TPTP Problem File: SLH0280^1.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 : Safe_Range_RC/0017_Preliminaries/prob_00065_002441__16348816_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1503 ( 477 unt; 225 typ; 0 def)
% Number of atoms : 3937 (1194 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11748 ( 413 ~; 74 |; 285 &;9151 @)
% ( 0 <=>;1825 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 37 ( 36 usr)
% Number of type conns : 930 ( 930 >; 0 *; 0 +; 0 <<)
% Number of symbols : 192 ( 189 usr; 13 con; 0-4 aty)
% Number of variables : 3818 ( 421 ^;3232 !; 165 ?;3818 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:20:57.927
%------------------------------------------------------------------------------
% Could-be-implicit typings (36)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mtf__a_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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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 (189)
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001tf__a,type,
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thf(sy_c_Fun_Ofun__upd_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001t__List__Olist_Itf__a_J_001tf__a,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_HOL_OUniq_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_HOL_OUniq_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001tf__a,type,
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thf(sy_c_Int_Oint__ge__less__than,type,
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thf(sy_c_Int_Oint__ge__less__than2,type,
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thf(sy_c_List_Odistinct_001t__Int__Oint,type,
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thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
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thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_List_Odistinct_001tf__a,type,
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thf(sy_c_List_Ofold_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Ofold_001t__Product____Type__Oprod_It__Int__Oint_Mtf__a_J_001_062_It__Int__Oint_Mtf__a_J,type,
fold_P8446539499127847477_int_a: ( product_prod_int_a > ( int > a ) > int > a ) > list_P2923742242954448599_int_a > ( int > a ) > int > a ).
thf(sy_c_List_Ofold_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
fold_P2653311671310692405_a_nat: ( produc424395135190311811_a_nat > ( list_a > nat ) > list_a > nat ) > list_P1129550237270585747_a_nat > ( list_a > nat ) > list_a > nat ).
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thf(sy_c_List_Ofold_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_List_Ofold_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001_062_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Ofold_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_List_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set_001t__Int__Oint_001t__Int__Oint,type,
linord2086799039702490726nt_int: ( int > int ) > set_int > list_int ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set_001t__Nat__Onat_001t__Nat__Onat,type,
linord1089935798310486446at_nat: ( nat > nat ) > set_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Int__Oint,type,
linord2612477271533052124et_int: set_int > list_int ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_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_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > 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__Int__Oint_J,type,
set_list_int2: list_list_int > set_list_int ).
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_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001tf__a,type,
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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__Nat__Onat,type,
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thf(sy_c_List_On__lists_001tf__a,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001t__Int__Oint,type,
rev_int: list_int > list_int ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
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thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001t__Int__Oint,type,
rotate1_int: list_int > list_int ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_Itf__a_J,type,
sorted_wrt_list_a: ( list_a > list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
thf(sy_c_List_Ozip_001t__Int__Oint_001tf__a,type,
zip_int_a: list_int > list_a > list_P2923742242954448599_int_a ).
thf(sy_c_List_Ozip_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
zip_list_a_nat: list_list_a > list_nat > list_P1129550237270585747_a_nat ).
thf(sy_c_List_Ozip_001t__List__Olist_Itf__a_J_001tf__a,type,
zip_list_a_a: list_list_a > list_a > list_P2210424090985720871st_a_a ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001tf__a,type,
zip_nat_a: list_nat > list_a > list_P2851791750731487283_nat_a ).
thf(sy_c_List_Ozip_001tf__a_001t__Nat__Onat,type,
zip_a_nat: list_a > list_nat > list_P3592885314253461005_a_nat ).
thf(sy_c_List_Ozip_001tf__a_001tf__a,type,
zip_a_a: list_a > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List__Index_Oinsert__nth_001t__Int__Oint,type,
list_insert_nth_int: nat > int > list_int > list_int ).
thf(sy_c_List__Index_Oinsert__nth_001t__List__Olist_Itf__a_J,type,
list_i9041641776701930082list_a: nat > list_a > list_list_a > list_list_a ).
thf(sy_c_List__Index_Oinsert__nth_001t__Nat__Onat,type,
list_insert_nth_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_List__Index_Oinsert__nth_001tf__a,type,
list_insert_nth_a: nat > a > list_a > list_a ).
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__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_list_a_o: ( list_a > $o ) > ( list_a > $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_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__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_eq_list_a_o: ( list_a > $o ) > ( list_a > $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_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_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Preliminaries_Oextend_001t__Nat__Onat,type,
extend_nat: set_nat > list_nat > list_nat > set_list_nat ).
thf(sy_c_Preliminaries_Oextend_001tf__a,type,
extend_a: set_nat > list_nat > list_a > set_list_a ).
thf(sy_c_Preliminaries_Olookup_001t__Int__Oint_001t__Int__Oint,type,
lookup_int_int: list_int > list_int > int > int ).
thf(sy_c_Preliminaries_Olookup_001t__Int__Oint_001t__Nat__Onat,type,
lookup_int_nat: list_int > list_nat > int > nat ).
thf(sy_c_Preliminaries_Olookup_001t__Nat__Onat_001t__Int__Oint,type,
lookup_nat_int: list_nat > list_int > nat > int ).
thf(sy_c_Preliminaries_Olookup_001t__Nat__Onat_001t__Nat__Onat,type,
lookup_nat_nat: list_nat > list_nat > nat > nat ).
thf(sy_c_Preliminaries_Olookup_001t__Nat__Onat_001tf__a,type,
lookup_nat_a: list_nat > list_a > nat > a ).
thf(sy_c_Preliminaries_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: set_nat > list_nat > list_nat > list_nat ).
thf(sy_c_Preliminaries_Orestrict_001t__Nat__Onat_001tf__a,type,
restrict_nat_a: set_nat > list_nat > list_a > list_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Nat__Onat_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
produc4172673786868812799nt_nat: ( int > nat > ( int > nat ) > int > nat ) > product_prod_int_nat > ( int > nat ) > int > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001tf__a_001_062_I_062_It__Int__Oint_Mtf__a_J_M_062_It__Int__Oint_Mtf__a_J_J,type,
produc2176213134075038027_int_a: ( int > a > ( int > a ) > int > a ) > product_prod_int_a > ( int > a ) > int > a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_Itf__a_J_001t__Nat__Onat_001_062_I_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J_M_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J_J,type,
produc6246542454500903119_a_nat: ( list_a > nat > ( list_a > nat ) > list_a > nat ) > produc424395135190311811_a_nat > ( list_a > nat ) > list_a > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_Itf__a_J_001tf__a_001_062_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_M_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
produc6787724560940851203st_a_a: ( list_a > a > ( list_a > a ) > list_a > a ) > produc2579390645249093025st_a_a > ( list_a > a ) > list_a > a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc8178142064113008363at_nat: ( nat > nat > ( nat > nat ) > nat > nat ) > product_prod_nat_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001tf__a_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
produc2909000522608705447_nat_a: ( nat > a > ( nat > a ) > nat > a ) > product_prod_nat_a > ( nat > a ) > nat > a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
produc4481717121449037155_nat_a: ( nat > a > nat > a ) > product_prod_nat_a > nat > a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001t__Nat__Onat_001_062_I_062_Itf__a_Mt__Nat__Onat_J_M_062_Itf__a_Mt__Nat__Onat_J_J,type,
produc7013214046051809481_a_nat: ( a > nat > ( a > nat ) > a > nat ) > product_prod_a_nat > ( a > nat ) > a > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__a_Mtf__a_J_J,type,
produc2369190251411148053_a_a_a: ( a > a > ( a > a ) > a > a ) > product_prod_a_a > ( a > a ) > a > a ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
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_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int2: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a2: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_Aa____,type,
aa: set_nat ).
thf(sy_v_a____,type,
a2: nat ).
thf(sy_v_x,type,
x: nat ).
thf(sy_v_xsa____,type,
xsa: list_nat ).
thf(sy_v_ysa____,type,
ysa: list_a ).
thf(sy_v_zsa____,type,
zsa: list_a ).
% Relevant facts (1266)
thf(fact_0_Cons_Oprems_I1_J,axiom,
member_nat @ x @ aa ).
% Cons.prems(1)
thf(fact_1_Cons_Oprems_I5_J,axiom,
( ( size_size_list_a @ ysa )
= ( finite_card_nat @ aa ) ) ).
% Cons.prems(5)
thf(fact_2_Cons_Oprems_I6_J,axiom,
member_list_a @ zsa @ ( extend_a @ aa @ ( cons_nat @ a2 @ xsa ) @ ysa ) ).
% Cons.prems(6)
thf(fact_3_Cons_Oprems_I2_J,axiom,
ord_less_eq_set_nat @ aa @ ( set_nat2 @ ( cons_nat @ a2 @ xsa ) ) ).
% Cons.prems(2)
thf(fact_4_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_5_List_Ofinite__set,axiom,
! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% List.finite_set
thf(fact_6_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N )
=> ( ord_less_eq_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_7_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_int,B: set_int] :
( ( ( linord2612477271533052124et_int @ A )
= ( linord2612477271533052124et_int @ B ) )
=> ( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B )
=> ( A = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_8_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( A = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_9_finite__subset,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( finite_finite_int @ B )
=> ( finite_finite_int @ A ) ) ) ).
% finite_subset
thf(fact_10_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_11_infinite__super,axiom,
! [S: set_int,T: set_int] :
( ( ord_less_eq_set_int @ S @ T )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T ) ) ) ).
% infinite_super
thf(fact_12_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_13_rev__finite__subset,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( finite_finite_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_14_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_15_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A2 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_16_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_17_finite__has__maximal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ A2 @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_18_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_19_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_20_finite__has__minimal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X2 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_21_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_22_finite__list,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ? [Xs2: list_int] :
( ( set_int2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_23_Cons_Oprems_I3_J,axiom,
distinct_nat @ ( cons_nat @ a2 @ xsa ) ).
% Cons.prems(3)
thf(fact_24_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_25_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_26_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_27_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( set_int2 @ ( linord2612477271533052124et_int @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_28_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A ) )
= ( finite_card_nat @ A ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_29_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: set_int] :
( ( size_size_list_int @ ( linord2612477271533052124et_int @ A ) )
= ( finite_card_int @ A ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_30_subset__code_I1_J,axiom,
! [Xs: list_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_31_subset__code_I1_J,axiom,
! [Xs: list_int,B: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B )
= ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( member_int @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_32_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_33_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_34_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_35_list_Oset__intros_I2_J,axiom,
! [Y: int,X22: list_int,X21: int] :
( ( member_int @ Y @ ( set_int2 @ X22 ) )
=> ( member_int @ Y @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_36_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_37_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_38_list_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_39_list_Oset__cases,axiom,
! [E: list_a,A2: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A2 ) )
=> ( ! [Z2: list_list_a] :
( A2
!= ( cons_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A2
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_40_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z2: list_nat] :
( A2
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_41_list_Oset__cases,axiom,
! [E: int,A2: list_int] :
( ( member_int @ E @ ( set_int2 @ A2 ) )
=> ( ! [Z2: list_int] :
( A2
!= ( cons_int @ E @ Z2 ) )
=> ~ ! [Z1: int,Z2: list_int] :
( ( A2
= ( cons_int @ Z1 @ Z2 ) )
=> ~ ( member_int @ E @ ( set_int2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_42_set__ConsD,axiom,
! [Y: list_a,X3: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_43_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_44_set__ConsD,axiom,
! [Y: int,X3: int,Xs: list_int] :
( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_45_card__length,axiom,
! [Xs: list_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( set_int2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ).
% card_length
thf(fact_46_card__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% card_length
thf(fact_47_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_48_not__Cons__self2,axiom,
! [X3: nat,Xs: list_nat] :
( ( cons_nat @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_49_not__Cons__self2,axiom,
! [X3: int,Xs: list_int] :
( ( cons_int @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_50_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X3: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_51_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X3: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_52_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X3: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_53_set__subset__Cons,axiom,
! [Xs: list_int,X3: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_54_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_55_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_56_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_57_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_58_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_59_length__extend,axiom,
! [Zs: list_nat,A: set_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ Zs @ ( extend_nat @ A @ Xs @ Ys ) )
=> ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Xs ) ) ) ).
% length_extend
thf(fact_60_length__extend,axiom,
! [Zs: list_a,A: set_nat,Xs: list_nat,Ys: list_a] :
( ( member_list_a @ Zs @ ( extend_a @ A @ Xs @ Ys ) )
=> ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Xs ) ) ) ).
% length_extend
thf(fact_61_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M2: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_62_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_int,C: nat] :
( ! [G: set_int] :
( ( ord_less_eq_set_int @ G @ F )
=> ( ( finite_finite_int @ G )
=> ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C ) ) )
=> ( ( finite_finite_int @ F )
& ( ord_less_eq_nat @ ( finite_card_int @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_63_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_64_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_int] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ S ) )
=> ~ ! [T2: set_int] :
( ( ord_less_eq_set_int @ T2 @ S )
=> ( ( ( finite_card_int @ T2 )
= N2 )
=> ~ ( finite_finite_int @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_65_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_66_exists__subset__between,axiom,
! [A: set_int,N2: nat,C: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ C ) )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( finite_finite_int @ C )
=> ? [B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
& ( ord_less_eq_set_int @ B2 @ C )
& ( ( finite_card_int @ B2 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_67_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
& ( ord_less_eq_set_nat @ B2 @ C )
& ( ( finite_card_nat @ B2 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_68_card__seteq,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B ) @ ( finite_card_int @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_69_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_70_card__mono,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B ) ) ) ) ).
% card_mono
thf(fact_71_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_72_card__le__if__inj__on__rel,axiom,
! [B: set_list_a,A: set_list_a,R: list_a > list_a > $o] :
( ( finite_finite_list_a @ B )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ? [B3: list_a] :
( ( member_list_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: list_a,A22: list_a,B4: list_a] :
( ( member_list_a @ A1 @ A )
=> ( ( member_list_a @ A22 @ A )
=> ( ( member_list_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ ( finite_card_list_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_73_card__le__if__inj__on__rel,axiom,
! [B: set_list_a,A: set_nat,R: nat > list_a > $o] :
( ( finite_finite_list_a @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B3: list_a] :
( ( member_list_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: list_a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_list_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_list_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_74_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_list_a,R: list_a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: list_a,A22: list_a,B4: nat] :
( ( member_list_a @ A1 @ A )
=> ( ( member_list_a @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_75_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_76_card__le__if__inj__on__rel,axiom,
! [B: set_int,A: set_list_a,R: list_a > int > $o] :
( ( finite_finite_int @ B )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ? [B3: int] :
( ( member_int @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: list_a,A22: list_a,B4: int] :
( ( member_list_a @ A1 @ A )
=> ( ( member_list_a @ A22 @ A )
=> ( ( member_int @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ ( finite_card_int @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_77_card__le__if__inj__on__rel,axiom,
! [B: set_int,A: set_nat,R: nat > int > $o] :
( ( finite_finite_int @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B3: int] :
( ( member_int @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: int] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_int @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_78_infinite__arbitrarily__large,axiom,
! [A: set_int,N2: nat] :
( ~ ( finite_finite_int @ A )
=> ? [B2: set_int] :
( ( finite_finite_int @ B2 )
& ( ( finite_card_int @ B2 )
= N2 )
& ( ord_less_eq_set_int @ B2 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_79_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ( finite_card_nat @ B2 )
= N2 )
& ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_80_card__subset__eq,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ( ( finite_card_int @ A )
= ( finite_card_int @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_81_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_82_card__distinct,axiom,
! [Xs: list_int] :
( ( ( finite_card_int @ ( set_int2 @ Xs ) )
= ( size_size_list_int @ Xs ) )
=> ( distinct_int @ Xs ) ) ).
% card_distinct
thf(fact_83_card__distinct,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) )
=> ( distinct_a @ Xs ) ) ).
% card_distinct
thf(fact_84_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_85_distinct__card,axiom,
! [Xs: list_int] :
( ( distinct_int @ Xs )
=> ( ( finite_card_int @ ( set_int2 @ Xs ) )
= ( size_size_list_int @ Xs ) ) ) ).
% distinct_card
thf(fact_86_distinct__card,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ) ).
% distinct_card
thf(fact_87_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_88_restrict__extend,axiom,
! [A: set_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( set_nat2 @ Xs ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( finite_card_nat @ A ) )
=> ( ( member_list_nat @ Zs @ ( extend_nat @ A @ Xs @ Ys ) )
=> ( ( restrict_nat_nat @ A @ Xs @ Zs )
= Ys ) ) ) ) ).
% restrict_extend
thf(fact_89_restrict__extend,axiom,
! [A: set_nat,Xs: list_nat,Ys: list_a,Zs: list_a] :
( ( ord_less_eq_set_nat @ A @ ( set_nat2 @ Xs ) )
=> ( ( ( size_size_list_a @ Ys )
= ( finite_card_nat @ A ) )
=> ( ( member_list_a @ Zs @ ( extend_a @ A @ Xs @ Ys ) )
=> ( ( restrict_nat_a @ A @ Xs @ Zs )
= Ys ) ) ) ) ).
% restrict_extend
thf(fact_90_sorted__simps_I2_J,axiom,
! [X3: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X3 @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X3 @ X ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_91_sorted__simps_I2_J,axiom,
! [X3: int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X3 @ Ys ) )
= ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Ys ) )
=> ( ord_less_eq_int @ X3 @ X ) )
& ( sorted_wrt_int @ ord_less_eq_int @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_92_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A2: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_101_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_102_Collect__cong,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X2: product_prod_int_int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec213857154873943460nt_int @ P )
= ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_cong
thf(fact_103_subsetI,axiom,
! [A: set_list_a,B: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% subsetI
thf(fact_104_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_105_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_106_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_107_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_int] : ( sorted_wrt_int @ ord_less_eq_int @ ( linord2612477271533052124et_int @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_108_Cons_Oprems_I4_J,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ a2 @ xsa ) ).
% Cons.prems(4)
thf(fact_109_distinct_Osimps_I2_J,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( distinct_list_a @ ( cons_list_a @ X3 @ Xs ) )
= ( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( distinct_list_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_110_distinct_Osimps_I2_J,axiom,
! [X3: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X3 @ Xs ) )
= ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_111_distinct_Osimps_I2_J,axiom,
! [X3: int,Xs: list_int] :
( ( distinct_int @ ( cons_int @ X3 @ Xs ) )
= ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
& ( distinct_int @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_112_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_113_finite__distinct__list,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ? [Xs2: list_int] :
( ( ( set_int2 @ Xs2 )
= A )
& ( distinct_int @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_114_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_115_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_116_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_117_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_118_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_119_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_120_finite__Collect__disjI,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( finite2998713641127702882nt_int
@ ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
& ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_121_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_122_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_123_finite__Collect__conjI,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
| ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) )
=> ( finite2998713641127702882nt_int
@ ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_124_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_125_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_126_finite__Collect__subsets,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_127_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_128_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_129_Cons_Ohyps,axiom,
! [A: set_nat,Ys: list_a,Zs: list_a,Sigma: nat > a] :
( ( member_nat @ x @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( set_nat2 @ xsa ) )
=> ( ( distinct_nat @ xsa )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ xsa )
=> ( ( ( size_size_list_a @ Ys )
= ( finite_card_nat @ A ) )
=> ( ( member_list_a @ Zs @ ( extend_a @ A @ xsa @ Ys ) )
=> ( ( fold_P5280602285094830901_nat_a
@ ( produc2909000522608705447_nat_a
@ ^ [X: nat,Y2: a,F2: nat > a] : ( fun_upd_nat_a @ F2 @ X @ Y2 ) )
@ ( zip_nat_a @ xsa @ Zs )
@ Sigma
@ x )
= ( fold_P5280602285094830901_nat_a
@ ( produc2909000522608705447_nat_a
@ ^ [X: nat,Y2: a,F2: nat > a] : ( fun_upd_nat_a @ F2 @ X @ Y2 ) )
@ ( zip_nat_a @ ( linord2614967742042102400et_nat @ A ) @ Ys )
@ Sigma
@ x ) ) ) ) ) ) ) ) ).
% Cons.hyps
thf(fact_130_finite__lists__distinct__length__eq,axiom,
! [A: set_int,N2: nat] :
( ( finite_finite_int @ A )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= N2 )
& ( distinct_int @ Xs3 )
& ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_131_finite__lists__distinct__length__eq,axiom,
! [A: set_a,N2: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= N2 )
& ( distinct_a @ Xs3 )
& ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_132_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N2: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N2 )
& ( distinct_nat @ Xs3 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_133_Collect__subset,axiom,
! [A: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_134_Collect__subset,axiom,
! [A: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_135_Collect__subset,axiom,
! [A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_136_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_137_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B5: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A4 )
@ ^ [X: list_a] : ( member_list_a @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_138_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_139_sorted__wrt__true,axiom,
! [Xs: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_140_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B: set_nat,R2: list_a > nat > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A5: list_a] :
( ( member_list_a @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_141_pigeonhole__infinite__rel,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_nat,R2: product_prod_int_int > nat > $o] :
( ~ ( finite2998713641127702882nt_int @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite2998713641127702882nt_int
@ ( collec213857154873943460nt_int
@ ^ [A5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_142_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B: set_int,R2: list_a > int > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A5: list_a] :
( ( member_list_a @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_143_pigeonhole__infinite__rel,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_int,R2: product_prod_int_int > int > $o] :
( ~ ( finite2998713641127702882nt_int @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite2998713641127702882nt_int
@ ( collec213857154873943460nt_int
@ ^ [A5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_144_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_145_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_int,R2: nat > int > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_146_pigeonhole__infinite__rel,axiom,
! [A: set_int,B: set_nat,R2: int > nat > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_147_pigeonhole__infinite__rel,axiom,
! [A: set_int,B: set_int,R2: int > int > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_148_not__finite__existsD,axiom,
! [P: product_prod_int_int > $o] :
( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
=> ? [X_1: product_prod_int_int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_149_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_150_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_151_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F3 @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F3 @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_152_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_153_finite__lists__length__eq,axiom,
! [A: set_int,N2: nat] :
( ( finite_finite_int @ A )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A )
& ( ( size_size_list_int @ Xs3 )
= N2 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_154_finite__lists__length__eq,axiom,
! [A: set_a,N2: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A )
& ( ( size_size_list_a @ Xs3 )
= N2 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_155_finite__lists__length__eq,axiom,
! [A: set_nat,N2: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ( size_size_list_nat @ Xs3 )
= N2 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_156_finite__lists__length__le,axiom,
! [A: set_int,N2: nat] :
( ( finite_finite_int @ A )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_int @ Xs3 ) @ N2 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_157_finite__lists__length__le,axiom,
! [A: set_a,N2: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs3 ) @ N2 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_158_finite__lists__length__le,axiom,
! [A: set_nat,N2: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N2 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_159_order__antisym__conv,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_160_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_161_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_162_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_163_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_164_ord__le__eq__subst,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_165_ord__le__eq__subst,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_166_ord__le__eq__subst,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > int,C2: int] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_167_ord__le__eq__subst,axiom,
! [A2: nat,B6: nat,F3: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_168_ord__le__eq__subst,axiom,
! [A2: nat,B6: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_169_ord__le__eq__subst,axiom,
! [A2: nat,B6: nat,F3: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_170_ord__le__eq__subst,axiom,
! [A2: int,B6: int,F3: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_171_ord__le__eq__subst,axiom,
! [A2: int,B6: int,F3: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_172_ord__le__eq__subst,axiom,
! [A2: int,B6: int,F3: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_173_ord__eq__le__subst,axiom,
! [A2: set_nat,F3: set_nat > set_nat,B6: set_nat,C2: set_nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_174_ord__eq__le__subst,axiom,
! [A2: nat,F3: set_nat > nat,B6: set_nat,C2: set_nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_175_ord__eq__le__subst,axiom,
! [A2: int,F3: set_nat > int,B6: set_nat,C2: set_nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_176_ord__eq__le__subst,axiom,
! [A2: set_nat,F3: nat > set_nat,B6: nat,C2: nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_177_ord__eq__le__subst,axiom,
! [A2: nat,F3: nat > nat,B6: nat,C2: nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_178_ord__eq__le__subst,axiom,
! [A2: int,F3: nat > int,B6: nat,C2: nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_179_ord__eq__le__subst,axiom,
! [A2: set_nat,F3: int > set_nat,B6: int,C2: int] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_180_ord__eq__le__subst,axiom,
! [A2: nat,F3: int > nat,B6: int,C2: int] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_181_ord__eq__le__subst,axiom,
! [A2: int,F3: int > int,B6: int,C2: int] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_182_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_183_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_184_order__eq__refl,axiom,
! [X3: set_nat,Y: set_nat] :
( ( X3 = Y )
=> ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_185_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_186_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_187_order__subst2,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_188_order__subst2,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_189_order__subst2,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > int,C2: int] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_eq_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_190_order__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_191_order__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_192_order__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_eq_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_193_order__subst2,axiom,
! [A2: int,B6: int,F3: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_194_order__subst2,axiom,
! [A2: int,B6: int,F3: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_195_order__subst2,axiom,
! [A2: int,B6: int,F3: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_196_order__subst1,axiom,
! [A2: set_nat,F3: set_nat > set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_197_order__subst1,axiom,
! [A2: set_nat,F3: nat > set_nat,B6: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_198_order__subst1,axiom,
! [A2: set_nat,F3: int > set_nat,B6: int,C2: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_199_order__subst1,axiom,
! [A2: nat,F3: set_nat > nat,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_200_order__subst1,axiom,
! [A2: nat,F3: nat > nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_201_order__subst1,axiom,
! [A2: nat,F3: int > nat,B6: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_202_order__subst1,axiom,
! [A2: int,F3: set_nat > int,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_203_order__subst1,axiom,
! [A2: int,F3: nat > int,B6: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_204_order__subst1,axiom,
! [A2: int,F3: int > int,B6: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_205_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_206_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_eq_nat @ A5 @ B7 )
& ( ord_less_eq_nat @ B7 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_207_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A5: int,B7: int] :
( ( ord_less_eq_int @ A5 @ B7 )
& ( ord_less_eq_int @ B7 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_208_antisym,axiom,
! [A2: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( A2 = B6 ) ) ) ).
% antisym
thf(fact_209_antisym,axiom,
! [A2: nat,B6: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ B6 @ A2 )
=> ( A2 = B6 ) ) ) ).
% antisym
thf(fact_210_antisym,axiom,
! [A2: int,B6: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ B6 @ A2 )
=> ( A2 = B6 ) ) ) ).
% antisym
thf(fact_211_dual__order_Otrans,axiom,
! [B6: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B6 )
=> ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_212_dual__order_Otrans,axiom,
! [B6: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B6 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B6 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_213_dual__order_Otrans,axiom,
! [B6: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B6 @ A2 )
=> ( ( ord_less_eq_int @ C2 @ B6 )
=> ( ord_less_eq_int @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_214_dual__order_Oantisym,axiom,
! [B6: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( A2 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_215_dual__order_Oantisym,axiom,
! [B6: nat,A2: nat] :
( ( ord_less_eq_nat @ B6 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B6 )
=> ( A2 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_216_dual__order_Oantisym,axiom,
! [B6: int,A2: int] :
( ( ord_less_eq_int @ B6 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B6 )
=> ( A2 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_217_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_218_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_eq_nat @ B7 @ A5 )
& ( ord_less_eq_nat @ A5 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_219_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A5: int,B7: int] :
( ( ord_less_eq_int @ B7 @ A5 )
& ( ord_less_eq_int @ A5 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_220_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B6: nat] :
( ! [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: nat,B4: nat] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A2 @ B6 ) ) ) ).
% linorder_wlog
thf(fact_221_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B6: int] :
( ! [A3: int,B4: int] :
( ( ord_less_eq_int @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: int,B4: int] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A2 @ B6 ) ) ) ).
% linorder_wlog
thf(fact_222_order__trans,axiom,
! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_223_order__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_224_order__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_225_order_Otrans,axiom,
! [A2: set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_226_order_Otrans,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_227_order_Otrans,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_228_order__antisym,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_229_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_230_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_231_ord__le__eq__trans,axiom,
! [A2: set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_232_ord__le__eq__trans,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_233_ord__le__eq__trans,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_234_ord__eq__le__trans,axiom,
! [A2: set_nat,B6: set_nat,C2: set_nat] :
( ( A2 = B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_235_ord__eq__le__trans,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( A2 = B6 )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_236_ord__eq__le__trans,axiom,
! [A2: int,B6: int,C2: int] :
( ( A2 = B6 )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_237_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
& ( ord_less_eq_set_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_238_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_239_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
& ( ord_less_eq_int @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_240_le__cases3,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_241_le__cases3,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_242_nle__le,axiom,
! [A2: nat,B6: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B6 ) )
= ( ( ord_less_eq_nat @ B6 @ A2 )
& ( B6 != A2 ) ) ) ).
% nle_le
thf(fact_243_nle__le,axiom,
! [A2: int,B6: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B6 ) )
= ( ( ord_less_eq_int @ B6 @ A2 )
& ( B6 != A2 ) ) ) ).
% nle_le
thf(fact_244_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X: int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_245_Collect__mono__iff,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
= ( ! [X: product_prod_int_int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_246_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_247_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_248_subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_249_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_250_Collect__mono,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X2: product_prod_int_int] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_mono
thf(fact_251_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_252_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_253_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B5: set_list_a] :
! [T3: list_a] :
( ( member_list_a @ T3 @ A4 )
=> ( member_list_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_254_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_255_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_256_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_257_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B5: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A4 )
=> ( member_list_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_258_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_259_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_260_subsetD,axiom,
! [A: set_list_a,B: set_list_a,C2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_261_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_262_in__mono,axiom,
! [A: set_list_a,B: set_list_a,X3: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ X3 @ A )
=> ( member_list_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_263_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_264_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_265_sorted__distinct__set__unique,axiom,
! [Xs: list_int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( distinct_int @ Xs )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
=> ( ( distinct_int @ Ys )
=> ( ( ( set_int2 @ Xs )
= ( set_int2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_266_sorted__wrt__mono__rel,axiom,
! [Xs: list_list_a,P: list_a > list_a > $o,Q: list_a > list_a > $o] :
( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( member_list_a @ Y3 @ ( set_list_a2 @ Xs ) )
=> ( ( P @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) ) ) )
=> ( ( sorted_wrt_list_a @ P @ Xs )
=> ( sorted_wrt_list_a @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_267_sorted__wrt__mono__rel,axiom,
! [Xs: list_int,P: int > int > $o,Q: int > int > $o] :
( ! [X2: int,Y3: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
=> ( ( P @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) ) ) )
=> ( ( sorted_wrt_int @ P @ Xs )
=> ( sorted_wrt_int @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_268_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_269_distinct__length__2__or__more,axiom,
! [A2: nat,B6: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ A2 @ ( cons_nat @ B6 @ Xs ) ) )
= ( ( A2 != B6 )
& ( distinct_nat @ ( cons_nat @ A2 @ Xs ) )
& ( distinct_nat @ ( cons_nat @ B6 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_270_distinct__length__2__or__more,axiom,
! [A2: int,B6: int,Xs: list_int] :
( ( distinct_int @ ( cons_int @ A2 @ ( cons_int @ B6 @ Xs ) ) )
= ( ( A2 != B6 )
& ( distinct_int @ ( cons_int @ A2 @ Xs ) )
& ( distinct_int @ ( cons_int @ B6 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_271_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( distinct_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_272_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: set_int] : ( distinct_int @ ( linord2612477271533052124et_int @ A ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_273_finite__sorted__distinct__unique,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [X2: list_nat] :
( ( ( set_nat2 @ X2 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X2 )
& ( distinct_nat @ X2 )
& ! [Y5: list_nat] :
( ( ( ( set_nat2 @ Y5 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
& ( distinct_nat @ Y5 ) )
=> ( Y5 = X2 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_274_finite__sorted__distinct__unique,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ? [X2: list_int] :
( ( ( set_int2 @ X2 )
= A )
& ( sorted_wrt_int @ ord_less_eq_int @ X2 )
& ( distinct_int @ X2 )
& ! [Y5: list_int] :
( ( ( ( set_int2 @ Y5 )
= A )
& ( sorted_wrt_int @ ord_less_eq_int @ Y5 )
& ( distinct_int @ Y5 ) )
=> ( Y5 = X2 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_275_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_276_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( distinct_int @ Xs )
=> ( ( linord2612477271533052124et_int @ ( set_int2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_277_sorted2,axiom,
! [X3: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_278_sorted2,axiom,
! [X3: int,Y: int,Zs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X3 @ ( cons_int @ Y @ Zs ) ) )
= ( ( ord_less_eq_int @ X3 @ Y )
& ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_279_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_280_finite__update__induct,axiom,
! [F3: nat > a,C2: a,P: ( nat > a ) > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( F3 @ A5 )
!= C2 ) ) )
=> ( ( P
@ ^ [A5: nat] : C2 )
=> ( ! [A3: nat,B4: a,F4: nat > a] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [C3: nat] :
( ( F4 @ C3 )
!= C2 ) ) )
=> ( ( ( F4 @ A3 )
= C2 )
=> ( ( B4 != C2 )
=> ( ( P @ F4 )
=> ( P @ ( fun_upd_nat_a @ F4 @ A3 @ B4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_update_induct
thf(fact_281_set__n__lists,axiom,
! [N2: nat,Xs: list_int] :
( ( set_list_int2 @ ( n_lists_int @ N2 @ Xs ) )
= ( collect_list_int
@ ^ [Ys2: list_int] :
( ( ( size_size_list_int @ Ys2 )
= N2 )
& ( ord_less_eq_set_int @ ( set_int2 @ Ys2 ) @ ( set_int2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_282_set__n__lists,axiom,
! [N2: nat,Xs: list_a] :
( ( set_list_a2 @ ( n_lists_a @ N2 @ Xs ) )
= ( collect_list_a
@ ^ [Ys2: list_a] :
( ( ( size_size_list_a @ Ys2 )
= N2 )
& ( ord_less_eq_set_a @ ( set_a2 @ Ys2 ) @ ( set_a2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_283_set__n__lists,axiom,
! [N2: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N2 @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N2 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_284_sorted__list__of__set__def,axiom,
( linord2614967742042102400et_nat
= ( linord1089935798310486446at_nat
@ ^ [X: nat] : X ) ) ).
% sorted_list_of_set_def
thf(fact_285_sorted__list__of__set__def,axiom,
( linord2612477271533052124et_int
= ( linord2086799039702490726nt_int
@ ^ [X: int] : X ) ) ).
% sorted_list_of_set_def
thf(fact_286_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: set_nat,L: list_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A ) ) )
= ( ( linord2614967742042102400et_nat @ A )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_287_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: set_int,L: list_int] :
( ( finite_finite_int @ A )
=> ( ( ( sorted_wrt_int @ ord_less_int @ L )
& ( ( set_int2 @ L )
= A )
& ( ( size_size_list_int @ L )
= ( finite_card_int @ A ) ) )
= ( ( linord2612477271533052124et_int @ A )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_288_card__lists__length__eq,axiom,
! [A: set_int,N2: nat] :
( ( finite_finite_int @ A )
=> ( ( finite_card_list_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A )
& ( ( size_size_list_int @ Xs3 )
= N2 ) ) ) )
= ( power_power_nat @ ( finite_card_int @ A ) @ N2 ) ) ) ).
% card_lists_length_eq
thf(fact_289_card__lists__length__eq,axiom,
! [A: set_a,N2: nat] :
( ( finite_finite_a @ A )
=> ( ( finite_card_list_a
@ ( collect_list_a
@ ^ [Xs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A )
& ( ( size_size_list_a @ Xs3 )
= N2 ) ) ) )
= ( power_power_nat @ ( finite_card_a @ A ) @ N2 ) ) ) ).
% card_lists_length_eq
thf(fact_290_card__lists__length__eq,axiom,
! [A: set_nat,N2: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ( size_size_list_nat @ Xs3 )
= N2 ) ) ) )
= ( power_power_nat @ ( finite_card_nat @ A ) @ N2 ) ) ) ).
% card_lists_length_eq
thf(fact_291_ex__lookup__extend,axiom,
! [X3: nat,A: set_nat,Xs: list_nat,Ys: list_a,D: a] :
( ~ ( member_nat @ X3 @ A )
=> ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( extend_a @ A @ Xs @ Ys ) )
& ( ( lookup_nat_a @ Xs @ X2 @ X3 )
= D ) ) ) ) ).
% ex_lookup_extend
thf(fact_292_fold__invariant,axiom,
! [Xs: list_P2851791750731487283_nat_a,Q: product_prod_nat_a > $o,P: ( nat > a ) > $o,S2: nat > a,F3: product_prod_nat_a > ( nat > a ) > nat > a] :
( ! [X2: product_prod_nat_a] :
( ( member8962352052110095674_nat_a @ X2 @ ( set_Pr4163146838226711502_nat_a @ Xs ) )
=> ( Q @ X2 ) )
=> ( ( P @ S2 )
=> ( ! [X2: product_prod_nat_a,S3: nat > a] :
( ( Q @ X2 )
=> ( ( P @ S3 )
=> ( P @ ( F3 @ X2 @ S3 ) ) ) )
=> ( P @ ( fold_P5280602285094830901_nat_a @ F3 @ Xs @ S2 ) ) ) ) ) ).
% fold_invariant
thf(fact_293_List_Ofold__cong,axiom,
! [A2: nat > a,B6: nat > a,Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a,F3: product_prod_nat_a > ( nat > a ) > nat > a,G2: product_prod_nat_a > ( nat > a ) > nat > a] :
( ( A2 = B6 )
=> ( ( Xs = Ys )
=> ( ! [X2: product_prod_nat_a] :
( ( member8962352052110095674_nat_a @ X2 @ ( set_Pr4163146838226711502_nat_a @ Xs ) )
=> ( ( F3 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( fold_P5280602285094830901_nat_a @ F3 @ Xs @ A2 )
= ( fold_P5280602285094830901_nat_a @ G2 @ Ys @ B6 ) ) ) ) ) ).
% List.fold_cong
thf(fact_294_fold__simps_I2_J,axiom,
! [F3: product_prod_nat_a > ( nat > a ) > nat > a,X3: product_prod_nat_a,Xs: list_P2851791750731487283_nat_a,S2: nat > a] :
( ( fold_P5280602285094830901_nat_a @ F3 @ ( cons_P8443330267410185325_nat_a @ X3 @ Xs ) @ S2 )
= ( fold_P5280602285094830901_nat_a @ F3 @ Xs @ ( F3 @ X3 @ S2 ) ) ) ).
% fold_simps(2)
thf(fact_295_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_296_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ~ ! [L2: list_int] :
( ( sorted_wrt_int @ ord_less_int @ L2 )
=> ( ( ( set_int2 @ L2 )
= A )
=> ( ( size_size_list_int @ L2 )
!= ( finite_card_int @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_297_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_298_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_299_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_300_fun__upds__notin,axiom,
! [Xs: list_list_a,Ys: list_a,X3: list_a,Sigma: list_a > a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( fold_P5788709195468586165st_a_a
@ ( produc6787724560940851203st_a_a
@ ^ [X: list_a,Y2: a,F2: list_a > a] : ( fun_upd_list_a_a @ F2 @ X @ Y2 ) )
@ ( zip_list_a_a @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_301_fun__upds__notin,axiom,
! [Xs: list_int,Ys: list_a,X3: int,Sigma: int > a] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( fold_P8446539499127847477_int_a
@ ( produc2176213134075038027_int_a
@ ^ [X: int,Y2: a,F2: int > a] : ( fun_upd_int_a @ F2 @ X @ Y2 ) )
@ ( zip_int_a @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_302_fun__upds__notin,axiom,
! [Xs: list_list_a,Ys: list_nat,X3: list_a,Sigma: list_a > nat] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( fold_P2653311671310692405_a_nat
@ ( produc6246542454500903119_a_nat
@ ^ [X: list_a,Y2: nat,F2: list_a > nat] : ( fun_upd_list_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_list_a_nat @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_303_fun__upds__notin,axiom,
! [Xs: list_int,Ys: list_nat,X3: int,Sigma: int > nat] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( fold_P6222384299469577773nt_nat
@ ( produc4172673786868812799nt_nat
@ ^ [X: int,Y2: nat,F2: int > nat] : ( fun_upd_int_nat @ F2 @ X @ Y2 ) )
@ ( zip_int_nat @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_304_fun__upds__notin,axiom,
! [Xs: list_a,Ys: list_a,X3: a,Sigma: a > a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( fold_P8422020818851269569_a_a_a
@ ( produc2369190251411148053_a_a_a
@ ^ [X: a,Y2: a,F2: a > a] : ( fun_upd_a_a @ F2 @ X @ Y2 ) )
@ ( zip_a_a @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_305_fun__upds__notin,axiom,
! [Xs: list_a,Ys: list_nat,X3: a,Sigma: a > nat] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( fold_P3994820982079749301_a_nat
@ ( produc7013214046051809481_a_nat
@ ^ [X: a,Y2: nat,F2: a > nat] : ( fun_upd_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_a_nat @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_306_fun__upds__notin,axiom,
! [Xs: list_nat,Ys: list_nat,X3: nat,Sigma: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( fold_P4014913454640143221at_nat
@ ( produc8178142064113008363at_nat
@ ^ [X: nat,Y2: nat,F2: nat > nat] : ( fun_upd_nat_nat @ F2 @ X @ Y2 ) )
@ ( zip_nat_nat @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_307_fun__upds__notin,axiom,
! [Xs: list_nat,Ys: list_a,X3: nat,Sigma: nat > a] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( fold_P5280602285094830901_nat_a
@ ( produc2909000522608705447_nat_a
@ ^ [X: nat,Y2: a,F2: nat > a] : ( fun_upd_nat_a @ F2 @ X @ Y2 ) )
@ ( zip_nat_a @ Xs @ Ys )
@ Sigma
@ X3 )
= ( Sigma @ X3 ) ) ) ) ).
% fun_upds_notin
thf(fact_308_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_309_order__less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_310_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_311_order__less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_312_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_313_order__less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_314_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_315_linorder__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_316_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_317_order__less__imp__triv,axiom,
! [X3: int,Y: int,P: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_318_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_319_order__less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_320_order__less__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_321_order__less__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_322_order__less__subst2,axiom,
! [A2: int,B6: int,F3: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_323_order__less__subst2,axiom,
! [A2: int,B6: int,F3: int > int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_324_order__less__subst1,axiom,
! [A2: nat,F3: nat > nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_325_order__less__subst1,axiom,
! [A2: nat,F3: int > nat,B6: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_326_order__less__subst1,axiom,
! [A2: int,F3: nat > int,B6: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_327_order__less__subst1,axiom,
! [A2: int,F3: int > int,B6: int,C2: int] :
( ( ord_less_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_328_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_329_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_330_ord__less__eq__subst,axiom,
! [A2: nat,B6: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_331_ord__less__eq__subst,axiom,
! [A2: nat,B6: nat,F3: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_332_ord__less__eq__subst,axiom,
! [A2: int,B6: int,F3: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_333_ord__less__eq__subst,axiom,
! [A2: int,B6: int,F3: int > int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ( F3 @ B6 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_334_ord__eq__less__subst,axiom,
! [A2: nat,F3: nat > nat,B6: nat,C2: nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_335_ord__eq__less__subst,axiom,
! [A2: int,F3: nat > int,B6: nat,C2: nat] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_336_ord__eq__less__subst,axiom,
! [A2: nat,F3: int > nat,B6: int,C2: int] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_337_ord__eq__less__subst,axiom,
! [A2: int,F3: int > int,B6: int,C2: int] :
( ( A2
= ( F3 @ B6 ) )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_338_order__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_339_order__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_340_order__less__asym_H,axiom,
! [A2: nat,B6: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ~ ( ord_less_nat @ B6 @ A2 ) ) ).
% order_less_asym'
thf(fact_341_order__less__asym_H,axiom,
! [A2: int,B6: int] :
( ( ord_less_int @ A2 @ B6 )
=> ~ ( ord_less_int @ B6 @ A2 ) ) ).
% order_less_asym'
thf(fact_342_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_343_linorder__neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_344_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_345_order__less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_346_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_347_linorder__neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_348_dual__order_Ostrict__implies__not__eq,axiom,
! [B6: nat,A2: nat] :
( ( ord_less_nat @ B6 @ A2 )
=> ( A2 != B6 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_349_dual__order_Ostrict__implies__not__eq,axiom,
! [B6: int,A2: int] :
( ( ord_less_int @ B6 @ A2 )
=> ( A2 != B6 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_350_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B6: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( A2 != B6 ) ) ).
% order.strict_implies_not_eq
thf(fact_351_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B6: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( A2 != B6 ) ) ).
% order.strict_implies_not_eq
thf(fact_352_dual__order_Ostrict__trans,axiom,
! [B6: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B6 @ A2 )
=> ( ( ord_less_nat @ C2 @ B6 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_353_dual__order_Ostrict__trans,axiom,
! [B6: int,A2: int,C2: int] :
( ( ord_less_int @ B6 @ A2 )
=> ( ( ord_less_int @ C2 @ B6 )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_354_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_355_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_356_order_Ostrict__trans,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_357_order_Ostrict__trans,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_358_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B6: nat] :
( ! [A3: nat,B4: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B4: nat] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A2 @ B6 ) ) ) ) ).
% linorder_less_wlog
thf(fact_359_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B6: int] :
( ! [A3: int,B4: int] :
( ( ord_less_int @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B4: int] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A2 @ B6 ) ) ) ) ).
% linorder_less_wlog
thf(fact_360_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N3 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_361_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_362_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_363_dual__order_Oasym,axiom,
! [B6: nat,A2: nat] :
( ( ord_less_nat @ B6 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B6 ) ) ).
% dual_order.asym
thf(fact_364_dual__order_Oasym,axiom,
! [B6: int,A2: int] :
( ( ord_less_int @ B6 @ A2 )
=> ~ ( ord_less_int @ A2 @ B6 ) ) ).
% dual_order.asym
thf(fact_365_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_366_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_367_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_368_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_369_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_370_ord__less__eq__trans,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_371_ord__less__eq__trans,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_372_ord__eq__less__trans,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( A2 = B6 )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_373_ord__eq__less__trans,axiom,
! [A2: int,B6: int,C2: int] :
( ( A2 = B6 )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_374_order_Oasym,axiom,
! [A2: nat,B6: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ~ ( ord_less_nat @ B6 @ A2 ) ) ).
% order.asym
thf(fact_375_order_Oasym,axiom,
! [A2: int,B6: int] :
( ( ord_less_int @ A2 @ B6 )
=> ~ ( ord_less_int @ B6 @ A2 ) ) ).
% order.asym
thf(fact_376_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_377_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_378_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_379_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_380_lt__ex,axiom,
! [X3: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X3 ) ).
% lt_ex
thf(fact_381_psubset__card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_382_psubset__card__mono,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_set_int @ A @ B )
=> ( ord_less_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_383_leD,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ~ ( ord_less_set_nat @ X3 @ Y ) ) ).
% leD
thf(fact_384_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_385_leD,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_int @ X3 @ Y ) ) ).
% leD
thf(fact_386_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_387_leI,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% leI
thf(fact_388_nless__le,axiom,
! [A2: set_nat,B6: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B6 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B6 )
| ( A2 = B6 ) ) ) ).
% nless_le
thf(fact_389_nless__le,axiom,
! [A2: nat,B6: nat] :
( ( ~ ( ord_less_nat @ A2 @ B6 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B6 )
| ( A2 = B6 ) ) ) ).
% nless_le
thf(fact_390_nless__le,axiom,
! [A2: int,B6: int] :
( ( ~ ( ord_less_int @ A2 @ B6 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B6 )
| ( A2 = B6 ) ) ) ).
% nless_le
thf(fact_391_antisym__conv1,axiom,
! [X3: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_392_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_393_antisym__conv1,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_394_antisym__conv2,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_395_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_396_antisym__conv2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_397_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
& ~ ( ord_less_eq_set_nat @ Y2 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_398_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_399_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_400_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_401_not__le__imp__less,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_eq_int @ Y @ X3 )
=> ( ord_less_int @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_402_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
( ( ord_less_set_nat @ A5 @ B7 )
| ( A5 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_403_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_nat @ A5 @ B7 )
| ( A5 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_404_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B7: int] :
( ( ord_less_int @ A5 @ B7 )
| ( A5 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_405_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B7 )
& ( A5 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_406_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_eq_nat @ A5 @ B7 )
& ( A5 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_407_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B7: int] :
( ( ord_less_eq_int @ A5 @ B7 )
& ( A5 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_408_order_Ostrict__trans1,axiom,
! [A2: set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_set_nat @ B6 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_409_order_Ostrict__trans1,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_410_order_Ostrict__trans1,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_411_order_Ostrict__trans2,axiom,
! [A2: set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_412_order_Ostrict__trans2,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_413_order_Ostrict__trans2,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_414_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B7 )
& ~ ( ord_less_eq_set_nat @ B7 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_415_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_eq_nat @ A5 @ B7 )
& ~ ( ord_less_eq_nat @ B7 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_416_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B7: int] :
( ( ord_less_eq_int @ A5 @ B7 )
& ~ ( ord_less_eq_int @ B7 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_417_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B7: set_nat,A5: set_nat] :
( ( ord_less_set_nat @ B7 @ A5 )
| ( A5 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_418_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B7: nat,A5: nat] :
( ( ord_less_nat @ B7 @ A5 )
| ( A5 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_419_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B7: int,A5: int] :
( ( ord_less_int @ B7 @ A5 )
| ( A5 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_420_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B7: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A5 )
& ( A5 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_421_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B7: nat,A5: nat] :
( ( ord_less_eq_nat @ B7 @ A5 )
& ( A5 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_422_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B7: int,A5: int] :
( ( ord_less_eq_int @ B7 @ A5 )
& ( A5 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_423_dual__order_Ostrict__trans1,axiom,
! [B6: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( ( ord_less_set_nat @ C2 @ B6 )
=> ( ord_less_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_424_dual__order_Ostrict__trans1,axiom,
! [B6: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B6 @ A2 )
=> ( ( ord_less_nat @ C2 @ B6 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_425_dual__order_Ostrict__trans1,axiom,
! [B6: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B6 @ A2 )
=> ( ( ord_less_int @ C2 @ B6 )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_426_dual__order_Ostrict__trans2,axiom,
! [B6: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ B6 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B6 )
=> ( ord_less_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_427_dual__order_Ostrict__trans2,axiom,
! [B6: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B6 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B6 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_428_dual__order_Ostrict__trans2,axiom,
! [B6: int,A2: int,C2: int] :
( ( ord_less_int @ B6 @ A2 )
=> ( ( ord_less_eq_int @ C2 @ B6 )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_429_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B7: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A5 )
& ~ ( ord_less_eq_set_nat @ A5 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_430_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B7: nat,A5: nat] :
( ( ord_less_eq_nat @ B7 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_431_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B7: int,A5: int] :
( ( ord_less_eq_int @ B7 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_432_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A2 @ B6 )
=> ( ord_less_eq_set_nat @ A2 @ B6 ) ) ).
% order.strict_implies_order
thf(fact_433_order_Ostrict__implies__order,axiom,
! [A2: nat,B6: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ord_less_eq_nat @ A2 @ B6 ) ) ).
% order.strict_implies_order
thf(fact_434_order_Ostrict__implies__order,axiom,
! [A2: int,B6: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ord_less_eq_int @ A2 @ B6 ) ) ).
% order.strict_implies_order
thf(fact_435_dual__order_Ostrict__implies__order,axiom,
! [B6: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B6 @ A2 )
=> ( ord_less_eq_set_nat @ B6 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_436_dual__order_Ostrict__implies__order,axiom,
! [B6: nat,A2: nat] :
( ( ord_less_nat @ B6 @ A2 )
=> ( ord_less_eq_nat @ B6 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_437_dual__order_Ostrict__implies__order,axiom,
! [B6: int,A2: int] :
( ( ord_less_int @ B6 @ A2 )
=> ( ord_less_eq_int @ B6 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_438_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_439_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_440_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_441_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_442_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_443_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_444_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_445_linorder__not__le,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y ) )
= ( ord_less_int @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_446_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_447_linorder__not__less,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_448_order__less__imp__le,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_449_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_450_order__less__imp__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_451_order__le__neq__trans,axiom,
! [A2: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( A2 != B6 )
=> ( ord_less_set_nat @ A2 @ B6 ) ) ) ).
% order_le_neq_trans
thf(fact_452_order__le__neq__trans,axiom,
! [A2: nat,B6: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( A2 != B6 )
=> ( ord_less_nat @ A2 @ B6 ) ) ) ).
% order_le_neq_trans
thf(fact_453_order__le__neq__trans,axiom,
! [A2: int,B6: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( A2 != B6 )
=> ( ord_less_int @ A2 @ B6 ) ) ) ).
% order_le_neq_trans
thf(fact_454_order__neq__le__trans,axiom,
! [A2: set_nat,B6: set_nat] :
( ( A2 != B6 )
=> ( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ord_less_set_nat @ A2 @ B6 ) ) ) ).
% order_neq_le_trans
thf(fact_455_order__neq__le__trans,axiom,
! [A2: nat,B6: nat] :
( ( A2 != B6 )
=> ( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ord_less_nat @ A2 @ B6 ) ) ) ).
% order_neq_le_trans
thf(fact_456_order__neq__le__trans,axiom,
! [A2: int,B6: int] :
( ( A2 != B6 )
=> ( ( ord_less_eq_int @ A2 @ B6 )
=> ( ord_less_int @ A2 @ B6 ) ) ) ).
% order_neq_le_trans
thf(fact_457_order__le__less__trans,axiom,
! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_458_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_459_order__le__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_460_order__less__le__trans,axiom,
! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_461_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_462_order__less__le__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_463_order__le__less__subst1,axiom,
! [A2: set_nat,F3: nat > set_nat,B6: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_464_order__le__less__subst1,axiom,
! [A2: set_nat,F3: int > set_nat,B6: int,C2: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_465_order__le__less__subst1,axiom,
! [A2: nat,F3: nat > nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_466_order__le__less__subst1,axiom,
! [A2: nat,F3: int > nat,B6: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_467_order__le__less__subst1,axiom,
! [A2: int,F3: nat > int,B6: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_468_order__le__less__subst1,axiom,
! [A2: int,F3: int > int,B6: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_469_order__le__less__subst2,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_470_order__le__less__subst2,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_471_order__le__less__subst2,axiom,
! [A2: set_nat,B6: set_nat,F3: set_nat > int,C2: int] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
=> ( ( ord_less_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_472_order__le__less__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_473_order__le__less__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_474_order__le__less__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_475_order__le__less__subst2,axiom,
! [A2: int,B6: int,F3: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_476_order__le__less__subst2,axiom,
! [A2: int,B6: int,F3: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_477_order__le__less__subst2,axiom,
! [A2: int,B6: int,F3: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_478_order__less__le__subst1,axiom,
! [A2: set_nat,F3: set_nat > set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_479_order__less__le__subst1,axiom,
! [A2: nat,F3: set_nat > nat,B6: set_nat,C2: set_nat] :
( ( ord_less_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_480_order__less__le__subst1,axiom,
! [A2: int,F3: set_nat > int,B6: set_nat,C2: set_nat] :
( ( ord_less_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_481_order__less__le__subst1,axiom,
! [A2: set_nat,F3: nat > set_nat,B6: nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_482_order__less__le__subst1,axiom,
! [A2: nat,F3: nat > nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_483_order__less__le__subst1,axiom,
! [A2: int,F3: nat > int,B6: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_484_order__less__le__subst1,axiom,
! [A2: set_nat,F3: int > set_nat,B6: int,C2: int] :
( ( ord_less_set_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_485_order__less__le__subst1,axiom,
! [A2: nat,F3: int > nat,B6: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_486_order__less__le__subst1,axiom,
! [A2: int,F3: int > int,B6: int,C2: int] :
( ( ord_less_int @ A2 @ ( F3 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_487_order__less__le__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_488_order__less__le__subst2,axiom,
! [A2: int,B6: int,F3: int > set_nat,C2: set_nat] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_489_order__less__le__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_490_order__less__le__subst2,axiom,
! [A2: int,B6: int,F3: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_491_order__less__le__subst2,axiom,
! [A2: nat,B6: nat,F3: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_eq_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_492_order__less__le__subst2,axiom,
! [A2: int,B6: int,F3: int > int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ ( F3 @ B6 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_int @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_493_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_494_linorder__le__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_495_order__le__imp__less__or__eq,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_set_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_496_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_497_order__le__imp__less__or__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_498_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B8: set_nat] :
( ( ord_less_set_nat @ B8 @ A6 )
=> ( P @ B8 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_499_finite__psubset__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ! [A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [B8: set_int] :
( ( ord_less_set_int @ B8 @ A6 )
=> ( P @ B8 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_500_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_501_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_502_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_503_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_504_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_505_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_506_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_507_finite__maxlen,axiom,
! [M2: set_list_a] :
( ( finite_finite_list_a @ M2 )
=> ? [N4: nat] :
! [X4: list_a] :
( ( member_list_a @ X4 @ M2 )
=> ( ord_less_nat @ ( size_size_list_a @ X4 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_508_finite__maxlen,axiom,
! [M2: set_list_nat] :
( ( finite8100373058378681591st_nat @ M2 )
=> ? [N4: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M2 )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_509_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_510_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_511_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N )
=> ( ord_less_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_512_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_513_card__psubset,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B ) )
=> ( ord_less_set_int @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_514_card__psubset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_515_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_516_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_517_strict__sorted__imp__sorted,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_518_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_519_strict__sorted__equal,axiom,
! [Xs: list_int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( ( sorted_wrt_int @ ord_less_int @ Ys )
=> ( ( ( set_int2 @ Ys )
= ( set_int2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_520_lookup_Osimps_I1_J,axiom,
! [X3: nat,Z3: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( X3 = Z3 )
=> ( ( lookup_nat_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z3 )
= Y ) )
& ( ( X3 != Z3 )
=> ( ( lookup_nat_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z3 )
= ( lookup_nat_nat @ Xs @ Ys @ Z3 ) ) ) ) ).
% lookup.simps(1)
thf(fact_521_lookup_Osimps_I1_J,axiom,
! [X3: nat,Z3: nat,Xs: list_nat,Y: int,Ys: list_int] :
( ( ( X3 = Z3 )
=> ( ( lookup_nat_int @ ( cons_nat @ X3 @ Xs ) @ ( cons_int @ Y @ Ys ) @ Z3 )
= Y ) )
& ( ( X3 != Z3 )
=> ( ( lookup_nat_int @ ( cons_nat @ X3 @ Xs ) @ ( cons_int @ Y @ Ys ) @ Z3 )
= ( lookup_nat_int @ Xs @ Ys @ Z3 ) ) ) ) ).
% lookup.simps(1)
thf(fact_522_lookup_Osimps_I1_J,axiom,
! [X3: int,Z3: int,Xs: list_int,Y: nat,Ys: list_nat] :
( ( ( X3 = Z3 )
=> ( ( lookup_int_nat @ ( cons_int @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z3 )
= Y ) )
& ( ( X3 != Z3 )
=> ( ( lookup_int_nat @ ( cons_int @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z3 )
= ( lookup_int_nat @ Xs @ Ys @ Z3 ) ) ) ) ).
% lookup.simps(1)
thf(fact_523_lookup_Osimps_I1_J,axiom,
! [X3: int,Z3: int,Xs: list_int,Y: int,Ys: list_int] :
( ( ( X3 = Z3 )
=> ( ( lookup_int_int @ ( cons_int @ X3 @ Xs ) @ ( cons_int @ Y @ Ys ) @ Z3 )
= Y ) )
& ( ( X3 != Z3 )
=> ( ( lookup_int_int @ ( cons_int @ X3 @ Xs ) @ ( cons_int @ Y @ Ys ) @ Z3 )
= ( lookup_int_int @ Xs @ Ys @ Z3 ) ) ) ) ).
% lookup.simps(1)
thf(fact_524_length__n__lists__elem,axiom,
! [Ys: list_a,N2: nat,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N2 @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= N2 ) ) ).
% length_n_lists_elem
thf(fact_525_length__n__lists__elem,axiom,
! [Ys: list_nat,N2: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N2 @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N2 ) ) ).
% length_n_lists_elem
thf(fact_526_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_527_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: set_int] : ( sorted_wrt_int @ ord_less_int @ ( linord2612477271533052124et_int @ A ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_528_fun__upds__twist__apply,axiom,
! [Xs: list_list_a,Ys: list_a,A2: list_a,B6: list_a,Sigma: list_a > a,X3: a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_list_a @ A2 @ ( set_list_a2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P5788709195468586165st_a_a
@ ( produc6787724560940851203st_a_a
@ ^ [X: list_a,Y2: a,F2: list_a > a] : ( fun_upd_list_a_a @ F2 @ X @ Y2 ) )
@ ( zip_list_a_a @ Xs @ Ys )
@ ( fun_upd_list_a_a @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P5788709195468586165st_a_a
@ ( produc6787724560940851203st_a_a
@ ^ [X: list_a,Y2: a,F2: list_a > a] : ( fun_upd_list_a_a @ F2 @ X @ Y2 ) )
@ ( zip_list_a_a @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_529_fun__upds__twist__apply,axiom,
! [Xs: list_int,Ys: list_a,A2: int,B6: int,Sigma: int > a,X3: a] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_int @ A2 @ ( set_int2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P8446539499127847477_int_a
@ ( produc2176213134075038027_int_a
@ ^ [X: int,Y2: a,F2: int > a] : ( fun_upd_int_a @ F2 @ X @ Y2 ) )
@ ( zip_int_a @ Xs @ Ys )
@ ( fun_upd_int_a @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P8446539499127847477_int_a
@ ( produc2176213134075038027_int_a
@ ^ [X: int,Y2: a,F2: int > a] : ( fun_upd_int_a @ F2 @ X @ Y2 ) )
@ ( zip_int_a @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_530_fun__upds__twist__apply,axiom,
! [Xs: list_list_a,Ys: list_nat,A2: list_a,B6: list_a,Sigma: list_a > nat,X3: nat] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_list_a @ A2 @ ( set_list_a2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P2653311671310692405_a_nat
@ ( produc6246542454500903119_a_nat
@ ^ [X: list_a,Y2: nat,F2: list_a > nat] : ( fun_upd_list_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_list_a_nat @ Xs @ Ys )
@ ( fun_upd_list_a_nat @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P2653311671310692405_a_nat
@ ( produc6246542454500903119_a_nat
@ ^ [X: list_a,Y2: nat,F2: list_a > nat] : ( fun_upd_list_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_list_a_nat @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_531_fun__upds__twist__apply,axiom,
! [Xs: list_int,Ys: list_nat,A2: int,B6: int,Sigma: int > nat,X3: nat] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_int @ A2 @ ( set_int2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P6222384299469577773nt_nat
@ ( produc4172673786868812799nt_nat
@ ^ [X: int,Y2: nat,F2: int > nat] : ( fun_upd_int_nat @ F2 @ X @ Y2 ) )
@ ( zip_int_nat @ Xs @ Ys )
@ ( fun_upd_int_nat @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P6222384299469577773nt_nat
@ ( produc4172673786868812799nt_nat
@ ^ [X: int,Y2: nat,F2: int > nat] : ( fun_upd_int_nat @ F2 @ X @ Y2 ) )
@ ( zip_int_nat @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_532_fun__upds__twist__apply,axiom,
! [Xs: list_a,Ys: list_a,A2: a,B6: a,Sigma: a > a,X3: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_a @ A2 @ ( set_a2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P8422020818851269569_a_a_a
@ ( produc2369190251411148053_a_a_a
@ ^ [X: a,Y2: a,F2: a > a] : ( fun_upd_a_a @ F2 @ X @ Y2 ) )
@ ( zip_a_a @ Xs @ Ys )
@ ( fun_upd_a_a @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P8422020818851269569_a_a_a
@ ( produc2369190251411148053_a_a_a
@ ^ [X: a,Y2: a,F2: a > a] : ( fun_upd_a_a @ F2 @ X @ Y2 ) )
@ ( zip_a_a @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_533_fun__upds__twist__apply,axiom,
! [Xs: list_a,Ys: list_nat,A2: a,B6: a,Sigma: a > nat,X3: nat] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_a @ A2 @ ( set_a2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P3994820982079749301_a_nat
@ ( produc7013214046051809481_a_nat
@ ^ [X: a,Y2: nat,F2: a > nat] : ( fun_upd_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_a_nat @ Xs @ Ys )
@ ( fun_upd_a_nat @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P3994820982079749301_a_nat
@ ( produc7013214046051809481_a_nat
@ ^ [X: a,Y2: nat,F2: a > nat] : ( fun_upd_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_a_nat @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_534_fun__upds__twist__apply,axiom,
! [Xs: list_nat,Ys: list_nat,A2: nat,B6: nat,Sigma: nat > nat,X3: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_nat @ A2 @ ( set_nat2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P4014913454640143221at_nat
@ ( produc8178142064113008363at_nat
@ ^ [X: nat,Y2: nat,F2: nat > nat] : ( fun_upd_nat_nat @ F2 @ X @ Y2 ) )
@ ( zip_nat_nat @ Xs @ Ys )
@ ( fun_upd_nat_nat @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P4014913454640143221at_nat
@ ( produc8178142064113008363at_nat
@ ^ [X: nat,Y2: nat,F2: nat > nat] : ( fun_upd_nat_nat @ F2 @ X @ Y2 ) )
@ ( zip_nat_nat @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_535_fun__upds__twist__apply,axiom,
! [Xs: list_nat,Ys: list_a,A2: nat,B6: nat,Sigma: nat > a,X3: a] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_nat @ A2 @ ( set_nat2 @ Xs ) )
=> ( ( A2 != B6 )
=> ( ( fold_P5280602285094830901_nat_a
@ ( produc2909000522608705447_nat_a
@ ^ [X: nat,Y2: a,F2: nat > a] : ( fun_upd_nat_a @ F2 @ X @ Y2 ) )
@ ( zip_nat_a @ Xs @ Ys )
@ ( fun_upd_nat_a @ Sigma @ A2 @ X3 )
@ B6 )
= ( fold_P5280602285094830901_nat_a
@ ( produc2909000522608705447_nat_a
@ ^ [X: nat,Y2: a,F2: nat > a] : ( fun_upd_nat_a @ F2 @ X @ Y2 ) )
@ ( zip_nat_a @ Xs @ Ys )
@ Sigma
@ B6 ) ) ) ) ) ).
% fun_upds_twist_apply
thf(fact_536_fun__upds__twist,axiom,
! [Xs: list_list_a,Ys: list_a,A2: list_a,Sigma: list_a > a,X3: a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_list_a @ A2 @ ( set_list_a2 @ Xs ) )
=> ( ( fold_P5788709195468586165st_a_a
@ ( produc6787724560940851203st_a_a
@ ^ [X: list_a,Y2: a,F2: list_a > a] : ( fun_upd_list_a_a @ F2 @ X @ Y2 ) )
@ ( zip_list_a_a @ Xs @ Ys )
@ ( fun_upd_list_a_a @ Sigma @ A2 @ X3 ) )
= ( fun_upd_list_a_a
@ ( fold_P5788709195468586165st_a_a
@ ( produc6787724560940851203st_a_a
@ ^ [X: list_a,Y2: a,F2: list_a > a] : ( fun_upd_list_a_a @ F2 @ X @ Y2 ) )
@ ( zip_list_a_a @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_537_fun__upds__twist,axiom,
! [Xs: list_int,Ys: list_a,A2: int,Sigma: int > a,X3: a] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_int @ A2 @ ( set_int2 @ Xs ) )
=> ( ( fold_P8446539499127847477_int_a
@ ( produc2176213134075038027_int_a
@ ^ [X: int,Y2: a,F2: int > a] : ( fun_upd_int_a @ F2 @ X @ Y2 ) )
@ ( zip_int_a @ Xs @ Ys )
@ ( fun_upd_int_a @ Sigma @ A2 @ X3 ) )
= ( fun_upd_int_a
@ ( fold_P8446539499127847477_int_a
@ ( produc2176213134075038027_int_a
@ ^ [X: int,Y2: a,F2: int > a] : ( fun_upd_int_a @ F2 @ X @ Y2 ) )
@ ( zip_int_a @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_538_fun__upds__twist,axiom,
! [Xs: list_list_a,Ys: list_nat,A2: list_a,Sigma: list_a > nat,X3: nat] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_list_a @ A2 @ ( set_list_a2 @ Xs ) )
=> ( ( fold_P2653311671310692405_a_nat
@ ( produc6246542454500903119_a_nat
@ ^ [X: list_a,Y2: nat,F2: list_a > nat] : ( fun_upd_list_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_list_a_nat @ Xs @ Ys )
@ ( fun_upd_list_a_nat @ Sigma @ A2 @ X3 ) )
= ( fun_upd_list_a_nat
@ ( fold_P2653311671310692405_a_nat
@ ( produc6246542454500903119_a_nat
@ ^ [X: list_a,Y2: nat,F2: list_a > nat] : ( fun_upd_list_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_list_a_nat @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_539_fun__upds__twist,axiom,
! [Xs: list_int,Ys: list_nat,A2: int,Sigma: int > nat,X3: nat] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_int @ A2 @ ( set_int2 @ Xs ) )
=> ( ( fold_P6222384299469577773nt_nat
@ ( produc4172673786868812799nt_nat
@ ^ [X: int,Y2: nat,F2: int > nat] : ( fun_upd_int_nat @ F2 @ X @ Y2 ) )
@ ( zip_int_nat @ Xs @ Ys )
@ ( fun_upd_int_nat @ Sigma @ A2 @ X3 ) )
= ( fun_upd_int_nat
@ ( fold_P6222384299469577773nt_nat
@ ( produc4172673786868812799nt_nat
@ ^ [X: int,Y2: nat,F2: int > nat] : ( fun_upd_int_nat @ F2 @ X @ Y2 ) )
@ ( zip_int_nat @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_540_fun__upds__twist,axiom,
! [Xs: list_a,Ys: list_a,A2: a,Sigma: a > a,X3: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_a @ A2 @ ( set_a2 @ Xs ) )
=> ( ( fold_P8422020818851269569_a_a_a
@ ( produc2369190251411148053_a_a_a
@ ^ [X: a,Y2: a,F2: a > a] : ( fun_upd_a_a @ F2 @ X @ Y2 ) )
@ ( zip_a_a @ Xs @ Ys )
@ ( fun_upd_a_a @ Sigma @ A2 @ X3 ) )
= ( fun_upd_a_a
@ ( fold_P8422020818851269569_a_a_a
@ ( produc2369190251411148053_a_a_a
@ ^ [X: a,Y2: a,F2: a > a] : ( fun_upd_a_a @ F2 @ X @ Y2 ) )
@ ( zip_a_a @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_541_fun__upds__twist,axiom,
! [Xs: list_a,Ys: list_nat,A2: a,Sigma: a > nat,X3: nat] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_a @ A2 @ ( set_a2 @ Xs ) )
=> ( ( fold_P3994820982079749301_a_nat
@ ( produc7013214046051809481_a_nat
@ ^ [X: a,Y2: nat,F2: a > nat] : ( fun_upd_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_a_nat @ Xs @ Ys )
@ ( fun_upd_a_nat @ Sigma @ A2 @ X3 ) )
= ( fun_upd_a_nat
@ ( fold_P3994820982079749301_a_nat
@ ( produc7013214046051809481_a_nat
@ ^ [X: a,Y2: nat,F2: a > nat] : ( fun_upd_a_nat @ F2 @ X @ Y2 ) )
@ ( zip_a_nat @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_542_fun__upds__twist,axiom,
! [Xs: list_nat,Ys: list_nat,A2: nat,Sigma: nat > nat,X3: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ~ ( member_nat @ A2 @ ( set_nat2 @ Xs ) )
=> ( ( fold_P4014913454640143221at_nat
@ ( produc8178142064113008363at_nat
@ ^ [X: nat,Y2: nat,F2: nat > nat] : ( fun_upd_nat_nat @ F2 @ X @ Y2 ) )
@ ( zip_nat_nat @ Xs @ Ys )
@ ( fun_upd_nat_nat @ Sigma @ A2 @ X3 ) )
= ( fun_upd_nat_nat
@ ( fold_P4014913454640143221at_nat
@ ( produc8178142064113008363at_nat
@ ^ [X: nat,Y2: nat,F2: nat > nat] : ( fun_upd_nat_nat @ F2 @ X @ Y2 ) )
@ ( zip_nat_nat @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_543_fun__upds__twist,axiom,
! [Xs: list_nat,Ys: list_a,A2: nat,Sigma: nat > a,X3: a] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ~ ( member_nat @ A2 @ ( set_nat2 @ Xs ) )
=> ( ( fold_P5280602285094830901_nat_a
@ ( produc2909000522608705447_nat_a
@ ^ [X: nat,Y2: a,F2: nat > a] : ( fun_upd_nat_a @ F2 @ X @ Y2 ) )
@ ( zip_nat_a @ Xs @ Ys )
@ ( fun_upd_nat_a @ Sigma @ A2 @ X3 ) )
= ( fun_upd_nat_a
@ ( fold_P5280602285094830901_nat_a
@ ( produc2909000522608705447_nat_a
@ ^ [X: nat,Y2: a,F2: nat > a] : ( fun_upd_nat_a @ F2 @ X @ Y2 ) )
@ ( zip_nat_a @ Xs @ Ys )
@ Sigma )
@ A2
@ X3 ) ) ) ) ).
% fun_upds_twist
thf(fact_544_strict__sorted__simps_I2_J,axiom,
! [X3: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X3 @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ord_less_nat @ X3 @ X ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).
% strict_sorted_simps(2)
thf(fact_545_strict__sorted__simps_I2_J,axiom,
! [X3: int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_int @ ( cons_int @ X3 @ Ys ) )
= ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Ys ) )
=> ( ord_less_int @ X3 @ X ) )
& ( sorted_wrt_int @ ord_less_int @ Ys ) ) ) ).
% strict_sorted_simps(2)
thf(fact_546_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_547_strict__sorted__iff,axiom,
! [L: list_int] :
( ( sorted_wrt_int @ ord_less_int @ L )
= ( ( sorted_wrt_int @ ord_less_eq_int @ L )
& ( distinct_int @ L ) ) ) ).
% strict_sorted_iff
thf(fact_548_distinct__zipI2,axiom,
! [Ys: list_a,Xs: list_nat] :
( ( distinct_a @ Ys )
=> ( distin7399675782055410666_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_549_distinct__zipI1,axiom,
! [Xs: list_nat,Ys: list_a] :
( ( distinct_nat @ Xs )
=> ( distin7399675782055410666_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_550_length__n__lists,axiom,
! [N2: nat,Xs: list_a] :
( ( size_s349497388124573686list_a @ ( n_lists_a @ N2 @ Xs ) )
= ( power_power_nat @ ( size_size_list_a @ Xs ) @ N2 ) ) ).
% length_n_lists
thf(fact_551_length__n__lists,axiom,
! [N2: nat,Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( n_lists_nat @ N2 @ Xs ) )
= ( power_power_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% length_n_lists
thf(fact_552_fun__upd__upd,axiom,
! [F3: nat > a,X3: nat,Y: a,Z3: a] :
( ( fun_upd_nat_a @ ( fun_upd_nat_a @ F3 @ X3 @ Y ) @ X3 @ Z3 )
= ( fun_upd_nat_a @ F3 @ X3 @ Z3 ) ) ).
% fun_upd_upd
thf(fact_553_fun__upd__triv,axiom,
! [F3: nat > a,X3: nat] :
( ( fun_upd_nat_a @ F3 @ X3 @ ( F3 @ X3 ) )
= F3 ) ).
% fun_upd_triv
thf(fact_554_fun__upd__apply,axiom,
( fun_upd_nat_a
= ( ^ [F2: nat > a,X: nat,Y2: a,Z4: nat] : ( if_a @ ( Z4 = X ) @ Y2 @ ( F2 @ Z4 ) ) ) ) ).
% fun_upd_apply
thf(fact_555_pred__subset__eq,axiom,
! [R2: set_list_a,S: set_list_a] :
( ( ord_less_eq_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ R2 )
@ ^ [X: list_a] : ( member_list_a @ X @ S ) )
= ( ord_le8861187494160871172list_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_556_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R2 )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_557_case__prod__app,axiom,
( produc2909000522608705447_nat_a
= ( ^ [F2: nat > a > ( nat > a ) > nat > a,X: product_prod_nat_a,Y2: nat > a] :
( produc4481717121449037155_nat_a
@ ^ [L3: nat,R3: a] : ( F2 @ L3 @ R3 @ Y2 )
@ X ) ) ) ).
% case_prod_app
thf(fact_558_distinct__n__lists,axiom,
! [Xs: list_nat,N2: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_list_nat @ ( n_lists_nat @ N2 @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_559_minf_I8_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ~ ( ord_less_eq_nat @ T4 @ X4 ) ) ).
% minf(8)
thf(fact_560_minf_I8_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ~ ( ord_less_eq_int @ T4 @ X4 ) ) ).
% minf(8)
thf(fact_561_minf_I6_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ord_less_eq_nat @ X4 @ T4 ) ) ).
% minf(6)
thf(fact_562_minf_I6_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ord_less_eq_int @ X4 @ T4 ) ) ).
% minf(6)
thf(fact_563_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_set_def
thf(fact_564_less__set__def,axiom,
( ord_less_set_list_a
= ( ^ [A4: set_list_a,B5: set_list_a] :
( ord_less_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A4 )
@ ^ [X: list_a] : ( member_list_a @ X @ B5 ) ) ) ) ).
% less_set_def
thf(fact_565_psubsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_566_psubsetD,axiom,
! [A: set_list_a,B: set_list_a,C2: list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_567_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_568_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_569_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_570_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_571_pinf_I3_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( X4 != T4 ) ) ).
% pinf(3)
thf(fact_572_pinf_I3_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( X4 != T4 ) ) ).
% pinf(3)
thf(fact_573_pinf_I4_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( X4 != T4 ) ) ).
% pinf(4)
thf(fact_574_pinf_I4_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( X4 != T4 ) ) ).
% pinf(4)
thf(fact_575_pinf_I5_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T4 ) ) ).
% pinf(5)
thf(fact_576_pinf_I5_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ~ ( ord_less_int @ X4 @ T4 ) ) ).
% pinf(5)
thf(fact_577_pinf_I7_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ord_less_nat @ T4 @ X4 ) ) ).
% pinf(7)
thf(fact_578_pinf_I7_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ord_less_int @ T4 @ X4 ) ) ).
% pinf(7)
thf(fact_579_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_580_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_581_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_582_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_583_minf_I3_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( X4 != T4 ) ) ).
% minf(3)
thf(fact_584_minf_I3_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( X4 != T4 ) ) ).
% minf(3)
thf(fact_585_minf_I4_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( X4 != T4 ) ) ).
% minf(4)
thf(fact_586_minf_I4_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( X4 != T4 ) ) ).
% minf(4)
thf(fact_587_minf_I5_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ord_less_nat @ X4 @ T4 ) ) ).
% minf(5)
thf(fact_588_minf_I5_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ord_less_int @ X4 @ T4 ) ) ).
% minf(5)
thf(fact_589_minf_I7_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ~ ( ord_less_nat @ T4 @ X4 ) ) ).
% minf(7)
thf(fact_590_minf_I7_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ~ ( ord_less_int @ T4 @ X4 ) ) ).
% minf(7)
thf(fact_591_fun__upd__idem__iff,axiom,
! [F3: nat > a,X3: nat,Y: a] :
( ( ( fun_upd_nat_a @ F3 @ X3 @ Y )
= F3 )
= ( ( F3 @ X3 )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_592_fun__upd__twist,axiom,
! [A2: nat,C2: nat,M4: nat > a,B6: a,D: a] :
( ( A2 != C2 )
=> ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ M4 @ A2 @ B6 ) @ C2 @ D )
= ( fun_upd_nat_a @ ( fun_upd_nat_a @ M4 @ C2 @ D ) @ A2 @ B6 ) ) ) ).
% fun_upd_twist
thf(fact_593_fun__upd__other,axiom,
! [Z3: nat,X3: nat,F3: nat > a,Y: a] :
( ( Z3 != X3 )
=> ( ( fun_upd_nat_a @ F3 @ X3 @ Y @ Z3 )
= ( F3 @ Z3 ) ) ) ).
% fun_upd_other
thf(fact_594_fun__upd__same,axiom,
! [F3: nat > a,X3: nat,Y: a] :
( ( fun_upd_nat_a @ F3 @ X3 @ Y @ X3 )
= Y ) ).
% fun_upd_same
thf(fact_595_fun__upd__idem,axiom,
! [F3: nat > a,X3: nat,Y: a] :
( ( ( F3 @ X3 )
= Y )
=> ( ( fun_upd_nat_a @ F3 @ X3 @ Y )
= F3 ) ) ).
% fun_upd_idem
thf(fact_596_fun__upd__eqD,axiom,
! [F3: nat > a,X3: nat,Y: a,G2: nat > a,Z3: a] :
( ( ( fun_upd_nat_a @ F3 @ X3 @ Y )
= ( fun_upd_nat_a @ G2 @ X3 @ Z3 ) )
=> ( Y = Z3 ) ) ).
% fun_upd_eqD
thf(fact_597_fun__upd__def,axiom,
( fun_upd_nat_a
= ( ^ [F2: nat > a,A5: nat,B7: a,X: nat] : ( if_a @ ( X = A5 ) @ B7 @ ( F2 @ X ) ) ) ) ).
% fun_upd_def
thf(fact_598_pinf_I6_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T4 ) ) ).
% pinf(6)
thf(fact_599_pinf_I6_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T4 ) ) ).
% pinf(6)
thf(fact_600_pinf_I8_J,axiom,
! [T4: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ord_less_eq_nat @ T4 @ X4 ) ) ).
% pinf(8)
thf(fact_601_pinf_I8_J,axiom,
! [T4: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ord_less_eq_int @ T4 @ X4 ) ) ).
% pinf(8)
thf(fact_602_distinct__product__lists,axiom,
! [Xss: list_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss ) )
=> ( distinct_nat @ X2 ) )
=> ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_603_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M4: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_604_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F3 @ I4 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_605_le__neq__implies__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( M4 != N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_606_less__or__eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_607_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N3: nat] :
( ( ord_less_nat @ M @ N3 )
| ( M = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_608_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_609_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_610_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_611_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_612_less__not__refl3,axiom,
! [S2: nat,T4: nat] :
( ( ord_less_nat @ S2 @ T4 )
=> ( S2 != T4 ) ) ).
% less_not_refl3
thf(fact_613_less__not__refl2,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ N2 @ M4 )
=> ( M4 != N2 ) ) ).
% less_not_refl2
thf(fact_614_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_615_nat__neq__iff,axiom,
! [M4: nat,N2: nat] :
( ( M4 != N2 )
= ( ( ord_less_nat @ M4 @ N2 )
| ( ord_less_nat @ N2 @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_616_size__neq__size__imp__neq,axiom,
! [X3: list_a,Y: list_a] :
( ( ( size_size_list_a @ X3 )
!= ( size_size_list_a @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_617_size__neq__size__imp__neq,axiom,
! [X3: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X3 )
!= ( size_size_list_nat @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_618_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_619_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_620_eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( M4 = N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% eq_imp_le
thf(fact_621_le__antisym,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M4 )
=> ( M4 = N2 ) ) ) ).
% le_antisym
thf(fact_622_nat__le__linear,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
| ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% nat_le_linear
thf(fact_623_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B6: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B6 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_624_in__set__product__lists__length,axiom,
! [Xs: list_a,Xss: list_list_a] :
( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss ) ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_625_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_626_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
& ( M != N3 ) ) ) ) ).
% nat_less_le
thf(fact_627_less__imp__le__nat,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_628_verit__comp__simplify1_I3_J,axiom,
! [B9: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B9 @ A7 ) )
= ( ord_less_nat @ A7 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_629_verit__comp__simplify1_I3_J,axiom,
! [B9: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B9 @ A7 ) )
= ( ord_less_int @ A7 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_630_complete__interval,axiom,
! [A2: nat,B6: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B6 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B6 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_631_complete__interval,axiom,
! [A2: int,B6: int,P: int > $o] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B6 )
=> ? [C4: int] :
( ( ord_less_eq_int @ A2 @ C4 )
& ( ord_less_eq_int @ C4 @ B6 )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A2 @ X4 )
& ( ord_less_int @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D2: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A2 @ X2 )
& ( ord_less_int @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D2 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_632_strict__sorted__equal__Uniq,axiom,
! [A: set_nat] :
( uniq_list_nat
@ ^ [Xs3: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs3 )
& ( ( set_nat2 @ Xs3 )
= A ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_633_strict__sorted__equal__Uniq,axiom,
! [A: set_int] :
( uniq_list_int
@ ^ [Xs3: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs3 )
& ( ( set_int2 @ Xs3 )
= A ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_634_distinct__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( distinct_nat @ Ys )
=> ( distin6923225563576452346at_nat @ ( product_nat_nat @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_635_not__in__set__insert,axiom,
! [X3: list_a,Xs: list_list_a] :
( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X3 @ Xs )
= ( cons_list_a @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_636_not__in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= ( cons_nat @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_637_not__in__set__insert,axiom,
! [X3: int,Xs: list_int] :
( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X3 @ Xs )
= ( cons_int @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_638_sorted__nth__mono,axiom,
! [Xs: list_nat,I2: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_639_sorted__nth__mono,axiom,
! [Xs: list_int,I2: nat,J: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_640_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_641_sorted__iff__nth__mono,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_642_in__set__insert,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_643_in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_644_in__set__insert,axiom,
! [X3: int,Xs: list_int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_645_distinct__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ( distinct_nat @ ( insert_nat @ X3 @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_insert
thf(fact_646_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
= ( ^ [Xs3: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I )
= ( nth_a @ Ys2 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_647_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [Xs3: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I )
= ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_648_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X6: a] : ( P @ I @ X6 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_a @ Xs3 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_649_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X6: nat] : ( P @ I @ X6 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_nat @ Xs3 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_650_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I4 )
= ( nth_a @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_651_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_652_all__set__conv__all__nth,axiom,
! [Xs: list_int,P: int > $o] :
( ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_653_all__set__conv__all__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_654_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_655_all__nth__imp__all__set,axiom,
! [Xs: list_list_a,P: list_a > $o,X3: list_a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P @ ( nth_list_a @ Xs @ I4 ) ) )
=> ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_656_all__nth__imp__all__set,axiom,
! [Xs: list_int,P: int > $o,X3: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I4 ) ) )
=> ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_657_all__nth__imp__all__set,axiom,
! [Xs: list_a,P: a > $o,X3: a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I4 ) ) )
=> ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_658_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X3: nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I4 ) ) )
=> ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_659_in__set__conv__nth,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_660_in__set__conv__nth,axiom,
! [X3: int,Xs: list_int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ I )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_661_in__set__conv__nth,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_662_in__set__conv__nth,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_663_list__ball__nth,axiom,
! [N2: nat,Xs: list_int,P: int > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_664_list__ball__nth,axiom,
! [N2: nat,Xs: list_a,P: a > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_a @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_665_list__ball__nth,axiom,
! [N2: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_666_nth__mem,axiom,
! [N2: nat,Xs: list_list_a] :
( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N2 ) @ ( set_list_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_667_nth__mem,axiom,
! [N2: nat,Xs: list_int] :
( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_668_nth__mem,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N2 ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_669_nth__mem,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_670_nth__eq__iff__index__eq,axiom,
! [Xs: list_a,I2: nat,J: nat] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_671_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I2: nat,J: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_672_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs3: list_a] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs3 ) )
=> ( ( I != J3 )
=> ( ( nth_a @ Xs3 @ I )
!= ( nth_a @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_673_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I != J3 )
=> ( ( nth_nat @ Xs3 @ I )
!= ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_674_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_a
= ( ^ [P3: a > a > $o,Xs3: list_a] :
! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs3 ) )
=> ( P3 @ ( nth_a @ Xs3 @ I ) @ ( nth_a @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_675_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs3: list_nat] :
! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( P3 @ ( nth_nat @ Xs3 @ I ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_676_sorted__wrt__nth__less,axiom,
! [P: a > a > $o,Xs: list_a,I2: nat,J: nat] :
( ( sorted_wrt_a @ P @ Xs )
=> ( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_677_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I2: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_678_verit__la__disequality,axiom,
! [A2: nat,B6: nat] :
( ( A2 = B6 )
| ~ ( ord_less_eq_nat @ A2 @ B6 )
| ~ ( ord_less_eq_nat @ B6 @ A2 ) ) ).
% verit_la_disequality
thf(fact_679_verit__la__disequality,axiom,
! [A2: int,B6: int] :
( ( A2 = B6 )
| ~ ( ord_less_eq_int @ A2 @ B6 )
| ~ ( ord_less_eq_int @ B6 @ A2 ) ) ).
% verit_la_disequality
thf(fact_680_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_681_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_682_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_683_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_684_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_685_List_Oinsert__def,axiom,
( insert_list_a
= ( ^ [X: list_a,Xs3: list_list_a] : ( if_list_list_a @ ( member_list_a @ X @ ( set_list_a2 @ Xs3 ) ) @ Xs3 @ ( cons_list_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_686_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_687_List_Oinsert__def,axiom,
( insert_int
= ( ^ [X: int,Xs3: list_int] : ( if_list_int @ ( member_int @ X @ ( set_int2 @ Xs3 ) ) @ Xs3 @ ( cons_int @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_688_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_689_sorted__iff__nth__mono__less,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_690_distinct__Ex1,axiom,
! [Xs: list_list_a,X3: list_a] :
( ( distinct_list_a @ Xs )
=> ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ X2 )
= X3 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ Y5 )
= X3 ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_691_distinct__Ex1,axiom,
! [Xs: list_int,X3: int] :
( ( distinct_int @ Xs )
=> ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ X2 )
= X3 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ Y5 )
= X3 ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_692_distinct__Ex1,axiom,
! [Xs: list_a,X3: a] :
( ( distinct_a @ Xs )
=> ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ X2 )
= X3 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ Y5 )
= X3 ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_693_distinct__Ex1,axiom,
! [Xs: list_nat,X3: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X2 )
= X3 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y5 )
= X3 ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_694_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I2: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_695_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_696_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
= ( ! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ J3 ) @ ( nth_int @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_697_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I2: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_698_sorted__rev__nth__mono,axiom,
! [Xs: list_int,I2: nat,J: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ J ) @ ( nth_int @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_699_nth__equal__first__eq,axiom,
! [X3: list_a,Xs: list_list_a,N2: nat] :
( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( nth_list_a @ ( cons_list_a @ X3 @ Xs ) @ N2 )
= X3 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_700_nth__equal__first__eq,axiom,
! [X3: int,Xs: list_int,N2: nat] :
( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N2 )
= X3 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_701_nth__equal__first__eq,axiom,
! [X3: a,Xs: list_a,N2: nat] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X3 @ Xs ) @ N2 )
= X3 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_702_nth__equal__first__eq,axiom,
! [X3: nat,Xs: list_nat,N2: nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N2 )
= X3 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_703_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ ( suc @ I ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_704_sorted__iff__nth__Suc,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ ( suc @ I ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_705_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_706_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_707_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_708_Suc__mono,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_709_Suc__less__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_less_eq
thf(fact_710_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_711_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_712_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_713_Suc__le__mono,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% Suc_le_mono
thf(fact_714_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_715_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_716_set__rev,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rev_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rev
thf(fact_717_set__rev,axiom,
! [Xs: list_int] :
( ( set_int2 @ ( rev_int @ Xs ) )
= ( set_int2 @ Xs ) ) ).
% set_rev
thf(fact_718_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_719_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_720_distinct__rev,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rev_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_rev
thf(fact_721_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_722_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_723_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_724_card_Oinfinite,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_card_int @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_725_nth__Cons__Suc,axiom,
! [X3: nat,Xs: list_nat,N2: nat] :
( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ ( suc @ N2 ) )
= ( nth_nat @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_726_nth__Cons__Suc,axiom,
! [X3: int,Xs: list_int,N2: nat] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ ( suc @ N2 ) )
= ( nth_int @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_727_nth__Cons__0,axiom,
! [X3: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_728_nth__Cons__0,axiom,
! [X3: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_729_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_730_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_731_Zero__not__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_not_Suc
thf(fact_732_Zero__neq__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_neq_Suc
thf(fact_733_Suc__neq__Zero,axiom,
! [M4: nat] :
( ( suc @ M4 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_734_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_735_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_736_diff__induct,axiom,
! [P: nat > nat > $o,M4: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M4 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_737_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_738_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_739_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_740_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_741_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_742_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_743_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_744_less__Suc__eq__0__disj,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ( M4 = zero_zero_nat )
| ? [J3: nat] :
( ( M4
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_745_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_746_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_747_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_748_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_749_zip__rev,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( zip_a_a @ ( rev_a @ Xs ) @ ( rev_a @ Ys ) )
= ( rev_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_750_zip__rev,axiom,
! [Xs: list_a,Ys: list_nat] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( zip_a_nat @ ( rev_a @ Xs ) @ ( rev_nat @ Ys ) )
= ( rev_Pr1328451580582734999_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_751_zip__rev,axiom,
! [Xs: list_nat,Ys: list_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( zip_nat_a @ ( rev_nat @ Xs ) @ ( rev_a @ Ys ) )
= ( rev_Pr4566615044306411965_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_752_zip__rev,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( zip_nat_nat @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys ) )
= ( rev_Pr6102188148953555047at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_753_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_754_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_755_Suc__lessD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_lessD
thf(fact_756_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_757_Suc__lessI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ( ( suc @ M4 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M4 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_758_less__SucE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M4 @ N2 )
=> ( M4 = N2 ) ) ) ).
% less_SucE
thf(fact_759_less__SucI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ M4 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_760_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ N2 )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_761_less__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_762_not__less__eq,axiom,
! [M4: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M4 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M4 ) ) ) ).
% not_less_eq
thf(fact_763_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_764_Suc__less__eq2,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M4 )
= ( ? [M6: nat] :
( ( M4
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_765_less__antisym,axiom,
! [N2: nat,M4: nat] :
( ~ ( ord_less_nat @ N2 @ M4 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M4 ) )
=> ( M4 = N2 ) ) ) ).
% less_antisym
thf(fact_766_Suc__less__SucD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_767_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_768_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I4 @ K3 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_769_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_770_not__less__less__Suc__eq,axiom,
! [N2: nat,M4: nat] :
( ~ ( ord_less_nat @ N2 @ M4 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M4 ) )
= ( N2 = M4 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_771_transitive__stepwise__le,axiom,
! [M4: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z5: nat] :
( ( R2 @ X2 @ Y3 )
=> ( ( R2 @ Y3 @ Z5 )
=> ( R2 @ X2 @ Z5 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M4 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_772_nat__induct__at__least,axiom,
! [M4: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( P @ M4 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_773_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_774_not__less__eq__eq,axiom,
! [M4: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_775_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_776_le__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M4 @ N2 )
| ( M4
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_777_Suc__le__D,axiom,
! [N2: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_778_le__SucI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_779_le__SucE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N2 )
=> ( M4
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_780_Suc__leD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_leD
thf(fact_781_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_782_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_783_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_784_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_785_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_786_gr__implies__not0,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_787_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_788_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_789_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_790_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_791_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_792_card__less__Suc2,axiom,
! [M2: set_nat,I2: nat] :
( ~ ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M2 )
& ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_793_card__less__Suc,axiom,
! [M2: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M2 )
& ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_794_card__less,axiom,
! [M2: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M2 )
& ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_795_sorted__wrt__rev,axiom,
! [P: nat > nat > $o,Xs: list_nat] :
( ( sorted_wrt_nat @ P @ ( rev_nat @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y2: nat] : ( P @ Y2 @ X )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_796_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( X = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_797_card__le__Suc0__iff__eq,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: int] :
( ( member_int @ X @ A )
=> ! [Y2: int] :
( ( member_int @ Y2 @ A )
=> ( X = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_798_lift__Suc__mono__le,axiom,
! [F3: nat > set_nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_set_nat @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_799_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_800_lift__Suc__mono__le,axiom,
! [F3: nat > int,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_int @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_801_lift__Suc__antimono__le,axiom,
! [F3: nat > set_nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_set_nat @ ( F3 @ N6 ) @ ( F3 @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_802_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F3 @ N6 ) @ ( F3 @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_803_lift__Suc__antimono__le,axiom,
! [F3: nat > int,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_int @ ( F3 @ N6 ) @ ( F3 @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_804_lift__Suc__mono__less__iff,axiom,
! [F3: nat > nat,N2: nat,M4: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F3 @ N2 ) @ ( F3 @ M4 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_805_lift__Suc__mono__less__iff,axiom,
! [F3: nat > int,N2: nat,M4: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_int @ ( F3 @ N2 ) @ ( F3 @ M4 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_806_lift__Suc__mono__less,axiom,
! [F3: nat > nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ord_less_nat @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_807_lift__Suc__mono__less,axiom,
! [F3: nat > int,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ord_less_int @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_808_Suc__length__conv,axiom,
! [N2: nat,Xs: list_int] :
( ( ( suc @ N2 )
= ( size_size_list_int @ Xs ) )
= ( ? [Y2: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_809_Suc__length__conv,axiom,
! [N2: nat,Xs: list_a] :
( ( ( suc @ N2 )
= ( size_size_list_a @ Xs ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_810_Suc__length__conv,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( suc @ N2 )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_811_length__Suc__conv,axiom,
! [Xs: list_int,N2: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_812_length__Suc__conv,axiom,
! [Xs: list_a,N2: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_813_length__Suc__conv,axiom,
! [Xs: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_814_le__imp__less__Suc,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_nat @ M4 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_815_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_816_less__Suc__eq__le,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_817_le__less__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M4 ) )
= ( N2 = M4 ) ) ) ).
% le_less_Suc_eq
thf(fact_818_Suc__le__lessD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_819_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_820_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_821_Suc__le__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_le_eq
thf(fact_822_Suc__leI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 ) ) ).
% Suc_leI
thf(fact_823_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_824_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I ) ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_825_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
= ( ! [I: nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ ( suc @ I ) ) @ ( nth_int @ Xs @ I ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_826_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_int @ Xs ) )
= ( ? [X: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N2 @ ( size_size_list_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_827_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_a @ Xs ) )
= ( ? [X: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N2 @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_828_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_829_length__pos__if__in__set,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_830_length__pos__if__in__set,axiom,
! [X3: int,Xs: list_int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_831_length__pos__if__in__set,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_832_length__pos__if__in__set,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_833_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_834_card__ge__0__finite,axiom,
! [A: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
=> ( finite_finite_int @ A ) ) ).
% card_ge_0_finite
thf(fact_835_power__mono__iff,axiom,
! [A2: nat,B6: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B6 @ N2 ) )
= ( ord_less_eq_nat @ A2 @ B6 ) ) ) ) ) ).
% power_mono_iff
thf(fact_836_power__mono__iff,axiom,
! [A2: int,B6: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ B6 @ N2 ) )
= ( ord_less_eq_int @ A2 @ B6 ) ) ) ) ) ).
% power_mono_iff
thf(fact_837_power__eq__0__iff,axiom,
! [A2: int,N2: nat] :
( ( ( power_power_int @ A2 @ N2 )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_838_power__eq__0__iff,axiom,
! [A2: nat,N2: nat] :
( ( ( power_power_nat @ A2 @ N2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_839_nat__zero__less__power__iff,axiom,
! [X3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_840_nat__power__eq__Suc__0__iff,axiom,
! [X3: nat,M4: nat] :
( ( ( power_power_nat @ X3 @ M4 )
= ( suc @ zero_zero_nat ) )
= ( ( M4 = zero_zero_nat )
| ( X3
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_841_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_842_power__Suc0__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_843_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_844_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_845_power__not__zero,axiom,
! [A2: int,N2: nat] :
( ( A2 != zero_zero_int )
=> ( ( power_power_int @ A2 @ N2 )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_846_power__not__zero,axiom,
! [A2: nat,N2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( power_power_nat @ A2 @ N2 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_847_power__mono,axiom,
! [A2: nat,B6: nat,N2: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B6 @ N2 ) ) ) ) ).
% power_mono
thf(fact_848_power__mono,axiom,
! [A2: int,B6: int,N2: nat] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ B6 @ N2 ) ) ) ) ).
% power_mono
thf(fact_849_zero__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% zero_le_power
thf(fact_850_zero__le__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N2 ) ) ) ).
% zero_le_power
thf(fact_851_zero__less__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% zero_less_power
thf(fact_852_zero__less__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N2 ) ) ) ).
% zero_less_power
thf(fact_853_nat__power__less__imp__less,axiom,
! [I2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M4 ) @ ( power_power_nat @ I2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_854_power__less__imp__less__base,axiom,
! [A2: nat,N2: nat,B6: nat] :
( ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B6 @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ord_less_nat @ A2 @ B6 ) ) ) ).
% power_less_imp_less_base
thf(fact_855_power__less__imp__less__base,axiom,
! [A2: int,N2: nat,B6: int] :
( ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ B6 @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B6 )
=> ( ord_less_int @ A2 @ B6 ) ) ) ).
% power_less_imp_less_base
thf(fact_856_power__le__imp__le__base,axiom,
! [A2: nat,N2: nat,B6: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ ( power_power_nat @ B6 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ord_less_eq_nat @ A2 @ B6 ) ) ) ).
% power_le_imp_le_base
thf(fact_857_power__le__imp__le__base,axiom,
! [A2: int,N2: nat,B6: int] :
( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ ( power_power_int @ B6 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B6 )
=> ( ord_less_eq_int @ A2 @ B6 ) ) ) ).
% power_le_imp_le_base
thf(fact_858_power__inject__base,axiom,
! [A2: nat,N2: nat,B6: nat] :
( ( ( power_power_nat @ A2 @ ( suc @ N2 ) )
= ( power_power_nat @ B6 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( A2 = B6 ) ) ) ) ).
% power_inject_base
thf(fact_859_power__inject__base,axiom,
! [A2: int,N2: nat,B6: int] :
( ( ( power_power_int @ A2 @ ( suc @ N2 ) )
= ( power_power_int @ B6 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B6 )
=> ( A2 = B6 ) ) ) ) ).
% power_inject_base
thf(fact_860_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ).
% zero_power
thf(fact_861_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_862_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_863_nat__one__le__power,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_864_power__eq__iff__eq__base,axiom,
! [N2: nat,A2: nat,B6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ( ( power_power_nat @ A2 @ N2 )
= ( power_power_nat @ B6 @ N2 ) )
= ( A2 = B6 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_865_power__eq__iff__eq__base,axiom,
! [N2: nat,A2: int,B6: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B6 )
=> ( ( ( power_power_int @ A2 @ N2 )
= ( power_power_int @ B6 @ N2 ) )
= ( A2 = B6 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_866_power__eq__imp__eq__base,axiom,
! [A2: nat,N2: nat,B6: nat] :
( ( ( power_power_nat @ A2 @ N2 )
= ( power_power_nat @ B6 @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A2 = B6 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_867_power__eq__imp__eq__base,axiom,
! [A2: int,N2: nat,B6: int] :
( ( ( power_power_int @ A2 @ N2 )
= ( power_power_int @ B6 @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A2 = B6 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_868_power__strict__mono,axiom,
! [A2: nat,B6: nat,N2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B6 @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_869_power__strict__mono,axiom,
! [A2: int,B6: int,N2: nat] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ B6 @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_870_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_871_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_872_length__Cons,axiom,
! [X3: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X3 @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_873_length__Cons,axiom,
! [X3: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X3 @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_874_length__Cons,axiom,
! [X3: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X3 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_875_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_876_gr__implies__not__zero,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_877_subset__Collect__iff,axiom,
! [B: set_list_a,A: set_list_a,P: list_a > $o] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ B
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_878_subset__Collect__iff,axiom,
! [B: set_int,A: set_int,P: int > $o] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ B
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: int] :
( ( member_int @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_879_subset__Collect__iff,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ B
@ ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_880_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_881_subset__CollectI,axiom,
! [B: set_list_a,A: set_list_a,Q: list_a > $o,P: list_a > $o] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ B )
& ( Q @ X ) ) )
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_882_subset__CollectI,axiom,
! [B: set_int,A: set_int,Q: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ B )
& ( Q @ X ) ) )
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_883_subset__CollectI,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,Q: product_prod_int_int > $o,P: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ B )
& ( Q @ X ) ) )
@ ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_884_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_885_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_886_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_887_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_888_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_889_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_890_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_891_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_892_power__decreasing__iff,axiom,
! [B6: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B6 )
=> ( ( ord_less_nat @ B6 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B6 @ M4 ) @ ( power_power_nat @ B6 @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_893_power__decreasing__iff,axiom,
! [B6: int,M4: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_int @ B6 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B6 @ M4 ) @ ( power_power_int @ B6 @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_894_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_895_power__one,axiom,
! [N2: nat] :
( ( power_power_int @ one_one_int @ N2 )
= one_one_int ) ).
% power_one
thf(fact_896_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_897_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_898_power__inject__exp,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ( power_power_nat @ A2 @ M4 )
= ( power_power_nat @ A2 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_899_power__inject__exp,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ( power_power_int @ A2 @ M4 )
= ( power_power_int @ A2 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_900_power__strict__increasing__iff,axiom,
! [B6: nat,X3: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B6 )
=> ( ( ord_less_nat @ ( power_power_nat @ B6 @ X3 ) @ ( power_power_nat @ B6 @ Y ) )
= ( ord_less_nat @ X3 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_901_power__strict__increasing__iff,axiom,
! [B6: int,X3: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B6 )
=> ( ( ord_less_int @ ( power_power_int @ B6 @ X3 ) @ ( power_power_int @ B6 @ Y ) )
= ( ord_less_nat @ X3 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_902_power__strict__decreasing__iff,axiom,
! [B6: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B6 )
=> ( ( ord_less_nat @ B6 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B6 @ M4 ) @ ( power_power_nat @ B6 @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_903_power__strict__decreasing__iff,axiom,
! [B6: int,M4: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_int @ B6 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B6 @ M4 ) @ ( power_power_int @ B6 @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_904_power__increasing__iff,axiom,
! [B6: nat,X3: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B6 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B6 @ X3 ) @ ( power_power_nat @ B6 @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_905_power__increasing__iff,axiom,
! [B6: int,X3: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B6 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B6 @ X3 ) @ ( power_power_int @ B6 @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_906_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_907_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_908_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_909_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_910_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_911_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_912_one__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_913_one__le__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_914_power__0,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_915_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_916_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_917_power__le__one,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_918_power__le__one,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_919_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_920_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_921_power__gt1,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_922_power__gt1,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_923_power__increasing,axiom,
! [N2: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_924_power__increasing,axiom,
! [N2: nat,N5: nat,A2: int] :
( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_925_power__strict__increasing,axiom,
! [N2: nat,N5: nat,A2: nat] :
( ( ord_less_nat @ N2 @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_926_power__strict__increasing,axiom,
! [N2: nat,N5: nat,A2: int] :
( ( ord_less_nat @ N2 @ N5 )
=> ( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_927_power__less__imp__less__exp,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_928_power__less__imp__less__exp,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_929_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_930_sorted__wrt01,axiom,
! [Xs: list_a,P: a > a > $o] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( sorted_wrt_a @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_931_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_932_power__Suc__le__self,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_933_power__Suc__le__self,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_934_power__Suc__less__one,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_935_power__Suc__less__one,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_936_power__decreasing,axiom,
! [N2: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N5 ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_937_power__decreasing,axiom,
! [N2: nat,N5: nat,A2: int] :
( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N5 ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_938_power__strict__decreasing,axiom,
! [N2: nat,N5: nat,A2: nat] :
( ( ord_less_nat @ N2 @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N5 ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_939_power__strict__decreasing,axiom,
! [N2: nat,N5: nat,A2: int] :
( ( ord_less_nat @ N2 @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ N5 ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_940_power__le__imp__le__exp,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_941_power__le__imp__le__exp,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_942_self__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_943_self__le__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_944_one__less__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% one_less_power
thf(fact_945_one__less__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% one_less_power
thf(fact_946_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_947_sorted01,axiom,
! [Xs: list_int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% sorted01
thf(fact_948_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_949_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_950_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_951_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_952_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_953_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_954_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_955_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_956_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_957_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_958_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_959_nth__Cons__pos,axiom,
! [N2: nat,X3: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_960_nth__Cons__pos,axiom,
! [N2: nat,X3: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N2 )
= ( nth_int @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_961_of__nat__zero__less__power__iff,axiom,
! [X3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_962_of__nat__zero__less__power__iff,axiom,
! [X3: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_963_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_964_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_965_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X )
@ ^ [X: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_966_of__nat__eq__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M4 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_967_Suc__diff__diff,axiom,
! [M4: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M4 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_968_diff__Suc__Suc,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_969_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ M4 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_970_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_971_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_972_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_973_set__rotate1,axiom,
! [Xs: list_int] :
( ( set_int2 @ ( rotate1_int @ Xs ) )
= ( set_int2 @ Xs ) ) ).
% set_rotate1
thf(fact_974_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_975_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_976_distinct1__rotate,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct1_rotate
thf(fact_977_diff__ge__0__iff__ge,axiom,
! [A2: int,B6: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B6 ) )
= ( ord_less_eq_int @ B6 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_978_diff__gt__0__iff__gt,axiom,
! [A2: int,B6: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B6 ) )
= ( ord_less_int @ B6 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_979_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_980_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_981_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_982_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_983_of__nat__eq__0__iff,axiom,
! [M4: nat] :
( ( ( semiri1316708129612266289at_nat @ M4 )
= zero_zero_nat )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_984_of__nat__eq__0__iff,axiom,
! [M4: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= zero_zero_int )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_985_of__nat__le__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_986_of__nat__le__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_987_of__nat__less__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_988_of__nat__less__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_989_zero__less__diff,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M4 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% zero_less_diff
thf(fact_990_diff__is__0__eq,axiom,
! [M4: nat,N2: nat] :
( ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_991_diff__is__0__eq_H,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_992_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_993_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_994_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_995_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_996_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_997_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_998_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_999_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X3: nat,B6: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X3 )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B6 ) @ W ) )
= ( X3
= ( power_power_nat @ B6 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1000_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X3: nat,B6: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X3 )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B6 ) @ W ) )
= ( X3
= ( power_power_nat @ B6 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1001_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B6: nat,W: nat,X3: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B6 ) @ W )
= ( semiri1316708129612266289at_nat @ X3 ) )
= ( ( power_power_nat @ B6 @ W )
= X3 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1002_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B6: nat,W: nat,X3: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B6 ) @ W )
= ( semiri1314217659103216013at_int @ X3 ) )
= ( ( power_power_nat @ B6 @ W )
= X3 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1003_of__nat__power,axiom,
! [M4: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M4 @ N2 ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ N2 ) ) ).
% of_nat_power
thf(fact_1004_of__nat__power,axiom,
! [M4: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M4 @ N2 ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M4 ) @ N2 ) ) ).
% of_nat_power
thf(fact_1005_of__nat__le__0__iff,axiom,
! [M4: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ zero_zero_nat )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1006_of__nat__le__0__iff,axiom,
! [M4: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ zero_zero_int )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1007_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1008_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1009_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1010_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1011_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B6: nat,W: nat,X3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B6 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B6 @ W ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1012_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B6: nat,W: nat,X3: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B6 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B6 @ W ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1013_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B6: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B6 ) @ W ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B6 @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1014_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B6: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B6 ) @ W ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B6 @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1015_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X3: nat,B6: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B6 ) @ W ) )
= ( ord_less_nat @ X3 @ ( power_power_nat @ B6 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1016_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X3: nat,B6: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B6 ) @ W ) )
= ( ord_less_nat @ X3 @ ( power_power_nat @ B6 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1017_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B6: nat,W: nat,X3: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B6 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
= ( ord_less_nat @ ( power_power_nat @ B6 @ W ) @ X3 ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1018_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B6: nat,W: nat,X3: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B6 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
= ( ord_less_nat @ ( power_power_nat @ B6 @ W ) @ X3 ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1019_diff__eq__diff__less__eq,axiom,
! [A2: int,B6: int,C2: int,D: int] :
( ( ( minus_minus_int @ A2 @ B6 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B6 )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1020_diff__right__mono,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B6 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1021_diff__left__mono,axiom,
! [B6: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B6 @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A2 ) @ ( minus_minus_int @ C2 @ B6 ) ) ) ).
% diff_left_mono
thf(fact_1022_diff__mono,axiom,
! [A2: int,B6: int,D: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B6 )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B6 @ D ) ) ) ) ).
% diff_mono
thf(fact_1023_diff__strict__right__mono,axiom,
! [A2: int,B6: int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B6 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1024_diff__strict__left__mono,axiom,
! [B6: int,A2: int,C2: int] :
( ( ord_less_int @ B6 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A2 ) @ ( minus_minus_int @ C2 @ B6 ) ) ) ).
% diff_strict_left_mono
thf(fact_1025_diff__eq__diff__less,axiom,
! [A2: int,B6: int,C2: int,D: int] :
( ( ( minus_minus_int @ A2 @ B6 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A2 @ B6 )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1026_diff__strict__mono,axiom,
! [A2: int,B6: int,D: int,C2: int] :
( ( ord_less_int @ A2 @ B6 )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B6 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1027_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1028_of__nat__diff,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ N2 @ M4 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M4 @ N2 ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_1029_of__nat__diff,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ N2 @ M4 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M4 @ N2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_1030_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_1031_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_1032_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_1033_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_1034_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_1035_diff__left__imp__eq,axiom,
! [A2: int,B6: int,C2: int] :
( ( ( minus_minus_int @ A2 @ B6 )
= ( minus_minus_int @ A2 @ C2 ) )
=> ( B6 = C2 ) ) ).
% diff_left_imp_eq
thf(fact_1036_diff__le__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1037_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B6: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B6 ) )
= ( ord_less_eq_nat @ B6 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1038_diff__le__self,axiom,
! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ).
% diff_le_self
thf(fact_1039_diff__le__mono,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1040_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1041_le__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1042_eq__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M4 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1043_diff__less__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ M4 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_1044_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1045_diffs0__imp__equal,axiom,
! [M4: nat,N2: nat] :
( ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M4 )
= zero_zero_nat )
=> ( M4 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1046_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% minus_nat.diff_0
thf(fact_1047_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1048_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1049_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1050_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_less_as_int
thf(fact_1051_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1052_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_1053_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_1054_of__nat__less__0__iff,axiom,
! [M4: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_1055_of__nat__less__0__iff,axiom,
! [M4: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_1056_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B7: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B7 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1057_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1058_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1059_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B7: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B7 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1060_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_1061_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_1062_less__imp__of__nat__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_1063_less__imp__of__nat__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_1064_of__nat__less__imp__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_1065_of__nat__less__imp__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_1066_Suc__diff__Suc,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ N2 @ M4 )
=> ( ( suc @ ( minus_minus_nat @ M4 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1067_diff__less__Suc,axiom,
! [M4: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M4 @ N2 ) @ ( suc @ M4 ) ) ).
% diff_less_Suc
thf(fact_1068_diff__less,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ) ) ).
% diff_less
thf(fact_1069_Suc__diff__le,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ N2 @ M4 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1070_less__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1071_diff__less__mono,axiom,
! [A2: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B6 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B6 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1072_diff__Suc__eq__diff__pred,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ M4 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1073_diff__Suc__less,axiom,
! [N2: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1074_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1075_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N2 )
= ( minus_minus_nat @ M4 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1076_nth__Cons_H,axiom,
! [N2: nat,X3: nat,Xs: list_nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N2 )
= X3 ) )
& ( ( N2 != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1077_nth__Cons_H,axiom,
! [N2: nat,X3: int,Xs: list_int] :
( ( ( N2 = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N2 )
= X3 ) )
& ( ( N2 != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N2 )
= ( nth_int @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1078_nth__non__equal__first__eq,axiom,
! [X3: nat,Y: nat,Xs: list_nat,N2: nat] :
( ( X3 != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N2 )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1079_nth__non__equal__first__eq,axiom,
! [X3: int,Y: int,Xs: list_int,N2: nat] :
( ( X3 != Y )
=> ( ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N2 )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1080_rev__nth,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rev_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_1081_rev__nth,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_1082_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1083_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1084_card__insert__le__m1,axiom,
! [N2: nat,Y: set_nat,X3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat2 @ X3 @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_1085_nth__insert__nth__back,axiom,
! [J: nat,I2: nat,Xs: list_a,X3: a] :
( ( ord_less_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_insert_nth_a @ J @ X3 @ Xs ) @ I2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) ) ) ).
% nth_insert_nth_back
thf(fact_1086_nth__insert__nth__back,axiom,
! [J: nat,I2: nat,Xs: list_nat,X3: nat] :
( ( ord_less_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_insert_nth_nat @ J @ X3 @ Xs ) @ I2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) ) ) ).
% nth_insert_nth_back
thf(fact_1087_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1088_finite__Diff2,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_Diff2
thf(fact_1089_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1090_finite__Diff,axiom,
! [A: set_int,B: set_int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1091_Diff__insert0,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X3 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1092_Diff__insert0,axiom,
! [X3: list_a,A: set_list_a,B: set_list_a] :
( ~ ( member_list_a @ X3 @ A )
=> ( ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X3 @ B ) )
= ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1093_insert__Diff1,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1094_insert__Diff1,axiom,
! [X3: list_a,B: set_list_a,A: set_list_a] :
( ( member_list_a @ X3 @ B )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X3 @ A ) @ B )
= ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1095_insert__iff,axiom,
! [A2: nat,B6: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat2 @ B6 @ A ) )
= ( ( A2 = B6 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1096_insert__iff,axiom,
! [A2: list_a,B6: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a2 @ B6 @ A ) )
= ( ( A2 = B6 )
| ( member_list_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1097_insertCI,axiom,
! [A2: nat,B: set_nat,B6: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B6 ) )
=> ( member_nat @ A2 @ ( insert_nat2 @ B6 @ B ) ) ) ).
% insertCI
thf(fact_1098_insertCI,axiom,
! [A2: list_a,B: set_list_a,B6: list_a] :
( ( ~ ( member_list_a @ A2 @ B )
=> ( A2 = B6 ) )
=> ( member_list_a @ A2 @ ( insert_list_a2 @ B6 @ B ) ) ) ).
% insertCI
thf(fact_1099_finite__interval__int4,axiom,
! [A2: int,B6: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A2 @ I )
& ( ord_less_int @ I @ B6 ) ) ) ) ).
% finite_interval_int4
thf(fact_1100_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1101_finite__Diff__insert,axiom,
! [A: set_int,A2: int,B: set_int] :
( ( finite_finite_int @ ( minus_minus_set_int @ A @ ( insert_int2 @ A2 @ B ) ) )
= ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1102_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_1103_finite__insert,axiom,
! [A2: int,A: set_int] :
( ( finite_finite_int @ ( insert_int2 @ A2 @ A ) )
= ( finite_finite_int @ A ) ) ).
% finite_insert
thf(fact_1104_insert__subset,axiom,
! [X3: list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a2 @ X3 @ A ) @ B )
= ( ( member_list_a @ X3 @ B )
& ( ord_le8861187494160871172list_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_1105_insert__subset,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X3 @ A ) @ B )
= ( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_1106_zle__diff1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1107_finite__interval__int3,axiom,
! [A2: int,B6: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A2 @ I )
& ( ord_less_eq_int @ I @ B6 ) ) ) ) ).
% finite_interval_int3
thf(fact_1108_finite__interval__int2,axiom,
! [A2: int,B6: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A2 @ I )
& ( ord_less_int @ I @ B6 ) ) ) ) ).
% finite_interval_int2
thf(fact_1109_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1110_list_Osimps_I15_J,axiom,
! [X21: int,X22: list_int] :
( ( set_int2 @ ( cons_int @ X21 @ X22 ) )
= ( insert_int2 @ X21 @ ( set_int2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1111_List_Oset__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X3 @ Xs ) )
= ( insert_nat2 @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1112_List_Oset__insert,axiom,
! [X3: int,Xs: list_int] :
( ( set_int2 @ ( insert_int @ X3 @ Xs ) )
= ( insert_int2 @ X3 @ ( set_int2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1113_card__insert__disjoint,axiom,
! [A: set_list_a,X3: list_a] :
( ( finite_finite_list_a @ A )
=> ( ~ ( member_list_a @ X3 @ A )
=> ( ( finite_card_list_a @ ( insert_list_a2 @ X3 @ A ) )
= ( suc @ ( finite_card_list_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1114_card__insert__disjoint,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X3 @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X3 @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1115_card__insert__disjoint,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X3 @ A )
=> ( ( finite_card_int @ ( insert_int2 @ X3 @ A ) )
= ( suc @ ( finite_card_int @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1116_card__Diff__insert,axiom,
! [A2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ~ ( member_list_a @ A2 @ B )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1117_card__Diff__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ A2 @ B )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1118_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1119_subset__Diff__insert,axiom,
! [A: set_list_a,B: set_list_a,X3: list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( minus_646659088055828811list_a @ B @ ( insert_list_a2 @ X3 @ C ) ) )
= ( ( ord_le8861187494160871172list_a @ A @ ( minus_646659088055828811list_a @ B @ C ) )
& ~ ( member_list_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1120_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X3: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat2 @ X3 @ C ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C ) )
& ~ ( member_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1121_insert__Diff__if,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X3 @ A ) @ B )
= ( insert_nat2 @ X3 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1122_insert__Diff__if,axiom,
! [X3: list_a,B: set_list_a,A: set_list_a] :
( ( ( member_list_a @ X3 @ B )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X3 @ A ) @ B )
= ( minus_646659088055828811list_a @ A @ B ) ) )
& ( ~ ( member_list_a @ X3 @ B )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X3 @ A ) @ B )
= ( insert_list_a2 @ X3 @ ( minus_646659088055828811list_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1123_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1124_psubset__imp__ex__mem,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ? [B4: list_a] : ( member_list_a @ B4 @ ( minus_646659088055828811list_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1125_Diff__mono,axiom,
! [A: set_nat,C: set_nat,D3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ D3 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1126_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1127_double__diff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1128_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1129_Diff__infinite__finite,axiom,
! [T: set_int,S: set_int] :
( ( finite_finite_int @ T )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1130_int__ops_I6_J,axiom,
! [A2: nat,B6: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B6 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B6 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B6 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B6 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1131_conj__le__cong,axiom,
! [X3: int,X7: int,P: $o,P4: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1132_imp__le__cong,axiom,
! [X3: int,X7: int,P: $o,P4: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1133_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1134_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat2 @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1135_finite_OinsertI,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( insert_int2 @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1136_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B6: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B6 @ B ) ) ) ).
% subset_insertI2
thf(fact_1137_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_1138_subset__insert,axiom,
! [X3: list_a,A: set_list_a,B: set_list_a] :
( ~ ( member_list_a @ X3 @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a2 @ X3 @ B ) )
= ( ord_le8861187494160871172list_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_1139_subset__insert,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1140_insert__mono,axiom,
! [C: set_nat,D3: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C @ D3 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C ) @ ( insert_nat2 @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_1141_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B2: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B2 ) )
& ~ ( member_nat @ A2 @ B2 ) ) ) ).
% mk_disjoint_insert
thf(fact_1142_mk__disjoint__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ? [B2: set_list_a] :
( ( A
= ( insert_list_a2 @ A2 @ B2 ) )
& ~ ( member_list_a @ A2 @ B2 ) ) ) ).
% mk_disjoint_insert
thf(fact_1143_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B6: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B6 @ B )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B6 @ B ) )
= ( ( ( A2 = B6 )
=> ( A = B ) )
& ( ( A2 != B6 )
=> ? [C5: set_nat] :
( ( A
= ( insert_nat2 @ B6 @ C5 ) )
& ~ ( member_nat @ B6 @ C5 )
& ( B
= ( insert_nat2 @ A2 @ C5 ) )
& ~ ( member_nat @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1144_insert__eq__iff,axiom,
! [A2: list_a,A: set_list_a,B6: list_a,B: set_list_a] :
( ~ ( member_list_a @ A2 @ A )
=> ( ~ ( member_list_a @ B6 @ B )
=> ( ( ( insert_list_a2 @ A2 @ A )
= ( insert_list_a2 @ B6 @ B ) )
= ( ( ( A2 = B6 )
=> ( A = B ) )
& ( ( A2 != B6 )
=> ? [C5: set_list_a] :
( ( A
= ( insert_list_a2 @ B6 @ C5 ) )
& ~ ( member_list_a @ B6 @ C5 )
& ( B
= ( insert_list_a2 @ A2 @ C5 ) )
& ~ ( member_list_a @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1145_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1146_insert__absorb,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ( insert_list_a2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1147_insert__ident,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ~ ( member_nat @ X3 @ B )
=> ( ( ( insert_nat2 @ X3 @ A )
= ( insert_nat2 @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1148_insert__ident,axiom,
! [X3: list_a,A: set_list_a,B: set_list_a] :
( ~ ( member_list_a @ X3 @ A )
=> ( ~ ( member_list_a @ X3 @ B )
=> ( ( ( insert_list_a2 @ X3 @ A )
= ( insert_list_a2 @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1149_Set_Oset__insert,axiom,
! [X3: nat,A: set_nat] :
( ( member_nat @ X3 @ A )
=> ~ ! [B2: set_nat] :
( ( A
= ( insert_nat2 @ X3 @ B2 ) )
=> ( member_nat @ X3 @ B2 ) ) ) ).
% Set.set_insert
thf(fact_1150_Set_Oset__insert,axiom,
! [X3: list_a,A: set_list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ! [B2: set_list_a] :
( ( A
= ( insert_list_a2 @ X3 @ B2 ) )
=> ( member_list_a @ X3 @ B2 ) ) ) ).
% Set.set_insert
thf(fact_1151_insertI2,axiom,
! [A2: nat,B: set_nat,B6: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat2 @ B6 @ B ) ) ) ).
% insertI2
thf(fact_1152_insertI2,axiom,
! [A2: list_a,B: set_list_a,B6: list_a] :
( ( member_list_a @ A2 @ B )
=> ( member_list_a @ A2 @ ( insert_list_a2 @ B6 @ B ) ) ) ).
% insertI2
thf(fact_1153_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_1154_insertI1,axiom,
! [A2: list_a,B: set_list_a] : ( member_list_a @ A2 @ ( insert_list_a2 @ A2 @ B ) ) ).
% insertI1
thf(fact_1155_insertE,axiom,
! [A2: nat,B6: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat2 @ B6 @ A ) )
=> ( ( A2 != B6 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_1156_insertE,axiom,
! [A2: list_a,B6: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a2 @ B6 @ A ) )
=> ( ( A2 != B6 )
=> ( member_list_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_1157_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat2 @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1158_insert__Collect,axiom,
! [A2: int,P: int > $o] :
( ( insert_int2 @ A2 @ ( collect_int @ P ) )
= ( collect_int
@ ^ [U2: int] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1159_insert__Collect,axiom,
! [A2: product_prod_int_int,P: product_prod_int_int > $o] :
( ( insert5033312907999012233nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) )
= ( collec213857154873943460nt_int
@ ^ [U2: product_prod_int_int] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1160_insert__compr,axiom,
( insert_list_a2
= ( ^ [A5: list_a,B5: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( X = A5 )
| ( member_list_a @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1161_insert__compr,axiom,
( insert_nat2
= ( ^ [A5: nat,B5: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A5 )
| ( member_nat @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1162_insert__compr,axiom,
( insert_int2
= ( ^ [A5: int,B5: set_int] :
( collect_int
@ ^ [X: int] :
( ( X = A5 )
| ( member_int @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1163_insert__compr,axiom,
( insert5033312907999012233nt_int
= ( ^ [A5: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( X = A5 )
| ( member5262025264175285858nt_int @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1164_set__insert__nth,axiom,
! [I2: nat,X3: nat,Xs: list_nat] :
( ( set_nat2 @ ( list_insert_nth_nat @ I2 @ X3 @ Xs ) )
= ( insert_nat2 @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% set_insert_nth
thf(fact_1165_set__insert__nth,axiom,
! [I2: nat,X3: int,Xs: list_int] :
( ( set_int2 @ ( list_insert_nth_int @ I2 @ X3 @ Xs ) )
= ( insert_int2 @ X3 @ ( set_int2 @ Xs ) ) ) ).
% set_insert_nth
thf(fact_1166_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1167_insert__nth_Osimps_I1_J,axiom,
! [X3: nat,Xs: list_nat] :
( ( list_insert_nth_nat @ zero_zero_nat @ X3 @ Xs )
= ( cons_nat @ X3 @ Xs ) ) ).
% insert_nth.simps(1)
thf(fact_1168_insert__nth_Osimps_I1_J,axiom,
! [X3: int,Xs: list_int] :
( ( list_insert_nth_int @ zero_zero_nat @ X3 @ Xs )
= ( cons_int @ X3 @ Xs ) ) ).
% insert_nth.simps(1)
thf(fact_1169_length__insert__nth,axiom,
! [N2: nat,X3: a,Xs: list_a] :
( ( size_size_list_a @ ( list_insert_nth_a @ N2 @ X3 @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_insert_nth
thf(fact_1170_length__insert__nth,axiom,
! [N2: nat,X3: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( list_insert_nth_nat @ N2 @ X3 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_insert_nth
thf(fact_1171_distinct__insert__nth,axiom,
! [Xs: list_list_a,X3: list_a,I2: nat] :
( ( distinct_list_a @ Xs )
=> ( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( distinct_list_a @ ( list_i9041641776701930082list_a @ I2 @ X3 @ Xs ) ) ) ) ).
% distinct_insert_nth
thf(fact_1172_distinct__insert__nth,axiom,
! [Xs: list_nat,X3: nat,I2: nat] :
( ( distinct_nat @ Xs )
=> ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( distinct_nat @ ( list_insert_nth_nat @ I2 @ X3 @ Xs ) ) ) ) ).
% distinct_insert_nth
thf(fact_1173_distinct__insert__nth,axiom,
! [Xs: list_int,X3: int,I2: nat] :
( ( distinct_int @ Xs )
=> ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( distinct_int @ ( list_insert_nth_int @ I2 @ X3 @ Xs ) ) ) ) ).
% distinct_insert_nth
thf(fact_1174_card__insert__le,axiom,
! [A: set_nat,X3: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat2 @ X3 @ A ) ) ) ).
% card_insert_le
thf(fact_1175_insert__nth__inverse,axiom,
! [J: nat,Xs: list_list_a,J4: nat,Xs4: list_list_a,X3: list_a] :
( ( ord_less_eq_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ J4 @ ( size_s349497388124573686list_a @ Xs4 ) )
=> ( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs4 ) )
=> ( ( ( list_i9041641776701930082list_a @ J @ X3 @ Xs )
= ( list_i9041641776701930082list_a @ J4 @ X3 @ Xs4 ) )
=> ( J = J4 ) ) ) ) ) ) ).
% insert_nth_inverse
thf(fact_1176_insert__nth__inverse,axiom,
! [J: nat,Xs: list_int,J4: nat,Xs4: list_int,X3: int] :
( ( ord_less_eq_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_eq_nat @ J4 @ ( size_size_list_int @ Xs4 ) )
=> ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int @ X3 @ ( set_int2 @ Xs4 ) )
=> ( ( ( list_insert_nth_int @ J @ X3 @ Xs )
= ( list_insert_nth_int @ J4 @ X3 @ Xs4 ) )
=> ( J = J4 ) ) ) ) ) ) ).
% insert_nth_inverse
thf(fact_1177_insert__nth__inverse,axiom,
! [J: nat,Xs: list_a,J4: nat,Xs4: list_a,X3: a] :
( ( ord_less_eq_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ J4 @ ( size_size_list_a @ Xs4 ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Xs4 ) )
=> ( ( ( list_insert_nth_a @ J @ X3 @ Xs )
= ( list_insert_nth_a @ J4 @ X3 @ Xs4 ) )
=> ( J = J4 ) ) ) ) ) ) ).
% insert_nth_inverse
thf(fact_1178_insert__nth__inverse,axiom,
! [J: nat,Xs: list_nat,J4: nat,Xs4: list_nat,X3: nat] :
( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ J4 @ ( size_size_list_nat @ Xs4 ) )
=> ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
=> ( ( ( list_insert_nth_nat @ J @ X3 @ Xs )
= ( list_insert_nth_nat @ J4 @ X3 @ Xs4 ) )
=> ( J = J4 ) ) ) ) ) ) ).
% insert_nth_inverse
thf(fact_1179_nth__insert__nth__index__eq,axiom,
! [I2: nat,Xs: list_a,X3: a] :
( ( ord_less_eq_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_insert_nth_a @ I2 @ X3 @ Xs ) @ I2 )
= X3 ) ) ).
% nth_insert_nth_index_eq
thf(fact_1180_nth__insert__nth__index__eq,axiom,
! [I2: nat,Xs: list_nat,X3: nat] :
( ( ord_less_eq_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_insert_nth_nat @ I2 @ X3 @ Xs ) @ I2 )
= X3 ) ) ).
% nth_insert_nth_index_eq
thf(fact_1181_card__insert__if,axiom,
! [A: set_list_a,X3: list_a] :
( ( finite_finite_list_a @ A )
=> ( ( ( member_list_a @ X3 @ A )
=> ( ( finite_card_list_a @ ( insert_list_a2 @ X3 @ A ) )
= ( finite_card_list_a @ A ) ) )
& ( ~ ( member_list_a @ X3 @ A )
=> ( ( finite_card_list_a @ ( insert_list_a2 @ X3 @ A ) )
= ( suc @ ( finite_card_list_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1182_card__insert__if,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X3 @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X3 @ A ) )
= ( finite_card_nat @ A ) ) )
& ( ~ ( member_nat @ X3 @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X3 @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1183_card__insert__if,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ X3 @ A )
=> ( ( finite_card_int @ ( insert_int2 @ X3 @ A ) )
= ( finite_card_int @ A ) ) )
& ( ~ ( member_int @ X3 @ A )
=> ( ( finite_card_int @ ( insert_int2 @ X3 @ A ) )
= ( suc @ ( finite_card_int @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1184_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X3 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1185_zle__int,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% zle_int
thf(fact_1186_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z4: int] :
( ( ord_less_eq_int @ D4 @ Z4 )
& ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_1187_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z4: int] :
( ( ord_less_eq_int @ D4 @ Z7 )
& ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_1188_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I ) @ Js @ ( upto_aux @ I @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1189_nat__ivt__aux,axiom,
! [N2: nat,F3: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F3 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F3 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
& ( ( F3 @ I4 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1190_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
= ( Z3 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1191_nat__intermed__int__val,axiom,
! [M4: nat,N2: nat,F3: nat > int,K: int] :
( ! [I4: nat] :
( ( ( ord_less_eq_nat @ M4 @ I4 )
& ( ord_less_nat @ I4 @ N2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_eq_int @ ( F3 @ M4 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F3 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ M4 @ I4 )
& ( ord_less_eq_nat @ I4 @ N2 )
& ( ( F3 @ I4 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1192_nat0__intermed__int__val,axiom,
! [N2: nat,F3: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F3 @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F3 @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F3 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F3 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
& ( ( F3 @ I4 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1193_add__Suc__right,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ M4 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M4 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1194_Nat_Oadd__0__right,axiom,
! [M4: nat] :
( ( plus_plus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% Nat.add_0_right
thf(fact_1195_add__is__0,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1196_nat__add__left__cancel__less,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1197_nat__add__left__cancel__le,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1198_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1199_add__gr__0,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M4 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1200_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1201_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1202_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1203_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1204_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1205_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1206_add__Suc,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M4 @ N2 ) ) ) ).
% add_Suc
thf(fact_1207_add__Suc__shift,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N2 )
= ( plus_plus_nat @ M4 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1208_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1209_trans__le__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_le_add2
thf(fact_1210_trans__le__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_le_add1
thf(fact_1211_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1212_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1213_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1214_add__leD2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1215_add__leD1,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% add_leD1
thf(fact_1216_le__add2,axiom,
! [N2: nat,M4: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M4 @ N2 ) ) ).
% le_add2
thf(fact_1217_le__add1,axiom,
! [N2: nat,M4: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M4 ) ) ).
% le_add1
thf(fact_1218_add__leE,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M4 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1219_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1220_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1221_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1222_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1223_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1224_trans__less__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_less_add1
thf(fact_1225_trans__less__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_less_add2
thf(fact_1226_less__add__eq__less,axiom,
! [K: nat,L: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M4 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1227_add__eq__self__zero,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= M4 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1228_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1229_diff__add__inverse2,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ N2 ) @ N2 )
= M4 ) ).
% diff_add_inverse2
thf(fact_1230_diff__add__inverse,axiom,
! [N2: nat,M4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M4 ) @ N2 )
= M4 ) ).
% diff_add_inverse
thf(fact_1231_diff__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% diff_cancel2
thf(fact_1232_Nat_Odiff__cancel,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1233_one__is__add,axiom,
! [M4: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M4 @ N2 ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1234_add__is__1,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1235_less__imp__Suc__add,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1236_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1237_less__add__Suc2,axiom,
! [I2: nat,M4: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M4 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1238_less__add__Suc1,axiom,
! [I2: nat,M4: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M4 ) ) ) ).
% less_add_Suc1
thf(fact_1239_less__natE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M4 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1240_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1241_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M4: nat,K: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F3 @ M3 ) @ ( F3 @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F3 @ M4 ) @ K ) @ ( F3 @ ( plus_plus_nat @ M4 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1242_diff__add__0,axiom,
! [N2: nat,M4: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M4 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1243_add__diff__inverse__nat,axiom,
! [M4: nat,N2: nat] :
( ~ ( ord_less_nat @ M4 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M4 @ N2 ) )
= M4 ) ) ).
% add_diff_inverse_nat
thf(fact_1244_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1245_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1246_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1247_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1248_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1249_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1250_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1251_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1252_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1253_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B6: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B6 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B6 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B6 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1254_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B6: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B6 ) )
= ( ( ( ord_less_nat @ A2 @ B6 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B6 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1255_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1256_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N3: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1257_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1258_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1259_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1260_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1261_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1262_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1263_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1264_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 )
= ( ord_less_int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1265_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y: a] :
( ( if_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y: a] :
( ( if_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X3: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X3: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_nat @ aa ).
%------------------------------------------------------------------------------