abs_abs_int @ ( minus_minus_int @ B2 @ A3 ) ) ) ).
% abs_minus_commute
thf(fact_960_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_961_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_962_mult__right__cancel,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A3 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_963_mult__right__cancel,axiom,
! [C: int,A3: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B2 @ C ) )
= ( A3 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_964_mult__left__cancel,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B2 ) )
= ( A3 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_965_mult__left__cancel,axiom,
! [C: int,A3: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B2 ) )
= ( A3 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_966_no__zero__divisors,axiom,
! [A3: nat,B2: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_967_no__zero__divisors,axiom,
! [A3: int,B2: int] :
( ( A3 != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A3 @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_968_divisors__zero,axiom,
! [A3: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ B2 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_969_divisors__zero,axiom,
! [A3: int,B2: int] :
( ( ( times_times_int @ A3 @ B2 )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_970_mult__not__zero,axiom,
! [A3: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ B2 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_971_mult__not__zero,axiom,
! [A3: int,B2: int] :
( ( ( times_times_int @ A3 @ B2 )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_972_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_973_mult_Ocomm__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.comm_neutral
thf(fact_974_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_975_comm__monoid__mult__class_Omult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_976_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_977_abs__mult__pos_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ X @ ( abs_abs_int @ Y ) )
= ( abs_abs_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_978_abs__eq__mult,axiom,
! [A3: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
| ( ord_less_eq_int @ A3 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B2 )
| ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A3 @ B2 ) )
= ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% abs_eq_mult
thf(fact_979_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_980_abs__eq__0__iff,axiom,
! [A3: int] :
( ( ( abs_abs_int @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_981_abs__ge__self,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ ( abs_abs_int @ A3 ) ) ).
% abs_ge_self
thf(fact_982_abs__le__D1,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B2 )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ).
% abs_le_D1
thf(fact_983_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B2 ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_984_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B2 ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_985_mult_Oassoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B2 ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_986_mult_Oassoc,axiom,
! [A3: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B2 ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_987_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B: nat] : ( times_times_nat @ B @ A4 ) ) ) ).
% mult.commute
thf(fact_988_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B: int] : ( times_times_int @ B @ A4 ) ) ) ).
% mult.commute
thf(fact_989_mult_Oleft__commute,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A3 @ C ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_990_mult_Oleft__commute,axiom,
! [B2: int,A3: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A3 @ C ) )
= ( times_times_int @ A3 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_991_abs__ge__zero,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A3 ) ) ).
% abs_ge_zero
thf(fact_992_abs__of__pos,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( abs_abs_int @ A3 )
= A3 ) ) ).
% abs_of_pos
thf(fact_993_abs__not__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( abs_abs_int @ A3 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_994_abs__triangle__ineq,axiom,
! [A3: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A3 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_triangle_ineq
thf(fact_995_abs__triangle__ineq2__sym,axiom,
! [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A3 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_996_abs__triangle__ineq3,axiom,
! [A3: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B2 ) ) ) ).
% abs_triangle_ineq3
thf(fact_997_abs__triangle__ineq2,axiom,
! [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B2 ) ) ) ).
% abs_triangle_ineq2
thf(fact_998_abs__ge__minus__self,axiom,
! [A3: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ ( abs_abs_int @ A3 ) ) ).
% abs_ge_minus_self
thf(fact_999_abs__le__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B2 )
= ( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B2 ) ) ) ).
% abs_le_iff
thf(fact_1000_abs__le__D2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B2 ) ) ).
% abs_le_D2
thf(fact_1001_abs__leI,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B2 ) ) ) ).
% abs_leI
thf(fact_1002_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1003_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_1004_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1005_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_1006_mult__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1007_mult__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1008_mult__mono_H,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1009_mult__mono_H,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1010_zero__le__square,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_1011_split__mult__pos__le,axiom,
! [A3: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_1012_mult__left__mono__neg,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_eq_int @ B2 @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1013_mult__nonpos__nonpos,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1014_mult__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1015_mult__left__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1016_mult__right__mono__neg,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_eq_int @ B2 @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1017_mult__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1018_mult__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1019_mult__le__0__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_1020_split__mult__neg__le,axiom,
! [A3: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1021_split__mult__neg__le,axiom,
! [A3: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1022_mult__nonneg__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1023_mult__nonneg__nonneg,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1024_mult__nonneg__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1025_mult__nonneg__nonpos,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1026_mult__nonpos__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1027_mult__nonpos__nonneg,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1028_mult__nonneg__nonpos2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1029_mult__nonneg__nonpos2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1030_zero__le__mult__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1031_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1032_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1033_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1034_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_1035_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1036_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_1037_mult__neg__neg,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_1038_not__square__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( times_times_int @ A3 @ A3 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1039_mult__less__0__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_1040_mult__neg__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1041_mult__neg__pos,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1042_mult__pos__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1043_mult__pos__neg,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1044_mult__pos__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1045_mult__pos__pos,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1046_mult__pos__neg2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1047_mult__pos__neg2,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A3 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1048_zero__less__mult__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1049_zero__less__mult__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1050_zero__less__mult__pos,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1051_zero__less__mult__pos2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1052_zero__less__mult__pos2,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1053_mult__less__cancel__left__neg,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1054_mult__less__cancel__left__pos,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A3 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1055_mult__strict__left__mono__neg,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1056_mult__strict__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1057_mult__strict__left__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1058_mult__less__cancel__left__disj,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1059_mult__strict__right__mono__neg,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1060_mult__strict__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1061_mult__strict__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1062_mult__less__cancel__right__disj,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1063_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1064_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1065_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1066_add__scale__eq__noteq,axiom,
! [R2: nat,A3: nat,B2: nat,C: nat,D2: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A3 = B2 )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A3 @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B2 @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1067_add__scale__eq__noteq,axiom,
! [R2: int,A3: int,B2: int,C: int,D2: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A3 = B2 )
& ( C != D2 ) )
=> ( ( plus_plus_int @ A3 @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B2 @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1068_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1069_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1070_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1071_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1072_last_Osimps,axiom,
! [Xs: list_fm,X: fm] :
( ( ( Xs = nil_fm )
=> ( ( last_fm @ ( cons_fm @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_fm )
=> ( ( last_fm @ ( cons_fm @ X @ Xs ) )
= ( last_fm @ Xs ) ) ) ) ).
% last.simps
thf(fact_1073_last_Osimps,axiom,
! [Xs: list_tm,X: tm] :
( ( ( Xs = nil_tm )
=> ( ( last_tm @ ( cons_tm @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_tm )
=> ( ( last_tm @ ( cons_tm @ X @ Xs ) )
= ( last_tm @ Xs ) ) ) ) ).
% last.simps
thf(fact_1074_last_Osimps,axiom,
! [Xs: list_int,X: int] :
( ( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_1075_last__ConsL,axiom,
! [Xs: list_fm,X: fm] :
( ( Xs = nil_fm )
=> ( ( last_fm @ ( cons_fm @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1076_last__ConsL,axiom,
! [Xs: list_tm,X: tm] :
( ( Xs = nil_tm )
=> ( ( last_tm @ ( cons_tm @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1077_last__ConsL,axiom,
! [Xs: list_int,X: int] :
( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1078_last__ConsR,axiom,
! [Xs: list_fm,X: fm] :
( ( Xs != nil_fm )
=> ( ( last_fm @ ( cons_fm @ X @ Xs ) )
= ( last_fm @ Xs ) ) ) ).
% last_ConsR
thf(fact_1079_last__ConsR,axiom,
! [Xs: list_tm,X: tm] :
( ( Xs != nil_tm )
=> ( ( last_tm @ ( cons_tm @ X @ Xs ) )
= ( last_tm @ Xs ) ) ) ).
% last_ConsR
thf(fact_1080_last__ConsR,axiom,
! [Xs: list_int,X: int] :
( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_1081_last__append,axiom,
! [Ys: list_tm,Xs: list_tm] :
( ( ( Ys = nil_tm )
=> ( ( last_tm @ ( append_tm @ Xs @ Ys ) )
= ( last_tm @ Xs ) ) )
& ( ( Ys != nil_tm )
=> ( ( last_tm @ ( append_tm @ Xs @ Ys ) )
= ( last_tm @ Ys ) ) ) ) ).
% last_append
thf(fact_1082_last__append,axiom,
! [Ys: list_fm,Xs: list_fm] :
( ( ( Ys = nil_fm )
=> ( ( last_fm @ ( append_fm @ Xs @ Ys ) )
= ( last_fm @ Xs ) ) )
& ( ( Ys != nil_fm )
=> ( ( last_fm @ ( append_fm @ Xs @ Ys ) )
= ( last_fm @ Ys ) ) ) ) ).
% last_append
thf(fact_1083_last__append,axiom,
! [Ys: list_int,Xs: list_int] :
( ( ( Ys = nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Xs ) ) )
& ( ( Ys != nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Ys ) ) ) ) ).
% last_append
thf(fact_1084_longest__common__suffix,axiom,
! [Xs: list_tm,Ys: list_tm] :
? [Ss: list_tm,Xs4: list_tm,Ys4: list_tm] :
( ( Xs
= ( append_tm @ Xs4 @ Ss ) )
& ( Ys
= ( append_tm @ Ys4 @ Ss ) )
& ( ( Xs4 = nil_tm )
| ( Ys4 = nil_tm )
| ( ( last_tm @ Xs4 )
!= ( last_tm @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_1085_longest__common__suffix,axiom,
! [Xs: list_fm,Ys: list_fm] :
? [Ss: list_fm,Xs4: list_fm,Ys4: list_fm] :
( ( Xs
= ( append_fm @ Xs4 @ Ss ) )
& ( Ys
= ( append_fm @ Ys4 @ Ss ) )
& ( ( Xs4 = nil_fm )
| ( Ys4 = nil_fm )
| ( ( last_fm @ Xs4 )
!= ( last_fm @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_1086_longest__common__suffix,axiom,
! [Xs: list_int,Ys: list_int] :
? [Ss: list_int,Xs4: list_int,Ys4: list_int] :
( ( Xs
= ( append_int @ Xs4 @ Ss ) )
& ( Ys
= ( append_int @ Ys4 @ Ss ) )
& ( ( Xs4 = nil_int )
| ( Ys4 = nil_int )
| ( ( last_int @ Xs4 )
!= ( last_int @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_1087_abs__minus__le__zero,axiom,
! [A3: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A3 ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_1088_eq__abs__iff_H,axiom,
! [A3: int,B2: int] :
( ( A3
= ( abs_abs_int @ B2 ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ( B2 = A3 )
| ( B2
= ( uminus_uminus_int @ A3 ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1089_abs__eq__iff_H,axiom,
! [A3: int,B2: int] :
( ( ( abs_abs_int @ A3 )
= B2 )
= ( ( ord_less_eq_int @ zero_zero_int @ B2 )
& ( ( A3 = B2 )
| ( A3
= ( uminus_uminus_int @ B2 ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1090_abs__diff__triangle__ineq,axiom,
! [A3: int,B2: int,C: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A3 @ B2 ) @ ( plus_plus_int @ C @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A3 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1091_abs__triangle__ineq4,axiom,
! [A3: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_triangle_ineq4
thf(fact_1092_abs__of__neg,axiom,
! [A3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( abs_abs_int @ A3 )
= ( uminus_uminus_int @ A3 ) ) ) ).
% abs_of_neg
thf(fact_1093_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_1094_abs__if,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_1095_zero__le__power__abs,axiom,
! [A3: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A3 ) @ N ) ) ).
% zero_le_power_abs
thf(fact_1096_mult__le__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1097_mult__le__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1098_mult__left__less__imp__less,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1099_mult__left__less__imp__less,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1100_mult__strict__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1101_mult__strict__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1102_mult__less__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1103_mult__right__less__imp__less,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1104_mult__right__less__imp__less,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1105_mult__strict__mono_H,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1106_mult__strict__mono_H,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1107_mult__less__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1108_mult__le__cancel__left__neg,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ B2 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1109_mult__le__cancel__left__pos,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ A3 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1110_mult__left__le__imp__le,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1111_mult__left__le__imp__le,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1112_mult__right__le__imp__le,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1113_mult__right__le__imp__le,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1114_mult__le__less__imp__less,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1115_mult__le__less__imp__less,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1116_mult__less__le__imp__less,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1117_mult__less__le__imp__less,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1118_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1119_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1120_mult__left__le,axiom,
! [C: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_1121_mult__left__le,axiom,
! [C: int,A3: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_1122_mult__le__one,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1123_mult__le__one,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B2 ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1124_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1125_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1126_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1127_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_1128_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1129_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1130_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1131_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_1132_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1133_mult__le__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1134_mult__le__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1135_mult__le__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1136_mult__le__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1137_mult__less__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1138_mult__less__cancel__left2,axiom,
! [C: int,A3: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1139_mult__less__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1140_mult__less__cancel__right2,axiom,
! [A3: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1141_convex__bound__le,axiom,
! [X: int,A3: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A3 )
=> ( ( ord_less_eq_int @ Y @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1142_power__Suc__less,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1143_power__Suc__less,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) @ ( power_power_int @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1144_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1145_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1146_convex__bound__lt,axiom,
! [X: int,A3: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A3 )
=> ( ( ord_less_int @ Y @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1147_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1148_power__eq__if,axiom,
( power_power_int
= ( ^ [P3: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1149_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1150_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1151_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1152_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1153_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1154_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1155_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_1156_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1157_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1158_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1159_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1160_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_1161_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_1162_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1163_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1164_power__Suc0__right,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_1165_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1166_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1167_nth__Cons__Suc,axiom,
! [X: fm,Xs: list_fm,N: nat] :
( ( nth_fm @ ( cons_fm @ X @ Xs ) @ ( suc @ N ) )
= ( nth_fm @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1168_nth__Cons__Suc,axiom,
! [X: tm,Xs: list_tm,N: nat] :
( ( nth_tm @ ( cons_tm @ X @ Xs ) @ ( suc @ N ) )
= ( nth_tm @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1169_nth__Cons__Suc,axiom,
! [X: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1170_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1171_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1172_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1173_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1174_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1175_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1176_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1177_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1178_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1179_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1180_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1181_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1182_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1183_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1184_Suc__length__conv,axiom,
! [N: nat,Xs: list_fm] :
( ( ( suc @ N )
= ( size_size_list_fm @ Xs ) )
= ( ? [Y3: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y3 @ Ys3 ) )
& ( ( size_size_list_fm @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1185_Suc__length__conv,axiom,
! [N: nat,Xs: list_tm] :
( ( ( suc @ N )
= ( size_size_list_tm @ Xs ) )
= ( ? [Y3: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y3 @ Ys3 ) )
& ( ( size_size_list_tm @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1186_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y3: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1187_length__Suc__conv,axiom,
! [Xs: list_fm,N: nat] :
( ( ( size_size_list_fm @ Xs )
= ( suc @ N ) )
= ( ? [Y3: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y3 @ Ys3 ) )
& ( ( size_size_list_fm @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1188_length__Suc__conv,axiom,
! [Xs: list_tm,N: nat] :
( ( ( size_size_list_tm @ Xs )
= ( suc @ N ) )
= ( ? [Y3: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y3 @ Ys3 ) )
& ( ( size_size_list_tm @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1189_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y3: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1190_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1191_replicate__Suc,axiom,
! [N: nat,X: fm] :
( ( replicate_fm @ ( suc @ N ) @ X )
= ( cons_fm @ X @ ( replicate_fm @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1192_replicate__Suc,axiom,
! [N: nat,X: tm] :
( ( replicate_tm @ ( suc @ N ) @ X )
= ( cons_tm @ X @ ( replicate_tm @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1193_replicate__Suc,axiom,
! [N: nat,X: int] :
( ( replicate_int @ ( suc @ N ) @ X )
= ( cons_int @ X @ ( replicate_int @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1194_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1195_old_Onat_Odistinct_I2_J,axiom,
! [Nat: nat] :
( ( suc @ Nat )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1196_old_Onat_Odistinct_I1_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
% old.nat.distinct(1)
thf(fact_1197_nat_OdiscI,axiom,
! [Nat2: nat,X23: nat] :
( ( Nat2
= ( suc @ X23 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1198_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1199_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1200_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P2 @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P2 @ X2 @ Y2 )
=> ( P2 @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1201_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1202_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1203_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1204_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1205_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1206_gen__length__code_I2_J,axiom,
! [N: nat,X: fm,Xs: list_fm] :
( ( gen_length_fm @ N @ ( cons_fm @ X @ Xs ) )
= ( gen_length_fm @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1207_gen__length__code_I2_J,axiom,
! [N: nat,X: tm,Xs: list_tm] :
( ( gen_length_tm @ N @ ( cons_tm @ X @ Xs ) )
= ( gen_length_tm @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1208_gen__length__code_I2_J,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1209_power__inject__base,axiom,
! [A3: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( power_power_nat @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% power_inject_base
thf(fact_1210_power__inject__base,axiom,
! [A3: int,N: nat,B2: int] :
( ( ( power_power_int @ A3 @ ( suc @ N ) )
= ( power_power_int @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% power_inject_base
thf(fact_1211_power__le__imp__le__base,axiom,
! [A3: nat,N: nat,B2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ) ).
% power_le_imp_le_base
thf(fact_1212_power__le__imp__le__base,axiom,
! [A3: int,N: nat,B2: int] :
( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ) ).
% power_le_imp_le_base
thf(fact_1213_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K2 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1214_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_fm] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_fm @ Xs ) )
= ( ? [X3: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_fm @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1215_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_tm] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_tm @ Xs ) )
= ( ? [X3: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_tm @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1216_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X3: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1217_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1218_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1219_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1220_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1221_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1222_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1223_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1224_power__Suc__le__self,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_1225_power__Suc__le__self,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_1226_power__Suc__less__one,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1227_power__Suc__less__one,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_1228_list_Osize_I4_J,axiom,
! [X21: fm,X22: list_fm] :
( ( size_size_list_fm @ ( cons_fm @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_fm @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1229_list_Osize_I4_J,axiom,
! [X21: tm,X22: list_tm] :
( ( size_size_list_tm @ ( cons_tm @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_tm @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1230_list_Osize_I4_J,axiom,
! [X21: int,X22: list_int] :
( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1231_length__Suc__conv__rev,axiom,
! [Xs: list_fm,N: nat] :
( ( ( size_size_list_fm @ Xs )
= ( suc @ N ) )
= ( ? [Y3: fm,Ys3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ Y3 @ nil_fm ) ) )
& ( ( size_size_list_fm @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1232_length__Suc__conv__rev,axiom,
! [Xs: list_tm,N: nat] :
( ( ( size_size_list_tm @ Xs )
= ( suc @ N ) )
= ( ? [Y3: tm,Ys3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ Y3 @ nil_tm ) ) )
& ( ( size_size_list_tm @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1233_length__Suc__conv__rev,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y3: int,Ys3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ Y3 @ nil_int ) ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1234_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1235_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1236_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1237_fm_Osize_I14_J,axiom,
! [X7: fm] :
( ( size_size_fm @ ( neg @ X7 ) )
= ( plus_plus_nat @ ( size_size_fm @ X7 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(14)
thf(fact_1238_fm_Osize__gen_I7_J,axiom,
! [X7: fm] :
( ( size_fm @ ( neg @ X7 ) )
= ( plus_plus_nat @ ( size_fm @ X7 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(7)
thf(fact_1239_fm_Osize_I9_J,axiom,
! [X21: fm,X22: fm] :
( ( size_size_fm @ ( imp @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X21 ) @ ( size_size_fm @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(9)
thf(fact_1240_fm_Osize__gen_I2_J,axiom,
! [X21: fm,X22: fm] :
( ( size_fm @ ( imp @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X21 ) @ ( size_fm @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(2)
thf(fact_1241_fm_Osize_I11_J,axiom,
! [X41: fm,X42: fm] :
( ( size_size_fm @ ( con @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X41 ) @ ( size_size_fm @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(11)
thf(fact_1242_fm_Osize_I10_J,axiom,
! [X31: fm,X32: fm] :
( ( size_size_fm @ ( dis @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X31 ) @ ( size_size_fm @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(10)
thf(fact_1243_fm_Osize_I13_J,axiom,
! [X62: fm] :
( ( size_size_fm @ ( uni @ X62 ) )
= ( plus_plus_nat @ ( size_size_fm @ X62 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(13)
thf(fact_1244_fm_Osize_I12_J,axiom,
! [X52: fm] :
( ( size_size_fm @ ( exi @ X52 ) )
= ( plus_plus_nat @ ( size_size_fm @ X52 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(12)
thf(fact_1245_fm_Osize__gen_I4_J,axiom,
! [X41: fm,X42: fm] :
( ( size_fm @ ( con @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X41 ) @ ( size_fm @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(4)
thf(fact_1246_fm_Osize__gen_I3_J,axiom,
! [X31: fm,X32: fm] :
( ( size_fm @ ( dis @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X31 ) @ ( size_fm @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(3)
thf(fact_1247_fm_Osize__gen_I6_J,axiom,
! [X62: fm] :
( ( size_fm @ ( uni @ X62 ) )
= ( plus_plus_nat @ ( size_fm @ X62 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(6)
thf(fact_1248_fm_Osize__gen_I5_J,axiom,
! [X52: fm] :
( ( size_fm @ ( exi @ X52 ) )
= ( plus_plus_nat @ ( size_fm @ X52 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(5)
thf(fact_1249_length__append__singleton,axiom,
! [Xs: list_tm,X: tm] :
( ( size_size_list_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) )
= ( suc @ ( size_size_list_tm @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1250_length__append__singleton,axiom,
! [Xs: list_int,X: int] :
( ( size_size_list_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1251_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_1252_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_1253_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_1254_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_1255_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons_int @ I @ nil_int ) ) ).
% upto_single
thf(fact_1256_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_1257_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_1258_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_1259_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_1260_upto__aux__def,axiom,
( upto_aux
= ( ^ [I4: int,J2: int] : ( append_int @ ( upto @ I4 @ J2 ) ) ) ) ).
% upto_aux_def
thf(fact_1261_upto__code,axiom,
( upto
= ( ^ [I4: int,J2: int] : ( upto_aux @ I4 @ J2 @ nil_int ) ) ) ).
% upto_code
thf(fact_1262_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_1263_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_1264_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_1265_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J2 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_1266_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_1267_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_1268_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_1269_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_1270_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_Suc_0_nat
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sequent_calculus @ ( cons_fm @ ( imp @ ( pre @ zero_zero_nat @ nil_tm ) @ ( pre @ zero_zero_nat @ nil_tm ) ) @ nil_fm ) ).
%------------------------------------------------------------------------------