TPTP Problem File: SLH0707^1.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_Calc3/0005_Semantics/prob_00057_001935__11827716_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1483 ( 592 unt; 205 typ; 0 def)
% Number of atoms : 3505 (1322 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10508 ( 359 ~; 67 |; 211 &;8360 @)
% ( 0 <=>;1511 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 937 ( 937 >; 0 *; 0 +; 0 <<)
% Number of symbols : 183 ( 180 usr; 17 con; 0-4 aty)
% Number of variables : 3766 ( 268 ^;3330 !; 168 ?;3766 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:24:45.752
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
list_list_list_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Syntax__Otm_J_J,type,
list_list_tm: $tType ).
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__Otm_J_J,type,
set_list_tm: $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__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_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__Otm_J,type,
set_tm: $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__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Syntax__Otm,type,
tm: $tType ).
thf(ty_n_t__Syntax__Ofm,type,
fm: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (180)
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Syntax__Otm,type,
fun_upd_nat_tm: ( nat > tm ) > nat > tm > nat > tm ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
fun_upd_nat_a: ( nat > a ) > nat > a > nat > a ).
thf(sy_c_Fun_Ofun__upd_001t__Syntax__Ofm_001t__List__Olist_It__Nat__Onat_J,type,
fun_upd_fm_list_nat: ( fm > list_nat ) > fm > list_nat > fm > list_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Syntax__Otm_001t__List__Olist_It__Nat__Onat_J,type,
fun_upd_tm_list_nat: ( tm > list_nat ) > tm > list_nat > tm > list_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Syntax__Otm_001t__Syntax__Otm,type,
fun_upd_tm_tm: ( tm > tm ) > tm > tm > tm > tm ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Syntax__Ofm_M_Eo_J,type,
uminus_uminus_fm_o: ( fm > $o ) > fm > $o ).
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__List__Olist_It__Nat__Onat_J_J,type,
uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).
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__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Syntax__Ofm_001t__Nat__Onat,type,
bind_fm_nat: list_fm > ( fm > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Syntax__Otm_001t__Nat__Onat,type,
bind_tm_nat: list_tm > ( tm > list_nat ) > list_nat ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Int__Oint_J,type,
concat_list_int: list_list_list_int > list_list_int ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__Syntax__Ofm,type,
concat_fm: list_list_fm > list_fm ).
thf(sy_c_List_Oconcat_001t__Syntax__Otm,type,
concat_tm: list_list_tm > list_tm ).
thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
gen_length_int: nat > list_int > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__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_OCons_001t__Syntax__Otm,type,
cons_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__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_ONil_001t__Syntax__Otm,type,
nil_tm: list_tm ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
map_int_int: ( int > int ) > list_int > list_int ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__List__Olist_It__Int__Oint_J,type,
map_int_list_int: ( int > list_int ) > list_int > list_list_int ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Nat__Onat,type,
map_int_nat: ( int > nat ) > list_int > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
map_li8902190837986183758st_int: ( list_int > list_list_int ) > list_list_int > list_list_list_int ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li7487060700005177878st_nat: ( list_fm > list_list_nat ) > list_list_fm > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Syntax__Otm_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li7696074350313601672st_nat: ( list_tm > list_list_nat ) > list_list_tm > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Syntax__Otm_J_001t__List__Olist_It__Syntax__Otm_J,type,
map_list_tm_list_tm: ( list_tm > list_tm ) > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
map_nat_int: ( nat > int ) > list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Syntax__Otm,type,
map_nat_tm: ( nat > tm ) > list_nat > list_tm ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Ofm_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_fm_list_list_nat: ( fm > list_list_nat ) > list_fm > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Ofm_001t__List__Olist_It__Nat__Onat_J,type,
map_fm_list_nat: ( fm > list_nat ) > list_fm > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Ofm_001t__Nat__Onat,type,
map_fm_nat: ( fm > nat ) > list_fm > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
map_fm_fm: ( fm > fm ) > list_fm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Otm_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_tm_list_list_nat: ( tm > list_list_nat ) > list_tm > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Otm_001t__List__Olist_It__Nat__Onat_J,type,
map_tm_list_nat: ( tm > list_nat ) > list_tm > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Otm_001t__List__Olist_It__Syntax__Otm_J,type,
map_tm_list_tm: ( tm > list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Otm_001t__Nat__Onat,type,
map_tm_nat: ( tm > nat ) > list_tm > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Otm_001t__Syntax__Otm,type,
map_tm_tm: ( tm > tm ) > list_tm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__Syntax__Otm_001tf__a,type,
map_tm_a: ( tm > a ) > list_tm > list_a ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Syntax__Ofm_J,type,
set_list_fm2: list_list_fm > set_list_fm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Syntax__Otm_J,type,
set_list_tm2: list_list_tm > set_list_tm ).
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_Oset_001t__Syntax__Otm,type,
set_tm2: list_tm > set_tm ).
thf(sy_c_List_Olist_Osize__list_001t__Int__Oint,type,
size_list_int: ( int > nat ) > list_int > nat ).
thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Syntax__Ofm,type,
size_list_fm: ( fm > nat ) > list_fm > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Syntax__Otm,type,
size_list_tm: ( tm > nat ) > list_tm > nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Syntax__Ofm_001t__List__Olist_It__Nat__Onat_J,type,
map_ta4685076891720636352st_nat: ( fm > list_nat ) > list_fm > list_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Syntax__Otm_001t__List__Olist_It__Nat__Onat_J,type,
map_ta1959924319307969074st_nat: ( tm > list_nat ) > list_tm > list_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Syntax__Otm_001t__Syntax__Otm,type,
map_tailrec_tm_tm: ( tm > tm ) > list_tm > list_tm ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Syntax__Ofm_001t__Nat__Onat,type,
maps_fm_nat: ( fm > list_nat ) > list_fm > list_nat ).
thf(sy_c_List_Omaps_001t__Syntax__Otm_001t__Nat__Onat,type,
maps_tm_nat: ( tm > list_nat ) > list_tm > list_nat ).
thf(sy_c_List_Omember_001t__Int__Oint,type,
member_int: list_int > int > $o ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Syntax__Ofm,type,
member_fm: list_fm > fm > $o ).
thf(sy_c_List_Omember_001t__Syntax__Otm,type,
member_tm: list_tm > tm > $o ).
thf(sy_c_List_On__lists_001t__Int__Oint,type,
n_lists_int: nat > list_int > list_list_int ).
thf(sy_c_List_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001t__Syntax__Ofm,type,
n_lists_fm: nat > list_fm > list_list_fm ).
thf(sy_c_List_On__lists_001t__Syntax__Otm,type,
n_lists_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Syntax__Ofm_J,type,
size_size_list_fm: list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Syntax__Otm_J,type,
size_size_list_tm: list_tm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Syntax__Ofm,type,
size_size_fm: fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Syntax__Otm,type,
size_size_tm: tm > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_less_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Syntax__Ofm_M_Eo_J,type,
ord_less_fm_o: ( fm > $o ) > ( fm > $o ) > $o ).
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__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_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_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Syntax__Ofm_M_Eo_J,type,
ord_less_eq_fm_o: ( fm > $o ) > ( fm > $o ) > $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__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__Syntax__Ofm_J,type,
ord_less_eq_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Otm_J,type,
ord_less_eq_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Semantics_Osemantics__fm_001t__List__Olist_It__Nat__Onat_J,type,
semant1976007830398715828st_nat: ( nat > list_nat ) > ( nat > list_list_nat > list_nat ) > ( nat > list_list_nat > $o ) > fm > $o ).
thf(sy_c_Semantics_Osemantics__fm_001t__Syntax__Otm,type,
semantics_fm_tm: ( nat > tm ) > ( nat > list_tm > tm ) > ( nat > list_tm > $o ) > fm > $o ).
thf(sy_c_Semantics_Osemantics__fm_001tf__a,type,
semantics_fm_a: ( nat > a ) > ( nat > list_a > a ) > ( nat > list_a > $o ) > fm > $o ).
thf(sy_c_Semantics_Osemantics__tm_001t__List__Olist_It__Nat__Onat_J,type,
semant8474227294840824358st_nat: ( nat > list_nat ) > ( nat > list_list_nat > list_nat ) > tm > list_nat ).
thf(sy_c_Semantics_Osemantics__tm_001t__Syntax__Otm,type,
semantics_tm_tm: ( nat > tm ) > ( nat > list_tm > tm ) > tm > tm ).
thf(sy_c_Semantics_Osemantics__tm_001tf__a,type,
semantics_tm_a: ( nat > a ) > ( nat > list_a > a ) > tm > a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Syntax__Ofm_J,type,
collect_list_fm: ( list_fm > $o ) > set_list_fm ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Syntax__Otm_J,type,
collect_list_tm: ( list_tm > $o ) > set_list_tm ).
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_Syntax_Oadd__env_001t__Syntax__Otm,type,
add_env_tm: tm > ( nat > tm ) > nat > tm ).
thf(sy_c_Syntax_Oadd__env_001tf__a,type,
add_env_a: a > ( nat > a ) > nat > a ).
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_Ofresh,type,
fresh: list_fm > nat ).
thf(sy_c_Syntax_Olift__tm,type,
lift_tm: tm > tm ).
thf(sy_c_Syntax_Omax__list,type,
max_list: list_nat > nat ).
thf(sy_c_Syntax_Osub__fm,type,
sub_fm: ( nat > tm ) > fm > fm ).
thf(sy_c_Syntax_Osub__tm,type,
sub_tm: ( nat > tm ) > tm > tm ).
thf(sy_c_Syntax_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_Syntax_Otm_OVar,type,
var: nat > tm ).
thf(sy_c_Syntax_Otm_Osize__tm,type,
size_tm: tm > 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_Syntax_Ovars__tm,type,
vars_tm: tm > list_nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int2: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Syntax__Ofm,type,
member_fm2: fm > set_fm > $o ).
thf(sy_c_member_001t__Syntax__Otm,type,
member_tm2: tm > set_tm > $o ).
thf(sy_v_Ea____,type,
ea: nat > a ).
thf(sy_v_F,type,
f: nat > list_a > a ).
thf(sy_v_G,type,
g: nat > list_a > $o ).
thf(sy_v_P____,type,
p: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_ts____,type,
ts: list_tm ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1266)
thf(fact_0__092_060open_062_092_060forall_062t_092_060in_062set_Ats_O_An_A_091_092_060notin_062_093_Avars__tm_At_092_060close_062,axiom,
! [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ ts ) )
=> ~ ( member_nat2 @ na @ ( set_nat2 @ ( vars_tm @ X ) ) ) ) ).
% \<open>\<forall>t\<in>set ts. n [\<notin>] vars_tm t\<close>
thf(fact_1_Pre,axiom,
ord_less_nat @ ( max_list @ ( vars_fm @ ( pre @ p @ ts ) ) ) @ na ).
% Pre
thf(fact_2_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_3_fun__upd__upd,axiom,
! [F: nat > a,X2: nat,Y: a,Z: a] :
( ( fun_upd_nat_a @ ( fun_upd_nat_a @ F @ X2 @ Y ) @ X2 @ Z )
= ( fun_upd_nat_a @ F @ X2 @ Z ) ) ).
% fun_upd_upd
thf(fact_4_fun__upd__triv,axiom,
! [F: nat > a,X2: nat] :
( ( fun_upd_nat_a @ F @ X2 @ ( F @ X2 ) )
= F ) ).
% fun_upd_triv
thf(fact_5_fun__upd__apply,axiom,
( fun_upd_nat_a
= ( ^ [F2: nat > a,X3: nat,Y2: a,Z2: nat] : ( if_a @ ( Z2 = X3 ) @ Y2 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_6_fun__upd__def,axiom,
( fun_upd_nat_a
= ( ^ [F2: nat > a,A: nat,B: a,X3: nat] : ( if_a @ ( X3 = A ) @ B @ ( F2 @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_7_fun__upd__eqD,axiom,
! [F: nat > a,X2: nat,Y: a,G: nat > a,Z: a] :
( ( ( fun_upd_nat_a @ F @ X2 @ Y )
= ( fun_upd_nat_a @ G @ X2 @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_8_fun__upd__idem,axiom,
! [F: nat > a,X2: nat,Y: a] :
( ( ( F @ X2 )
= Y )
=> ( ( fun_upd_nat_a @ F @ X2 @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_9_fun__upd__same,axiom,
! [F: nat > a,X2: nat,Y: a] :
( ( fun_upd_nat_a @ F @ X2 @ Y @ X2 )
= Y ) ).
% fun_upd_same
thf(fact_10_fun__upd__other,axiom,
! [Z: nat,X2: nat,F: nat > a,Y: a] :
( ( Z != X2 )
=> ( ( fun_upd_nat_a @ F @ X2 @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_11_fun__upd__twist,axiom,
! [A2: nat,C: nat,M: nat > a,B2: a,D: a] :
( ( A2 != C )
=> ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ M @ A2 @ B2 ) @ C @ D )
= ( fun_upd_nat_a @ ( fun_upd_nat_a @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_12_fun__upd__idem__iff,axiom,
! [F: nat > a,X2: nat,Y: a] :
( ( ( fun_upd_nat_a @ F @ X2 @ Y )
= F )
= ( ( F @ X2 )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_13__092_060open_062n_A_091_092_060notin_062_093_Aconcat_A_Imap_Avars__tm_Ats_J_092_060close_062,axiom,
~ ( member_nat2 @ na @ ( set_nat2 @ ( concat_nat @ ( map_tm_list_nat @ vars_tm @ ts ) ) ) ) ).
% \<open>n [\<notin>] concat (map vars_tm ts)\<close>
thf(fact_14__092_060open_062max__list_A_Iconcat_A_Imap_Avars__tm_Ats_J_J_A_060_An_092_060close_062,axiom,
ord_less_nat @ ( max_list @ ( concat_nat @ ( map_tm_list_nat @ vars_tm @ ts ) ) ) @ na ).
% \<open>max_list (concat (map vars_tm ts)) < n\<close>
thf(fact_15_vars__fm_Osimps_I2_J,axiom,
! [Uu: nat,Ts: list_tm] :
( ( vars_fm @ ( pre @ Uu @ Ts ) )
= ( concat_nat @ ( map_tm_list_nat @ vars_tm @ Ts ) ) ) ).
% vars_fm.simps(2)
thf(fact_16_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_17_map__fun__upd,axiom,
! [Y: tm,Xs: list_tm,F: tm > list_nat,V: list_nat] :
( ~ ( member_tm2 @ Y @ ( set_tm2 @ Xs ) )
=> ( ( map_tm_list_nat @ ( fun_upd_tm_list_nat @ F @ Y @ V ) @ Xs )
= ( map_tm_list_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_18_map__fun__upd,axiom,
! [Y: tm,Xs: list_tm,F: tm > tm,V: tm] :
( ~ ( member_tm2 @ Y @ ( set_tm2 @ Xs ) )
=> ( ( map_tm_tm @ ( fun_upd_tm_tm @ F @ Y @ V ) @ Xs )
= ( map_tm_tm @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_19_map__fun__upd,axiom,
! [Y: nat,Xs: list_nat,F: nat > a,V: a] :
( ~ ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ( map_nat_a @ ( fun_upd_nat_a @ F @ Y @ V ) @ Xs )
= ( map_nat_a @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_20_map__fun__upd,axiom,
! [Y: nat,Xs: list_nat,F: nat > nat,V: nat] :
( ~ ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ( map_nat_nat @ ( fun_upd_nat_nat @ F @ Y @ V ) @ Xs )
= ( map_nat_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_21_map__fun__upd,axiom,
! [Y: fm,Xs: list_fm,F: fm > list_nat,V: list_nat] :
( ~ ( member_fm2 @ Y @ ( set_fm2 @ Xs ) )
=> ( ( map_fm_list_nat @ ( fun_upd_fm_list_nat @ F @ Y @ V ) @ Xs )
= ( map_fm_list_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_22_map__eq__conv,axiom,
! [F: tm > list_nat,Xs: list_tm,G: tm > list_nat] :
( ( ( map_tm_list_nat @ F @ Xs )
= ( map_tm_list_nat @ G @ Xs ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_23_map__eq__conv,axiom,
! [F: tm > tm,Xs: list_tm,G: tm > tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Xs ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_24_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_25_map__eq__conv,axiom,
! [F: fm > list_nat,Xs: list_fm,G: fm > list_nat] :
( ( ( map_fm_list_nat @ F @ Xs )
= ( map_fm_list_nat @ G @ Xs ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_26_map__ident,axiom,
( ( map_tm_tm
@ ^ [X3: tm] : X3 )
= ( ^ [Xs2: list_tm] : Xs2 ) ) ).
% map_ident
thf(fact_27_map__ident,axiom,
( ( map_nat_nat
@ ^ [X3: nat] : X3 )
= ( ^ [Xs2: list_nat] : Xs2 ) ) ).
% map_ident
thf(fact_28_upd__vars__tm,axiom,
! [N: nat,T: tm,E: nat > a,X2: a,F3: nat > list_a > a] :
( ~ ( member_nat2 @ N @ ( set_nat2 @ ( vars_tm @ T ) ) )
=> ( ( semantics_tm_a @ ( fun_upd_nat_a @ E @ N @ X2 ) @ F3 @ T )
= ( semantics_tm_a @ E @ F3 @ T ) ) ) ).
% upd_vars_tm
thf(fact_29_map__concat,axiom,
! [F: tm > list_nat,Xs: list_list_tm] :
( ( map_tm_list_nat @ F @ ( concat_tm @ Xs ) )
= ( concat_list_nat @ ( map_li7696074350313601672st_nat @ ( map_tm_list_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_30_map__concat,axiom,
! [F: fm > list_nat,Xs: list_list_fm] :
( ( map_fm_list_nat @ F @ ( concat_fm @ Xs ) )
= ( concat_list_nat @ ( map_li7487060700005177878st_nat @ ( map_fm_list_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_31_map__concat,axiom,
! [F: tm > tm,Xs: list_list_tm] :
( ( map_tm_tm @ F @ ( concat_tm @ Xs ) )
= ( concat_tm @ ( map_list_tm_list_tm @ ( map_tm_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_32_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_33_vars__tm_Osimps_I2_J,axiom,
! [Uu: nat,Ts: list_tm] :
( ( vars_tm @ ( fun @ Uu @ Ts ) )
= ( concat_nat @ ( map_tm_list_nat @ vars_tm @ Ts ) ) ) ).
% vars_tm.simps(2)
thf(fact_34_list_Omap__cong,axiom,
! [X2: list_tm,Ya: list_tm,F: tm > list_nat,G: tm > list_nat] :
( ( X2 = Ya )
=> ( ! [Z3: tm] :
( ( member_tm2 @ Z3 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_tm_list_nat @ F @ X2 )
= ( map_tm_list_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_35_list_Omap__cong,axiom,
! [X2: list_tm,Ya: list_tm,F: tm > tm,G: tm > tm] :
( ( X2 = Ya )
=> ( ! [Z3: tm] :
( ( member_tm2 @ Z3 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_tm_tm @ F @ X2 )
= ( map_tm_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_36_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_37_list_Omap__cong,axiom,
! [X2: list_fm,Ya: list_fm,F: fm > list_nat,G: fm > list_nat] :
( ( X2 = Ya )
=> ( ! [Z3: fm] :
( ( member_fm2 @ Z3 @ ( set_fm2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_fm_list_nat @ F @ X2 )
= ( map_fm_list_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_38_list_Omap__cong0,axiom,
! [X2: list_tm,F: tm > list_nat,G: tm > list_nat] :
( ! [Z3: tm] :
( ( member_tm2 @ Z3 @ ( set_tm2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_tm_list_nat @ F @ X2 )
= ( map_tm_list_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_39_list_Omap__cong0,axiom,
! [X2: list_tm,F: tm > tm,G: tm > tm] :
( ! [Z3: tm] :
( ( member_tm2 @ Z3 @ ( set_tm2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_tm_tm @ F @ X2 )
= ( map_tm_tm @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_40_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_41_list_Omap__cong0,axiom,
! [X2: list_fm,F: fm > list_nat,G: fm > list_nat] :
( ! [Z3: fm] :
( ( member_fm2 @ Z3 @ ( set_fm2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_fm_list_nat @ F @ X2 )
= ( map_fm_list_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_42_list_Oinj__map__strong,axiom,
! [X2: list_tm,Xa: list_tm,F: tm > list_nat,Fa: tm > list_nat] :
( ! [Z3: tm,Za: tm] :
( ( member_tm2 @ Z3 @ ( set_tm2 @ X2 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_tm_list_nat @ F @ X2 )
= ( map_tm_list_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_43_list_Oinj__map__strong,axiom,
! [X2: list_tm,Xa: list_tm,F: tm > tm,Fa: tm > tm] :
( ! [Z3: tm,Za: tm] :
( ( member_tm2 @ Z3 @ ( set_tm2 @ X2 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_tm_tm @ F @ X2 )
= ( map_tm_tm @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_44_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_45_list_Oinj__map__strong,axiom,
! [X2: list_fm,Xa: list_fm,F: fm > list_nat,Fa: fm > list_nat] :
( ! [Z3: fm,Za: fm] :
( ( member_fm2 @ Z3 @ ( set_fm2 @ X2 ) )
=> ( ( member_fm2 @ Za @ ( set_fm2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_fm_list_nat @ F @ X2 )
= ( map_fm_list_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_46_list_Omap__ident__strong,axiom,
! [T: list_tm,F: tm > tm] :
( ! [Z3: tm] :
( ( member_tm2 @ Z3 @ ( set_tm2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_tm_tm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_47_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_48_list_Omap__ident__strong,axiom,
! [T: list_fm,F: fm > fm] :
( ! [Z3: fm] :
( ( member_fm2 @ Z3 @ ( set_fm2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_fm_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_49_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat2 @ Z3 @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_50_map__ext,axiom,
! [Xs: list_tm,F: tm > list_nat,G: tm > list_nat] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_list_nat @ F @ Xs )
= ( map_tm_list_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_51_map__ext,axiom,
! [Xs: list_tm,F: tm > tm,G: tm > tm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_52_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_53_map__ext,axiom,
! [Xs: list_fm,F: fm > list_nat,G: fm > list_nat] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_list_nat @ F @ Xs )
= ( map_fm_list_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_54_map__idI,axiom,
! [Xs: list_tm,F: tm > tm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_tm_tm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_55_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_56_map__idI,axiom,
! [Xs: list_fm,F: fm > fm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_fm_fm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_57_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_58_tm_Oinject_I2_J,axiom,
! [X21: nat,X22: list_tm,Y21: nat,Y22: list_tm] :
( ( ( fun @ X21 @ X22 )
= ( fun @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% tm.inject(2)
thf(fact_59_semantics__tm_Osimps_I2_J,axiom,
! [E: nat > list_nat,F3: nat > list_list_nat > list_nat,F: nat,Ts: list_tm] :
( ( semant8474227294840824358st_nat @ E @ F3 @ ( fun @ F @ Ts ) )
= ( F3 @ F @ ( map_tm_list_nat @ ( semant8474227294840824358st_nat @ E @ F3 ) @ Ts ) ) ) ).
% semantics_tm.simps(2)
thf(fact_60_semantics__tm_Osimps_I2_J,axiom,
! [E: nat > tm,F3: nat > list_tm > tm,F: nat,Ts: list_tm] :
( ( semantics_tm_tm @ E @ F3 @ ( fun @ F @ Ts ) )
= ( F3 @ F @ ( map_tm_tm @ ( semantics_tm_tm @ E @ F3 ) @ Ts ) ) ) ).
% semantics_tm.simps(2)
thf(fact_61_list_Omap__ident,axiom,
! [T: list_tm] :
( ( map_tm_tm
@ ^ [X3: tm] : X3
@ T )
= T ) ).
% list.map_ident
thf(fact_62_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X3: nat] : X3
@ T )
= T ) ).
% list.map_ident
thf(fact_63_semantics__fm_Osimps_I2_J,axiom,
! [E: nat > list_nat,F3: nat > list_list_nat > list_nat,G2: nat > list_list_nat > $o,P: nat,Ts: list_tm] :
( ( semant1976007830398715828st_nat @ E @ F3 @ G2 @ ( pre @ P @ Ts ) )
= ( G2 @ P @ ( map_tm_list_nat @ ( semant8474227294840824358st_nat @ E @ F3 ) @ Ts ) ) ) ).
% semantics_fm.simps(2)
thf(fact_64_semantics__fm_Osimps_I2_J,axiom,
! [E: nat > tm,F3: nat > list_tm > tm,G2: nat > list_tm > $o,P: nat,Ts: list_tm] :
( ( semantics_fm_tm @ E @ F3 @ G2 @ ( pre @ P @ Ts ) )
= ( G2 @ P @ ( map_tm_tm @ ( semantics_tm_tm @ E @ F3 ) @ Ts ) ) ) ).
% semantics_fm.simps(2)
thf(fact_65_semantics__fm_Osimps_I2_J,axiom,
! [E: nat > a,F3: nat > list_a > a,G2: nat > list_a > $o,P: nat,Ts: list_tm] :
( ( semantics_fm_a @ E @ F3 @ G2 @ ( pre @ P @ Ts ) )
= ( G2 @ P @ ( map_tm_a @ ( semantics_tm_a @ E @ F3 ) @ Ts ) ) ) ).
% semantics_fm.simps(2)
thf(fact_66_ex__map__conv,axiom,
! [Ys: list_tm,F: tm > tm] :
( ( ? [Xs2: list_tm] :
( Ys
= ( map_tm_tm @ F @ Xs2 ) ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Ys ) )
=> ? [Y2: tm] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_67_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_68_ex__map__conv,axiom,
! [Ys: list_list_nat,F: tm > list_nat] :
( ( ? [Xs2: list_tm] :
( Ys
= ( map_tm_list_nat @ F @ Xs2 ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ? [Y2: tm] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_69_ex__map__conv,axiom,
! [Ys: list_list_nat,F: fm > list_nat] :
( ( ? [Xs2: list_fm] :
( Ys
= ( map_fm_list_nat @ F @ Xs2 ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ? [Y2: fm] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_70_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > list_nat,G: tm > list_nat] :
( ( Xs = Ys )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_list_nat @ F @ Xs )
= ( map_tm_list_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_71_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > tm,G: tm > tm] :
( ( Xs = Ys )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_72_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_73_map__cong,axiom,
! [Xs: list_fm,Ys: list_fm,F: fm > list_nat,G: fm > list_nat] :
( ( Xs = Ys )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_list_nat @ F @ Xs )
= ( map_fm_list_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_74_vars__fms__def,axiom,
( vars_fms
= ( ^ [A3: list_fm] : ( concat_nat @ ( map_fm_list_nat @ vars_fm @ A3 ) ) ) ) ).
% vars_fms_def
thf(fact_75_sub__fm__semantics,axiom,
! [E: nat > a,F3: nat > list_a > a,G2: nat > list_a > $o,S: nat > tm,P2: fm] :
( ( semantics_fm_a @ E @ F3 @ G2 @ ( sub_fm @ S @ P2 ) )
= ( semantics_fm_a
@ ^ [N2: nat] : ( semantics_tm_a @ E @ F3 @ ( S @ N2 ) )
@ F3
@ G2
@ P2 ) ) ).
% sub_fm_semantics
thf(fact_76_concat__map__maps,axiom,
! [F: tm > list_nat,Xs: list_tm] :
( ( concat_nat @ ( map_tm_list_nat @ F @ Xs ) )
= ( maps_tm_nat @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_77_concat__map__maps,axiom,
! [F: fm > list_nat,Xs: list_fm] :
( ( concat_nat @ ( map_fm_list_nat @ F @ Xs ) )
= ( maps_fm_nat @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_78_maps__def,axiom,
( maps_tm_nat
= ( ^ [F2: tm > list_nat,Xs2: list_tm] : ( concat_nat @ ( map_tm_list_nat @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_79_maps__def,axiom,
( maps_fm_nat
= ( ^ [F2: fm > list_nat,Xs2: list_fm] : ( concat_nat @ ( map_fm_list_nat @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_80_List_Obind__def,axiom,
( bind_tm_nat
= ( ^ [Xs2: list_tm,F2: tm > list_nat] : ( concat_nat @ ( map_tm_list_nat @ F2 @ Xs2 ) ) ) ) ).
% List.bind_def
thf(fact_81_List_Obind__def,axiom,
( bind_fm_nat
= ( ^ [Xs2: list_fm,F2: fm > list_nat] : ( concat_nat @ ( map_fm_list_nat @ F2 @ Xs2 ) ) ) ) ).
% List.bind_def
thf(fact_82_map__eq__map__tailrec,axiom,
map_tm_list_nat = map_ta1959924319307969074st_nat ).
% map_eq_map_tailrec
thf(fact_83_map__eq__map__tailrec,axiom,
map_fm_list_nat = map_ta4685076891720636352st_nat ).
% map_eq_map_tailrec
thf(fact_84_map__eq__map__tailrec,axiom,
map_tm_tm = map_tailrec_tm_tm ).
% map_eq_map_tailrec
thf(fact_85_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_86_lift__tm_Osimps_I2_J,axiom,
! [F: nat,Ts: list_tm] :
( ( lift_tm @ ( fun @ F @ Ts ) )
= ( fun @ F @ ( map_tm_tm @ lift_tm @ Ts ) ) ) ).
% lift_tm.simps(2)
thf(fact_87_sub__tm_Osimps_I2_J,axiom,
! [S: nat > tm,F: nat,Ts: list_tm] :
( ( sub_tm @ S @ ( fun @ F @ Ts ) )
= ( fun @ F @ ( map_tm_tm @ ( sub_tm @ S ) @ Ts ) ) ) ).
% sub_tm.simps(2)
thf(fact_88_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
! [A2: fm,P: fm > $o] :
( ( member_fm2 @ A2 @ ( collect_fm @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat2 @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A4: set_fm] :
( ( collect_fm
@ ^ [X3: fm] : ( member_fm2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_94_tm_Oinject_I1_J,axiom,
! [X1: nat,Y1: nat] :
( ( ( var @ X1 )
= ( var @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tm.inject(1)
thf(fact_95_sub__tm_Osimps_I1_J,axiom,
! [S: nat > tm,N: nat] :
( ( sub_tm @ S @ ( var @ N ) )
= ( S @ N ) ) ).
% sub_tm.simps(1)
thf(fact_96_sub__fm_Osimps_I2_J,axiom,
! [S: nat > tm,P: nat,Ts: list_tm] :
( ( sub_fm @ S @ ( pre @ P @ Ts ) )
= ( pre @ P @ ( map_tm_tm @ ( sub_tm @ S ) @ Ts ) ) ) ).
% sub_fm.simps(2)
thf(fact_97_tm_Oexhaust,axiom,
! [Y: tm] :
( ! [X12: nat] :
( Y
!= ( var @ X12 ) )
=> ~ ! [X212: nat,X222: list_tm] :
( Y
!= ( fun @ X212 @ X222 ) ) ) ).
% tm.exhaust
thf(fact_98_tm_Odistinct_I1_J,axiom,
! [X1: nat,X21: nat,X22: list_tm] :
( ( var @ X1 )
!= ( fun @ X21 @ X22 ) ) ).
% tm.distinct(1)
thf(fact_99_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_100_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_101_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_102_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_103_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_104_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_105_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_106_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_107_lift__lemma,axiom,
! [X2: tm,E: nat > tm,F3: nat > list_tm > tm,T: tm] :
( ( semantics_tm_tm @ ( add_env_tm @ X2 @ E ) @ F3 @ ( lift_tm @ T ) )
= ( semantics_tm_tm @ E @ F3 @ T ) ) ).
% lift_lemma
thf(fact_108_max__list__vars__fms,axiom,
! [A4: list_fm,N: nat,P2: fm] :
( ( ord_less_eq_nat @ ( max_list @ ( vars_fms @ A4 ) ) @ N )
=> ( ( member_fm2 @ P2 @ ( set_fm2 @ A4 ) )
=> ( ord_less_eq_nat @ ( max_list @ ( vars_fm @ P2 ) ) @ N ) ) ) ).
% max_list_vars_fms
thf(fact_109_vars__fms__member,axiom,
! [P2: fm,A4: list_fm] :
( ( member_fm2 @ P2 @ ( set_fm2 @ A4 ) )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( vars_fm @ P2 ) ) @ ( set_nat2 @ ( vars_fms @ A4 ) ) ) ) ).
% vars_fms_member
thf(fact_110_in__set__member,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
= ( member_tm @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_111_in__set__member,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_112_in__set__member,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
= ( member_fm @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_113_in__set__member,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( member_list_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_114_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ~ ( ord_less_nat @ T @ X ) ) ).
% minf(7)
thf(fact_115_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ~ ( ord_less_int @ T @ X ) ) ).
% minf(7)
thf(fact_116_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ord_less_nat @ X @ T ) ) ).
% minf(5)
thf(fact_117_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ord_less_int @ X @ T ) ) ).
% minf(5)
thf(fact_118_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( X != T ) ) ).
% minf(4)
thf(fact_119_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( X != T ) ) ).
% minf(4)
thf(fact_120_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( X != T ) ) ).
% minf(3)
thf(fact_121_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( X != T ) ) ).
% minf(3)
thf(fact_122_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P3 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_123_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P3 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_124_add__env__semantics,axiom,
! [E: nat > tm,F3: nat > list_tm > tm,T: tm,S: nat > tm,N: nat] :
( ( semantics_tm_tm @ E @ F3 @ ( add_env_tm @ T @ S @ N ) )
= ( add_env_tm @ ( semantics_tm_tm @ E @ F3 @ T )
@ ^ [M3: nat] : ( semantics_tm_tm @ E @ F3 @ ( S @ M3 ) )
@ N ) ) ).
% add_env_semantics
thf(fact_125_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_126_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_127_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_128_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_129_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_130_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_131_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ T ) ) ).
% pinf(6)
thf(fact_132_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ T ) ) ).
% pinf(6)
thf(fact_133_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ T @ X ) ) ).
% pinf(8)
thf(fact_134_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ord_less_eq_int @ T @ X ) ) ).
% pinf(8)
thf(fact_135_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_136_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ord_less_eq_int @ X @ T ) ) ).
% minf(6)
thf(fact_137_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_138_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X ) ) ).
% minf(8)
thf(fact_139_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_140_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_141_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_142_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_143_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_144_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_145_subset__code_I1_J,axiom,
! [Xs: list_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B3 )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( member_tm2 @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_146_subset__code_I1_J,axiom,
! [Xs: list_fm,B3: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B3 )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_147_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B3 )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat2 @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_148_subset__code_I1_J,axiom,
! [Xs: list_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_149_max__list__mono,axiom,
! [A4: list_nat,B3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ A4 ) @ ( set_nat2 @ B3 ) )
=> ( ord_less_eq_nat @ ( max_list @ A4 ) @ ( max_list @ B3 ) ) ) ).
% max_list_mono
thf(fact_150_max__list__concat,axiom,
! [Xs: list_nat,Xss: list_list_nat] :
( ( member_list_nat2 @ Xs @ ( set_list_nat2 @ Xss ) )
=> ( ord_less_eq_nat @ ( max_list @ Xs ) @ ( max_list @ ( concat_nat @ Xss ) ) ) ) ).
% max_list_concat
thf(fact_151_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P3 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_152_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P3 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_153_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P3 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_154_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P3 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_155_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_156_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_157_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_158_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_159_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ~ ( ord_less_nat @ X @ T ) ) ).
% pinf(5)
thf(fact_160_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ~ ( ord_less_int @ X @ T ) ) ).
% pinf(5)
thf(fact_161_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ord_less_nat @ T @ X ) ) ).
% pinf(7)
thf(fact_162_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ord_less_int @ T @ X ) ) ).
% pinf(7)
thf(fact_163_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P3 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_164_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P3 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_165_psubsetI,axiom,
! [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_set_nat @ A4 @ B3 ) ) ) ).
% psubsetI
thf(fact_166_subset__antisym,axiom,
! [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_167_subsetI,axiom,
! [A4: set_fm,B3: set_fm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ A4 )
=> ( member_fm2 @ X4 @ B3 ) )
=> ( ord_less_eq_set_fm @ A4 @ B3 ) ) ).
% subsetI
thf(fact_168_subsetI,axiom,
! [A4: set_list_nat,B3: set_list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat2 @ X4 @ A4 )
=> ( member_list_nat2 @ X4 @ B3 ) )
=> ( ord_le6045566169113846134st_nat @ A4 @ B3 ) ) ).
% subsetI
thf(fact_169_subsetI,axiom,
! [A4: set_nat,B3: set_nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ A4 )
=> ( member_nat2 @ X4 @ B3 ) )
=> ( ord_less_eq_set_nat @ A4 @ B3 ) ) ).
% subsetI
thf(fact_170_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_171_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_172_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_173_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_174_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_175_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_176_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_177_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_178_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_179_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_180_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_181_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_182_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_183_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_184_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_185_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > set_nat,C: set_nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_186_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_187_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_188_psubsetD,axiom,
! [A4: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A4 @ B3 )
=> ( ( member_nat2 @ C @ A4 )
=> ( member_nat2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_189_psubsetD,axiom,
! [A4: set_fm,B3: set_fm,C: fm] :
( ( ord_less_set_fm @ A4 @ B3 )
=> ( ( member_fm2 @ C @ A4 )
=> ( member_fm2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_190_psubsetD,axiom,
! [A4: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A4 @ B3 )
=> ( ( member_list_nat2 @ C @ A4 )
=> ( member_list_nat2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_191_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_192_less__set__def,axiom,
( ord_less_set_fm
= ( ^ [A3: set_fm,B4: set_fm] :
( ord_less_fm_o
@ ^ [X3: fm] : ( member_fm2 @ X3 @ A3 )
@ ^ [X3: fm] : ( member_fm2 @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_193_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ A3 )
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_194_less__eq__set__def,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B4: set_fm] :
( ord_less_eq_fm_o
@ ^ [X3: fm] : ( member_fm2 @ X3 @ A3 )
@ ^ [X3: fm] : ( member_fm2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_195_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ A3 )
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_196_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_197_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_198_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_199_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_200_le__cases3,axiom,
! [X2: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_201_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_202_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z5: set_nat] : ( Y5 = Z5 ) )
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_203_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_204_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_205_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_206_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_207_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_208_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_209_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_210_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_211_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_212_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_213_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_214_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_215_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_216_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_217_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_218_order__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_219_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: 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 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_220_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: 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 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_221_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_222_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z5: set_nat] : ( Y5 = Z5 ) )
= ( ^ [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_223_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_224_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_225_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_226_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_227_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_228_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_229_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_230_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_231_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_232_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_233_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_234_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z5: set_nat] : ( Y5 = Z5 ) )
= ( ^ [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_235_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_236_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_237_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_238_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_239_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_240_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_241_order__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B2: int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_242_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_243_order__subst1,axiom,
! [A2: int,F: set_nat > int,B2: set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_244_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_245_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_246_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_247_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_248_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_249_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_250_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_251_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_252_order__subst2,axiom,
! [A2: int,B2: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_253_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_254_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_255_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_256_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_257_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_258_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_259_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_260_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_261_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_262_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_263_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_264_ord__eq__le__subst,axiom,
! [A2: int,F: set_nat > int,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_265_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_266_ord__eq__le__subst,axiom,
! [A2: set_nat,F: int > set_nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_267_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_268_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_269_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_270_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_271_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_272_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_273_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_274_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_275_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_276_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_277_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_278_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_279_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_280_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_281_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_282_lt__ex,axiom,
! [X2: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X2 ) ).
% lt_ex
thf(fact_283_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_284_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_285_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_286_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_287_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_288_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_289_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_290_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_291_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_292_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_293_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_294_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_295_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_296_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_297_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_298_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_299_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_300_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_301_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_302_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ( P5 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_303_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: 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 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_304_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: 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 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_305_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_306_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_307_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_308_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_309_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_310_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_311_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_312_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_313_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_314_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_315_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_316_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_317_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_318_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_319_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_320_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_321_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_322_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_323_order__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_324_order__less__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_325_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_326_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_327_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_328_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_329_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_330_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_331_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_332_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_333_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_334_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_335_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_336_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_337_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_338_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_339_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_340_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_341_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_342_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_343_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_344_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_345_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_346_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_347_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_348_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_349_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_350_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_351_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_352_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_353_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_354_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_355_in__mono,axiom,
! [A4: set_fm,B3: set_fm,X2: fm] :
( ( ord_less_eq_set_fm @ A4 @ B3 )
=> ( ( member_fm2 @ X2 @ A4 )
=> ( member_fm2 @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_356_in__mono,axiom,
! [A4: set_list_nat,B3: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B3 )
=> ( ( member_list_nat2 @ X2 @ A4 )
=> ( member_list_nat2 @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_357_in__mono,axiom,
! [A4: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ( member_nat2 @ X2 @ A4 )
=> ( member_nat2 @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_358_subsetD,axiom,
! [A4: set_fm,B3: set_fm,C: fm] :
( ( ord_less_eq_set_fm @ A4 @ B3 )
=> ( ( member_fm2 @ C @ A4 )
=> ( member_fm2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_359_subsetD,axiom,
! [A4: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B3 )
=> ( ( member_list_nat2 @ C @ A4 )
=> ( member_list_nat2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_360_subsetD,axiom,
! [A4: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ( member_nat2 @ C @ A4 )
=> ( member_nat2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_361_equalityE,axiom,
! [A4: set_nat,B3: set_nat] :
( ( A4 = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ).
% equalityE
thf(fact_362_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B4: set_fm] :
! [X3: fm] :
( ( member_fm2 @ X3 @ A3 )
=> ( member_fm2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_363_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] :
! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A3 )
=> ( member_list_nat2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_364_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
=> ( member_nat2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_365_equalityD1,axiom,
! [A4: set_nat,B3: set_nat] :
( ( A4 = B3 )
=> ( ord_less_eq_set_nat @ A4 @ B3 ) ) ).
% equalityD1
thf(fact_366_equalityD2,axiom,
! [A4: set_nat,B3: set_nat] :
( ( A4 = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A4 ) ) ).
% equalityD2
thf(fact_367_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B4: set_fm] :
! [T2: fm] :
( ( member_fm2 @ T2 @ A3 )
=> ( member_fm2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_368_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat2 @ T2 @ A3 )
=> ( member_list_nat2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_369_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A3 )
=> ( member_nat2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_370_subset__refl,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_371_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_372_subset__trans,axiom,
! [A4: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_373_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z5: set_nat] : ( Y5 = Z5 ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_374_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_375_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_376_subset__psubset__trans,axiom,
! [A4: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_377_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_378_psubset__subset__trans,axiom,
! [A4: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_379_psubset__imp__subset,axiom,
! [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
=> ( ord_less_eq_set_nat @ A4 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_380_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_381_psubsetE,axiom,
! [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ).
% psubsetE
thf(fact_382_Collect__subset,axiom,
! [A4: set_fm,P: fm > $o] :
( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X3: fm] :
( ( member_fm2 @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_383_Collect__subset,axiom,
! [A4: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_384_Collect__subset,axiom,
! [A4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_385_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_386_leD,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_387_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_388_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_389_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_390_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_391_nless__le,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_392_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_393_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_394_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_395_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_396_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_397_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_398_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_399_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_400_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_401_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_402_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_403_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_404_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_405_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_406_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( ord_less_int @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_407_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_408_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_409_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_410_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_411_order_Ostrict__trans1,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_412_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_413_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_414_order_Ostrict__trans2,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_415_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_416_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ~ ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_417_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
& ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_418_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ~ ( ord_less_eq_int @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_419_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_420_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_421_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( ord_less_int @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_422_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_423_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_424_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_425_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_426_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_427_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_428_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_429_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_430_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_431_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_432_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
& ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_433_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ~ ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_434_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_435_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_436_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_437_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_438_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_439_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_440_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_441_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_442_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_443_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_444_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_445_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_446_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_447_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_448_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_449_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_450_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_451_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_452_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_453_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_454_order__le__neq__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_455_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_456_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_457_order__neq__le__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_458_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_459_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_460_order__le__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_461_order__le__less__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_462_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_463_order__less__le__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_464_order__less__le__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_465_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_466_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_467_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_468_order__le__less__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B2: int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_469_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_470_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_471_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_472_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_473_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_474_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_475_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_476_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_477_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_478_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_479_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_480_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_481_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_482_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_483_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_484_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_485_order__less__le__subst1,axiom,
! [A2: int,F: set_nat > int,B2: set_nat,C: set_nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_486_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_487_order__less__le__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B2: int,C: int] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_488_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_489_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_490_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_491_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B2 )
& ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ C3 ) )
=> ( P @ X ) )
& ! [D2: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_492_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A2 @ C3 )
& ( ord_less_eq_int @ C3 @ B2 )
& ! [X: int] :
( ( ( ord_less_eq_int @ A2 @ X )
& ( ord_less_int @ X @ C3 ) )
=> ( P @ X ) )
& ! [D2: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A2 @ X4 )
& ( ord_less_int @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_493_pred__subset__eq,axiom,
! [R: set_fm,S2: set_fm] :
( ( ord_less_eq_fm_o
@ ^ [X3: fm] : ( member_fm2 @ X3 @ R )
@ ^ [X3: fm] : ( member_fm2 @ X3 @ S2 ) )
= ( ord_less_eq_set_fm @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_494_pred__subset__eq,axiom,
! [R: set_list_nat,S2: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ R )
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ S2 ) )
= ( ord_le6045566169113846134st_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_495_pred__subset__eq,axiom,
! [R: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ R )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ S2 ) )
= ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_496_conj__subset__def,axiom,
! [A4: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A4
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_497_prop__restrict,axiom,
! [X2: fm,Z6: set_fm,X6: set_fm,P: fm > $o] :
( ( member_fm2 @ X2 @ Z6 )
=> ( ( ord_less_eq_set_fm @ Z6
@ ( collect_fm
@ ^ [X3: fm] :
( ( member_fm2 @ X3 @ X6 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_498_prop__restrict,axiom,
! [X2: list_nat,Z6: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
( ( member_list_nat2 @ X2 @ Z6 )
=> ( ( ord_le6045566169113846134st_nat @ Z6
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat2 @ X3 @ X6 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_499_prop__restrict,axiom,
! [X2: nat,Z6: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat2 @ X2 @ Z6 )
=> ( ( ord_less_eq_set_nat @ Z6
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ X6 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_500_Collect__restrict,axiom,
! [X6: set_fm,P: fm > $o] :
( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X3: fm] :
( ( member_fm2 @ X3 @ X6 )
& ( P @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_501_Collect__restrict,axiom,
! [X6: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat2 @ X3 @ X6 )
& ( P @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_502_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ X6 )
& ( P @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_503_subset__CollectI,axiom,
! [B3: set_fm,A4: set_fm,Q: fm > $o,P: fm > $o] :
( ( ord_less_eq_set_fm @ B3 @ A4 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B3 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X3: fm] :
( ( member_fm2 @ X3 @ B3 )
& ( Q @ X3 ) ) )
@ ( collect_fm
@ ^ [X3: fm] :
( ( member_fm2 @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_504_subset__CollectI,axiom,
! [B3: set_list_nat,A4: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A4 )
=> ( ! [X4: list_nat] :
( ( member_list_nat2 @ X4 @ B3 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat2 @ X3 @ B3 )
& ( Q @ X3 ) ) )
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_505_subset__CollectI,axiom,
! [B3: set_nat,A4: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B3 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ B3 )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_506_subset__Collect__iff,axiom,
! [B3: set_fm,A4: set_fm,P: fm > $o] :
( ( ord_less_eq_set_fm @ B3 @ A4 )
=> ( ( ord_less_eq_set_fm @ B3
@ ( collect_fm
@ ^ [X3: fm] :
( ( member_fm2 @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_507_subset__Collect__iff,axiom,
! [B3: set_list_nat,A4: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ B3
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_508_subset__Collect__iff,axiom,
! [B3: set_nat,A4: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_509_sub__fm_Osimps_I4_J,axiom,
! [S: nat > tm,P2: fm] :
( ( sub_fm @ S @ ( uni @ P2 ) )
= ( uni
@ ( sub_fm
@ ( add_env_tm @ ( var @ zero_zero_nat )
@ ^ [N2: nat] : ( lift_tm @ ( S @ N2 ) ) )
@ P2 ) ) ) ).
% sub_fm.simps(4)
thf(fact_510_fm_Oinject_I3_J,axiom,
! [X42: fm,Y42: fm] :
( ( ( uni @ X42 )
= ( uni @ Y42 ) )
= ( X42 = Y42 ) ) ).
% fm.inject(3)
thf(fact_511_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_512_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_513_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_514_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_515_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_516_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_517_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_518_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_519_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_520_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_521_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_522_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_523_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_524_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_525_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_526_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_527_add__env_Osimps_I1_J,axiom,
! [T: tm,S: nat > tm] :
( ( add_env_tm @ T @ S @ zero_zero_nat )
= T ) ).
% add_env.simps(1)
thf(fact_528_fm_Odistinct_I9_J,axiom,
! [X21: nat,X22: list_tm,X42: fm] :
( ( pre @ X21 @ X22 )
!= ( uni @ X42 ) ) ).
% fm.distinct(9)
thf(fact_529_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_530_vars__fm_Osimps_I4_J,axiom,
! [P2: fm] :
( ( vars_fm @ ( uni @ P2 ) )
= ( vars_fm @ P2 ) ) ).
% vars_fm.simps(4)
thf(fact_531_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_532_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_533_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_534_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_535_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_536_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_537_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_538_semantics__fm_Osimps_I4_J,axiom,
! [E: nat > tm,F3: nat > list_tm > tm,G2: nat > list_tm > $o,P2: fm] :
( ( semantics_fm_tm @ E @ F3 @ G2 @ ( uni @ P2 ) )
= ( ! [X3: tm] : ( semantics_fm_tm @ ( add_env_tm @ X3 @ E ) @ F3 @ G2 @ P2 ) ) ) ).
% semantics_fm.simps(4)
thf(fact_539_semantics__fm_Osimps_I4_J,axiom,
! [E: nat > a,F3: nat > list_a > a,G2: nat > list_a > $o,P2: fm] :
( ( semantics_fm_a @ E @ F3 @ G2 @ ( uni @ P2 ) )
= ( ! [X3: a] : ( semantics_fm_a @ ( add_env_a @ X3 @ E ) @ F3 @ G2 @ P2 ) ) ) ).
% semantics_fm.simps(4)
thf(fact_540_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_541_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_542_tm_Osize__gen_I1_J,axiom,
! [X1: nat] :
( ( size_tm @ ( var @ X1 ) )
= zero_zero_nat ) ).
% tm.size_gen(1)
thf(fact_543_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_544_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_545_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_546_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_547_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_548_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_549_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_550_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_551_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_552_tm_Osize_I3_J,axiom,
! [X1: nat] :
( ( size_size_tm @ ( var @ X1 ) )
= zero_zero_nat ) ).
% tm.size(3)
thf(fact_553_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
@ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_554_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_555_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_556_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_557_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_558_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_559_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_560_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_561_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_562_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_563_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_564_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_565_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_566_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_567_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_568_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_569_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_570_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A2 @ Top ) ) ).
% ordering_top.extremum
thf(fact_571_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A2 ) ) ).
% ordering_top.extremum_strict
thf(fact_572_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
= ( A2 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_573_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A2 != Top )
= ( Less @ A2 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_574_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
=> ( A2 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_575_size__neq__size__imp__neq,axiom,
! [X2: tm,Y: tm] :
( ( ( size_size_tm @ X2 )
!= ( size_size_tm @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_576_size__neq__size__imp__neq,axiom,
! [X2: fm,Y: fm] :
( ( ( size_size_fm @ X2 )
!= ( size_size_fm @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_577_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_578_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_579_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_580_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_less_as_int
thf(fact_581_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_leq_as_int
thf(fact_582_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_583_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_584_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_585_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_586_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_587_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_588_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_589_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_590_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_591_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_592_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_593_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_594_fm_Osize__gen_I1_J,axiom,
( ( size_fm @ falsity )
= zero_zero_nat ) ).
% fm.size_gen(1)
thf(fact_595_fm_Osize_I6_J,axiom,
! [X21: nat,X22: list_tm] :
( ( size_size_fm @ ( pre @ X21 @ X22 ) )
= zero_zero_nat ) ).
% fm.size(6)
thf(fact_596_add__upd__commute,axiom,
! [Y: a,E: nat > a,N: nat,X2: a,M: nat] :
( ( add_env_a @ Y @ ( fun_upd_nat_a @ E @ N @ X2 ) @ M )
= ( fun_upd_nat_a @ ( add_env_a @ Y @ E ) @ ( suc @ N ) @ X2 @ M ) ) ).
% add_upd_commute
thf(fact_597_add__upd__commute,axiom,
! [Y: tm,E: nat > tm,N: nat,X2: tm,M: nat] :
( ( add_env_tm @ Y @ ( fun_upd_nat_tm @ E @ N @ X2 ) @ M )
= ( fun_upd_nat_tm @ ( add_env_tm @ Y @ E ) @ ( suc @ N ) @ X2 @ M ) ) ).
% add_upd_commute
thf(fact_598_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_599_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_600_length__map,axiom,
! [F: tm > list_nat,Xs: list_tm] :
( ( size_s3023201423986296836st_nat @ ( map_tm_list_nat @ F @ Xs ) )
= ( size_size_list_tm @ Xs ) ) ).
% length_map
thf(fact_601_length__map,axiom,
! [F: fm > list_nat,Xs: list_fm] :
( ( size_s3023201423986296836st_nat @ ( map_fm_list_nat @ F @ Xs ) )
= ( size_size_list_fm @ Xs ) ) ).
% length_map
thf(fact_602_length__map,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( size_size_list_tm @ ( map_tm_tm @ F @ Xs ) )
= ( size_size_list_tm @ Xs ) ) ).
% length_map
thf(fact_603_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_604_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_605_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_606_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_607_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_608_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_609_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_610_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_611_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_612_verit__la__generic,axiom,
! [A2: int,X2: int] :
( ( ord_less_eq_int @ A2 @ X2 )
| ( A2 = X2 )
| ( ord_less_eq_int @ X2 @ A2 ) ) ).
% verit_la_generic
thf(fact_613_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_614_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_615_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_616_imp__le__cong,axiom,
! [X2: int,X7: int,P: $o,P3: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_617_conj__le__cong,axiom,
! [X2: int,X7: int,P: $o,P3: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_618_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_619_fm_Osize_I5_J,axiom,
( ( size_size_fm @ falsity )
= zero_zero_nat ) ).
% fm.size(5)
thf(fact_620_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_621_map__eq__imp__length__eq,axiom,
! [F: tm > tm,Xs: list_tm,G: tm > tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_622_map__eq__imp__length__eq,axiom,
! [F: tm > tm,Xs: list_tm,G: nat > tm,Ys: list_nat] :
( ( ( map_tm_tm @ F @ Xs )
= ( map_nat_tm @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_623_map__eq__imp__length__eq,axiom,
! [F: nat > tm,Xs: list_nat,G: tm > tm,Ys: list_tm] :
( ( ( map_nat_tm @ F @ Xs )
= ( map_tm_tm @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_tm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_624_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_625_map__eq__imp__length__eq,axiom,
! [F: tm > list_nat,Xs: list_tm,G: tm > list_nat,Ys: list_tm] :
( ( ( map_tm_list_nat @ F @ Xs )
= ( map_tm_list_nat @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_626_map__eq__imp__length__eq,axiom,
! [F: tm > list_nat,Xs: list_tm,G: fm > list_nat,Ys: list_fm] :
( ( ( map_tm_list_nat @ F @ Xs )
= ( map_fm_list_nat @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_627_map__eq__imp__length__eq,axiom,
! [F: fm > list_nat,Xs: list_fm,G: tm > list_nat,Ys: list_tm] :
( ( ( map_fm_list_nat @ F @ Xs )
= ( map_tm_list_nat @ G @ Ys ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_628_map__eq__imp__length__eq,axiom,
! [F: fm > list_nat,Xs: list_fm,G: fm > list_nat,Ys: list_fm] :
( ( ( map_fm_list_nat @ F @ Xs )
= ( map_fm_list_nat @ G @ Ys ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_629_map__eq__imp__length__eq,axiom,
! [F: tm > list_nat,Xs: list_tm,G: nat > list_nat,Ys: list_nat] :
( ( ( map_tm_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_630_map__eq__imp__length__eq,axiom,
! [F: fm > list_nat,Xs: list_fm,G: nat > list_nat,Ys: list_nat] :
( ( ( map_fm_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_631_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_632_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_633_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_634_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_635_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_636_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_637_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_638_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_639_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_640_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_641_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_642_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_643_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_644_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_645_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_646_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_647_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_648_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_649_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_650_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_651_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_652_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_653_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_654_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_655_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_656_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_657_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_658_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_659_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_660_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_661_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_662_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_663_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_664_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_665_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_666_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_667_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_668_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_669_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_670_add__env_Osimps_I2_J,axiom,
! [T: tm,S: nat > tm,N: nat] :
( ( add_env_tm @ T @ S @ ( suc @ N ) )
= ( S @ N ) ) ).
% add_env.simps(2)
thf(fact_671_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_672_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_673_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_674_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_675_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_676_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_677_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_678_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_679_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_680_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_681_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_682_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_683_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_684_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_685_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_686_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_687_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_688_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_689_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_690_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_691_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_692_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_693_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_694_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_695_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_696_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_697_semantics__fm_Osimps_I1_J,axiom,
! [Uu: nat > a,Uv: nat > list_a > a,Uw: nat > list_a > $o] :
~ ( semantics_fm_a @ Uu @ Uv @ Uw @ falsity ) ).
% semantics_fm.simps(1)
thf(fact_698_fm_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_tm] :
( falsity
!= ( pre @ X21 @ X22 ) ) ).
% fm.distinct(1)
thf(fact_699_fm_Odistinct_I5_J,axiom,
! [X42: fm] :
( falsity
!= ( uni @ X42 ) ) ).
% fm.distinct(5)
thf(fact_700_sub__fm_Osimps_I1_J,axiom,
! [Uu: nat > tm] :
( ( sub_fm @ Uu @ falsity )
= falsity ) ).
% sub_fm.simps(1)
thf(fact_701_length__pos__if__in__set,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_702_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_703_length__pos__if__in__set,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_704_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_705_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_706_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_707_fresh__def,axiom,
( fresh
= ( ^ [A3: list_fm] : ( suc @ ( max_list @ ( vars_fms @ A3 ) ) ) ) ) ).
% fresh_def
thf(fact_708_set__n__lists,axiom,
! [N: nat,Xs: list_tm] :
( ( set_list_tm2 @ ( n_lists_tm @ N @ Xs ) )
= ( collect_list_tm
@ ^ [Ys3: list_tm] :
( ( ( size_size_list_tm @ Ys3 )
= N )
& ( ord_less_eq_set_tm @ ( set_tm2 @ Ys3 ) @ ( set_tm2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_709_set__n__lists,axiom,
! [N: nat,Xs: list_fm] :
( ( set_list_fm2 @ ( n_lists_fm @ N @ Xs ) )
= ( collect_list_fm
@ ^ [Ys3: list_fm] :
( ( ( size_size_list_fm @ Ys3 )
= N )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Ys3 ) @ ( set_fm2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_710_set__n__lists,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys3 )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys3 ) @ ( set_list_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_711_set__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
= ( collect_list_nat
@ ^ [Ys3: list_nat] :
( ( ( size_size_list_nat @ Ys3 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_712_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_713_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_714_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_715_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_716_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_717_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_718_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_719_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_720_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_721_add__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_722_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_723_add__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_724_neg__le__iff__le,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_725_neg__less__iff__less,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_726_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_727_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_728_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_729_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_730_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_731_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_732_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_733_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_734_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_735_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_736_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_737_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_738_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_739_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_740_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_741_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_742_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_743_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_744_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_745_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_746_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_747_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_748_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_760_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_761_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_762_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_763_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_764_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_765_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_766_le__imp__neg__le,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_767_minus__le__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_768_le__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_769_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_770_add__le__imp__le__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_771_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_772_add__le__imp__le__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_773_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C4: nat] :
( B
= ( plus_plus_nat @ A @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_774_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_775_add__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_776_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C3: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_777_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_778_add__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_779_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_780_add__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_781_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_782_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_783_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_784_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_785_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_786_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_787_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_788_minus__less__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_less_iff
thf(fact_789_less__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_790_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_791_add__less__imp__less__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_792_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_793_add__less__imp__less__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_794_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_795_add__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_796_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_797_add__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_798_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_799_add__strict__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_800_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_801_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_802_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_803_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_804_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_805_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_806_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A2: nat] :
( ( A4
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_807_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_808_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_809_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_810_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_811_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_812_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_813_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_814_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_815_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_816_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_817_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_818_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_819_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_820_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_821_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_822_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_823_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_824_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_825_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_826_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_827_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_828_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_829_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_830_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_831_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_832_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_833_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_834_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_835_add__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_836_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_837_add__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_838_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_839_add__increasing2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_840_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_841_add__decreasing2,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_842_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_843_add__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_844_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_845_add__decreasing,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_846_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_847_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_848_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_849_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_850_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_851_add__le__less__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_852_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_853_add__less__le__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_854_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_855_add__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_856_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_857_add__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_858_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C3: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_859_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_860_pos__add__strict,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_861_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_862_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_863_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_864_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_865_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_866_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_867_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_868_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_869_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_870_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_871_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_872_semantics__fm_Osimps_I3_J,axiom,
! [E: nat > a,F3: nat > list_a > a,G2: nat > list_a > $o,P2: fm,Q4: fm] :
( ( semantics_fm_a @ E @ F3 @ G2 @ ( imp @ P2 @ Q4 ) )
= ( ( semantics_fm_a @ E @ F3 @ G2 @ P2 )
=> ( semantics_fm_a @ E @ F3 @ G2 @ Q4 ) ) ) ).
% semantics_fm.simps(3)
thf(fact_873_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_874_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_875_fm_Odistinct_I11_J,axiom,
! [X312: fm,X322: fm,X42: fm] :
( ( imp @ X312 @ X322 )
!= ( uni @ X42 ) ) ).
% fm.distinct(11)
thf(fact_876_fm_Odistinct_I3_J,axiom,
! [X312: fm,X322: fm] :
( falsity
!= ( imp @ X312 @ X322 ) ) ).
% fm.distinct(3)
thf(fact_877_sub__fm_Osimps_I3_J,axiom,
! [S: nat > tm,P2: fm,Q4: fm] :
( ( sub_fm @ S @ ( imp @ P2 @ Q4 ) )
= ( imp @ ( sub_fm @ S @ P2 ) @ ( sub_fm @ S @ Q4 ) ) ) ).
% sub_fm.simps(3)
thf(fact_878_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_879_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_880_add__neg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_881_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_882_add__nonneg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_883_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_884_add__nonpos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_885_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_886_add__pos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_887_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_888_add__strict__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_889_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_890_add__strict__increasing2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_891_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_892_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_893_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_894_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_895_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_896_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_897_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_898_negD,axiom,
! [X2: int] :
( ( ord_less_int @ X2 @ zero_zero_int )
=> ? [N3: nat] :
( X2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_899_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_900_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_901_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_902_Compl__anti__mono,axiom,
! [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B3 ) @ ( uminus5710092332889474511et_nat @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_903_Compl__subset__Compl__iff,axiom,
! [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( uminus5710092332889474511et_nat @ B3 ) )
= ( ord_less_eq_set_nat @ B3 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_904_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z2: int] :
? [N2: nat] :
( Z2
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_905_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W: int,Z2: int] :
? [N2: nat] :
( Z2
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_906_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_907_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_908_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_909_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_910_tm_Osize__gen_I2_J,axiom,
! [X21: nat,X22: list_tm] :
( ( size_tm @ ( fun @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_list_tm @ size_tm @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% tm.size_gen(2)
thf(fact_911_tm_Osize_I4_J,axiom,
! [X21: nat,X22: list_tm] :
( ( size_size_tm @ ( fun @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_list_tm @ size_size_tm @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% tm.size(4)
thf(fact_912_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs2: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_913_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_914_ComplI,axiom,
! [C: nat,A4: set_nat] :
( ~ ( member_nat2 @ C @ A4 )
=> ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A4 ) ) ) ).
% ComplI
thf(fact_915_ComplI,axiom,
! [C: fm,A4: set_fm] :
( ~ ( member_fm2 @ C @ A4 )
=> ( member_fm2 @ C @ ( uminus_uminus_set_fm @ A4 ) ) ) ).
% ComplI
thf(fact_916_ComplI,axiom,
! [C: list_nat,A4: set_list_nat] :
( ~ ( member_list_nat2 @ C @ A4 )
=> ( member_list_nat2 @ C @ ( uminus3195874150345416415st_nat @ A4 ) ) ) ).
% ComplI
thf(fact_917_Compl__iff,axiom,
! [C: nat,A4: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A4 ) )
= ( ~ ( member_nat2 @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_918_Compl__iff,axiom,
! [C: fm,A4: set_fm] :
( ( member_fm2 @ C @ ( uminus_uminus_set_fm @ A4 ) )
= ( ~ ( member_fm2 @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_919_Compl__iff,axiom,
! [C: list_nat,A4: set_list_nat] :
( ( member_list_nat2 @ C @ ( uminus3195874150345416415st_nat @ A4 ) )
= ( ~ ( member_list_nat2 @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_920_ComplD,axiom,
! [C: nat,A4: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A4 ) )
=> ~ ( member_nat2 @ C @ A4 ) ) ).
% ComplD
thf(fact_921_ComplD,axiom,
! [C: fm,A4: set_fm] :
( ( member_fm2 @ C @ ( uminus_uminus_set_fm @ A4 ) )
=> ~ ( member_fm2 @ C @ A4 ) ) ).
% ComplD
thf(fact_922_ComplD,axiom,
! [C: list_nat,A4: set_list_nat] :
( ( member_list_nat2 @ C @ ( uminus3195874150345416415st_nat @ A4 ) )
=> ~ ( member_list_nat2 @ C @ A4 ) ) ).
% ComplD
thf(fact_923_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_924_uminus__set__def,axiom,
( uminus_uminus_set_fm
= ( ^ [A3: set_fm] :
( collect_fm
@ ( uminus_uminus_fm_o
@ ^ [X3: fm] : ( member_fm2 @ X3 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_925_uminus__set__def,axiom,
( uminus3195874150345416415st_nat
= ( ^ [A3: set_list_nat] :
( collect_list_nat
@ ( uminus5770388063884162150_nat_o
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_926_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ^ [X3: nat] :
~ ( member_nat2 @ X3 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_927_Compl__eq,axiom,
( uminus_uminus_set_fm
= ( ^ [A3: set_fm] :
( collect_fm
@ ^ [X3: fm] :
~ ( member_fm2 @ X3 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_928_Compl__eq,axiom,
( uminus3195874150345416415st_nat
= ( ^ [A3: set_list_nat] :
( collect_list_nat
@ ^ [X3: list_nat] :
~ ( member_list_nat2 @ X3 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_929_size__list__estimation,axiom,
! [X2: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_930_size__list__estimation,axiom,
! [X2: fm,Xs: list_fm,Y: nat,F: fm > nat] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_nat @ Y @ ( size_list_fm @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_931_size__list__estimation,axiom,
! [X2: list_nat,Xs: list_list_nat,Y: nat,F: list_nat > nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_nat @ Y @ ( size_list_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_932_size__list__estimation,axiom,
! [X2: tm,Xs: list_tm,Y: nat,F: tm > nat] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_nat @ Y @ ( size_list_tm @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_933_size__list__pointwise,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_934_size__list__pointwise,axiom,
! [Xs: list_fm,F: fm > nat,G: fm > nat] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_nat @ ( size_list_fm @ F @ Xs ) @ ( size_list_fm @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_935_size__list__pointwise,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_nat @ ( size_list_list_nat @ F @ Xs ) @ ( size_list_list_nat @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_936_size__list__pointwise,axiom,
! [Xs: list_tm,F: tm > nat,G: tm > nat] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_nat @ ( size_list_tm @ F @ Xs ) @ ( size_list_tm @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_937_size__list__estimation_H,axiom,
! [X2: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_938_size__list__estimation_H,axiom,
! [X2: fm,Xs: list_fm,Y: nat,F: fm > nat] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_fm @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_939_size__list__estimation_H,axiom,
! [X2: list_nat,Xs: list_list_nat,Y: nat,F: list_nat > nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_940_size__list__estimation_H,axiom,
! [X2: tm,Xs: list_tm,Y: nat,F: tm > nat] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_tm @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_941_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_942_list_Osize__gen_I2_J,axiom,
! [X2: nat > nat,X21: nat,X22: list_nat] :
( ( size_list_nat @ X2 @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X21 ) @ ( size_list_nat @ X2 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_943_list_Osize__gen_I2_J,axiom,
! [X2: int > nat,X21: int,X22: list_int] :
( ( size_list_int @ X2 @ ( cons_int @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X21 ) @ ( size_list_int @ X2 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_944_list_Osize__gen_I2_J,axiom,
! [X2: tm > nat,X21: tm,X22: list_tm] :
( ( size_list_tm @ X2 @ ( cons_tm @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X21 ) @ ( size_list_tm @ X2 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_945_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_946_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_947_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_948_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_949_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_950_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_951_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_952_set__ConsD,axiom,
! [Y: tm,X2: tm,Xs: list_tm] :
( ( member_tm2 @ Y @ ( set_tm2 @ ( cons_tm @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_tm2 @ Y @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_953_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_954_set__ConsD,axiom,
! [Y: list_nat,X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_nat2 @ Y @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_955_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_956_set__ConsD,axiom,
! [Y: int,X2: int,Xs: list_int] :
( ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_int2 @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_957_list_Oset__cases,axiom,
! [E2: tm,A2: list_tm] :
( ( member_tm2 @ E2 @ ( set_tm2 @ A2 ) )
=> ( ! [Z22: list_tm] :
( A2
!= ( cons_tm @ E2 @ Z22 ) )
=> ~ ! [Z1: tm,Z22: list_tm] :
( ( A2
= ( cons_tm @ Z1 @ Z22 ) )
=> ~ ( member_tm2 @ E2 @ ( set_tm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_958_list_Oset__cases,axiom,
! [E2: fm,A2: list_fm] :
( ( member_fm2 @ E2 @ ( set_fm2 @ A2 ) )
=> ( ! [Z22: list_fm] :
( A2
!= ( cons_fm @ E2 @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A2
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm2 @ E2 @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_959_list_Oset__cases,axiom,
! [E2: list_nat,A2: list_list_nat] :
( ( member_list_nat2 @ E2 @ ( set_list_nat2 @ A2 ) )
=> ( ! [Z22: list_list_nat] :
( A2
!= ( cons_list_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A2
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat2 @ E2 @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_960_list_Oset__cases,axiom,
! [E2: nat,A2: list_nat] :
( ( member_nat2 @ E2 @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_961_list_Oset__cases,axiom,
! [E2: int,A2: list_int] :
( ( member_int2 @ E2 @ ( set_int2 @ A2 ) )
=> ( ! [Z22: list_int] :
( A2
!= ( cons_int @ E2 @ Z22 ) )
=> ~ ! [Z1: int,Z22: list_int] :
( ( A2
= ( cons_int @ Z1 @ Z22 ) )
=> ~ ( member_int2 @ E2 @ ( set_int2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_962_list_Oset__intros_I1_J,axiom,
! [X21: tm,X22: list_tm] : ( member_tm2 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_963_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_964_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] : ( member_list_nat2 @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_965_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_966_list_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_int] : ( member_int2 @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_967_list_Oset__intros_I2_J,axiom,
! [Y: tm,X22: list_tm,X21: tm] :
( ( member_tm2 @ Y @ ( set_tm2 @ X22 ) )
=> ( member_tm2 @ Y @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_968_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_969_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X22: list_list_nat,X21: list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ X22 ) )
=> ( member_list_nat2 @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_970_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_971_list_Oset__intros_I2_J,axiom,
! [Y: int,X22: list_int,X21: int] :
( ( member_int2 @ Y @ ( set_int2 @ X22 ) )
=> ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_972_map__eq__Cons__conv,axiom,
! [F: tm > list_nat,Xs: list_tm,Y: list_nat,Ys: list_list_nat] :
( ( ( map_tm_list_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
= ( ? [Z2: tm,Zs: list_tm] :
( ( Xs
= ( cons_tm @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_tm_list_nat @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_973_map__eq__Cons__conv,axiom,
! [F: fm > list_nat,Xs: list_fm,Y: list_nat,Ys: list_list_nat] :
( ( ( map_fm_list_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
= ( ? [Z2: fm,Zs: list_fm] :
( ( Xs
= ( cons_fm @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_fm_list_nat @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_974_map__eq__Cons__conv,axiom,
! [F: tm > tm,Xs: list_tm,Y: tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( cons_tm @ Y @ Ys ) )
= ( ? [Z2: tm,Zs: list_tm] :
( ( Xs
= ( cons_tm @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_tm_tm @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_975_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_976_map__eq__Cons__conv,axiom,
! [F: int > nat,Xs: list_int,Y: nat,Ys: list_nat] :
( ( ( map_int_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: int,Zs: list_int] :
( ( Xs
= ( cons_int @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_int_nat @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_977_map__eq__Cons__conv,axiom,
! [F: nat > int,Xs: list_nat,Y: int,Ys: list_int] :
( ( ( map_nat_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z2: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_int @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_978_map__eq__Cons__conv,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z2: int,Zs: list_int] :
( ( Xs
= ( cons_int @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_int_int @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_979_Cons__eq__map__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,F: tm > list_nat,Ys: list_tm] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_tm_list_nat @ F @ Ys ) )
= ( ? [Z2: tm,Zs: list_tm] :
( ( Ys
= ( cons_tm @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_tm_list_nat @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_980_Cons__eq__map__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,F: fm > list_nat,Ys: list_fm] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_fm_list_nat @ F @ Ys ) )
= ( ? [Z2: fm,Zs: list_fm] :
( ( Ys
= ( cons_fm @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_fm_list_nat @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_981_Cons__eq__map__conv,axiom,
! [X2: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X2 @ Xs )
= ( map_tm_tm @ F @ Ys ) )
= ( ? [Z2: tm,Zs: list_tm] :
( ( Ys
= ( cons_tm @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_tm_tm @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_982_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z2: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_983_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: int > nat,Ys: list_int] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_int_nat @ F @ Ys ) )
= ( ? [Z2: int,Zs: list_int] :
( ( Ys
= ( cons_int @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_int_nat @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_984_Cons__eq__map__conv,axiom,
! [X2: int,Xs: list_int,F: nat > int,Ys: list_nat] :
( ( ( cons_int @ X2 @ Xs )
= ( map_nat_int @ F @ Ys ) )
= ( ? [Z2: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_int @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_985_Cons__eq__map__conv,axiom,
! [X2: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( map_int_int @ F @ Ys ) )
= ( ? [Z2: int,Zs: list_int] :
( ( Ys
= ( cons_int @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_int_int @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_986_map__eq__Cons__D,axiom,
! [F: tm > list_nat,Xs: list_tm,Y: list_nat,Ys: list_list_nat] :
( ( ( map_tm_list_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
=> ? [Z3: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_list_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_987_map__eq__Cons__D,axiom,
! [F: fm > list_nat,Xs: list_fm,Y: list_nat,Ys: list_list_nat] :
( ( ( map_fm_list_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
=> ? [Z3: fm,Zs2: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_list_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_988_map__eq__Cons__D,axiom,
! [F: tm > tm,Xs: list_tm,Y: tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( cons_tm @ Y @ Ys ) )
=> ? [Z3: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_tm @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_989_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_990_map__eq__Cons__D,axiom,
! [F: int > nat,Xs: list_int,Y: nat,Ys: list_nat] :
( ( ( map_int_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_int_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_991_map__eq__Cons__D,axiom,
! [F: nat > int,Xs: list_nat,Y: int,Ys: list_int] :
( ( ( map_nat_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_992_map__eq__Cons__D,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_int_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_993_Cons__eq__map__D,axiom,
! [X2: list_nat,Xs: list_list_nat,F: tm > list_nat,Ys: list_tm] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_tm_list_nat @ F @ Ys ) )
=> ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_list_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_994_Cons__eq__map__D,axiom,
! [X2: list_nat,Xs: list_list_nat,F: fm > list_nat,Ys: list_fm] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_fm_list_nat @ F @ Ys ) )
=> ? [Z3: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_list_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_995_Cons__eq__map__D,axiom,
! [X2: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X2 @ Xs )
= ( map_tm_tm @ F @ Ys ) )
=> ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_tm @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_996_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_997_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: int > nat,Ys: list_int] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_int_nat @ F @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_int_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_998_Cons__eq__map__D,axiom,
! [X2: int,Xs: list_int,F: nat > int,Ys: list_nat] :
( ( ( cons_int @ X2 @ Xs )
= ( map_nat_int @ F @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_999_Cons__eq__map__D,axiom,
! [X2: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( map_int_int @ F @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_int_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1000_list_Osimps_I9_J,axiom,
! [F: tm > list_nat,X21: tm,X22: list_tm] :
( ( map_tm_list_nat @ F @ ( cons_tm @ X21 @ X22 ) )
= ( cons_list_nat @ ( F @ X21 ) @ ( map_tm_list_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1001_list_Osimps_I9_J,axiom,
! [F: fm > list_nat,X21: fm,X22: list_fm] :
( ( map_fm_list_nat @ F @ ( cons_fm @ X21 @ X22 ) )
= ( cons_list_nat @ ( F @ X21 ) @ ( map_fm_list_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1002_list_Osimps_I9_J,axiom,
! [F: tm > tm,X21: tm,X22: list_tm] :
( ( map_tm_tm @ F @ ( cons_tm @ X21 @ X22 ) )
= ( cons_tm @ ( F @ X21 ) @ ( map_tm_tm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1003_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1004_list_Osimps_I9_J,axiom,
! [F: nat > int,X21: nat,X22: list_nat] :
( ( map_nat_int @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_nat_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1005_list_Osimps_I9_J,axiom,
! [F: int > nat,X21: int,X22: list_int] :
( ( map_int_nat @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_int_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1006_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X22: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1007_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1008_not__Cons__self2,axiom,
! [X2: int,Xs: list_int] :
( ( cons_int @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1009_member__rec_I1_J,axiom,
! [X2: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_1010_member__rec_I1_J,axiom,
! [X2: int,Xs: list_int,Y: int] :
( ( member_int @ ( cons_int @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_int @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_1011_nat__mono,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1012_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P5 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_1013_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P5 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_1014_eq__nat__nat__iff,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z7 ) )
= ( Z = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1015_set__subset__Cons,axiom,
! [Xs: list_tm,X2: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1016_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_1017_set__subset__Cons,axiom,
! [Xs: list_list_nat,X2: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1018_set__subset__Cons,axiom,
! [Xs: list_int,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1019_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_1020_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y2: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1021_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1022_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y2: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1023_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1024_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X2: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1025_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1026_gen__length__code_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1027_gen__length__code_I2_J,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X2 @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1028_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1029_nat__le__iff,axiom,
! [X2: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
= ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1030_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1031_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1032_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1033_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X3: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1034_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1035_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A: nat,B: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1036_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1037_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1038_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1039_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1040_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1041_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1042_nat__add__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1043_list_Osize_I4_J,axiom,
! [X21: int,X22: list_int] :
( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1044_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1045_n__lists_Osimps_I2_J,axiom,
! [N: nat,Xs: list_nat] :
( ( n_lists_nat @ ( suc @ N ) @ Xs )
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [Ys3: list_nat] :
( map_nat_list_nat
@ ^ [Y2: nat] : ( cons_nat @ Y2 @ Ys3 )
@ Xs )
@ ( n_lists_nat @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_1046_n__lists_Osimps_I2_J,axiom,
! [N: nat,Xs: list_int] :
( ( n_lists_int @ ( suc @ N ) @ Xs )
= ( concat_list_int
@ ( map_li8902190837986183758st_int
@ ^ [Ys3: list_int] :
( map_int_list_int
@ ^ [Y2: int] : ( cons_int @ Y2 @ Ys3 )
@ Xs )
@ ( n_lists_int @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_1047_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1048_concat__map__singleton,axiom,
! [F: tm > list_nat,Xs: list_tm] :
( ( concat_list_nat
@ ( map_tm_list_list_nat
@ ^ [X3: tm] : ( cons_list_nat @ ( F @ X3 ) @ nil_list_nat )
@ Xs ) )
= ( map_tm_list_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1049_concat__map__singleton,axiom,
! [F: fm > list_nat,Xs: list_fm] :
( ( concat_list_nat
@ ( map_fm_list_list_nat
@ ^ [X3: fm] : ( cons_list_nat @ ( F @ X3 ) @ nil_list_nat )
@ Xs ) )
= ( map_fm_list_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1050_concat__map__singleton,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( concat_tm
@ ( map_tm_list_tm
@ ^ [X3: tm] : ( cons_tm @ ( F @ X3 ) @ nil_tm )
@ Xs ) )
= ( map_tm_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1051_concat__map__singleton,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( concat_nat
@ ( map_nat_list_nat
@ ^ [X3: nat] : ( cons_nat @ ( F @ X3 ) @ nil_nat )
@ Xs ) )
= ( map_nat_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1052_concat__map__singleton,axiom,
! [F: tm > nat,Xs: list_tm] :
( ( concat_nat
@ ( map_tm_list_nat
@ ^ [X3: tm] : ( cons_nat @ ( F @ X3 ) @ nil_nat )
@ Xs ) )
= ( map_tm_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1053_concat__map__singleton,axiom,
! [F: fm > nat,Xs: list_fm] :
( ( concat_nat
@ ( map_fm_list_nat
@ ^ [X3: fm] : ( cons_nat @ ( F @ X3 ) @ nil_nat )
@ Xs ) )
= ( map_fm_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1054_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1055_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1056_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1057_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1058_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1059_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1060_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1061_list_Omap__disc__iff,axiom,
! [F: tm > list_nat,A2: list_tm] :
( ( ( map_tm_list_nat @ F @ A2 )
= nil_list_nat )
= ( A2 = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_1062_list_Omap__disc__iff,axiom,
! [F: fm > list_nat,A2: list_fm] :
( ( ( map_fm_list_nat @ F @ A2 )
= nil_list_nat )
= ( A2 = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_1063_list_Omap__disc__iff,axiom,
! [F: tm > tm,A2: list_tm] :
( ( ( map_tm_tm @ F @ A2 )
= nil_tm )
= ( A2 = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_1064_list_Omap__disc__iff,axiom,
! [F: nat > nat,A2: list_nat] :
( ( ( map_nat_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1065_Nil__is__map__conv,axiom,
! [F: tm > list_nat,Xs: list_tm] :
( ( nil_list_nat
= ( map_tm_list_nat @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_1066_Nil__is__map__conv,axiom,
! [F: fm > list_nat,Xs: list_fm] :
( ( nil_list_nat
= ( map_fm_list_nat @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_1067_Nil__is__map__conv,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( nil_tm
= ( map_tm_tm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_1068_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1069_map__is__Nil__conv,axiom,
! [F: tm > list_nat,Xs: list_tm] :
( ( ( map_tm_list_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_1070_map__is__Nil__conv,axiom,
! [F: fm > list_nat,Xs: list_fm] :
( ( ( map_fm_list_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_1071_map__is__Nil__conv,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_1072_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1073_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_1074_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1075_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_1076_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_1077_Nil__eq__concat__conv,axiom,
! [Xss: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss ) )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xss ) )
=> ( X3 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1078_concat__eq__Nil__conv,axiom,
! [Xss: list_list_nat] :
( ( ( concat_nat @ Xss )
= nil_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xss ) )
=> ( X3 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1079_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_1080_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_1081_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_1082_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_1083_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1084_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1085_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_1086_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_1087_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1088_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_1089_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_1090_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_1091_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_1092_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= ( ring_1_of_int_int @ Z ) ) ) ).
% of_nat_nat
thf(fact_1093_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_1094_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1095_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X4: int] : ( P @ ( cons_int @ X4 @ nil_int ) )
=> ( ! [X4: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1096_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1097_list__induct2_H,axiom,
! [P: list_nat > list_int > $o,Xs: list_nat,Ys: list_int] :
( ( P @ nil_nat @ nil_int )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_nat @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: nat,Xs3: list_nat,Y3: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1098_list__induct2_H,axiom,
! [P: list_int > list_nat > $o,Xs: list_int,Ys: list_nat] :
( ( P @ nil_int @ nil_nat )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_int @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs3: list_int,Y3: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1099_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_int @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs3: list_int,Y3: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1100_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y2: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1101_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y2: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1102_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X4: nat] :
( X2
!= ( cons_nat @ X4 @ nil_nat ) )
=> ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1103_remdups__adj_Ocases,axiom,
! [X2: list_int] :
( ( X2 != nil_int )
=> ( ! [X4: int] :
( X2
!= ( cons_int @ X4 @ nil_int ) )
=> ~ ! [X4: int,Y3: int,Xs3: list_int] :
( X2
!= ( cons_int @ X4 @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1104_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss2: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss2 ) )
=> ~ ! [X4: nat,Xs3: list_nat,Xss2: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_1105_transpose_Ocases,axiom,
! [X2: list_list_int] :
( ( X2 != nil_list_int )
=> ( ! [Xss2: list_list_int] :
( X2
!= ( cons_list_int @ nil_int @ Xss2 ) )
=> ~ ! [X4: int,Xs3: list_int,Xss2: list_list_int] :
( X2
!= ( cons_list_int @ ( cons_int @ X4 @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_1106_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X4: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X4 @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_1107_min__list_Ocases,axiom,
! [X2: list_int] :
( ! [X4: int,Xs3: list_int] :
( X2
!= ( cons_int @ X4 @ Xs3 ) )
=> ( X2 = nil_int ) ) ).
% min_list.cases
thf(fact_1108_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_1109_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_1110_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_1111_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_1112_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_1113_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_1114_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1115_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1116_list_Osimps_I8_J,axiom,
! [F: tm > list_nat] :
( ( map_tm_list_nat @ F @ nil_tm )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_1117_list_Osimps_I8_J,axiom,
! [F: fm > list_nat] :
( ( map_fm_list_nat @ F @ nil_fm )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_1118_list_Osimps_I8_J,axiom,
! [F: tm > tm] :
( ( map_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% list.simps(8)
thf(fact_1119_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1120_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_1121_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1122_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1123_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_1124_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_1125_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_1126_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1127_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1128_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1129_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1130_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1131_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1132_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1133_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1134_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1135_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1136_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1137_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1138_add__mono1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1139_add__mono1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1140_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_1141_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_1142_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1143_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1144_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_1145_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_1146_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1147_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs3: list_int,P: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P @ nil_int @ nil_int @ nil_int )
=> ( ! [X4: int,Xs3: list_int,Y3: int,Ys4: list_int,Z3: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1148_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs3: list_nat,P: list_int > list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_int @ nil_int @ nil_nat )
=> ( ! [X4: int,Xs3: list_int,Y3: int,Ys4: list_int,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1149_list__induct3,axiom,
! [Xs: list_int,Ys: list_nat,Zs3: list_int,P: list_int > list_nat > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P @ nil_int @ nil_nat @ nil_int )
=> ( ! [X4: int,Xs3: list_int,Y3: nat,Ys4: list_nat,Z3: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1150_list__induct3,axiom,
! [Xs: list_int,Ys: list_nat,Zs3: list_nat,P: list_int > list_nat > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_int @ nil_nat @ nil_nat )
=> ( ! [X4: int,Xs3: list_int,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1151_list__induct3,axiom,
! [Xs: list_nat,Ys: list_int,Zs3: list_int,P: list_nat > list_int > list_int > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_int @ nil_int )
=> ( ! [X4: nat,Xs3: list_nat,Y3: int,Ys4: list_int,Z3: int,Zs2: list_int] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1152_list__induct3,axiom,
! [Xs: list_nat,Ys: list_int,Zs3: list_nat,P: list_nat > list_int > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_int @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y3: int,Ys4: list_int,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1153_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_int,P: list_nat > list_nat > list_int > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_int )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: int,Zs2: list_int] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1154_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: 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 @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1155_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_int @ nil_int )
=> ( ! [X4: int,Xs3: list_int,Y3: int,Ys4: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1156_list__induct2,axiom,
! [Xs: list_int,Ys: list_nat,P: list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_int @ nil_nat )
=> ( ! [X4: int,Xs3: list_int,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1157_list__induct2,axiom,
! [Xs: list_nat,Ys: list_int,P: list_nat > list_int > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_nat @ nil_int )
=> ( ! [X4: nat,Xs3: list_nat,Y3: int,Ys4: list_int] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1158_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1159_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1160_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_1161_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1162_max__list_Osimps_I1_J,axiom,
( ( max_list @ nil_nat )
= zero_zero_nat ) ).
% max_list.simps(1)
thf(fact_1163_vars__fm_Osimps_I1_J,axiom,
( ( vars_fm @ falsity )
= nil_nat ) ).
% vars_fm.simps(1)
thf(fact_1164_vars__tm_Osimps_I1_J,axiom,
! [N: nat] :
( ( vars_tm @ ( var @ N ) )
= ( cons_nat @ N @ nil_nat ) ) ).
% vars_tm.simps(1)
thf(fact_1165_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1166_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1167_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1168_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_1169_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1170_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1171_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1172_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1173_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1174_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1175_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1176_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_1177_lift__tm_Osimps_I1_J,axiom,
! [N: nat] :
( ( lift_tm @ ( var @ N ) )
= ( var @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% lift_tm.simps(1)
thf(fact_1178_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M7: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1179_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_1180_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1181_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1182_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1183_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1184_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1185_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1186_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1187_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1188_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1189_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1190_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1191_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1192_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1193_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_1194_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1195_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1196_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1197_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1198_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1199_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1200_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1201_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1202_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1203_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1204_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1205_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1206_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1207_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1208_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1209_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1210_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1211_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1212_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1213_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1214_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1215_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1216_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1217_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1218_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1219_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1220_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1221_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1222_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1223_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1224_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1225_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1226_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1227_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1228_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1229_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1230_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1231_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1232_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1233_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1234_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1235_diff__nat__eq__if,axiom,
! [Z7: int,Z: int] :
( ( ( ord_less_int @ Z7 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_1236_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1237_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1238_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1239_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1240_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A: nat,B: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_1241_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1242_nat__diff__distrib_H,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X2 @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1243_nat__diff__distrib,axiom,
! [Z7: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ord_less_eq_int @ Z7 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1244_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_1245_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1246_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1247_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1248_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1249_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1250_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1251_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1252_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q4 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1253_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_1254_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1255_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1256_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1257_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_1258_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_1259_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1260_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X2: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X2 )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1261_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1262_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1263_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1264_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1265_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
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__Int__Oint_J_T,axiom,
! [X2: list_int,Y: list_int] :
( ( if_list_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X2: list_int,Y: list_int] :
( ( if_list_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( semantics_fm_a @ ( fun_upd_nat_a @ ea @ na @ x ) @ f @ g @ ( pre @ p @ ts ) )
= ( semantics_fm_a @ ea @ f @ g @ ( pre @ p @ ts ) ) ) ).
%------------------------------------------------------------------------------