TPTP Problem File: SLH0706^1.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_Calc3/0004_Syntax/prob_00071_001869__11784712_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1450 ( 613 unt; 168 typ; 0 def)
% Number of atoms : 3498 (1309 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9550 ( 395 ~; 59 |; 199 &;7503 @)
% ( 0 <=>;1394 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 517 ( 517 >; 0 *; 0 +; 0 <<)
% Number of symbols : 156 ( 153 usr; 17 con; 0-3 aty)
% Number of variables : 3333 ( 208 ^;3045 !; 80 ?;3333 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:21:41.637
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__List__Olist_It__List__Olist_It__Syntax__Ofm_J_J,type,
list_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Syntax__Ofm_J_J,type,
set_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Syntax__Ofm_J_J,type,
set_set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Syntax__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__List__Olist_It__Syntax__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Ofm_J,type,
set_fm: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Syntax__Ofm,type,
fm: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (153)
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Syntax__Ofm,type,
finite_Fpow_fm: set_fm > set_set_fm ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Syntax__Ofm,type,
finite_card_fm: set_fm > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
gcd_Gcd_int: set_int > int ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_Groups_Ogroup_001t__Int__Oint,type,
group_int: ( int > int > int ) > int > ( int > int ) > $o ).
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_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Syntax__Ofm_J,type,
minus_minus_set_fm: set_fm > set_fm > set_fm ).
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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Syntax__Ofm_J,type,
uminus_uminus_set_fm: set_fm > set_fm ).
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__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_It__Syntax__Ofm_J,type,
if_list_fm: $o > list_fm > list_fm > list_fm ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_List_Oarg__min__list_001t__Nat__Onat_001t__Nat__Onat,type,
arg_min_list_nat_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Syntax__Ofm,type,
can_select_fm: ( fm > $o ) > set_fm > $o ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Syntax__Ofm,type,
coset_fm: list_fm > set_fm ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Syntax__Ofm,type,
distinct_fm: list_fm > $o ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Syntax__Ofm,type,
drop_fm: nat > list_fm > list_fm ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Syntax__Ofm,type,
insert_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Syntax__Ofm,type,
last_fm: list_fm > fm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Syntax__Ofm,type,
cons_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Syntax__Ofm,type,
nil_fm: list_fm ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Syntax__Ofm,type,
hd_fm: list_fm > fm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Syntax__Ofm_J,type,
set_list_fm2: list_list_fm > set_list_fm ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Syntax__Ofm,type,
set_fm2: list_fm > set_fm ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Syntax__Ofm,type,
list_ex1_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Syntax__Ofm,type,
list_ex_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Syntax__Ofm,type,
list_update_fm: list_fm > nat > fm > list_fm ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Syntax__Ofm,type,
member_fm: list_fm > fm > $o ).
thf(sy_c_List_Omin__list_001t__Nat__Onat,type,
min_list_nat: list_nat > nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Syntax__Ofm,type,
nth_fm: list_fm > nat > fm ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001t__Syntax__Ofm,type,
nths_fm: list_fm > set_nat > list_fm ).
thf(sy_c_List_Onull_001t__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Onull_001t__Syntax__Ofm,type,
null_fm: list_fm > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__Nat__Onat,type,
lexordp_eq_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001t__Syntax__Ofm,type,
remdups_adj_fm: list_fm > list_fm ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001t__Syntax__Ofm,type,
remove1_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Syntax__Ofm,type,
replicate_fm: nat > fm > list_fm ).
thf(sy_c_List_Osplice_001t__Nat__Onat,type,
splice_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Syntax__Ofm,type,
subseqs_fm: list_fm > list_list_fm ).
thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Syntax__Ofm_J,type,
size_size_list_fm: list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Syntax__Ofm,type,
size_size_fm: fm > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Syntax__Ofm_M_Eo_J,type,
bot_bot_fm_o: fm > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Syntax__Ofm_J_J,type,
bot_bot_set_set_fm: set_set_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Ofm_J,type,
bot_bot_set_fm: set_fm ).
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_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Syntax__Ofm_J,type,
ord_less_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Int__Oint_J,type,
ord_less_eq_o_int: ( $o > int ) > ( $o > int ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Syntax__Ofm_J_J,type,
ord_less_eq_o_set_fm: ( $o > set_fm ) > ( $o > set_fm ) > $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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Syntax__Ofm_J_J,type,
ord_le1461404734466536732set_fm: set_set_fm > set_set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Ofm_J,type,
ord_less_eq_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Syntax__Ofm_J,type,
order_6179083242224974829set_fm: ( set_fm > $o ) > set_fm ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Syntax__Ofm,type,
collect_fm: ( fm > $o ) > set_fm ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Syntax__Ofm,type,
pow_fm: set_fm > set_set_fm ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Syntax__Ofm_J_001t__Set__Oset_It__Syntax__Ofm_J,type,
image_list_fm_set_fm: ( list_fm > set_fm ) > set_list_fm > set_set_fm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Syntax__Ofm,type,
image_nat_fm: ( nat > fm ) > set_nat > set_fm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Syntax__Ofm_001t__Nat__Onat,type,
image_fm_nat: ( fm > nat ) > set_fm > set_nat ).
thf(sy_c_Set_Oimage_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
image_fm_fm: ( fm > fm ) > set_fm > set_fm ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Syntax__Ofm_J,type,
insert_set_fm: set_fm > set_set_fm > set_set_fm ).
thf(sy_c_Set_Oinsert_001t__Syntax__Ofm,type,
insert_fm2: fm > set_fm > set_fm ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Syntax__Ofm,type,
is_empty_fm: set_fm > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Syntax__Ofm,type,
is_singleton_fm: set_fm > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Syntax__Ofm,type,
remove_fm: fm > set_fm > set_fm ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Syntax__Ofm,type,
the_elem_fm: set_fm > fm ).
thf(sy_c_Syntax_Ofm_OFalsity,type,
falsity: fm ).
thf(sy_c_Syntax_Ofm_OImp,type,
imp: fm > fm > fm ).
thf(sy_c_Syntax_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_Syntax_Ofm_OUni,type,
uni: fm > fm ).
thf(sy_c_Syntax_Ofm_Osize__fm,type,
size_fm: fm > nat ).
thf(sy_c_Syntax_Omax__list,type,
max_list: list_nat > nat ).
thf(sy_c_Syntax_Ovars__fm,type,
vars_fm: fm > list_nat ).
thf(sy_c_Syntax_Ovars__fms,type,
vars_fms: list_fm > list_nat ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Syntax__Ofm_J,type,
member_set_fm: set_fm > set_set_fm > $o ).
thf(sy_c_member_001t__Syntax__Ofm,type,
member_fm2: fm > set_fm > $o ).
thf(sy_v_A,type,
a: list_fm ).
thf(sy_v_p,type,
p: fm ).
% Relevant facts (1273)
thf(fact_0_subsetI,axiom,
! [A: set_fm,B: set_fm] :
( ! [X: fm] :
( ( member_fm2 @ X @ A )
=> ( member_fm2 @ X @ B ) )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% subsetI
thf(fact_1_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_2_subset__antisym,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_3_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_4_order__refl,axiom,
! [X2: set_fm] : ( ord_less_eq_set_fm @ X2 @ X2 ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_6_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_7_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_8_dual__order_Orefl,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_9_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_11_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_12_subset__code_I1_J,axiom,
! [Xs: list_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_13_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_14_in__mono,axiom,
! [A: set_fm,B: set_fm,X2: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm2 @ X2 @ A )
=> ( member_fm2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_15_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_16_subsetD,axiom,
! [A: set_fm,B: set_fm,C: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ B ) ) ) ).
% subsetD
thf(fact_17_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% subsetD
thf(fact_18_equalityE,axiom,
! [A: set_fm,B: set_fm] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_fm @ A @ B )
=> ~ ( ord_less_eq_set_fm @ B @ A ) ) ) ).
% equalityE
thf(fact_19_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_20_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B2: set_fm] :
! [X3: fm] :
( ( member_fm2 @ X3 @ A3 )
=> ( member_fm2 @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_21_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
=> ( member_nat2 @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_22_equalityD1,axiom,
! [A: set_fm,B: set_fm] :
( ( A = B )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% equalityD1
thf(fact_23_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_24_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_25_equalityD2,axiom,
! [A: set_fm,B: set_fm] :
( ( A = B )
=> ( ord_less_eq_set_fm @ B @ A ) ) ).
% equalityD2
thf(fact_26_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat2 @ T @ A3 )
=> ( member_nat2 @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_27_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B2: set_fm] :
! [T: fm] :
( ( member_fm2 @ T @ A3 )
=> ( member_fm2 @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_28_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_29_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_30_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_31_order__antisym__conv,axiom,
! [Y: set_fm,X2: set_fm] :
( ( ord_less_eq_set_fm @ Y @ X2 )
=> ( ( ord_less_eq_set_fm @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_32_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_33_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_34_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_35_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_36_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_37_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_38_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_39_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_41_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_42_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_43_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > set_fm,C: set_fm] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_45_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_46_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_47_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_48_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_49_ord__eq__le__subst,axiom,
! [A2: int,F: set_nat > int,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_50_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_ord__eq__le__subst,axiom,
! [A2: set_fm,F: nat > set_fm,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_ord__eq__le__subst,axiom,
! [A2: set_nat,F: int > set_nat,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_53_ord__eq__le__subst,axiom,
! [A2: set_fm,F: int > set_fm,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_55_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_56_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_57_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_58_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_59_order__eq__refl,axiom,
! [X2: set_fm,Y: set_fm] :
( ( X2 = Y )
=> ( ord_less_eq_set_fm @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_60_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_61_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_62_order__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_63_order__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_64_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_65_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_66_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_67_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_fm @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_68_order__subst2,axiom,
! [A2: int,B3: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_69_order__subst2,axiom,
! [A2: int,B3: int,F: int > set_fm,C: set_fm] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_fm @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_70_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_71_order__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_72_order__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_73_order__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_74_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_order__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B3: int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_76_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_77_order__subst1,axiom,
! [A2: nat,F: set_fm > nat,B3: set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_fm @ B3 @ C )
=> ( ! [X: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_78_order__subst1,axiom,
! [A2: int,F: set_nat > int,B3: set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_79_order__subst1,axiom,
! [A2: int,F: set_fm > int,B3: set_fm,C: set_fm] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_fm @ B3 @ C )
=> ( ! [X: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_80_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_81_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_82_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_83_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_fm,Z: set_fm] : ( Y3 = Z ) )
= ( ^ [A4: set_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
& ( ord_less_eq_set_fm @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_84_antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_85_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_86_antisym,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_87_antisym,axiom,
! [A2: set_fm,B3: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B3 )
=> ( ( ord_less_eq_set_fm @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_88_dual__order_Otrans,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_89_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_90_dual__order_Otrans,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_91_dual__order_Otrans,axiom,
! [B3: set_fm,A2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ B3 @ A2 )
=> ( ( ord_less_eq_set_fm @ C @ B3 )
=> ( ord_less_eq_set_fm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_92_dual__order_Oantisym,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_93_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_94_dual__order_Oantisym,axiom,
! [B3: int,A2: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_95_dual__order_Oantisym,axiom,
! [B3: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ B3 @ A2 )
=> ( ( ord_less_eq_set_fm @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_96_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_97_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_98_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_99_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_fm,Z: set_fm] : ( Y3 = Z ) )
= ( ^ [A4: set_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ B4 @ A4 )
& ( ord_less_eq_set_fm @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_100_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_101_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B3: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_102_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_103_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_104_order__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_105_order__trans,axiom,
! [X2: set_fm,Y: set_fm,Z2: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y )
=> ( ( ord_less_eq_set_fm @ Y @ Z2 )
=> ( ord_less_eq_set_fm @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_106_order_Otrans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_107_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_108_order_Otrans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_109_order_Otrans,axiom,
! [A2: set_fm,B3: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B3 )
=> ( ( ord_less_eq_set_fm @ B3 @ C )
=> ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_110_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_111_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_112_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_113_order__antisym,axiom,
! [X2: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y )
=> ( ( ord_less_eq_set_fm @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_114_ord__le__eq__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_115_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_116_ord__le__eq__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_117_ord__le__eq__trans,axiom,
! [A2: set_fm,B3: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_118_ord__eq__le__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_119_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_120_ord__eq__le__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( A2 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_121_ord__eq__le__trans,axiom,
! [A2: set_fm,B3: set_fm,C: set_fm] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_fm @ B3 @ C )
=> ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_122_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_123_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_124_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_125_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_fm,Z: set_fm] : ( Y3 = Z ) )
= ( ^ [X3: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y4 )
& ( ord_less_eq_set_fm @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_126_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_127_le__cases3,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_128_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_129_nle__le,axiom,
! [A2: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_130_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_131_Collect__mono__iff,axiom,
! [P: fm > $o,Q: fm > $o] :
( ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q ) )
= ( ! [X3: fm] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_132_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_133_set__eq__subset,axiom,
( ( ^ [Y3: set_fm,Z: set_fm] : ( Y3 = Z ) )
= ( ^ [A3: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A3 @ B2 )
& ( ord_less_eq_set_fm @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_134_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_135_subset__trans,axiom,
! [A: set_fm,B: set_fm,C2: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C2 )
=> ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_136_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_137_Collect__mono,axiom,
! [P: fm > $o,Q: fm > $o] :
( ! [X: fm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q ) ) ) ).
% Collect_mono
thf(fact_138_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_139_subset__refl,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).
% subset_refl
thf(fact_140_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_141_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_142_Greatest__equality,axiom,
! [P: int > $o,X2: int] :
( ( P @ X2 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( order_Greatest_int @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_143_Greatest__equality,axiom,
! [P: set_fm > $o,X2: set_fm] :
( ( P @ X2 )
=> ( ! [Y2: set_fm] :
( ( P @ Y2 )
=> ( ord_less_eq_set_fm @ Y2 @ X2 ) )
=> ( ( order_6179083242224974829set_fm @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_144_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_145_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_146_GreatestI2__order,axiom,
! [P: int > $o,X2: int,Q: int > $o] :
( ( P @ X2 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ! [X: int] :
( ( P @ X )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_147_GreatestI2__order,axiom,
! [P: set_fm > $o,X2: set_fm,Q: set_fm > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_fm] :
( ( P @ Y2 )
=> ( ord_less_eq_set_fm @ Y2 @ X2 ) )
=> ( ! [X: set_fm] :
( ( P @ X )
=> ( ! [Y5: set_fm] :
( ( P @ Y5 )
=> ( ord_less_eq_set_fm @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_6179083242224974829set_fm @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_148_mem__Collect__eq,axiom,
! [A2: fm,P: fm > $o] :
( ( member_fm2 @ A2 @ ( collect_fm @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_fm] :
( ( collect_fm
@ ^ [X3: fm] : ( member_fm2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_152_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X3 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_153_subset__code_I2_J,axiom,
! [A: set_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ A @ ( coset_fm @ Ys ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Ys ) )
=> ~ ( member_fm2 @ X3 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_154_in__set__member,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
= ( member_fm @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_155_in__set__member,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_156_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X4: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_157_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_158_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_int
= ( ^ [X4: $o > int,Y6: $o > int] :
( ( ord_less_eq_int @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_int @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_159_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_fm
= ( ^ [X4: $o > set_fm,Y6: $o > set_fm] :
( ( ord_less_eq_set_fm @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_fm @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_160_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_161_verit__la__disequality,axiom,
! [A2: int,B3: int] :
( ( A2 = B3 )
| ~ ( ord_less_eq_int @ A2 @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_162_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_163_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_164_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_165_verit__comp__simplify1_I2_J,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_166_in__set__insert,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_167_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_168_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P2: fm > $o,Xs2: list_fm] :
? [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y4: fm] :
( ( ( member_fm2 @ Y4 @ ( set_fm2 @ Xs2 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_169_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y4: nat] :
( ( ( member_nat2 @ Y4 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_170_set__nths__subset,axiom,
! [Xs: list_nat,I: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) @ ( set_nat2 @ Xs ) ) ).
% set_nths_subset
thf(fact_171_set__nths__subset,axiom,
! [Xs: list_fm,I: set_nat] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( nths_fm @ Xs @ I ) ) @ ( set_fm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_172_vars__fm_Osimps_I4_J,axiom,
! [P3: fm] :
( ( vars_fm @ ( uni @ P3 ) )
= ( vars_fm @ P3 ) ) ).
% vars_fm.simps(4)
thf(fact_173_set__remove1__subset,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X2 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_174_set__remove1__subset,axiom,
! [X2: fm,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( remove1_fm @ X2 @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_175_fm_Oinject_I3_J,axiom,
! [X42: fm,Y42: fm] :
( ( ( uni @ X42 )
= ( uni @ Y42 ) )
= ( X42 = Y42 ) ) ).
% fm.inject(3)
thf(fact_176_in__set__remove1,axiom,
! [A2: fm,B3: fm,Xs: list_fm] :
( ( A2 != B3 )
=> ( ( member_fm2 @ A2 @ ( set_fm2 @ ( remove1_fm @ B3 @ Xs ) ) )
= ( member_fm2 @ A2 @ ( set_fm2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_177_in__set__remove1,axiom,
! [A2: nat,B3: nat,Xs: list_nat] :
( ( A2 != B3 )
=> ( ( member_nat2 @ A2 @ ( set_nat2 @ ( remove1_nat @ B3 @ Xs ) ) )
= ( member_nat2 @ A2 @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_178_remove1__idem,axiom,
! [X2: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( remove1_fm @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_179_remove1__idem,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_180_notin__set__remove1,axiom,
! [X2: fm,Xs: list_fm,Y: fm] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ~ ( member_fm2 @ X2 @ ( set_fm2 @ ( remove1_fm @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_181_notin__set__remove1,axiom,
! [X2: nat,Xs: list_nat,Y: nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X2 @ ( set_nat2 @ ( remove1_nat @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_182_in__set__nthsD,axiom,
! [X2: fm,Xs: list_fm,I: set_nat] :
( ( member_fm2 @ X2 @ ( set_fm2 @ ( nths_fm @ Xs @ I ) ) )
=> ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_183_in__set__nthsD,axiom,
! [X2: nat,Xs: list_nat,I: set_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_184_notin__set__nthsI,axiom,
! [X2: fm,Xs: list_fm,I: set_nat] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ~ ( member_fm2 @ X2 @ ( set_fm2 @ ( nths_fm @ Xs @ I ) ) ) ) ).
% notin_set_nthsI
thf(fact_185_notin__set__nthsI,axiom,
! [X2: nat,Xs: list_nat,I: set_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) ) ) ).
% notin_set_nthsI
thf(fact_186_can__select__set__list__ex1,axiom,
! [P: fm > $o,A: list_fm] :
( ( can_select_fm @ P @ ( set_fm2 @ A ) )
= ( list_ex1_fm @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_187_can__select__set__list__ex1,axiom,
! [P: nat > $o,A: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_188_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_189_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_fm @ ( coset_fm @ nil_fm ) @ ( set_fm2 @ nil_fm ) ) ).
% subset_code(3)
thf(fact_190_not__in__set__insert,axiom,
! [X2: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X2 @ Xs )
= ( cons_fm @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_191_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_192_List_Oset__insert,axiom,
! [X2: fm,Xs: list_fm] :
( ( set_fm2 @ ( insert_fm @ X2 @ Xs ) )
= ( insert_fm2 @ X2 @ ( set_fm2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_193_List_Oset__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X2 @ Xs ) )
= ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_194_fm_Odistinct_I5_J,axiom,
! [X42: fm] :
( falsity
!= ( uni @ X42 ) ) ).
% fm.distinct(5)
thf(fact_195_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X3: fm,Xs2: list_fm] : ( if_list_fm @ ( member_fm2 @ X3 @ ( set_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_fm @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_196_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_197_compl__coset,axiom,
! [Xs: list_fm] :
( ( uminus_uminus_set_fm @ ( coset_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% compl_coset
thf(fact_198_compl__coset,axiom,
! [Xs: list_nat] :
( ( uminus5710092332889474511et_nat @ ( coset_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% compl_coset
thf(fact_199_coset__def,axiom,
( coset_fm
= ( ^ [Xs2: list_fm] : ( uminus_uminus_set_fm @ ( set_fm2 @ Xs2 ) ) ) ) ).
% coset_def
thf(fact_200_coset__def,axiom,
( coset_nat
= ( ^ [Xs2: list_nat] : ( uminus5710092332889474511et_nat @ ( set_nat2 @ Xs2 ) ) ) ) ).
% coset_def
thf(fact_201_verit__minus__simplify_I4_J,axiom,
! [B3: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_202_insert__absorb2,axiom,
! [X2: nat,A: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ X2 @ A ) )
= ( insert_nat2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_203_insert__iff,axiom,
! [A2: fm,B3: fm,A: set_fm] :
( ( member_fm2 @ A2 @ ( insert_fm2 @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_fm2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_204_insert__iff,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_205_insertCI,axiom,
! [A2: fm,B: set_fm,B3: fm] :
( ( ~ ( member_fm2 @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_fm2 @ A2 @ ( insert_fm2 @ B3 @ B ) ) ) ).
% insertCI
thf(fact_206_insertCI,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( ~ ( member_nat2 @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B3 @ B ) ) ) ).
% insertCI
thf(fact_207_Compl__iff,axiom,
! [C: fm,A: set_fm] :
( ( member_fm2 @ C @ ( uminus_uminus_set_fm @ A ) )
= ( ~ ( member_fm2 @ C @ A ) ) ) ).
% Compl_iff
thf(fact_208_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat2 @ C @ A ) ) ) ).
% Compl_iff
thf(fact_209_ComplI,axiom,
! [C: fm,A: set_fm] :
( ~ ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ ( uminus_uminus_set_fm @ A ) ) ) ).
% ComplI
thf(fact_210_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_211_member__remove,axiom,
! [X2: fm,Y: fm,A: set_fm] :
( ( member_fm2 @ X2 @ ( remove_fm @ Y @ A ) )
= ( ( member_fm2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_212_member__remove,axiom,
! [X2: nat,Y: nat,A: set_nat] :
( ( member_nat2 @ X2 @ ( remove_nat @ Y @ A ) )
= ( ( member_nat2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_213_insert__subset,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ A ) @ B )
= ( ( member_nat2 @ X2 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_214_insert__subset,axiom,
! [X2: fm,A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( insert_fm2 @ X2 @ A ) @ B )
= ( ( member_fm2 @ X2 @ B )
& ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% insert_subset
thf(fact_215_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_216_Compl__subset__Compl__iff,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( uminus_uminus_set_fm @ A ) @ ( uminus_uminus_set_fm @ B ) )
= ( ord_less_eq_set_fm @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_217_Compl__anti__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_218_Compl__anti__mono,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ ( uminus_uminus_set_fm @ B ) @ ( uminus_uminus_set_fm @ A ) ) ) ).
% Compl_anti_mono
thf(fact_219_nths__nil,axiom,
! [A: set_nat] :
( ( nths_nat @ nil_nat @ A )
= nil_nat ) ).
% nths_nil
thf(fact_220_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_221_list_Osimps_I15_J,axiom,
! [X21: fm,X22: list_fm] :
( ( set_fm2 @ ( cons_fm @ X21 @ X22 ) )
= ( insert_fm2 @ X21 @ ( set_fm2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_222_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_223_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_224_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B3: int] :
( ( A2 = B3 )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B3 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_225_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_226_mk__disjoint__insert,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ A )
=> ? [B6: set_fm] :
( ( A
= ( insert_fm2 @ A2 @ B6 ) )
& ~ ( member_fm2 @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_227_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ? [B6: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B6 ) )
& ~ ( member_nat2 @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_228_insert__commute,axiom,
! [X2: nat,Y: nat,A: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ Y @ A ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_229_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X @ Xs3 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X: nat,Xs3: list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_230_insert__eq__iff,axiom,
! [A2: fm,A: set_fm,B3: fm,B: set_fm] :
( ~ ( member_fm2 @ A2 @ A )
=> ( ~ ( member_fm2 @ B3 @ B )
=> ( ( ( insert_fm2 @ A2 @ A )
= ( insert_fm2 @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_fm] :
( ( A
= ( insert_fm2 @ B3 @ C3 ) )
& ~ ( member_fm2 @ B3 @ C3 )
& ( B
= ( insert_fm2 @ A2 @ C3 ) )
& ~ ( member_fm2 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_231_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B3: nat,B: set_nat] :
( ~ ( member_nat2 @ A2 @ A )
=> ( ~ ( member_nat2 @ B3 @ B )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat2 @ B3 @ C3 ) )
& ~ ( member_nat2 @ B3 @ C3 )
& ( B
= ( insert_nat2 @ A2 @ C3 ) )
& ~ ( member_nat2 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_232_insert__absorb,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ A )
=> ( ( insert_fm2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_233_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_234_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_235_insert__ident,axiom,
! [X2: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm2 @ X2 @ A )
=> ( ~ ( member_fm2 @ X2 @ B )
=> ( ( ( insert_fm2 @ X2 @ A )
= ( insert_fm2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_236_insert__ident,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ A )
= ( insert_nat2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_237_Set_Oset__insert,axiom,
! [X2: fm,A: set_fm] :
( ( member_fm2 @ X2 @ A )
=> ~ ! [B6: set_fm] :
( ( A
= ( insert_fm2 @ X2 @ B6 ) )
=> ( member_fm2 @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_238_Set_Oset__insert,axiom,
! [X2: nat,A: set_nat] :
( ( member_nat2 @ X2 @ A )
=> ~ ! [B6: set_nat] :
( ( A
= ( insert_nat2 @ X2 @ B6 ) )
=> ( member_nat2 @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_239_insertI2,axiom,
! [A2: fm,B: set_fm,B3: fm] :
( ( member_fm2 @ A2 @ B )
=> ( member_fm2 @ A2 @ ( insert_fm2 @ B3 @ B ) ) ) ).
% insertI2
thf(fact_240_insertI2,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( member_nat2 @ A2 @ B )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B3 @ B ) ) ) ).
% insertI2
thf(fact_241_insertI1,axiom,
! [A2: fm,B: set_fm] : ( member_fm2 @ A2 @ ( insert_fm2 @ A2 @ B ) ) ).
% insertI1
thf(fact_242_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_243_insertE,axiom,
! [A2: fm,B3: fm,A: set_fm] :
( ( member_fm2 @ A2 @ ( insert_fm2 @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_fm2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_244_insertE,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_245_ComplD,axiom,
! [C: fm,A: set_fm] :
( ( member_fm2 @ C @ ( uminus_uminus_set_fm @ A ) )
=> ~ ( member_fm2 @ C @ A ) ) ).
% ComplD
thf(fact_246_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat2 @ C @ A ) ) ).
% ComplD
thf(fact_247_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y2: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_248_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_249_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_250_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_251_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_252_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_253_vars__fm_Osimps_I1_J,axiom,
( ( vars_fm @ falsity )
= nil_nat ) ).
% vars_fm.simps(1)
thf(fact_254_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_255_subset__insertI2,axiom,
! [A: set_fm,B: set_fm,B3: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_256_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_257_subset__insertI,axiom,
! [B: set_fm,A2: fm] : ( ord_less_eq_set_fm @ B @ ( insert_fm2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_258_subset__insert,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_259_subset__insert,axiom,
! [X2: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm2 @ X2 @ A )
=> ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X2 @ B ) )
= ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% subset_insert
thf(fact_260_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C2 ) @ ( insert_nat2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_261_insert__mono,axiom,
! [C2: set_fm,D: set_fm,A2: fm] :
( ( ord_less_eq_set_fm @ C2 @ D )
=> ( ord_less_eq_set_fm @ ( insert_fm2 @ A2 @ C2 ) @ ( insert_fm2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_262_set__ConsD,axiom,
! [Y: fm,X2: fm,Xs: list_fm] :
( ( member_fm2 @ Y @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_fm2 @ Y @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_263_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_264_list_Oset__cases,axiom,
! [E: fm,A2: list_fm] :
( ( member_fm2 @ E @ ( set_fm2 @ A2 ) )
=> ( ! [Z22: list_fm] :
( A2
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A2
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm2 @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_265_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_266_list_Oset__intros_I1_J,axiom,
! [X21: fm,X22: list_fm] : ( member_fm2 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_267_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_268_list_Oset__intros_I2_J,axiom,
! [Y: fm,X22: list_fm,X21: fm] :
( ( member_fm2 @ Y @ ( set_fm2 @ X22 ) )
=> ( member_fm2 @ Y @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_269_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_270_can__select__def,axiom,
( can_select_fm
= ( ^ [P2: fm > $o,A3: set_fm] :
? [X3: fm] :
( ( member_fm2 @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y4: fm] :
( ( ( member_fm2 @ Y4 @ A3 )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_271_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y4: nat] :
( ( ( member_nat2 @ Y4 @ A3 )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_272_remove1_Osimps_I1_J,axiom,
! [X2: nat] :
( ( remove1_nat @ X2 @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_273_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_274_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_275_set__subset__Cons,axiom,
! [Xs: list_fm,X2: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_276_compl__le__compl__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_277_compl__le__compl__iff,axiom,
! [X2: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ ( uminus_uminus_set_fm @ X2 ) @ ( uminus_uminus_set_fm @ Y ) )
= ( ord_less_eq_set_fm @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_278_neg__le__iff__le,axiom,
! [B3: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_279_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_280_neg__equal__iff__equal,axiom,
! [A2: int,B3: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B3 ) )
= ( A2 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_281_the__elem__set,axiom,
! [X2: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X2 @ nil_fm ) ) )
= X2 ) ).
% the_elem_set
thf(fact_282_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_283_insert__subsetI,axiom,
! [X2: nat,A: set_nat,X5: set_nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_284_insert__subsetI,axiom,
! [X2: fm,A: set_fm,X5: set_fm] :
( ( member_fm2 @ X2 @ A )
=> ( ( ord_less_eq_set_fm @ X5 @ A )
=> ( ord_less_eq_set_fm @ ( insert_fm2 @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_285_compl__le__swap2,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_286_compl__le__swap2,axiom,
! [Y: set_fm,X2: set_fm] :
( ( ord_less_eq_set_fm @ ( uminus_uminus_set_fm @ Y ) @ X2 )
=> ( ord_less_eq_set_fm @ ( uminus_uminus_set_fm @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_287_minus__equation__iff,axiom,
! [A2: int,B3: int] :
( ( ( uminus_uminus_int @ A2 )
= B3 )
= ( ( uminus_uminus_int @ B3 )
= A2 ) ) ).
% minus_equation_iff
thf(fact_288_equation__minus__iff,axiom,
! [A2: int,B3: int] :
( ( A2
= ( uminus_uminus_int @ B3 ) )
= ( B3
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_289_le__minus__iff,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_290_minus__le__iff,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_291_le__imp__neg__le,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_292_compl__mono,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X2 ) ) ) ).
% compl_mono
thf(fact_293_compl__mono,axiom,
! [X2: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y )
=> ( ord_less_eq_set_fm @ ( uminus_uminus_set_fm @ Y ) @ ( uminus_uminus_set_fm @ X2 ) ) ) ).
% compl_mono
thf(fact_294_compl__le__swap1,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X2 ) )
=> ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_295_compl__le__swap1,axiom,
! [Y: set_fm,X2: set_fm] :
( ( ord_less_eq_set_fm @ Y @ ( uminus_uminus_set_fm @ X2 ) )
=> ( ord_less_eq_set_fm @ X2 @ ( uminus_uminus_set_fm @ Y ) ) ) ).
% compl_le_swap1
thf(fact_296_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_297_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_298_nths__singleton,axiom,
! [A: set_nat,X2: nat] :
( ( ( member_nat2 @ zero_zero_nat @ A )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A )
= ( cons_nat @ X2 @ nil_nat ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_299_subset__Compl__singleton,axiom,
! [A: set_nat,B3: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat2 @ B3 @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat2 @ B3 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_300_subset__Compl__singleton,axiom,
! [A: set_fm,B3: fm] :
( ( ord_less_eq_set_fm @ A @ ( uminus_uminus_set_fm @ ( insert_fm2 @ B3 @ bot_bot_set_fm ) ) )
= ( ~ ( member_fm2 @ B3 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_301_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: nat > nat > $o,X2: nat,Xs: list_nat] :
~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% ord.lexordp_eq_simps(3)
thf(fact_302_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_303_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_304_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_305_all__not__in__conv,axiom,
! [A: set_fm] :
( ( ! [X3: fm] :
~ ( member_fm2 @ X3 @ A ) )
= ( A = bot_bot_set_fm ) ) ).
% all_not_in_conv
thf(fact_306_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat2 @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_307_empty__iff,axiom,
! [C: fm] :
~ ( member_fm2 @ C @ bot_bot_set_fm ) ).
% empty_iff
thf(fact_308_empty__iff,axiom,
! [C: nat] :
~ ( member_nat2 @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_309_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_310_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_311_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_312_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_313_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_314_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_315_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_316_empty__subsetI,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A ) ).
% empty_subsetI
thf(fact_317_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_318_subset__empty,axiom,
! [A: set_fm] :
( ( ord_less_eq_set_fm @ A @ bot_bot_set_fm )
= ( A = bot_bot_set_fm ) ) ).
% subset_empty
thf(fact_319_singletonI,axiom,
! [A2: fm] : ( member_fm2 @ A2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ).
% singletonI
thf(fact_320_singletonI,axiom,
! [A2: nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_321_nths__empty,axiom,
! [Xs: list_nat] :
( ( nths_nat @ Xs @ bot_bot_set_nat )
= nil_nat ) ).
% nths_empty
thf(fact_322_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_323_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs: list_nat] :
( ( lexordp_eq_nat @ Less @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_324_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_325_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_326_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_327_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_328_singleton__insert__inj__eq,axiom,
! [B3: nat,A2: nat,A: set_nat] :
( ( ( insert_nat2 @ B3 @ bot_bot_set_nat )
= ( insert_nat2 @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_329_singleton__insert__inj__eq,axiom,
! [B3: fm,A2: fm,A: set_fm] :
( ( ( insert_fm2 @ B3 @ bot_bot_set_fm )
= ( insert_fm2 @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B3 @ bot_bot_set_fm ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_330_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B3: nat] :
( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B3 @ bot_bot_set_nat ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_331_singleton__insert__inj__eq_H,axiom,
! [A2: fm,A: set_fm,B3: fm] :
( ( ( insert_fm2 @ A2 @ A )
= ( insert_fm2 @ B3 @ bot_bot_set_fm ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B3 @ bot_bot_set_fm ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_332_set__empty,axiom,
! [Xs: list_fm] :
( ( ( set_fm2 @ Xs )
= bot_bot_set_fm )
= ( Xs = nil_fm ) ) ).
% set_empty
thf(fact_333_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_334_set__empty2,axiom,
! [Xs: list_fm] :
( ( bot_bot_set_fm
= ( set_fm2 @ Xs ) )
= ( Xs = nil_fm ) ) ).
% set_empty2
thf(fact_335_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_336_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_337_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_338_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_339_ex__in__conv,axiom,
! [A: set_fm] :
( ( ? [X3: fm] : ( member_fm2 @ X3 @ A ) )
= ( A != bot_bot_set_fm ) ) ).
% ex_in_conv
thf(fact_340_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat2 @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_341_equals0I,axiom,
! [A: set_fm] :
( ! [Y2: fm] :
~ ( member_fm2 @ Y2 @ A )
=> ( A = bot_bot_set_fm ) ) ).
% equals0I
thf(fact_342_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_343_equals0D,axiom,
! [A: set_fm,A2: fm] :
( ( A = bot_bot_set_fm )
=> ~ ( member_fm2 @ A2 @ A ) ) ).
% equals0D
thf(fact_344_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_345_emptyE,axiom,
! [A2: fm] :
~ ( member_fm2 @ A2 @ bot_bot_set_fm ) ).
% emptyE
thf(fact_346_emptyE,axiom,
! [A2: nat] :
~ ( member_nat2 @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_347_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_348_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_349_bot_Oextremum__uniqueI,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
=> ( A2 = bot_bot_set_fm ) ) ).
% bot.extremum_uniqueI
thf(fact_350_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_351_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_352_bot_Oextremum__unique,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
= ( A2 = bot_bot_set_fm ) ) ).
% bot.extremum_unique
thf(fact_353_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_354_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_355_bot_Oextremum,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A2 ) ).
% bot.extremum
thf(fact_356_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_357_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X: nat] :
~ ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_358_subset__emptyI,axiom,
! [A: set_fm] :
( ! [X: fm] :
~ ( member_fm2 @ X @ A )
=> ( ord_less_eq_set_fm @ A @ bot_bot_set_fm ) ) ).
% subset_emptyI
thf(fact_359_singleton__inject,axiom,
! [A2: nat,B3: nat] :
( ( ( insert_nat2 @ A2 @ bot_bot_set_nat )
= ( insert_nat2 @ B3 @ bot_bot_set_nat ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_360_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_361_doubleton__eq__iff,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B3 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_362_singleton__iff,axiom,
! [B3: fm,A2: fm] :
( ( member_fm2 @ B3 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_363_singleton__iff,axiom,
! [B3: nat,A2: nat] :
( ( member_nat2 @ B3 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_364_singletonD,axiom,
! [B3: fm,A2: fm] :
( ( member_fm2 @ B3 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_365_singletonD,axiom,
! [B3: nat,A2: nat] :
( ( member_nat2 @ B3 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_366_ord_Olexordp__eq_ONil,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_367_subset__singletonD,axiom,
! [A: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_368_subset__singletonD,axiom,
! [A: set_fm,X2: fm] :
( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X2 @ bot_bot_set_fm ) )
=> ( ( A = bot_bot_set_fm )
| ( A
= ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ) ).
% subset_singletonD
thf(fact_369_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_370_subset__singleton__iff,axiom,
! [X5: set_fm,A2: fm] :
( ( ord_less_eq_set_fm @ X5 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
= ( ( X5 = bot_bot_set_fm )
| ( X5
= ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ) ) ).
% subset_singleton_iff
thf(fact_371_empty__set,axiom,
( bot_bot_set_fm
= ( set_fm2 @ nil_fm ) ) ).
% empty_set
thf(fact_372_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_373_subset__Compl__self__eq,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_374_subset__Compl__self__eq,axiom,
! [A: set_fm] :
( ( ord_less_eq_set_fm @ A @ ( uminus_uminus_set_fm @ A ) )
= ( A = bot_bot_set_fm ) ) ).
% subset_Compl_self_eq
thf(fact_375_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_nat
= ( ^ [Less2: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ? [Ys3: list_nat] :
( ( A1 = nil_nat )
& ( A22 = Ys3 ) )
| ? [X3: nat,Y4: nat,Xs2: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( Less2 @ X3 @ Y4 ) )
| ? [X3: nat,Y4: nat,Xs2: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y4 )
& ~ ( Less2 @ Y4 @ X3 )
& ( lexordp_eq_nat @ Less2 @ Xs2 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_376_ord_Olexordp__eq_Ocases,axiom,
! [Less: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ( lexordp_eq_nat @ Less @ A12 @ A23 )
=> ( ( A12 != nil_nat )
=> ( ! [X: nat] :
( ? [Xs3: list_nat] :
( A12
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y2: nat] :
( ? [Ys2: list_nat] :
( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( Less @ X @ Y2 ) ) )
=> ~ ! [X: nat,Y2: nat,Xs3: list_nat] :
( ( A12
= ( cons_nat @ X @ Xs3 ) )
=> ! [Ys2: list_nat] :
( ( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ~ ( lexordp_eq_nat @ Less @ Xs3 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_377_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_378_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_379_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_380_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_381_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat2 @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_382_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_383_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_384_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_385_fm_Osize__gen_I1_J,axiom,
( ( size_fm @ falsity )
= zero_zero_nat ) ).
% fm.size_gen(1)
thf(fact_386_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_387_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_388_is__singletonI_H,axiom,
! [A: set_fm] :
( ( A != bot_bot_set_fm )
=> ( ! [X: fm,Y2: fm] :
( ( member_fm2 @ X @ A )
=> ( ( member_fm2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_fm @ A ) ) ) ).
% is_singletonI'
thf(fact_389_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] :
( ( member_nat2 @ X @ A )
=> ( ( member_nat2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_390_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_391_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_392_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_393_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_394_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X: nat] :
( A
!= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_395_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X3: nat] :
( A3
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_396_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_397_set__replicate,axiom,
! [N: nat,X2: fm] :
( ( N != zero_zero_nat )
=> ( ( set_fm2 @ ( replicate_fm @ N @ X2 ) )
= ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ).
% set_replicate
thf(fact_398_set__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_399_max__list_Osimps_I1_J,axiom,
( ( max_list @ nil_nat )
= zero_zero_nat ) ).
% max_list.simps(1)
thf(fact_400_subset__subseqs,axiom,
! [X5: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat @ X5 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_401_subset__subseqs,axiom,
! [X5: set_fm,Xs: list_fm] :
( ( ord_less_eq_set_fm @ X5 @ ( set_fm2 @ Xs ) )
=> ( member_set_fm @ X5 @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_402_fm_Osize_I5_J,axiom,
( ( size_size_fm @ falsity )
= zero_zero_nat ) ).
% fm.size(5)
thf(fact_403_transpose__empty,axiom,
! [Xs: list_list_nat] :
( ( ( transpose_nat @ Xs )
= nil_list_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( X3 = nil_nat ) ) ) ) ).
% transpose_empty
thf(fact_404_image__eqI,axiom,
! [B3: fm,F: fm > fm,X2: fm,A: set_fm] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_fm2 @ X2 @ A )
=> ( member_fm2 @ B3 @ ( image_fm_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_405_image__eqI,axiom,
! [B3: nat,F: fm > nat,X2: fm,A: set_fm] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_fm2 @ X2 @ A )
=> ( member_nat2 @ B3 @ ( image_fm_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_406_image__eqI,axiom,
! [B3: fm,F: nat > fm,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_fm2 @ B3 @ ( image_nat_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_407_image__eqI,axiom,
! [B3: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_408_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_409_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_410_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_411_insert__image,axiom,
! [X2: fm,A: set_fm,F: fm > nat] :
( ( member_fm2 @ X2 @ A )
=> ( ( insert_nat2 @ ( F @ X2 ) @ ( image_fm_nat @ F @ A ) )
= ( image_fm_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_412_insert__image,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( insert_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_413_image__insert,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_414_empty__replicate,axiom,
! [N: nat,X2: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_415_replicate__empty,axiom,
! [N: nat,X2: nat] :
( ( ( replicate_nat @ N @ X2 )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_416_Ball__set__replicate,axiom,
! [N: nat,A2: fm,P: fm > $o] :
( ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ ( replicate_fm @ N @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_417_Ball__set__replicate,axiom,
! [N: nat,A2: nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_418_Bex__set__replicate,axiom,
! [N: nat,A2: fm,P: fm > $o] :
( ( ? [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ ( replicate_fm @ N @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_419_Bex__set__replicate,axiom,
! [N: nat,A2: nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_420_in__set__replicate,axiom,
! [X2: fm,N: nat,Y: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ ( replicate_fm @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_421_in__set__replicate,axiom,
! [X2: nat,N: nat,Y: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_422_rev__image__eqI,axiom,
! [X2: fm,A: set_fm,B3: fm,F: fm > fm] :
( ( member_fm2 @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_fm2 @ B3 @ ( image_fm_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_423_rev__image__eqI,axiom,
! [X2: fm,A: set_fm,B3: nat,F: fm > nat] :
( ( member_fm2 @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat2 @ B3 @ ( image_fm_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_424_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: fm,F: nat > fm] :
( ( member_nat2 @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_fm2 @ B3 @ ( image_nat_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_425_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat2 @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_426_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat2 @ X @ ( image_nat_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X6: nat] :
( ( member_nat2 @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_427_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N2 )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_428_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( image_nat_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X: nat] :
( ( member_nat2 @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_429_image__iff,axiom,
! [Z2: nat,F: nat > nat,A: set_nat] :
( ( member_nat2 @ Z2 @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_430_imageI,axiom,
! [X2: fm,A: set_fm,F: fm > fm] :
( ( member_fm2 @ X2 @ A )
=> ( member_fm2 @ ( F @ X2 ) @ ( image_fm_fm @ F @ A ) ) ) ).
% imageI
thf(fact_431_imageI,axiom,
! [X2: fm,A: set_fm,F: fm > nat] :
( ( member_fm2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_fm_nat @ F @ A ) ) ) ).
% imageI
thf(fact_432_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > fm] :
( ( member_nat2 @ X2 @ A )
=> ( member_fm2 @ ( F @ X2 ) @ ( image_nat_fm @ F @ A ) ) ) ).
% imageI
thf(fact_433_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_434_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_435_subset__image__iff,axiom,
! [B: set_nat,F: fm > nat,A: set_fm] :
( ( ord_less_eq_set_nat @ B @ ( image_fm_nat @ F @ A ) )
= ( ? [AA: set_fm] :
( ( ord_less_eq_set_fm @ AA @ A )
& ( B
= ( image_fm_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_436_subset__image__iff,axiom,
! [B: set_fm,F: nat > fm,A: set_nat] :
( ( ord_less_eq_set_fm @ B @ ( image_nat_fm @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_437_subset__image__iff,axiom,
! [B: set_fm,F: fm > fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_fm_fm @ F @ A ) )
= ( ? [AA: set_fm] :
( ( ord_less_eq_set_fm @ AA @ A )
& ( B
= ( image_fm_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_438_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( member_nat2 @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_439_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_440_subset__imageE,axiom,
! [B: set_nat,F: fm > nat,A: set_fm] :
( ( ord_less_eq_set_nat @ B @ ( image_fm_nat @ F @ A ) )
=> ~ ! [C4: set_fm] :
( ( ord_less_eq_set_fm @ C4 @ A )
=> ( B
!= ( image_fm_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_441_subset__imageE,axiom,
! [B: set_fm,F: nat > fm,A: set_nat] :
( ( ord_less_eq_set_fm @ B @ ( image_nat_fm @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_fm @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_442_subset__imageE,axiom,
! [B: set_fm,F: fm > fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_fm_fm @ F @ A ) )
=> ~ ! [C4: set_fm] :
( ( ord_less_eq_set_fm @ C4 @ A )
=> ( B
!= ( image_fm_fm @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_443_image__subsetI,axiom,
! [A: set_fm,F: fm > nat,B: set_nat] :
( ! [X: fm] :
( ( member_fm2 @ X @ A )
=> ( member_nat2 @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_fm_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_444_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_445_image__subsetI,axiom,
! [A: set_fm,F: fm > fm,B: set_fm] :
( ! [X: fm] :
( ( member_fm2 @ X @ A )
=> ( member_fm2 @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_fm @ ( image_fm_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_446_image__subsetI,axiom,
! [A: set_nat,F: nat > fm,B: set_fm] :
( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( member_fm2 @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_fm @ ( image_nat_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_447_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_448_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > fm] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_fm @ ( image_nat_fm @ F @ A ) @ ( image_nat_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_449_image__mono,axiom,
! [A: set_fm,B: set_fm,F: fm > nat] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_fm_nat @ F @ A ) @ ( image_fm_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_450_image__mono,axiom,
! [A: set_fm,B: set_fm,F: fm > fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ ( image_fm_fm @ F @ A ) @ ( image_fm_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_451_replicate__0,axiom,
! [X2: nat] :
( ( replicate_nat @ zero_zero_nat @ X2 )
= nil_nat ) ).
% replicate_0
thf(fact_452_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > nat,X2: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_453_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_454_set__replicate__conv__if,axiom,
! [N: nat,X2: fm] :
( ( ( N = zero_zero_nat )
=> ( ( set_fm2 @ ( replicate_fm @ N @ X2 ) )
= bot_bot_set_fm ) )
& ( ( N != zero_zero_nat )
=> ( ( set_fm2 @ ( replicate_fm @ N @ X2 ) )
= ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ) ).
% set_replicate_conv_if
thf(fact_455_set__replicate__conv__if,axiom,
! [N: nat,X2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_456_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_457_bot__empty__eq,axiom,
( bot_bot_fm_o
= ( ^ [X3: fm] : ( member_fm2 @ X3 @ bot_bot_set_fm ) ) ) ).
% bot_empty_eq
thf(fact_458_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_459_is__empty__set,axiom,
! [Xs: list_fm] :
( ( is_empty_fm @ ( set_fm2 @ Xs ) )
= ( null_fm @ Xs ) ) ).
% is_empty_set
thf(fact_460_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_461_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_462_all__subset__image,axiom,
! [F: fm > nat,A: set_fm,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_fm_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( P @ ( image_fm_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_463_all__subset__image,axiom,
! [F: nat > fm,A: set_nat,P: set_fm > $o] :
( ( ! [B2: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ ( image_nat_fm @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_fm @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_464_all__subset__image,axiom,
! [F: fm > fm,A: set_fm,P: set_fm > $o] :
( ( ! [B2: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ ( image_fm_fm @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( P @ ( image_fm_fm @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_465_subseqs__powset,axiom,
! [Xs: list_fm] :
( ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
= ( pow_fm @ ( set_fm2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_466_subseqs__powset,axiom,
! [Xs: list_nat] :
( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
= ( pow_nat @ ( set_nat2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_467_f__arg__min__list__f,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( F @ ( arg_min_list_nat_nat @ F @ Xs ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_468_remdups__adj__replicate,axiom,
! [N: nat,X2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X2 ) )
= ( cons_nat @ X2 @ nil_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_469_Pow__singleton__iff,axiom,
! [X5: set_nat,Y7: set_nat] :
( ( ( pow_nat @ X5 )
= ( insert_set_nat @ Y7 @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
& ( Y7 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_470_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_471_PowI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( member_set_nat @ A @ ( pow_nat @ B ) ) ) ).
% PowI
thf(fact_472_PowI,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( member_set_fm @ A @ ( pow_fm @ B ) ) ) ).
% PowI
thf(fact_473_Pow__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_474_Pow__iff,axiom,
! [A: set_fm,B: set_fm] :
( ( member_set_fm @ A @ ( pow_fm @ B ) )
= ( ord_less_eq_set_fm @ A @ B ) ) ).
% Pow_iff
thf(fact_475_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_476_remdups__adj__set,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( remdups_adj_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_477_remdups__adj__set,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( remdups_adj_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_478_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_479_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_480_Pow__bottom,axiom,
! [B: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B ) ) ).
% Pow_bottom
thf(fact_481_PowD,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% PowD
thf(fact_482_PowD,axiom,
! [A: set_fm,B: set_fm] :
( ( member_set_fm @ A @ ( pow_fm @ B ) )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% PowD
thf(fact_483_Pow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B ) ) ) ).
% Pow_mono
thf(fact_484_Pow__mono,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_le1461404734466536732set_fm @ ( pow_fm @ A ) @ ( pow_fm @ B ) ) ) ).
% Pow_mono
thf(fact_485_remdups__adj_Osimps_I2_J,axiom,
! [X2: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_486_remdups__adj_Oelims,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X2 )
= Y )
=> ( ( ( X2 = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X: nat] :
( ( X2
= ( cons_nat @ X @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X @ nil_nat ) ) )
=> ~ ! [X: nat,Y2: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs3 ) ) )
=> ~ ( ( ( X = Y2 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs3 ) ) ) )
& ( ( X != Y2 )
=> ( Y
= ( cons_nat @ X @ ( remdups_adj_nat @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_487_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_488_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_489_eq__Nil__null,axiom,
! [Xs: list_nat] :
( ( Xs = nil_nat )
= ( null_nat @ Xs ) ) ).
% eq_Nil_null
thf(fact_490_Pow__set_I1_J,axiom,
( ( pow_fm @ ( set_fm2 @ nil_fm ) )
= ( insert_set_fm @ bot_bot_set_fm @ bot_bot_set_set_fm ) ) ).
% Pow_set(1)
thf(fact_491_Pow__set_I1_J,axiom,
( ( pow_nat @ ( set_nat2 @ nil_nat ) )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_set(1)
thf(fact_492_Min__singleton,axiom,
! [X2: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Min_singleton
thf(fact_493_min__list__Min,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( min_list_nat @ Xs )
= ( lattic8721135487736765967in_nat @ ( set_nat2 @ Xs ) ) ) ) ).
% min_list_Min
thf(fact_494_remdups__adj__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X2 @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 ) ) ) ).
% remdups_adj_singleton
thf(fact_495_set__replicate__Suc,axiom,
! [N: nat,X2: fm] :
( ( set_fm2 @ ( replicate_fm @ ( suc @ N ) @ X2 ) )
= ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ).
% set_replicate_Suc
thf(fact_496_set__replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_497_fm_Osize__gen_I2_J,axiom,
! [X21: nat,X22: list_tm] :
( ( size_fm @ ( pre @ X21 @ X22 ) )
= zero_zero_nat ) ).
% fm.size_gen(2)
thf(fact_498_fm_Osize_I6_J,axiom,
! [X21: nat,X22: list_tm] :
( ( size_size_fm @ ( pre @ X21 @ X22 ) )
= zero_zero_nat ) ).
% fm.size(6)
thf(fact_499_fm_Oinject_I1_J,axiom,
! [X21: nat,X22: list_tm,Y21: nat,Y22: list_tm] :
( ( ( pre @ X21 @ X22 )
= ( pre @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% fm.inject(1)
thf(fact_500_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_501_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_502_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_503_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_504_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_505_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_506_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X: nat,Y2: nat] :
( ( P @ X @ Y2 )
=> ( P @ ( suc @ X ) @ ( suc @ Y2 ) ) )
=> ( P @ M3 @ N ) ) ) ) ).
% diff_induct
thf(fact_507_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_508_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
% old.nat.exhaust
thf(fact_509_nat_OdiscI,axiom,
! [Nat2: nat,X23: nat] :
( ( Nat2
= ( suc @ X23 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_510_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_511_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_512_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_513_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_514_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_515_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y2: nat,Ys2: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_516_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_517_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_518_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_519_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_520_lift__Suc__mono__le,axiom,
! [F: nat > set_fm,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_fm @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_fm @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_521_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_522_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_523_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_524_lift__Suc__antimono__le,axiom,
! [F: nat > set_fm,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_fm @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_fm @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_525_replicate__eqI,axiom,
! [Xs: list_fm,N: nat,X2: fm] :
( ( ( size_size_list_fm @ Xs )
= N )
=> ( ! [Y2: fm] :
( ( member_fm2 @ Y2 @ ( set_fm2 @ Xs ) )
=> ( Y2 = X2 ) )
=> ( Xs
= ( replicate_fm @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_526_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X2: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) )
=> ( Y2 = X2 ) )
=> ( Xs
= ( replicate_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_527_replicate__length__same,axiom,
! [Xs: list_fm,X2: fm] :
( ! [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( X = X2 ) )
=> ( ( replicate_fm @ ( size_size_list_fm @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_528_replicate__length__same,axiom,
! [Xs: list_nat,X2: nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( X = X2 ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_529_fm_Odistinct_I9_J,axiom,
! [X21: nat,X22: list_tm,X42: fm] :
( ( pre @ X21 @ X22 )
!= ( uni @ X42 ) ) ).
% fm.distinct(9)
thf(fact_530_fm_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_tm] :
( falsity
!= ( pre @ X21 @ X22 ) ) ).
% fm.distinct(1)
thf(fact_531_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_532_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_533_fm_Oexhaust,axiom,
! [Y: fm] :
( ( Y != falsity )
=> ( ! [X212: nat,X222: list_tm] :
( Y
!= ( pre @ X212 @ X222 ) )
=> ( ! [X31: fm,X32: fm] :
( Y
!= ( imp @ X31 @ X32 ) )
=> ~ ! [X43: fm] :
( Y
!= ( uni @ X43 ) ) ) ) ) ).
% fm.exhaust
thf(fact_534_fm_Oinject_I2_J,axiom,
! [X312: fm,X322: fm,Y31: fm,Y32: fm] :
( ( ( imp @ X312 @ X322 )
= ( imp @ Y31 @ Y32 ) )
= ( ( X312 = Y31 )
& ( X322 = Y32 ) ) ) ).
% fm.inject(2)
thf(fact_535_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_536_fm_Odistinct_I11_J,axiom,
! [X312: fm,X322: fm,X42: fm] :
( ( imp @ X312 @ X322 )
!= ( uni @ X42 ) ) ).
% fm.distinct(11)
thf(fact_537_fm_Odistinct_I7_J,axiom,
! [X21: nat,X22: list_tm,X312: fm,X322: fm] :
( ( pre @ X21 @ X22 )
!= ( imp @ X312 @ X322 ) ) ).
% fm.distinct(7)
thf(fact_538_fm_Odistinct_I3_J,axiom,
! [X312: fm,X322: fm] :
( falsity
!= ( imp @ X312 @ X322 ) ) ).
% fm.distinct(3)
thf(fact_539_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_540_Fpow__mono,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_le1461404734466536732set_fm @ ( finite_Fpow_fm @ A ) @ ( finite_Fpow_fm @ B ) ) ) ).
% Fpow_mono
thf(fact_541_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_542_card__set__1__iff__replicate,axiom,
! [Xs: list_fm] :
( ( ( finite_card_fm @ ( set_fm2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_fm )
& ? [X3: fm] :
( Xs
= ( replicate_fm @ ( size_size_list_fm @ Xs ) @ X3 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_543_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X3: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X3 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_544_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_545_card__insert__le,axiom,
! [A: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat2 @ X2 @ A ) ) ) ).
% card_insert_le
thf(fact_546_list_Oset__sel_I1_J,axiom,
! [A2: list_fm] :
( ( A2 != nil_fm )
=> ( member_fm2 @ ( hd_fm @ A2 ) @ ( set_fm2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_547_list_Oset__sel_I1_J,axiom,
! [A2: list_nat] :
( ( A2 != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A2 ) @ ( set_nat2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_548_hd__in__set,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( member_fm2 @ ( hd_fm @ Xs ) @ ( set_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_549_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_550_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_551_card__length,axiom,
! [Xs: list_fm] : ( ord_less_eq_nat @ ( finite_card_fm @ ( set_fm2 @ Xs ) ) @ ( size_size_list_fm @ Xs ) ) ).
% card_length
thf(fact_552_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_553_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: nat] :
( A
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_554_card__eq__SucD,axiom,
! [A: set_fm,K: nat] :
( ( ( finite_card_fm @ A )
= ( suc @ K ) )
=> ? [B5: fm,B6: set_fm] :
( ( A
= ( insert_fm2 @ B5 @ B6 ) )
& ~ ( member_fm2 @ B5 @ B6 )
& ( ( finite_card_fm @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_fm ) ) ) ) ).
% card_eq_SucD
thf(fact_555_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B5: nat,B6: set_nat] :
( ( A
= ( insert_nat2 @ B5 @ B6 ) )
& ~ ( member_nat2 @ B5 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_556_card__Suc__eq,axiom,
! [A: set_fm,K: nat] :
( ( ( finite_card_fm @ A )
= ( suc @ K ) )
= ( ? [B4: fm,B2: set_fm] :
( ( A
= ( insert_fm2 @ B4 @ B2 ) )
& ~ ( member_fm2 @ B4 @ B2 )
& ( ( finite_card_fm @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bot_set_fm ) ) ) ) ) ).
% card_Suc_eq
thf(fact_557_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B4: nat,B2: set_nat] :
( ( A
= ( insert_nat2 @ B4 @ B2 ) )
& ~ ( member_nat2 @ B4 @ B2 )
& ( ( finite_card_nat @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_558_fm_Osize_I8_J,axiom,
! [X42: fm] :
( ( size_size_fm @ ( uni @ X42 ) )
= ( plus_plus_nat @ ( size_size_fm @ X42 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(8)
thf(fact_559_fm_Osize__gen_I4_J,axiom,
! [X42: fm] :
( ( size_fm @ ( uni @ X42 ) )
= ( plus_plus_nat @ ( size_fm @ X42 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(4)
thf(fact_560_fm_Osize__gen_I3_J,axiom,
! [X312: fm,X322: fm] :
( ( size_fm @ ( imp @ X312 @ X322 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X312 ) @ ( size_fm @ X322 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(3)
thf(fact_561_fm_Osize_I7_J,axiom,
! [X312: fm,X322: fm] :
( ( size_size_fm @ ( imp @ X312 @ X322 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X312 ) @ ( size_size_fm @ X322 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(7)
thf(fact_562_nth__equal__first__eq,axiom,
! [X2: fm,Xs: list_fm,N: nat] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( ( ( nth_fm @ ( cons_fm @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_563_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_564_add__left__cancel,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_565_add__left__cancel,axiom,
! [A2: int,B3: int,C: int] :
( ( ( plus_plus_int @ A2 @ B3 )
= ( plus_plus_int @ A2 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_566_add__right__cancel,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_567_add__right__cancel,axiom,
! [B3: int,A2: int,C: int] :
( ( ( plus_plus_int @ B3 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_568_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_569_add__le__cancel__left,axiom,
! [C: int,A2: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
= ( ord_less_eq_int @ A2 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_570_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_571_add__le__cancel__right,axiom,
! [A2: int,C: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
= ( ord_less_eq_int @ A2 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_572_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_573_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_574_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_575_add__cancel__left__left,axiom,
! [B3: nat,A2: nat] :
( ( ( plus_plus_nat @ B3 @ A2 )
= A2 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_576_add__cancel__left__left,axiom,
! [B3: int,A2: int] :
( ( ( plus_plus_int @ B3 @ A2 )
= A2 )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_577_add__cancel__left__right,axiom,
! [A2: nat,B3: nat] :
( ( ( plus_plus_nat @ A2 @ B3 )
= A2 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_578_add__cancel__left__right,axiom,
! [A2: int,B3: int] :
( ( ( plus_plus_int @ A2 @ B3 )
= A2 )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_579_add__cancel__right__left,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( plus_plus_nat @ B3 @ A2 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_580_add__cancel__right__left,axiom,
! [A2: int,B3: int] :
( ( A2
= ( plus_plus_int @ B3 @ A2 ) )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_581_add__cancel__right__right,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_582_add__cancel__right__right,axiom,
! [A2: int,B3: int] :
( ( A2
= ( plus_plus_int @ A2 @ B3 ) )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_583_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_584_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_585_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_586_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_587_minus__add__distrib,axiom,
! [A2: int,B3: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_588_minus__add__cancel,axiom,
! [A2: int,B3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_589_add__minus__cancel,axiom,
! [A2: int,B3: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_590_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_591_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_592_add__le__same__cancel1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_593_add__le__same__cancel1,axiom,
! [B3: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_594_add__le__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_595_add__le__same__cancel2,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_596_le__add__same__cancel1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_597_le__add__same__cancel1,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
= ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_598_le__add__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_599_le__add__same__cancel2,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_600_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_601_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_602_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_603_image__add__0,axiom,
! [S: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S )
= S ) ).
% image_add_0
thf(fact_604_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_605_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_606_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_607_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B3: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_608_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_609_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_610_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B3: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_611_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B3: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_612_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B3: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_613_group__cancel_Oadd2,axiom,
! [B: int,K: int,B3: int,A2: int] :
( ( B
= ( plus_plus_int @ K @ B3 ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_614_add_Oassoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_615_add_Oassoc,axiom,
! [A2: int,B3: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_616_add_Oleft__cancel,axiom,
! [A2: int,B3: int,C: int] :
( ( ( plus_plus_int @ A2 @ B3 )
= ( plus_plus_int @ A2 @ C ) )
= ( B3 = C ) ) ).
% add.left_cancel
thf(fact_617_add_Oright__cancel,axiom,
! [B3: int,A2: int,C: int] :
( ( ( plus_plus_int @ B3 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B3 = C ) ) ).
% add.right_cancel
thf(fact_618_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_619_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_620_add_Oleft__commute,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_621_add_Oleft__commute,axiom,
! [B3: int,A2: int,C: int] :
( ( plus_plus_int @ B3 @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_622_add__left__imp__eq,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_623_add__left__imp__eq,axiom,
! [A2: int,B3: int,C: int] :
( ( ( plus_plus_int @ A2 @ B3 )
= ( plus_plus_int @ A2 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_624_add__right__imp__eq,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_625_add__right__imp__eq,axiom,
! [B3: int,A2: int,C: int] :
( ( ( plus_plus_int @ B3 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_626_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_627_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_628_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_629_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_630_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_631_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_632_add__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_mono
thf(fact_633_add__mono,axiom,
! [A2: int,B3: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D2 ) ) ) ) ).
% add_mono
thf(fact_634_add__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_635_add__left__mono,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_636_less__eqE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ~ ! [C5: nat] :
( B3
!= ( plus_plus_nat @ A2 @ C5 ) ) ) ).
% less_eqE
thf(fact_637_add__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_638_add__right__mono,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_639_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C6: nat] :
( B4
= ( plus_plus_nat @ A4 @ C6 ) ) ) ) ).
% le_iff_add
thf(fact_640_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_641_add__le__imp__le__left,axiom,
! [C: int,A2: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
=> ( ord_less_eq_int @ A2 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_642_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_643_add__le__imp__le__right,axiom,
! [A2: int,C: int,B3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
=> ( ord_less_eq_int @ A2 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_644_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_645_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_646_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_647_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_648_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_649_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_650_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_651_is__num__normalize_I8_J,axiom,
! [A2: int,B3: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_652_add_Oinverse__distrib__swap,axiom,
! [A2: int,B3: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_653_group__cancel_Oneg1,axiom,
! [A: int,K: int,A2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_654_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_655_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_656_add__decreasing,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_657_add__decreasing,axiom,
! [A2: int,C: int,B3: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_658_add__increasing,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_659_add__increasing,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_660_add__decreasing2,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_661_add__decreasing2,axiom,
! [C: int,A2: int,B3: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_662_add__increasing2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_663_add__increasing2,axiom,
! [C: int,B3: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B3 @ A2 )
=> ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_664_add__nonneg__nonneg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_665_add__nonneg__nonneg,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_666_add__nonpos__nonpos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_667_add__nonpos__nonpos,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_668_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_669_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_670_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_671_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_672_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B3: int] :
( ( ( uminus_uminus_int @ A2 )
= B3 )
= ( ( plus_plus_int @ A2 @ B3 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_673_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B3: int] :
( ( A2
= ( uminus_uminus_int @ B3 ) )
= ( ( plus_plus_int @ A2 @ B3 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_674_add_Oinverse__unique,axiom,
! [A2: int,B3: int] :
( ( ( plus_plus_int @ A2 @ B3 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B3 ) ) ).
% add.inverse_unique
thf(fact_675_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_676_add__eq__0__iff,axiom,
! [A2: int,B3: int] :
( ( ( plus_plus_int @ A2 @ B3 )
= zero_zero_int )
= ( B3
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_677_one__is__add,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_678_add__is__1,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_679_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_680_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_681_translation__Compl,axiom,
! [A2: int,T2: set_int] :
( ( image_int_int @ ( plus_plus_int @ A2 ) @ ( uminus1532241313380277803et_int @ T2 ) )
= ( uminus1532241313380277803et_int @ ( image_int_int @ ( plus_plus_int @ A2 ) @ T2 ) ) ) ).
% translation_Compl
thf(fact_682_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_683_add__0__iff,axiom,
! [B3: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ B3 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_684_add__0__iff,axiom,
! [B3: int,A2: int] :
( ( B3
= ( plus_plus_int @ B3 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_685_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_686_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_687_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_688_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_689_in__set__dropD,axiom,
! [X2: fm,N: nat,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ ( drop_fm @ N @ Xs ) ) )
=> ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_690_in__set__dropD,axiom,
! [X2: nat,N: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_691_set__drop__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_drop_subset
thf(fact_692_set__drop__subset,axiom,
! [N: nat,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( drop_fm @ N @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).
% set_drop_subset
thf(fact_693_set__drop__subset__set__drop,axiom,
! [N: nat,M3: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M3 @ Xs ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_694_set__drop__subset__set__drop,axiom,
! [N: nat,M3: nat,Xs: list_fm] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( drop_fm @ M3 @ Xs ) ) @ ( set_fm2 @ ( drop_fm @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_695_Gcd__0__iff,axiom,
! [A: set_int] :
( ( ( gcd_Gcd_int @ A )
= zero_zero_int )
= ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% Gcd_0_iff
thf(fact_696_Gcd__0__iff,axiom,
! [A: set_nat] :
( ( ( gcd_Gcd_nat @ A )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_0_iff
thf(fact_697_add_Ogroup__axioms,axiom,
group_int @ plus_plus_int @ zero_zero_int @ uminus_uminus_int ).
% add.group_axioms
thf(fact_698_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_699_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_700_add__less__cancel__left,axiom,
! [C: int,A2: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
= ( ord_less_int @ A2 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_701_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_702_add__less__cancel__right,axiom,
! [A2: int,C: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
= ( ord_less_int @ A2 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_703_neg__less__iff__less,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_704_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_705_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_706_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_707_add__less__same__cancel1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_708_add__less__same__cancel1,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_709_add__less__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_710_add__less__same__cancel2,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_711_less__add__same__cancel1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_712_less__add__same__cancel1,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_713_less__add__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_714_less__add__same__cancel2,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_715_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_716_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_717_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_718_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_719_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_720_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_721_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_722_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_723_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_724_Gcd__empty,axiom,
( ( gcd_Gcd_int @ bot_bot_set_int )
= zero_zero_int ) ).
% Gcd_empty
thf(fact_725_Gcd__empty,axiom,
( ( gcd_Gcd_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Gcd_empty
thf(fact_726_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_727_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_728_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_729_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_730_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_731_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_732_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_733_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_734_add__less__imp__less__right,axiom,
! [A2: int,C: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
=> ( ord_less_int @ A2 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_735_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_nat @ A2 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_736_add__less__imp__less__left,axiom,
! [C: int,A2: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
=> ( ord_less_int @ A2 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_737_add__strict__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_738_add__strict__right__mono,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_739_add__strict__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_740_add__strict__left__mono,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_741_add__strict__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_742_add__strict__mono,axiom,
! [A2: int,B3: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_743_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_744_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_745_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_746_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_747_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_748_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_749_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_750_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_751_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_752_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_753_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_754_less__minus__iff,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_755_minus__less__iff,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B3 )
= ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).
% minus_less_iff
thf(fact_756_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_757_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_758_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_759_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_760_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_761_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_762_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_763_lt__ex,axiom,
! [X2: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X2 ) ).
% lt_ex
thf(fact_764_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_765_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_766_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_767_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_768_order_Oasym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_769_order_Oasym,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ~ ( ord_less_int @ B3 @ A2 ) ) ).
% order.asym
thf(fact_770_ord__eq__less__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_771_ord__eq__less__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( A2 = B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_772_ord__less__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_773_ord__less__eq__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_774_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_775_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_776_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_777_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_778_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_779_dual__order_Oasym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_780_dual__order_Oasym,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
=> ~ ( ord_less_int @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_781_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_782_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_783_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P2: nat > $o] :
? [N5: nat] :
( ( P2 @ N5 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ~ ( P2 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_784_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_785_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B3: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_786_order_Ostrict__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_787_order_Ostrict__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_788_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_789_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_790_dual__order_Ostrict__trans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_791_dual__order_Ostrict__trans,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_792_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_793_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_794_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_795_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_796_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_797_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_798_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_799_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_800_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_801_linorder__neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_802_order__less__asym_H,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_803_order__less__asym_H,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ~ ( ord_less_int @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_804_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_805_order__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_806_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_807_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_808_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_809_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_810_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_811_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_812_ord__less__eq__subst,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_813_ord__less__eq__subst,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_814_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_815_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_816_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_817_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_818_order__less__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_819_order__less__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_820_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_821_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_822_order__less__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_823_order__less__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_824_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_825_order__less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_826_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_827_order__less__imp__triv,axiom,
! [X2: int,Y: int,P: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_828_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_829_linorder__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_830_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_831_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_832_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_833_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_834_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_835_order__less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_836_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_837_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_838_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_839_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_840_order__le__imp__less__or__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_841_order__le__imp__less__or__eq,axiom,
! [X2: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y )
=> ( ( ord_less_set_fm @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_842_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_843_linorder__le__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_844_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_845_order__less__le__subst2,axiom,
! [A2: int,B3: int,F: int > set_nat,C: set_nat] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_846_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_847_order__less__le__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_848_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_849_order__less__le__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_850_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_fm @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_851_order__less__le__subst2,axiom,
! [A2: int,B3: int,F: int > set_fm,C: set_fm] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_fm @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_852_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_853_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_854_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_855_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_856_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_857_order__less__le__subst1,axiom,
! [A2: int,F: set_nat > int,B3: set_nat,C: set_nat] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_858_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_859_order__less__le__subst1,axiom,
! [A2: set_fm,F: nat > set_fm,B3: nat,C: nat] :
( ( ord_less_set_fm @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_860_order__less__le__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B3: int,C: int] :
( ( ord_less_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_861_order__less__le__subst1,axiom,
! [A2: set_fm,F: int > set_fm,B3: int,C: int] :
( ( ord_less_set_fm @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_862_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_863_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_864_order__le__less__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_865_order__le__less__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_866_order__le__less__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_867_order__le__less__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_868_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_869_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_set_fm @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_870_order__le__less__subst2,axiom,
! [A2: int,B3: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_871_order__le__less__subst2,axiom,
! [A2: int,B3: int,F: int > set_fm,C: set_fm] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_set_fm @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_872_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_873_order__le__less__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B3: int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_874_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_875_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_876_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_877_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_878_order__le__less__subst1,axiom,
! [A2: set_fm,F: nat > set_fm,B3: nat,C: nat] :
( ( ord_less_eq_set_fm @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_879_order__le__less__subst1,axiom,
! [A2: set_fm,F: int > set_fm,B3: int,C: int] :
( ( ord_less_eq_set_fm @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_880_order__less__le__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_881_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_882_order__less__le__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_883_order__less__le__trans,axiom,
! [X2: set_fm,Y: set_fm,Z2: set_fm] :
( ( ord_less_set_fm @ X2 @ Y )
=> ( ( ord_less_eq_set_fm @ Y @ Z2 )
=> ( ord_less_set_fm @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_884_order__le__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_885_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_886_order__le__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_887_order__le__less__trans,axiom,
! [X2: set_fm,Y: set_fm,Z2: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y )
=> ( ( ord_less_set_fm @ Y @ Z2 )
=> ( ord_less_set_fm @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_888_order__neq__le__trans,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_889_order__neq__le__trans,axiom,
! [A2: nat,B3: nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_890_order__neq__le__trans,axiom,
! [A2: int,B3: int] :
( ( A2 != B3 )
=> ( ( ord_less_eq_int @ A2 @ B3 )
=> ( ord_less_int @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_891_order__neq__le__trans,axiom,
! [A2: set_fm,B3: set_fm] :
( ( A2 != B3 )
=> ( ( ord_less_eq_set_fm @ A2 @ B3 )
=> ( ord_less_set_fm @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_892_order__le__neq__trans,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_893_order__le__neq__trans,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_894_order__le__neq__trans,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_int @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_895_order__le__neq__trans,axiom,
! [A2: set_fm,B3: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_fm @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_896_order__less__imp__le,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_897_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_898_order__less__imp__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_899_order__less__imp__le,axiom,
! [X2: set_fm,Y: set_fm] :
( ( ord_less_set_fm @ X2 @ Y )
=> ( ord_less_eq_set_fm @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_900_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_901_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_902_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_903_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_904_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_905_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_906_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_907_order__less__le,axiom,
( ord_less_set_fm
= ( ^ [X3: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_908_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_909_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_910_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_911_order__le__less,axiom,
( ord_less_eq_set_fm
= ( ^ [X3: set_fm,Y4: set_fm] :
( ( ord_less_set_fm @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_912_dual__order_Ostrict__implies__order,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B3 @ A2 )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_913_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_914_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ord_less_eq_int @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_915_dual__order_Ostrict__implies__order,axiom,
! [B3: set_fm,A2: set_fm] :
( ( ord_less_set_fm @ B3 @ A2 )
=> ( ord_less_eq_set_fm @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_916_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_917_order_Ostrict__implies__order,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_918_order_Ostrict__implies__order,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ord_less_eq_int @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_919_order_Ostrict__implies__order,axiom,
! [A2: set_fm,B3: set_fm] :
( ( ord_less_set_fm @ A2 @ B3 )
=> ( ord_less_eq_set_fm @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_920_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_921_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_922_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_923_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [B4: set_fm,A4: set_fm] :
( ( ord_less_eq_set_fm @ B4 @ A4 )
& ~ ( ord_less_eq_set_fm @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_924_dual__order_Ostrict__trans2,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_925_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_926_dual__order_Ostrict__trans2,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_927_dual__order_Ostrict__trans2,axiom,
! [B3: set_fm,A2: set_fm,C: set_fm] :
( ( ord_less_set_fm @ B3 @ A2 )
=> ( ( ord_less_eq_set_fm @ C @ B3 )
=> ( ord_less_set_fm @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_928_dual__order_Ostrict__trans1,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_929_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_930_dual__order_Ostrict__trans1,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_931_dual__order_Ostrict__trans1,axiom,
! [B3: set_fm,A2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ B3 @ A2 )
=> ( ( ord_less_set_fm @ C @ B3 )
=> ( ord_less_set_fm @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_932_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_933_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_934_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_935_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [B4: set_fm,A4: set_fm] :
( ( ord_less_eq_set_fm @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_936_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_937_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_938_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_939_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [B4: set_fm,A4: set_fm] :
( ( ord_less_set_fm @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_940_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_941_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_942_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_943_order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [A4: set_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
& ~ ( ord_less_eq_set_fm @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_944_order_Ostrict__trans2,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_945_order_Ostrict__trans2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_946_order_Ostrict__trans2,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_947_order_Ostrict__trans2,axiom,
! [A2: set_fm,B3: set_fm,C: set_fm] :
( ( ord_less_set_fm @ A2 @ B3 )
=> ( ( ord_less_eq_set_fm @ B3 @ C )
=> ( ord_less_set_fm @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_948_order_Ostrict__trans1,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_949_order_Ostrict__trans1,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_950_order_Ostrict__trans1,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_951_order_Ostrict__trans1,axiom,
! [A2: set_fm,B3: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B3 )
=> ( ( ord_less_set_fm @ B3 @ C )
=> ( ord_less_set_fm @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_952_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_953_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_954_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_955_order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [A4: set_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_956_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_957_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_958_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_959_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B4: set_fm] :
( ( ord_less_set_fm @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_960_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_961_not__le__imp__less,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_eq_int @ Y @ X2 )
=> ( ord_less_int @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_962_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_963_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_964_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_965_less__le__not__le,axiom,
( ord_less_set_fm
= ( ^ [X3: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y4 )
& ~ ( ord_less_eq_set_fm @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_966_antisym__conv2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_967_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_968_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_969_antisym__conv2,axiom,
! [X2: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y )
=> ( ( ~ ( ord_less_set_fm @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_970_antisym__conv1,axiom,
! [X2: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_971_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_972_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_973_antisym__conv1,axiom,
! [X2: set_fm,Y: set_fm] :
( ~ ( ord_less_set_fm @ X2 @ Y )
=> ( ( ord_less_eq_set_fm @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_974_nless__le,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_975_nless__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_976_nless__le,axiom,
! [A2: int,B3: int] :
( ( ~ ( ord_less_int @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_977_nless__le,axiom,
! [A2: set_fm,B3: set_fm] :
( ( ~ ( ord_less_set_fm @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_set_fm @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_978_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_979_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_980_leD,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_981_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_982_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_983_leD,axiom,
! [Y: set_fm,X2: set_fm] :
( ( ord_less_eq_set_fm @ Y @ X2 )
=> ~ ( ord_less_set_fm @ X2 @ Y ) ) ).
% leD
thf(fact_984_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_985_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_986_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_987_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_988_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_989_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_990_add__le__less__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_991_add__le__less__mono,axiom,
! [A2: int,B3: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_992_add__less__le__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_993_add__less__le__mono,axiom,
! [A2: int,B3: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_994_add__neg__neg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_995_add__neg__neg,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_996_add__pos__pos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_997_add__pos__pos,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_998_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ! [C5: nat] :
( ( B3
= ( plus_plus_nat @ A2 @ C5 ) )
=> ( C5 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_999_pos__add__strict,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1000_pos__add__strict,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1001_less__Suc__eq__0__disj,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( M3 = zero_zero_nat )
| ? [J2: nat] :
( ( M3
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1002_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_1003_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1004_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1005_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1006_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1007_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1008_max__list__in,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_nat @ ( max_list @ Xs ) @ N )
=> ~ ( member_nat2 @ N @ ( set_nat2 @ Xs ) ) ) ).
% max_list_in
thf(fact_1009_add__strict__increasing2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1010_add__strict__increasing2,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1011_add__strict__increasing,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1012_add__strict__increasing,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1013_add__pos__nonneg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1014_add__pos__nonneg,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1015_add__nonpos__neg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1016_add__nonpos__neg,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1017_add__nonneg__pos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1018_add__nonneg__pos,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1019_add__neg__nonpos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1020_add__neg__nonpos,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1021_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1022_length__pos__if__in__set,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1023_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1024_all__set__conv__all__nth,axiom,
! [Xs: list_fm,P: fm > $o] :
( ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
=> ( P @ ( nth_fm @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1025_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1026_all__nth__imp__all__set,axiom,
! [Xs: list_fm,P: fm > $o,X2: fm] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_fm @ Xs ) )
=> ( P @ ( nth_fm @ Xs @ I5 ) ) )
=> ( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1027_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I5 ) ) )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1028_in__set__conv__nth,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1029_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1030_list__ball__nth,axiom,
! [N: nat,Xs: list_fm,P: fm > $o] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( ! [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_fm @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1031_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1032_nth__mem,axiom,
! [N: nat,Xs: list_fm] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( member_fm2 @ ( nth_fm @ Xs @ N ) @ ( set_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1033_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1034_add__less__zeroD,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1035_complete__interval,axiom,
! [A2: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A2 @ C5 )
& ( ord_less_eq_nat @ C5 @ B3 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C5 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1036_complete__interval,axiom,
! [A2: int,B3: int,P: int > $o] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C5: int] :
( ( ord_less_eq_int @ A2 @ C5 )
& ( ord_less_eq_int @ C5 @ B3 )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A2 @ X6 )
& ( ord_less_int @ X6 @ C5 ) )
=> ( P @ X6 ) )
& ! [D3: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A2 @ X )
& ( ord_less_int @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1037_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_1038_pinf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_1039_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_1040_psubsetI,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_fm @ A @ B ) ) ) ).
% psubsetI
thf(fact_1041_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1042_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1043_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B2: set_fm] :
( ( ord_less_set_fm @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1044_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1045_subset__psubset__trans,axiom,
! [A: set_fm,B: set_fm,C2: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_set_fm @ B @ C2 )
=> ( ord_less_set_fm @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1046_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1047_subset__not__subset__eq,axiom,
( ord_less_set_fm
= ( ^ [A3: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A3 @ B2 )
& ~ ( ord_less_eq_set_fm @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1048_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1049_psubset__subset__trans,axiom,
! [A: set_fm,B: set_fm,C2: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C2 )
=> ( ord_less_set_fm @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1050_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1051_psubset__imp__subset,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1052_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_1053_psubset__eq,axiom,
( ord_less_set_fm
= ( ^ [A3: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_1054_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_1055_psubsetE,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ~ ( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ B @ A ) ) ) ).
% psubsetE
thf(fact_1056_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_1057_minf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_1058_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_1059_minf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_1060_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_1061_pinf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_1062_set__swap,axiom,
! [I2: nat,Xs: list_fm,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_fm @ Xs ) )
=> ( ( set_fm2 @ ( list_update_fm @ ( list_update_fm @ Xs @ I2 @ ( nth_fm @ Xs @ J ) ) @ J @ ( nth_fm @ Xs @ I2 ) ) )
= ( set_fm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1063_set__swap,axiom,
! [I2: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I2 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1064_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_1065_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_1066_negative__eq__positive,axiom,
! [N: nat,M3: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M3 ) )
= ( ( N = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1067_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X2: nat] :
( ( ( list_update_nat @ Xs @ K @ X2 )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_1068_list__ex__simps_I2_J,axiom,
! [P: nat > $o] :
~ ( list_ex_nat @ P @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_1069_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1316708129612266289at_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1070_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1071_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_1072_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_1073_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1074_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1075_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_1076_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_1077_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1078_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1079_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1080_negD,axiom,
! [X2: int] :
( ( ord_less_int @ X2 @ zero_zero_int )
=> ? [N3: nat] :
( X2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1081_psubsetD,axiom,
! [A: set_fm,B: set_fm,C: fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_1082_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_1083_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1084_list__ex__cong,axiom,
! [Xs: list_fm,Ys: list_fm,F: fm > $o,G: fm > $o] :
( ( Xs = Ys )
=> ( ! [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_ex_fm @ F @ Xs )
= ( list_ex_fm @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_1085_list__ex__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
( ( Xs = Ys )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_ex_nat @ F @ Xs )
= ( list_ex_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_1086_int__ops_I5_J,axiom,
! [A2: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(5)
thf(fact_1087_int__plus,axiom,
! [N: nat,M3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% int_plus
thf(fact_1088_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1089_int__zle__neg,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) )
= ( ( N = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1090_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1091_list__update_Osimps_I1_J,axiom,
! [I2: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I2 @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_1092_list__update__code_I1_J,axiom,
! [I2: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I2 @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_1093_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1094_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_1095_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_1096_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_1097_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_1098_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1099_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1100_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_1101_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_1102_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1103_int__cases4,axiom,
! [M3: int] :
( ! [N3: nat] :
( M3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1104_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1105_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1106_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1107_set__update__subsetI,axiom,
! [Xs: list_nat,A: set_nat,X2: nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I2 @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1108_set__update__subsetI,axiom,
! [Xs: list_fm,A: set_fm,X2: fm,I2: nat] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ A )
=> ( ( member_fm2 @ X2 @ A )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( list_update_fm @ Xs @ I2 @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1109_set__update__subset__insert,axiom,
! [Xs: list_nat,I2: nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I2 @ X2 ) ) @ ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1110_set__update__subset__insert,axiom,
! [Xs: list_fm,I2: nat,X2: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( list_update_fm @ Xs @ I2 @ X2 ) ) @ ( insert_fm2 @ X2 @ ( set_fm2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1111_set__update__memI,axiom,
! [N: nat,Xs: list_fm,X2: fm] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( member_fm2 @ X2 @ ( set_fm2 @ ( list_update_fm @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1112_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1113_split__Nil__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( splice_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% split_Nil_iff
thf(fact_1114_splice__Nil2,axiom,
! [Xs: list_nat] :
( ( splice_nat @ Xs @ nil_nat )
= Xs ) ).
% splice_Nil2
thf(fact_1115_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1116_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1117_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1118_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1119_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1120_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1121_int__if,axiom,
! [P: $o,A2: nat,B3: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B3 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B3 ) )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% int_if
thf(fact_1122_distinct_Osimps_I2_J,axiom,
! [X2: fm,Xs: list_fm] :
( ( distinct_fm @ ( cons_fm @ X2 @ Xs ) )
= ( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( distinct_fm @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1123_distinct_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1124_Gcd__int__greater__eq__0,axiom,
! [K3: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K3 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_1125_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_1126_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1127_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1128_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1129_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1130_distinct__singleton,axiom,
! [X2: nat] : ( distinct_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_singleton
thf(fact_1131_splice_Osimps_I1_J,axiom,
! [Ys: list_nat] :
( ( splice_nat @ nil_nat @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_1132_imp__le__cong,axiom,
! [X2: int,X8: int,P: $o,P5: $o] :
( ( X2 = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1133_conj__le__cong,axiom,
! [X2: int,X8: int,P: $o,P5: $o] :
( ( X2 = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1134_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_1135_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_1136_last_Osimps,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_1137_last__in__set,axiom,
! [As: list_fm] :
( ( As != nil_fm )
=> ( member_fm2 @ ( last_fm @ As ) @ ( set_fm2 @ As ) ) ) ).
% last_in_set
thf(fact_1138_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat2 @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_1139_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_1140_distinct__card,axiom,
! [Xs: list_fm] :
( ( distinct_fm @ Xs )
=> ( ( finite_card_fm @ ( set_fm2 @ Xs ) )
= ( size_size_list_fm @ Xs ) ) ) ).
% distinct_card
thf(fact_1141_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_1142_card__distinct,axiom,
! [Xs: list_fm] :
( ( ( finite_card_fm @ ( set_fm2 @ Xs ) )
= ( size_size_list_fm @ Xs ) )
=> ( distinct_fm @ Xs ) ) ).
% card_distinct
thf(fact_1143_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_1144_splice_Oelims,axiom,
! [X2: list_nat,Xa: list_nat,Y: list_nat] :
( ( ( splice_nat @ X2 @ Xa )
= Y )
=> ( ( ( X2 = nil_nat )
=> ( Y != Xa ) )
=> ~ ! [X: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X @ Xs3 ) )
=> ( Y
!= ( cons_nat @ X @ ( splice_nat @ Xa @ Xs3 ) ) ) ) ) ) ).
% splice.elims
thf(fact_1145_distinct__Ex1,axiom,
! [Xs: list_fm,X2: fm] :
( ( distinct_fm @ Xs )
=> ( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ X )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1146_distinct__Ex1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1147_set__update__distinct,axiom,
! [Xs: list_fm,N: nat,X2: fm] :
( ( distinct_fm @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( ( set_fm2 @ ( list_update_fm @ Xs @ N @ X2 ) )
= ( insert_fm2 @ X2 @ ( minus_minus_set_fm @ ( set_fm2 @ Xs ) @ ( insert_fm2 @ ( nth_fm @ Xs @ N ) @ bot_bot_set_fm ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1148_set__update__distinct,axiom,
! [Xs: list_nat,N: nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) )
= ( insert_nat2 @ X2 @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1149_DiffI,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ A )
=> ( ~ ( member_fm2 @ C @ B )
=> ( member_fm2 @ C @ ( minus_minus_set_fm @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1150_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ A )
=> ( ~ ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1151_Diff__iff,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ ( minus_minus_set_fm @ A @ B ) )
= ( ( member_fm2 @ C @ A )
& ~ ( member_fm2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1152_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat2 @ C @ A )
& ~ ( member_nat2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1153_Diff__idemp,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_1154_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_1155_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_1156_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1157_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_1158_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_1159_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1160_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1161_add__diff__cancel,axiom,
! [A2: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
= A2 ) ).
% add_diff_cancel
thf(fact_1162_diff__add__cancel,axiom,
! [A2: int,B3: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
= A2 ) ).
% diff_add_cancel
thf(fact_1163_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( minus_minus_nat @ A2 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_1164_add__diff__cancel__left,axiom,
! [C: int,A2: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
= ( minus_minus_int @ A2 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_1165_add__diff__cancel__left_H,axiom,
! [A2: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ A2 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_1166_add__diff__cancel__left_H,axiom,
! [A2: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ A2 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_1167_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( minus_minus_nat @ A2 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_1168_add__diff__cancel__right,axiom,
! [A2: int,C: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
= ( minus_minus_int @ A2 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_1169_add__diff__cancel__right_H,axiom,
! [A2: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_1170_add__diff__cancel__right_H,axiom,
! [A2: int,B3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_1171_minus__diff__eq,axiom,
! [A2: int,B3: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B3 ) )
= ( minus_minus_int @ B3 @ A2 ) ) ).
% minus_diff_eq
thf(fact_1172_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1173_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1174_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_1175_insert__Diff1,axiom,
! [X2: fm,B: set_fm,A: set_fm] :
( ( member_fm2 @ X2 @ B )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X2 @ A ) @ B )
= ( minus_minus_set_fm @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1176_insert__Diff1,axiom,
! [X2: nat,B: set_nat,A: set_nat] :
( ( member_nat2 @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1177_Diff__insert0,axiom,
! [X2: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm2 @ X2 @ A )
=> ( ( minus_minus_set_fm @ A @ ( insert_fm2 @ X2 @ B ) )
= ( minus_minus_set_fm @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1178_Diff__insert0,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1179_diff__ge__0__iff__ge,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
= ( ord_less_eq_int @ B3 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1180_diff__gt__0__iff__gt,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
= ( ord_less_int @ B3 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1181_le__add__diff__inverse2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ B3 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1182_le__add__diff__inverse2,axiom,
! [B3: int,A2: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1183_le__add__diff__inverse,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1184_le__add__diff__inverse,axiom,
! [B3: int,A2: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A2 @ B3 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1185_diff__add__zero,axiom,
! [A2: nat,B3: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1186_verit__minus__simplify_I3_J,axiom,
! [B3: int] :
( ( minus_minus_int @ zero_zero_int @ B3 )
= ( uminus_uminus_int @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_1187_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_1188_uminus__add__conv__diff,axiom,
! [A2: int,B3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 )
= ( minus_minus_int @ B3 @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1189_diff__minus__eq__add,axiom,
! [A2: int,B3: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B3 ) )
= ( plus_plus_int @ A2 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_1190_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1191_Diff__eq__empty__iff,axiom,
! [A: set_fm,B: set_fm] :
( ( ( minus_minus_set_fm @ A @ B )
= bot_bot_set_fm )
= ( ord_less_eq_set_fm @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1192_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1193_set__remove1__eq,axiom,
! [Xs: list_fm,X2: fm] :
( ( distinct_fm @ Xs )
=> ( ( set_fm2 @ ( remove1_fm @ X2 @ Xs ) )
= ( minus_minus_set_fm @ ( set_fm2 @ Xs ) @ ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ) ).
% set_remove1_eq
thf(fact_1194_set__remove1__eq,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( set_nat2 @ ( remove1_nat @ X2 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% set_remove1_eq
thf(fact_1195_psubset__imp__ex__mem,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ? [B5: fm] : ( member_fm2 @ B5 @ ( minus_minus_set_fm @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1196_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B5: nat] : ( member_nat2 @ B5 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1197_diff__strict__mono,axiom,
! [A2: int,B3: int,D2: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_1198_diff__eq__diff__less,axiom,
! [A2: int,B3: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B3 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A2 @ B3 )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_1199_diff__strict__left__mono,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_1200_diff__strict__right__mono,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1201_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1202_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1203_diff__le__eq,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% diff_le_eq
thf(fact_1204_le__diff__eq,axiom,
! [A2: int,C: int,B3: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% le_diff_eq
thf(fact_1205_diff__add,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ A2 )
= B3 ) ) ).
% diff_add
thf(fact_1206_le__add__diff,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1207_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1208_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1209_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1210_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1211_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1212_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B3 @ A2 ) )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1214_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( minus_minus_nat @ B3 @ A2 )
= C )
= ( B3
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1215_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1216_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1217_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1218_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1219_minus__diff__commute,axiom,
! [B3: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B3 ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_1220_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1221_diff__mono,axiom,
! [A2: int,B3: int,D2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_1222_diff__left__mono,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% diff_left_mono
thf(fact_1223_diff__right__mono,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% diff_right_mono
thf(fact_1224_diff__eq__diff__less__eq,axiom,
! [A2: int,B3: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B3 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A2 @ B3 )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1225_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1226_Diff__mono,axiom,
! [A: set_fm,C2: set_fm,D: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ C2 )
=> ( ( ord_less_eq_set_fm @ D @ B )
=> ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B ) @ ( minus_minus_set_fm @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1227_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1228_Diff__subset,axiom,
! [A: set_fm,B: set_fm] : ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1229_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1230_double__diff,axiom,
! [A: set_fm,B: set_fm,C2: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C2 )
=> ( ( minus_minus_set_fm @ B @ ( minus_minus_set_fm @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1231_diff__diff__eq,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1232_diff__diff__eq,axiom,
! [A2: int,B3: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1233_add__implies__diff,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B3 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_1234_add__implies__diff,axiom,
! [C: int,B3: int,A2: int] :
( ( ( plus_plus_int @ C @ B3 )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_1235_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B3: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B3 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1236_diff__add__eq,axiom,
! [A2: int,B3: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).
% diff_add_eq
thf(fact_1237_diff__diff__eq2,axiom,
! [A2: int,B3: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).
% diff_diff_eq2
thf(fact_1238_add__diff__eq,axiom,
! [A2: int,B3: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% add_diff_eq
thf(fact_1239_eq__diff__eq,axiom,
! [A2: int,C: int,B3: int] :
( ( A2
= ( minus_minus_int @ C @ B3 ) )
= ( ( plus_plus_int @ A2 @ B3 )
= C ) ) ).
% eq_diff_eq
thf(fact_1240_diff__eq__eq,axiom,
! [A2: int,B3: int,C: int] :
( ( ( minus_minus_int @ A2 @ B3 )
= C )
= ( A2
= ( plus_plus_int @ C @ B3 ) ) ) ).
% diff_eq_eq
thf(fact_1241_group__cancel_Osub1,axiom,
! [A: int,K: int,A2: int,B3: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).
% group_cancel.sub1
thf(fact_1242_diff__shunt__var,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1243_diff__shunt__var,axiom,
! [X2: set_fm,Y: set_fm] :
( ( ( minus_minus_set_fm @ X2 @ Y )
= bot_bot_set_fm )
= ( ord_less_eq_set_fm @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1244_less__diff__eq,axiom,
! [A2: int,C: int,B3: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% less_diff_eq
thf(fact_1245_diff__less__eq,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% diff_less_eq
thf(fact_1246_group__cancel_Osub2,axiom,
! [B: int,K: int,B3: int,A2: int] :
( ( B
= ( plus_plus_int @ K @ B3 ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).
% group_cancel.sub2
thf(fact_1247_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1248_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1249_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1250_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1251_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_1252_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_1253_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1254_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_1255_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X2 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1256_int__ops_I6_J,axiom,
! [A2: nat,B3: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1257_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1258_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1259_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1260_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1261_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_1262_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1263_Gcd__remove0__nat,axiom,
! [M: set_nat] :
( ( finite_finite_nat @ M )
=> ( ( gcd_Gcd_nat @ M )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_1264_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1265_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
= ( ( ( ord_less_nat @ A2 @ B3 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B3 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1266_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B3 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B3 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1267_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1268_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_1269_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1270_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1271_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1272_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
! [X2: list_fm,Y: list_fm] :
( ( if_list_fm @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
! [X2: list_fm,Y: list_fm] :
( ( if_list_fm @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
member_fm2 @ p @ ( set_fm2 @ a ) ).
thf(conj_1,conjecture,
ord_less_eq_set_nat @ ( set_nat2 @ ( vars_fm @ p ) ) @ ( set_nat2 @ ( vars_fms @ a ) ) ).
%------------------------------------------------------------------------------