TPTP Problem File: SLH0559^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00356_013333__14827616_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1401 ( 431 unt; 137 typ; 0 def)
% Number of atoms : 4214 (1054 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12122 ( 513 ~; 78 |; 299 &;9000 @)
% ( 0 <=>;2232 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 548 ( 548 >; 0 *; 0 +; 0 <<)
% Number of symbols : 123 ( 122 usr; 17 con; 0-3 aty)
% Number of variables : 3753 ( 186 ^;3327 !; 240 ?;3753 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:45:32.527
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
list_Extended_enat: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
set_Extended_enat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (122)
thf(sy_c_Finite__Set_Ocard_001t__Extended____Nat__Oenat,type,
finite121521170596916366d_enat: set_Extended_enat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
finite4001608067531595151d_enat: set_Extended_enat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Extended____Nat__Oenat,type,
infini7641415182203889163d_enat: set_Extended_enat > nat > extended_enat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_List_Odistinct_001t__Extended____Nat__Oenat,type,
distin4523846830085650399d_enat: list_Extended_enat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Real__Oreal,type,
distinct_real: list_real > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Extended____Nat__Oenat,type,
linord1591021928418041270d_enat: set_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Real__Oreal,type,
linord4252657396651189596t_real: set_real > list_real ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001tf__a,type,
linord3083462915744475214_set_a: set_a > list_a ).
thf(sy_c_List_Olist_OCons_001t__Extended____Nat__Oenat,type,
cons_Extended_enat: extended_enat > list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
set_Extended_enat2: list_Extended_enat > set_Extended_enat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__Extended____Nat__Oenat,type,
tl_Extended_enat: list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001t__Real__Oreal,type,
tl_real: list_real > list_real ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001tf__a,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
nth_Extended_enat: list_Extended_enat > nat > extended_enat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oremdups__adj_001t__Extended____Nat__Oenat,type,
remdup6152102037098707618d_enat: list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001t__Real__Oreal,type,
remdups_adj_real: list_real > list_real ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_Orev_001t__Extended____Nat__Oenat,type,
rev_Extended_enat: list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Real__Oreal,type,
rev_real: list_real > list_real ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__Extended____Nat__Oenat,type,
sorted143172755617435219d_enat: ( extended_enat > extended_enat > $o ) > list_Extended_enat > $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_001t__Real__Oreal,type,
sorted_wrt_real: ( real > real > $o ) > list_real > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_Median_Odown__ray_001t__Extended____Nat__Oenat,type,
down_r2452194630239499347d_enat: set_Extended_enat > $o ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001t__Real__Oreal,type,
down_ray_real: set_real > $o ).
thf(sy_c_Median_Odown__ray_001tf__a,type,
down_ray_a: set_a > $o ).
thf(sy_c_Median_Ointerval_001t__Extended____Nat__Oenat,type,
interv132053279953483690d_enat: set_Extended_enat > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001t__Real__Oreal,type,
interval_real: set_real > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Osort__map_001t__Extended____Nat__Oenat,type,
sort_m8616768064977705053d_enat: ( nat > extended_enat ) > nat > nat > extended_enat ).
thf(sy_c_Median_Osort__map_001t__Nat__Onat,type,
sort_map_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Median_Osort__map_001t__Real__Oreal,type,
sort_map_real: ( nat > real ) > nat > nat > real ).
thf(sy_c_Median_Osort__map_001tf__a,type,
sort_map_a: ( nat > a ) > nat > nat > a ).
thf(sy_c_Median_Oup__ray_001t__Extended____Nat__Oenat,type,
up_ray_Extended_enat: set_Extended_enat > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001t__Real__Oreal,type,
up_ray_real: set_real > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
size_s3941691890525107288d_enat: list_Extended_enat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > 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_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
bot_bo7653980558646680370d_enat: set_Extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $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__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $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__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
insert_Extended_enat: extended_enat > set_Extended_enat > set_Extended_enat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Extended____Nat__Oenat,type,
member_Extended_enat: extended_enat > set_Extended_enat > $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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_i,type,
i2: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1263)
thf(fact_0_assms_I1_J,axiom,
interval_a @ i ).
% assms(1)
thf(fact_1_assms_I6_J,axiom,
member_a @ ( nth_a @ xs @ i2 ) @ i ).
% assms(6)
thf(fact_2_assms_I7_J,axiom,
member_a @ ( nth_a @ xs @ k ) @ i ).
% assms(7)
thf(fact_3_assms_I4_J,axiom,
ord_less_eq_nat @ i2 @ j ).
% assms(4)
thf(fact_4_assms_I5_J,axiom,
ord_less_eq_nat @ j @ k ).
% assms(5)
thf(fact_5_assms_I2_J,axiom,
sorted_wrt_a @ ord_less_eq_a @ xs ).
% assms(2)
thf(fact_6_assms_I3_J,axiom,
ord_less_nat @ k @ ( size_size_list_a @ xs ) ).
% assms(3)
thf(fact_7_list__update__id,axiom,
! [Xs: list_a,I: nat] :
( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_8_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_9_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_a,X: a] :
( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_10_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_11_list__update__overwrite,axiom,
! [Xs: list_nat,I: nat,X: nat,Y: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I @ Y )
= ( list_update_nat @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_12_list__update__overwrite,axiom,
! [Xs: list_a,I: nat,X: a,Y: a] :
( ( list_update_a @ ( list_update_a @ Xs @ I @ X ) @ I @ Y )
= ( list_update_a @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_13_length__list__update,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_14_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_15_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_16_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_17_nth__list__update__eq,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_18_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_19_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_20_sorted__nth__mono,axiom,
! [Xs: list_a,I: nat,J: nat] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I ) @ ( nth_a @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_21_sorted__nth__mono,axiom,
! [Xs: list_real,I: nat,J: nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I ) @ ( nth_real @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_22_sorted__nth__mono,axiom,
! [Xs: list_Extended_enat,I: nat,J: nat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I ) @ ( nth_Extended_enat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_23_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_24_sorted__iff__nth__mono,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_25_sorted__iff__nth__mono,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I2 ) @ ( nth_real @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_26_sorted__iff__nth__mono,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I2 ) @ ( nth_Extended_enat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_27_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_28_sorted__iff__nth__mono__less,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_29_sorted__iff__nth__mono__less,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I2 ) @ ( nth_real @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_30_sorted__iff__nth__mono__less,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I2 ) @ ( nth_Extended_enat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_31_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_32_strict__sorted__imp__sorted,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_a @ Xs )
=> ( sorted_wrt_a @ ord_less_eq_a @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_33_strict__sorted__imp__sorted,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_34_strict__sorted__imp__sorted,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_35_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_36_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_37_nth__equalityI,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_38_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_39_interval__def,axiom,
( interval_nat
= ( ^ [I4: set_nat] :
! [X2: nat,Y2: nat,Z: nat] :
( ( member_nat @ X2 @ I4 )
=> ( ( member_nat @ Z @ I4 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( member_nat @ Y2 @ I4 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_40_interval__def,axiom,
( interval_a
= ( ^ [I4: set_a] :
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ I4 )
=> ( ( member_a @ Z @ I4 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( member_a @ Y2 @ I4 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_41_interval__def,axiom,
( interval_real
= ( ^ [I4: set_real] :
! [X2: real,Y2: real,Z: real] :
( ( member_real @ X2 @ I4 )
=> ( ( member_real @ Z @ I4 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z )
=> ( member_real @ Y2 @ I4 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_42_interval__def,axiom,
( interv132053279953483690d_enat
= ( ^ [I4: set_Extended_enat] :
! [X2: extended_enat,Y2: extended_enat,Z: extended_enat] :
( ( member_Extended_enat @ X2 @ I4 )
=> ( ( member_Extended_enat @ Z @ I4 )
=> ( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
=> ( ( ord_le2932123472753598470d_enat @ Y2 @ Z )
=> ( member_Extended_enat @ Y2 @ I4 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_43_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: a] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_44_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: nat] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_45_nth__list__update,axiom,
! [I: nat,Xs: list_a,J: nat,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_46_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_47_interval__rule,axiom,
! [I5: set_nat,A: nat,X: nat,B: nat] :
( ( interval_nat @ I5 )
=> ( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ( member_nat @ A @ I5 )
=> ( ( member_nat @ B @ I5 )
=> ( member_nat @ X @ I5 ) ) ) ) ) ) ).
% interval_rule
thf(fact_48_interval__rule,axiom,
! [I5: set_a,A: a,X: a,B: a] :
( ( interval_a @ I5 )
=> ( ( ord_less_eq_a @ A @ X )
=> ( ( ord_less_eq_a @ X @ B )
=> ( ( member_a @ A @ I5 )
=> ( ( member_a @ B @ I5 )
=> ( member_a @ X @ I5 ) ) ) ) ) ) ).
% interval_rule
thf(fact_49_interval__rule,axiom,
! [I5: set_real,A: real,X: real,B: real] :
( ( interval_real @ I5 )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( member_real @ A @ I5 )
=> ( ( member_real @ B @ I5 )
=> ( member_real @ X @ I5 ) ) ) ) ) ) ).
% interval_rule
thf(fact_50_interval__rule,axiom,
! [I5: set_Extended_enat,A: extended_enat,X: extended_enat,B: extended_enat] :
( ( interv132053279953483690d_enat @ I5 )
=> ( ( ord_le2932123472753598470d_enat @ A @ X )
=> ( ( ord_le2932123472753598470d_enat @ X @ B )
=> ( ( member_Extended_enat @ A @ I5 )
=> ( ( member_Extended_enat @ B @ I5 )
=> ( member_Extended_enat @ X @ I5 ) ) ) ) ) ) ).
% interval_rule
thf(fact_51_list__update__swap,axiom,
! [I: nat,I6: nat,Xs: list_nat,X: nat,X4: nat] :
( ( I != I6 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I6 @ X4 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I6 @ X4 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_52_list__update__swap,axiom,
! [I: nat,I6: nat,Xs: list_a,X: a,X4: a] :
( ( I != I6 )
=> ( ( list_update_a @ ( list_update_a @ Xs @ I @ X ) @ I6 @ X4 )
= ( list_update_a @ ( list_update_a @ Xs @ I6 @ X4 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_53_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_54_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_55_neq__if__length__neq,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_56_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_57_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_a,Z2: list_a] : ( Y3 = Z2 ) )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I2 )
= ( nth_a @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_58_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z2: list_nat] : ( Y3 = Z2 ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_59_sorted__wrt__nth__less,axiom,
! [P: a > a > $o,Xs: list_a,I: nat,J: nat] :
( ( sorted_wrt_a @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I ) @ ( nth_a @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_60_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_61_list__update__same__conv,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I @ X )
= Xs )
= ( ( nth_a @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_62_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_63_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_a
= ( ^ [P2: a > a > $o,Xs3: list_a] :
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs3 ) )
=> ( P2 @ ( nth_a @ Xs3 @ I2 ) @ ( nth_a @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_64_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P2: nat > nat > $o,Xs3: list_nat] :
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
=> ( P2 @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_65_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_66_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_67_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_68_order__refl,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ X ) ).
% order_refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_73_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_74_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_75_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_76_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_77_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_78_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_79_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I7: nat] :
( ( ord_less_nat @ K2 @ I7 )
=> ( P @ I7 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_80_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_81_minf_I8_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z3 )
=> ~ ( ord_less_eq_a @ T @ X5 ) ) ).
% minf(8)
thf(fact_82_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_83_minf_I8_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ~ ( ord_le2932123472753598470d_enat @ T @ X5 ) ) ).
% minf(8)
thf(fact_84_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_85_minf_I6_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z3 )
=> ( ord_less_eq_a @ X5 @ T ) ) ).
% minf(6)
thf(fact_86_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_87_minf_I6_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ord_le2932123472753598470d_enat @ X5 @ T ) ) ).
% minf(6)
thf(fact_88_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_89_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_90_order__antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_91_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_92_order__antisym__conv,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_93_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_94_linorder__le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_95_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_96_linorder__le__cases,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_97_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_101_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > extended_enat,C: extended_enat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_ord__eq__le__subst,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_115_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_116_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_117_ord__eq__le__subst,axiom,
! [A: real,F: a > real,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_ord__eq__le__subst,axiom,
! [A: extended_enat,F: a > extended_enat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_119_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_120_ord__eq__le__subst,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_121_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_122_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_123_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_124_linorder__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
| ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_linear
thf(fact_125_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_126_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_127_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_128_order__eq__refl,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X = Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_129_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_130_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_131_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_132_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_134_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_135_order__subst2,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_136_order__subst2,axiom,
! [A: a,B: a,F: a > extended_enat,C: extended_enat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_137_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_140_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: a,F: extended_enat > a,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: real,F: a > real,B: a,C: a] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_150_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_151_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_152_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: extended_enat,Z2: extended_enat] : ( Y3 = Z2 ) )
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
& ( ord_le2932123472753598470d_enat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_153_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_154_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_155_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_156_antisym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_157_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_158_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_159_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_160_dual__order_Otrans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_161_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_162_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_163_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_164_dual__order_Oantisym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_165_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_166_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_167_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_168_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: extended_enat,Z2: extended_enat] : ( Y3 = Z2 ) )
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
& ( ord_le2932123472753598470d_enat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_169_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_170_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: a,B3: a] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_171_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_172_linorder__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extended_enat,B3: extended_enat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_173_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_174_order__trans,axiom,
! [X: a,Y: a,Z4: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z4 )
=> ( ord_less_eq_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_175_order__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z4 )
=> ( ord_less_eq_real @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_176_order__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ Z4 )
=> ( ord_le2932123472753598470d_enat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_177_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_178_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_179_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_180_order_Otrans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% order.trans
thf(fact_181_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_182_order__antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_183_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_184_order__antisym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_185_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_186_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_187_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_188_ord__le__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_189_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_190_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_191_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_192_ord__eq__le__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_193_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_194_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ( ord_less_eq_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_195_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_196_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: extended_enat,Z2: extended_enat] : ( Y3 = Z2 ) )
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
& ( ord_le2932123472753598470d_enat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_197_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_198_le__cases3,axiom,
! [X: a,Y: a,Z4: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z4 ) )
=> ( ( ( ord_less_eq_a @ X @ Z4 )
=> ~ ( ord_less_eq_a @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_a @ Z4 @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z4 )
=> ~ ( ord_less_eq_a @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_a @ Z4 @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_199_le__cases3,axiom,
! [X: real,Y: real,Z4: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z4 ) )
=> ( ( ( ord_less_eq_real @ X @ Z4 )
=> ~ ( ord_less_eq_real @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z4 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z4 )
=> ~ ( ord_less_eq_real @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z4 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_200_le__cases3,axiom,
! [X: extended_enat,Y: extended_enat,Z4: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ Z4 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Z4 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ X @ Z4 )
=> ~ ( ord_le2932123472753598470d_enat @ Z4 @ Y ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Z4 @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ X ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ Z4 )
=> ~ ( ord_le2932123472753598470d_enat @ Z4 @ X ) )
=> ~ ( ( ord_le2932123472753598470d_enat @ Z4 @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_201_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_202_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_203_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_204_nle__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_205_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_206_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_207_order__less__imp__not__less,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_208_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_209_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_210_order__less__imp__not__eq2,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_211_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_212_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_213_order__less__imp__not__eq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_214_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_215_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_216_linorder__less__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
| ( X = Y )
| ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_217_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_218_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_219_order__less__imp__triv,axiom,
! [X: extended_enat,Y: extended_enat,P: $o] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_220_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_221_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_222_order__less__not__sym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_223_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_224_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_225_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_226_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_227_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_228_order__less__subst2,axiom,
! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_229_order__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_230_order__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_231_order__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_232_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_233_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_234_order__less__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_235_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_236_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_237_order__less__subst1,axiom,
! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_238_order__less__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_239_order__less__subst1,axiom,
! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_240_order__less__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_241_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_242_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_243_order__less__irrefl,axiom,
! [X: extended_enat] :
~ ( ord_le72135733267957522d_enat @ X @ X ) ).
% order_less_irrefl
thf(fact_244_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_245_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_246_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_247_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_248_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_249_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_250_ord__less__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_251_ord__less__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_252_ord__less__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_253_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_254_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_255_ord__eq__less__subst,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_256_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_257_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_258_ord__eq__less__subst,axiom,
! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_259_ord__eq__less__subst,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_260_ord__eq__less__subst,axiom,
! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_261_ord__eq__less__subst,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_262_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_263_order__less__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_264_order__less__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ Y @ Z4 )
=> ( ord_le72135733267957522d_enat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_265_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_266_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_267_order__less__asym_H,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).
% order_less_asym'
thf(fact_268_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_269_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_270_linorder__neq__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X != Y )
= ( ( ord_le72135733267957522d_enat @ X @ Y )
| ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_271_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_272_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_273_order__less__asym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% order_less_asym
thf(fact_274_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_275_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_276_linorder__neqE,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X != Y )
=> ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_277_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_278_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_279_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_280_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_281_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_282_order_Ostrict__implies__not__eq,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_283_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_284_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_285_dual__order_Ostrict__trans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_286_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_287_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_288_not__less__iff__gr__or__eq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( ( ord_le72135733267957522d_enat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_289_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_290_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_291_order_Ostrict__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_292_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_293_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_294_linorder__less__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A4: extended_enat,B3: extended_enat] :
( ( ord_le72135733267957522d_enat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extended_enat] : ( P @ A4 @ A4 )
=> ( ! [A4: extended_enat,B3: extended_enat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_295_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P2: nat > $o] :
? [N2: nat] :
( ( P2 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P2 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_296_exists__least__iff,axiom,
( ( ^ [P3: extended_enat > $o] :
? [X7: extended_enat] : ( P3 @ X7 ) )
= ( ^ [P2: extended_enat > $o] :
? [N2: extended_enat] :
( ( P2 @ N2 )
& ! [M: extended_enat] :
( ( ord_le72135733267957522d_enat @ M @ N2 )
=> ~ ( P2 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_297_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_298_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_299_dual__order_Oirrefl,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% dual_order.irrefl
thf(fact_300_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_301_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_302_dual__order_Oasym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ).
% dual_order.asym
thf(fact_303_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_304_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_305_linorder__cases,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( X != Y )
=> ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_306_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_307_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_308_antisym__conv3,axiom,
! [Y: extended_enat,X: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ Y @ X )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_309_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X6: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X6 )
=> ( P @ Y5 ) )
=> ( P @ X6 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_310_less__induct,axiom,
! [P: extended_enat > $o,A: extended_enat] :
( ! [X6: extended_enat] :
( ! [Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Y5 @ X6 )
=> ( P @ Y5 ) )
=> ( P @ X6 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_311_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_312_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_313_ord__less__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_314_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_315_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_316_ord__eq__less__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_317_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_318_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_319_order_Oasym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).
% order.asym
thf(fact_320_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_321_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_322_less__imp__neq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_323_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_324_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_325_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_326_lt__ex,axiom,
! [X: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% lt_ex
thf(fact_327_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_328_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ Z5 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_329_pinf_I1_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_330_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_331_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ Z5 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_332_pinf_I2_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z5 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_333_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_334_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_335_pinf_I3_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_336_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_337_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_338_pinf_I4_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_339_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_340_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_341_pinf_I5_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ~ ( ord_le72135733267957522d_enat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_342_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_343_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_344_pinf_I7_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ord_le72135733267957522d_enat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_345_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_346_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z5 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_347_minf_I1_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Z5 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_348_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_349_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z5 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_350_minf_I2_J,axiom,
! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: extended_enat] :
! [X6: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Z5 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_351_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_352_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_353_minf_I3_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_354_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_355_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_356_minf_I4_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_357_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_358_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_359_minf_I5_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X5 @ T ) ) ).
% minf(5)
thf(fact_360_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_361_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_362_minf_I7_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ~ ( ord_le72135733267957522d_enat @ T @ X5 ) ) ).
% minf(7)
thf(fact_363_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X6: nat] :
( ( P @ X6 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_364_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_365_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_366_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_367_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_368_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_369_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_370_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_371_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_372_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_373_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_374_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_375_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_376_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_377_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_378_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_379_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_380_order__le__imp__less__or__eq,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_381_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_382_order__le__imp__less__or__eq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_383_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_384_linorder__le__less__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_a @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_385_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_386_linorder__le__less__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
| ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_387_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_388_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_389_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_390_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_391_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_392_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > a,C: a] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_393_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_394_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_395_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_396_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_397_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_398_order__less__le__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_399_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_400_order__less__le__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_401_order__less__le__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_402_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_403_order__less__le__subst1,axiom,
! [A: real,F: a > real,B: a,C: a] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_404_order__less__le__subst1,axiom,
! [A: extended_enat,F: a > extended_enat,B: a,C: a] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_405_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_406_order__less__le__subst1,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_407_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_408_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_409_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_410_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_411_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_412_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_413_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_414_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > extended_enat,C: extended_enat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X6: a,Y4: a] :
( ( ord_less_eq_a @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_415_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_416_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_eq_real @ X6 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_417_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_418_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_419_order__le__less__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_420_order__le__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_421_order__le__less__subst1,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_422_order__le__less__subst1,axiom,
! [A: a,F: extended_enat > a,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_423_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_424_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X6: real,Y4: real] :
( ( ord_less_real @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_425_order__le__less__subst1,axiom,
! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_426_order__le__less__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_427_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_428_order__less__le__trans,axiom,
! [X: a,Y: a,Z4: a] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z4 )
=> ( ord_less_a @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_429_order__less__le__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_430_order__less__le__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ Z4 )
=> ( ord_le72135733267957522d_enat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_431_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_432_order__le__less__trans,axiom,
! [X: a,Y: a,Z4: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ Y @ Z4 )
=> ( ord_less_a @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_433_order__le__less__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_434_order__le__less__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ Y @ Z4 )
=> ( ord_le72135733267957522d_enat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_435_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_436_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_437_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_438_order__neq__le__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A != B )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_439_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_440_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_441_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_442_order__le__neq__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( A != B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_443_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_444_order__less__imp__le,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_445_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_446_order__less__imp__le,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_447_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_448_linorder__not__less,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_not_less
thf(fact_449_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_450_linorder__not__less,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_451_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_452_linorder__not__le,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X @ Y ) )
= ( ord_less_a @ Y @ X ) ) ).
% linorder_not_le
thf(fact_453_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_454_linorder__not__le,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ X @ Y ) )
= ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_455_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_456_order__less__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_457_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_458_order__less__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_459_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_460_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_461_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_462_order__le__less,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le72135733267957522d_enat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_463_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_464_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_465_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_466_dual__order_Ostrict__implies__order,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_467_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_468_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_469_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_470_order_Ostrict__implies__order,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_471_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_472_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A3: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ~ ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_473_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_474_dual__order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B2: extended_enat,A3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
& ~ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_475_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_476_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_477_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_478_dual__order_Ostrict__trans2,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_479_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_480_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_481_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_482_dual__order_Ostrict__trans1,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_483_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_484_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A3: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_485_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_486_dual__order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B2: extended_enat,A3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_487_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_488_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A3: a] :
( ( ord_less_a @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_489_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_490_dual__order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B2: extended_enat,A3: extended_enat] :
( ( ord_le72135733267957522d_enat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_491_dense__le__bounded,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z4 ) ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ).
% dense_le_bounded
thf(fact_492_dense__ge__bounded,axiom,
! [Z4: real,X: real,Y: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z4 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ).
% dense_ge_bounded
thf(fact_493_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_494_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_495_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_496_order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
& ~ ( ord_le2932123472753598470d_enat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_497_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_498_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_499_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_500_order_Ostrict__trans2,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_501_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_502_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_503_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_504_order_Ostrict__trans1,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_505_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_506_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_507_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_508_order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_509_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_510_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B2: a] :
( ( ord_less_a @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_511_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_512_order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_513_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_514_not__le__imp__less,axiom,
! [Y: a,X: a] :
( ~ ( ord_less_eq_a @ Y @ X )
=> ( ord_less_a @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_515_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_516_not__le__imp__less,axiom,
! [Y: extended_enat,X: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ord_le72135733267957522d_enat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_517_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_518_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ~ ( ord_less_eq_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_519_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_520_less__le__not__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
& ~ ( ord_le2932123472753598470d_enat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_521_dense__le,axiom,
! [Y: real,Z4: real] :
( ! [X6: real] :
( ( ord_less_real @ X6 @ Y )
=> ( ord_less_eq_real @ X6 @ Z4 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ).
% dense_le
thf(fact_522_dense__ge,axiom,
! [Z4: real,Y: real] :
( ! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ord_less_eq_real @ Y @ X6 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ).
% dense_ge
thf(fact_523_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_524_antisym__conv2,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_525_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_526_antisym__conv2,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_527_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_528_antisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_529_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_530_antisym__conv1,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_531_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_532_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_533_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_534_nless__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ A @ B ) )
= ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_535_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_536_leI,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% leI
thf(fact_537_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_538_leI,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% leI
thf(fact_539_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_540_leD,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_a @ X @ Y ) ) ).
% leD
thf(fact_541_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_542_leD,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le72135733267957522d_enat @ X @ Y ) ) ).
% leD
thf(fact_543_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_544_pinf_I6_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ Z3 @ X5 )
=> ~ ( ord_less_eq_a @ X5 @ T ) ) ).
% pinf(6)
thf(fact_545_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_546_pinf_I6_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ~ ( ord_le2932123472753598470d_enat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_547_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_548_pinf_I8_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ Z3 @ X5 )
=> ( ord_less_eq_a @ T @ X5 ) ) ).
% pinf(8)
thf(fact_549_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_550_pinf_I8_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ord_le2932123472753598470d_enat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_551_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D: nat] :
( ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ D ) )
=> ( P @ X6 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_552_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A @ X5 )
& ( ord_less_real @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D: real] :
( ! [X6: real] :
( ( ( ord_less_eq_real @ A @ X6 )
& ( ord_less_real @ X6 @ D ) )
=> ( P @ X6 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_553_complete__interval,axiom,
! [A: extended_enat,B: extended_enat,P: extended_enat > $o] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ C2 )
& ( ord_le2932123472753598470d_enat @ C2 @ B )
& ! [X5: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A @ X5 )
& ( ord_le72135733267957522d_enat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D: extended_enat] :
( ! [X6: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A @ X6 )
& ( ord_le72135733267957522d_enat @ X6 @ D ) )
=> ( P @ X6 ) )
=> ( ord_le2932123472753598470d_enat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_554_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_555_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_556_verit__comp__simplify1_I3_J,axiom,
! [B4: a,A5: a] :
( ( ~ ( ord_less_eq_a @ B4 @ A5 ) )
= ( ord_less_a @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_557_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_558_verit__comp__simplify1_I3_J,axiom,
! [B4: extended_enat,A5: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ B4 @ A5 ) )
= ( ord_le72135733267957522d_enat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_559_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_560_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ J2 ) @ ( nth_a @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_561_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs ) )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ J2 ) @ ( nth_real @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_562_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( rev_Extended_enat @ Xs ) )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ J2 ) @ ( nth_Extended_enat @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_563_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_564_sorted__rev__nth__mono,axiom,
! [Xs: list_a,I: nat,J: nat] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ J ) @ ( nth_a @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_565_sorted__rev__nth__mono,axiom,
! [Xs: list_real,I: nat,J: nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ J ) @ ( nth_real @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_566_sorted__rev__nth__mono,axiom,
! [Xs: list_Extended_enat,I: nat,J: nat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( rev_Extended_enat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ J ) @ ( nth_Extended_enat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_567_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_568_sorted__iff__nth__Suc,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_569_sorted__iff__nth__Suc,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I2 ) @ ( nth_real @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_570_sorted__iff__nth__Suc,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I2 ) @ ( nth_Extended_enat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_571_set__swap,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_572_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_573_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N @ I ) @ ( sort_map_nat @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_574_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > a] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_a @ ( sort_map_a @ F @ N @ I ) @ ( sort_map_a @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_575_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > real] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N @ I ) @ ( sort_map_real @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_576_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > extended_enat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le2932123472753598470d_enat @ ( sort_m8616768064977705053d_enat @ F @ N @ I ) @ ( sort_m8616768064977705053d_enat @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_577_list__ex__length,axiom,
( list_ex_a
= ( ^ [P2: a > $o,Xs3: list_a] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs3 ) )
& ( P2 @ ( nth_a @ Xs3 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_578_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P2: nat > $o,Xs3: list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs3 ) )
& ( P2 @ ( nth_nat @ Xs3 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_579_distinct__swap,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( distinct_a @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
= ( distinct_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_580_distinct__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_581_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I4: set_nat] :
! [X2: nat,Y2: nat] :
( ( member_nat @ Y2 @ I4 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( member_nat @ X2 @ I4 ) ) ) ) ) ).
% down_ray_def
thf(fact_582_down__ray__def,axiom,
( down_ray_a
= ( ^ [I4: set_a] :
! [X2: a,Y2: a] :
( ( member_a @ Y2 @ I4 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
=> ( member_a @ X2 @ I4 ) ) ) ) ) ).
% down_ray_def
thf(fact_583_down__ray__def,axiom,
( down_ray_real
= ( ^ [I4: set_real] :
! [X2: real,Y2: real] :
( ( member_real @ Y2 @ I4 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( member_real @ X2 @ I4 ) ) ) ) ) ).
% down_ray_def
thf(fact_584_down__ray__def,axiom,
( down_r2452194630239499347d_enat
= ( ^ [I4: set_Extended_enat] :
! [X2: extended_enat,Y2: extended_enat] :
( ( member_Extended_enat @ Y2 @ I4 )
=> ( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
=> ( member_Extended_enat @ X2 @ I4 ) ) ) ) ) ).
% down_ray_def
thf(fact_585_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_586_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_587_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_588_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_589_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_590_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_591_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_592_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_593_subset__code_I1_J,axiom,
! [Xs: list_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_594_subset__code_I1_J,axiom,
! [Xs: list_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_595_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_596_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_597_list__ex__cong,axiom,
! [Xs: list_a,Ys2: list_a,F: a > $o,G: a > $o] :
( ( Xs = Ys2 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Ys2 ) )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( list_ex_a @ F @ Xs )
= ( list_ex_a @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_598_list__ex__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > $o,G: nat > $o] :
( ( Xs = Ys2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( list_ex_nat @ F @ Xs )
= ( list_ex_nat @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_599_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys2 )
=> ( ( distinct_nat @ Ys2 )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_600_sorted__distinct__set__unique,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
=> ( ( distinct_a @ Xs )
=> ( ( sorted_wrt_a @ ord_less_eq_a @ Ys2 )
=> ( ( distinct_a @ Ys2 )
=> ( ( ( set_a2 @ Xs )
= ( set_a2 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_601_sorted__distinct__set__unique,axiom,
! [Xs: list_real,Ys2: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( ( distinct_real @ Xs )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Ys2 )
=> ( ( distinct_real @ Ys2 )
=> ( ( ( set_real2 @ Xs )
= ( set_real2 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_602_sorted__distinct__set__unique,axiom,
! [Xs: list_Extended_enat,Ys2: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( ( distin4523846830085650399d_enat @ Xs )
=> ( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Ys2 )
=> ( ( distin4523846830085650399d_enat @ Ys2 )
=> ( ( ( set_Extended_enat2 @ Xs )
= ( set_Extended_enat2 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_603_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_604_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_605_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_606_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_607_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_608_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_609_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_610_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_611_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_612_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X6: nat] : ( R @ X6 @ X6 )
=> ( ! [X6: nat,Y4: nat,Z3: nat] :
( ( R @ X6 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X6 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_613_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_614_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_615_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_616_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_617_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_618_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_619_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_620_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_621_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_622_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_623_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_624_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_625_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_626_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_627_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_628_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_629_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_630_sorted__wrt__mono__rel,axiom,
! [Xs: list_a,P: a > a > $o,Q: a > a > $o] :
( ! [X6: a,Y4: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y4 @ ( set_a2 @ Xs ) )
=> ( ( P @ X6 @ Y4 )
=> ( Q @ X6 @ Y4 ) ) ) )
=> ( ( sorted_wrt_a @ P @ Xs )
=> ( sorted_wrt_a @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_631_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X6: nat,Y4: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y4 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X6 @ Y4 )
=> ( Q @ X6 @ Y4 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_632_set__update__subsetI,axiom,
! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_633_set__update__subsetI,axiom,
! [Xs: list_a,A2: set_a,X: a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_634_distinct__Ex1,axiom,
! [Xs: list_a,X: a] :
( ( distinct_a @ Xs )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [X6: nat] :
( ( ord_less_nat @ X6 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ X6 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ Y5 )
= X ) )
=> ( Y5 = X6 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_635_distinct__Ex1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [X6: nat] :
( ( ord_less_nat @ X6 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X6 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y5 )
= X ) )
=> ( Y5 = X6 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_636_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_637_lift__Suc__mono__le,axiom,
! [F: nat > a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_638_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_639_lift__Suc__mono__le,axiom,
! [F: nat > extended_enat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le2932123472753598470d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_640_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_641_lift__Suc__antimono__le,axiom,
! [F: nat > a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_642_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_643_lift__Suc__antimono__le,axiom,
! [F: nat > extended_enat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le2932123472753598470d_enat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_644_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_645_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_646_lift__Suc__mono__less,axiom,
! [F: nat > extended_enat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_le72135733267957522d_enat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_647_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_648_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_649_lift__Suc__mono__less__iff,axiom,
! [F: nat > extended_enat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_le72135733267957522d_enat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_650_strict__sorted__equal,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( sorted_wrt_a @ ord_less_a @ Xs )
=> ( ( sorted_wrt_a @ ord_less_a @ Ys2 )
=> ( ( ( set_a2 @ Ys2 )
= ( set_a2 @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_651_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys2 )
=> ( ( ( set_nat2 @ Ys2 )
= ( set_nat2 @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_652_strict__sorted__equal,axiom,
! [Xs: list_real,Ys2: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
=> ( ( sorted_wrt_real @ ord_less_real @ Ys2 )
=> ( ( ( set_real2 @ Ys2 )
= ( set_real2 @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_653_strict__sorted__equal,axiom,
! [Xs: list_Extended_enat,Ys2: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ Xs )
=> ( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ Ys2 )
=> ( ( ( set_Extended_enat2 @ Ys2 )
= ( set_Extended_enat2 @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_654_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_655_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_656_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_657_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_658_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_659_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_660_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_661_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_662_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_663_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I2 ) ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_664_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ ( suc @ I2 ) ) @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_665_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( rev_real @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ ( suc @ I2 ) ) @ ( nth_real @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_666_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( rev_Extended_enat @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ ( suc @ I2 ) ) @ ( nth_Extended_enat @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_667_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_668_strict__sorted__iff,axiom,
! [L: list_a] :
( ( sorted_wrt_a @ ord_less_a @ L )
= ( ( sorted_wrt_a @ ord_less_eq_a @ L )
& ( distinct_a @ L ) ) ) ).
% strict_sorted_iff
thf(fact_669_strict__sorted__iff,axiom,
! [L: list_real] :
( ( sorted_wrt_real @ ord_less_real @ L )
= ( ( sorted_wrt_real @ ord_less_eq_real @ L )
& ( distinct_real @ L ) ) ) ).
% strict_sorted_iff
thf(fact_670_strict__sorted__iff,axiom,
! [L: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ L )
= ( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ L )
& ( distin4523846830085650399d_enat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_671_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs3: list_a] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs3 ) )
=> ( ( I2 != J2 )
=> ( ( nth_a @ Xs3 @ I2 )
!= ( nth_a @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_672_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I2 != J2 )
=> ( ( nth_nat @ Xs3 @ I2 )
!= ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_673_nth__eq__iff__index__eq,axiom,
! [Xs: list_a,I: nat,J: nat] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ Xs @ I )
= ( nth_a @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_674_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_675_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_676_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_677_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_678_verit__comp__simplify1_I2_J,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_679_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_680_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_681_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_682_verit__la__disequality,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = B )
| ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ~ ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_683_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_684_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_685_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
=> ( P @ X6 ) )
=> ( P @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_686_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
=> ( P @ X6 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_687_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_688_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_689_all__nth__imp__all__set,axiom,
! [Xs: list_a,P: a > $o,X: a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I3 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_690_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_691_all__set__conv__all__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_692_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_693_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_694_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_695_verit__comp__simplify1_I1_J,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_696_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_697_set__update__memI,axiom,
! [N: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_698_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_699_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N @ I ) @ ( sort_map_nat @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_700_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > a] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_a @ ( sort_map_a @ F @ N @ I ) @ ( sort_map_a @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_701_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > real] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N @ I ) @ ( sort_map_real @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_702_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > extended_enat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_le2932123472753598470d_enat @ ( sort_m8616768064977705053d_enat @ F @ N @ I ) @ ( sort_m8616768064977705053d_enat @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_703_rev__nth,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rev_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_704_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_705_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_706_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
=> ( ( distinct_a @ Xs )
=> ( ( linord3083462915744475214_set_a @ ( set_a2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_707_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( ( distinct_real @ Xs )
=> ( ( linord4252657396651189596t_real @ ( set_real2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_708_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( ( distin4523846830085650399d_enat @ Xs )
=> ( ( linord1591021928418041270d_enat @ ( set_Extended_enat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_709_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I4: set_nat] :
! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ I4 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( member_nat @ Y2 @ I4 ) ) ) ) ) ).
% up_ray_def
thf(fact_710_up__ray__def,axiom,
( up_ray_a
= ( ^ [I4: set_a] :
! [X2: a,Y2: a] :
( ( member_a @ X2 @ I4 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
=> ( member_a @ Y2 @ I4 ) ) ) ) ) ).
% up_ray_def
thf(fact_711_up__ray__def,axiom,
( up_ray_real
= ( ^ [I4: set_real] :
! [X2: real,Y2: real] :
( ( member_real @ X2 @ I4 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( member_real @ Y2 @ I4 ) ) ) ) ) ).
% up_ray_def
thf(fact_712_up__ray__def,axiom,
( up_ray_Extended_enat
= ( ^ [I4: set_Extended_enat] :
! [X2: extended_enat,Y2: extended_enat] :
( ( member_Extended_enat @ X2 @ I4 )
=> ( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
=> ( member_Extended_enat @ Y2 @ I4 ) ) ) ) ) ).
% up_ray_def
thf(fact_713_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_714_sorted01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( sorted_wrt_a @ ord_less_eq_a @ Xs ) ) ).
% sorted01
thf(fact_715_sorted01,axiom,
! [Xs: list_real] :
( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ one_one_nat )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).
% sorted01
thf(fact_716_sorted01,axiom,
! [Xs: list_Extended_enat] :
( ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs ) @ one_one_nat )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs ) ) ).
% sorted01
thf(fact_717_finite__sorted__distinct__unique,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [X6: list_nat] :
( ( ( set_nat2 @ X6 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X6 )
& ( distinct_nat @ X6 )
& ! [Y5: list_nat] :
( ( ( ( set_nat2 @ Y5 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
& ( distinct_nat @ Y5 ) )
=> ( Y5 = X6 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_718_finite__sorted__distinct__unique,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [X6: list_a] :
( ( ( set_a2 @ X6 )
= A2 )
& ( sorted_wrt_a @ ord_less_eq_a @ X6 )
& ( distinct_a @ X6 )
& ! [Y5: list_a] :
( ( ( ( set_a2 @ Y5 )
= A2 )
& ( sorted_wrt_a @ ord_less_eq_a @ Y5 )
& ( distinct_a @ Y5 ) )
=> ( Y5 = X6 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_719_finite__sorted__distinct__unique,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ? [X6: list_real] :
( ( ( set_real2 @ X6 )
= A2 )
& ( sorted_wrt_real @ ord_less_eq_real @ X6 )
& ( distinct_real @ X6 )
& ! [Y5: list_real] :
( ( ( ( set_real2 @ Y5 )
= A2 )
& ( sorted_wrt_real @ ord_less_eq_real @ Y5 )
& ( distinct_real @ Y5 ) )
=> ( Y5 = X6 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_720_finite__sorted__distinct__unique,axiom,
! [A2: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A2 )
=> ? [X6: list_Extended_enat] :
( ( ( set_Extended_enat2 @ X6 )
= A2 )
& ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ X6 )
& ( distin4523846830085650399d_enat @ X6 )
& ! [Y5: list_Extended_enat] :
( ( ( ( set_Extended_enat2 @ Y5 )
= A2 )
& ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Y5 )
& ( distin4523846830085650399d_enat @ Y5 ) )
=> ( Y5 = X6 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_721_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) )
=> ( ( nth_a @ ( remdups_adj_a @ Xs ) @ I )
!= ( nth_a @ ( remdups_adj_a @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_722_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_723_nth__tl,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( tl_a @ Xs ) ) )
=> ( ( nth_a @ ( tl_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_724_nth__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_725_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_726_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_727_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_728_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_729_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_730_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_731_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_732_length__tl,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_733_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_734_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B5 ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B5 )
=> ( A2 = B5 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_735_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_736_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_737_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_738_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_739_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_740_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_741_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_742_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_743_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_744_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_745_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_746_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_747_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_748_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_749_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_750_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_751_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_752_sorted__tl,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( tl_nat @ Xs ) ) ) ).
% sorted_tl
thf(fact_753_sorted__tl,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
=> ( sorted_wrt_a @ ord_less_eq_a @ ( tl_a @ Xs ) ) ) ).
% sorted_tl
thf(fact_754_sorted__tl,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ ( tl_real @ Xs ) ) ) ).
% sorted_tl
thf(fact_755_sorted__tl,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( tl_Extended_enat @ Xs ) ) ) ).
% sorted_tl
thf(fact_756_rev__update,axiom,
! [K: nat,Xs: list_a,Y: a] :
( ( ord_less_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( rev_a @ ( list_update_a @ Xs @ K @ Y ) )
= ( list_update_a @ ( rev_a @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_757_rev__update,axiom,
! [K: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_758_remdups__adj__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% remdups_adj_length
thf(fact_759_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_760_finite__maxlen,axiom,
! [M7: set_list_a] :
( ( finite_finite_list_a @ M7 )
=> ? [N3: nat] :
! [X5: list_a] :
( ( member_list_a @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_a @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_761_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N3: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_762_sorted__remdups__adj,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_adj_nat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_763_sorted__remdups__adj,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
=> ( sorted_wrt_a @ ord_less_eq_a @ ( remdups_adj_a @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_764_sorted__remdups__adj,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ ( remdups_adj_real @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_765_sorted__remdups__adj,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( remdup6152102037098707618d_enat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_766_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A2 )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_767_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_768_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A2: set_a] : ( sorted_wrt_a @ ord_less_eq_a @ ( linord3083462915744475214_set_a @ A2 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_769_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A2: set_real] : ( sorted_wrt_real @ ord_less_eq_real @ ( linord4252657396651189596t_real @ A2 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_770_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A2: set_Extended_enat] : ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( linord1591021928418041270d_enat @ A2 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_771_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A2: set_a] : ( sorted_wrt_a @ ord_less_a @ ( linord3083462915744475214_set_a @ A2 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_772_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_773_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A2: set_real] : ( sorted_wrt_real @ ord_less_real @ ( linord4252657396651189596t_real @ A2 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_774_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A2: set_Extended_enat] : ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ ( linord1591021928418041270d_enat @ A2 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_775_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_776_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_777_subsetI,axiom,
! [A2: set_a,B5: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ( member_a @ X6 @ B5 ) )
=> ( ord_less_eq_set_a @ A2 @ B5 ) ) ).
% subsetI
thf(fact_778_subsetI,axiom,
! [A2: set_nat,B5: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( member_nat @ X6 @ B5 ) )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% subsetI
thf(fact_779_finite__Diff2,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_780_finite__Diff,axiom,
! [A2: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).
% finite_Diff
thf(fact_781_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_782_psubset__imp__ex__mem,axiom,
! [A2: set_a,B5: set_a] :
( ( ord_less_set_a @ A2 @ B5 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B5 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_783_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_784_psubsetD,axiom,
! [A2: set_a,B5: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B5 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_785_psubsetD,axiom,
! [A2: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_786_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_787_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_788_one__reorient,axiom,
! [X: extended_enat] :
( ( one_on7984719198319812577d_enat = X )
= ( X = one_on7984719198319812577d_enat ) ) ).
% one_reorient
thf(fact_789_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_790_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_791_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_792_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A6 )
=> ( member_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_793_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_794_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_795_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_796_subsetD,axiom,
! [A2: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_797_subsetD,axiom,
! [A2: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_798_in__mono,axiom,
! [A2: set_a,B5: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_799_in__mono,axiom,
! [A2: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_800_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_801_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_802_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_803_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_804_diff__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D2 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_805_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_806_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_807_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_808_diff__strict__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D2 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_809_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_nat @ X6 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_810_finite__has__minimal2,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_less_eq_a @ X6 @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A2 )
=> ( ( ord_less_eq_a @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_811_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X6: real] :
( ( member_real @ X6 @ A2 )
& ( ord_less_eq_real @ X6 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_812_finite__has__minimal2,axiom,
! [A2: set_Extended_enat,A: extended_enat] :
( ( finite4001608067531595151d_enat @ A2 )
=> ( ( member_Extended_enat @ A @ A2 )
=> ? [X6: extended_enat] :
( ( member_Extended_enat @ X6 @ A2 )
& ( ord_le2932123472753598470d_enat @ X6 @ A )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_813_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_nat @ A @ X6 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_814_finite__has__maximal2,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_less_eq_a @ A @ X6 )
& ! [Xa: a] :
( ( member_a @ Xa @ A2 )
=> ( ( ord_less_eq_a @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_815_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X6: real] :
( ( member_real @ X6 @ A2 )
& ( ord_less_eq_real @ A @ X6 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_816_finite__has__maximal2,axiom,
! [A2: set_Extended_enat,A: extended_enat] :
( ( finite4001608067531595151d_enat @ A2 )
=> ( ( member_Extended_enat @ A @ A2 )
=> ? [X6: extended_enat] :
( ( member_Extended_enat @ X6 @ A2 )
& ( ord_le2932123472753598470d_enat @ A @ X6 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_817_rev__finite__subset,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_818_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_819_finite__subset,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_820_finite__indexed__bound,axiom,
! [A2: set_a,P: a > nat > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ? [X_12: nat] : ( P @ X6 @ X_12 ) )
=> ? [M5: nat] :
! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_821_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ? [X_12: nat] : ( P @ X6 @ X_12 ) )
=> ? [M5: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_822_finite__indexed__bound,axiom,
! [A2: set_a,P: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ? [X_12: a] : ( P @ X6 @ X_12 ) )
=> ? [M5: a] :
! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [K2: a] :
( ( ord_less_eq_a @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_823_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > a > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ? [X_12: a] : ( P @ X6 @ X_12 ) )
=> ? [M5: a] :
! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ? [K2: a] :
( ( ord_less_eq_a @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_824_finite__indexed__bound,axiom,
! [A2: set_a,P: a > real > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ? [X_12: real] : ( P @ X6 @ X_12 ) )
=> ? [M5: real] :
! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_825_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ? [X_12: real] : ( P @ X6 @ X_12 ) )
=> ? [M5: real] :
! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_826_finite__indexed__bound,axiom,
! [A2: set_a,P: a > extended_enat > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ? [X_12: extended_enat] : ( P @ X6 @ X_12 ) )
=> ? [M5: extended_enat] :
! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [K2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_827_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > extended_enat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ? [X_12: extended_enat] : ( P @ X6 @ X_12 ) )
=> ? [M5: extended_enat] :
! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ? [K2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ K2 @ M5 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_828_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_829_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_830_less__numeral__extra_I4_J,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).
% less_numeral_extra(4)
thf(fact_831_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_832_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_833_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_834_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A2: set_a,L: list_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( sorted_wrt_a @ ord_less_a @ L )
& ( ( set_a2 @ L )
= A2 )
& ( ( size_size_list_a @ L )
= ( finite_card_a @ A2 ) ) )
= ( ( linord3083462915744475214_set_a @ A2 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_835_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A2: set_nat,L: list_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A2 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A2 ) ) )
= ( ( linord2614967742042102400et_nat @ A2 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_836_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A2: set_real,L: list_real] :
( ( finite_finite_real @ A2 )
=> ( ( ( sorted_wrt_real @ ord_less_real @ L )
& ( ( set_real2 @ L )
= A2 )
& ( ( size_size_list_real @ L )
= ( finite_card_real @ A2 ) ) )
= ( ( linord4252657396651189596t_real @ A2 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_837_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A2: set_Extended_enat,L: list_Extended_enat] :
( ( finite4001608067531595151d_enat @ A2 )
=> ( ( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ L )
& ( ( set_Extended_enat2 @ L )
= A2 )
& ( ( size_s3941691890525107288d_enat @ L )
= ( finite121521170596916366d_enat @ A2 ) ) )
= ( ( linord1591021928418041270d_enat @ A2 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_838_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_839_Diff__iff,axiom,
! [C: a,A2: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B5 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_840_Diff__iff,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_841_DiffI,axiom,
! [C: a,A2: set_a,B5: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_842_DiffI,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_843_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_844_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_845_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_846_not__gr__zero,axiom,
! [N: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_847_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_848_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_849_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_850_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_851_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_852_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_853_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_854_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_855_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_856_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_857_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_858_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_859_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_860_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_861_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_862_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_863_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_864_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_865_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_866_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_867_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_868_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_869_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_870_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_871_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A2: set_a] :
( ( size_size_list_a @ ( linord3083462915744475214_set_a @ A2 ) )
= ( finite_card_a @ A2 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_872_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A2 ) )
= ( finite_card_nat @ A2 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_873_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_874_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_875_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_876_less__numeral__extra_I3_J,axiom,
~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).
% less_numeral_extra(3)
thf(fact_877_DiffD2,axiom,
! [C: a,A2: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B5 ) )
=> ~ ( member_a @ C @ B5 ) ) ).
% DiffD2
thf(fact_878_DiffD2,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ~ ( member_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_879_DiffD1,axiom,
! [C: a,A2: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B5 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_880_DiffD1,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_881_DiffE,axiom,
! [C: a,A2: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B5 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B5 ) ) ) ).
% DiffE
thf(fact_882_DiffE,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_883_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_884_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_885_zero__reorient,axiom,
! [X: extended_enat] :
( ( zero_z5237406670263579293d_enat = X )
= ( X = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_886_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_887_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_888_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_889_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_890_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_891_zero__le,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X ) ).
% zero_le
thf(fact_892_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_893_zero__less__iff__neq__zero,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% zero_less_iff_neq_zero
thf(fact_894_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_895_gr__implies__not__zero,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N )
=> ( N != zero_z5237406670263579293d_enat ) ) ).
% gr_implies_not_zero
thf(fact_896_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_897_not__less__zero,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_less_zero
thf(fact_898_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_899_gr__zeroI,axiom,
! [N: extended_enat] :
( ( N != zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ).
% gr_zeroI
thf(fact_900_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_901_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_902_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_903_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_904_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_905_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_906_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_907_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X6: nat] : ( P @ X6 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X6: nat,Y4: nat] :
( ( P @ X6 @ Y4 )
=> ( P @ ( suc @ X6 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_908_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_909_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_910_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_911_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_912_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_913_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_914_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_915_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_916_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_917_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_918_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_919_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_920_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_921_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_922_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_923_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_924_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_925_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_926_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_927_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_928_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_929_less__numeral__extra_I1_J,axiom,
ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% less_numeral_extra(1)
thf(fact_930_card__le__Suc0__iff__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_931_card__subset__eq,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B5 ) )
=> ( A2 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_932_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N )
& ( ord_less_eq_set_nat @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_933_card__le__sym__Diff,axiom,
! [A2: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_934_card__less__sym__Diff,axiom,
! [A2: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_935_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_936_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_937_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_938_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_939_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_940_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_941_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_942_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I7: nat] :
( ( ord_less_nat @ I7 @ K2 )
=> ~ ( P @ I7 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_943_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_944_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_945_card__mono,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ) ).
% card_mono
thf(fact_946_card__seteq,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B5 ) ) ) ) ).
% card_seteq
thf(fact_947_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C3 )
& ( ( finite_card_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_948_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_949_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C3: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_950_card__Diff__subset,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B5 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_951_diff__card__le__card__Diff,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_952_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_953_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_954_psubset__card__mono,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_set_nat @ A2 @ B5 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_955_distinct__card,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ) ).
% distinct_card
thf(fact_956_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_957_card__distinct,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) )
=> ( distinct_a @ Xs ) ) ).
% card_distinct
thf(fact_958_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_959_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I7: nat] :
( ( ord_less_eq_nat @ I7 @ K2 )
=> ~ ( P @ I7 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_960_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_961_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_962_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_963_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_964_card__psubset,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) )
=> ( ord_less_set_nat @ A2 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_965_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_966_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_967_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ~ ! [L2: list_a] :
( ( sorted_wrt_a @ ord_less_a @ L2 )
=> ( ( ( set_a2 @ L2 )
= A2 )
=> ( ( size_size_list_a @ L2 )
!= ( finite_card_a @ A2 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_968_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A2 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A2 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_969_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ~ ! [L2: list_real] :
( ( sorted_wrt_real @ ord_less_real @ L2 )
=> ( ( ( set_real2 @ L2 )
= A2 )
=> ( ( size_size_list_real @ L2 )
!= ( finite_card_real @ A2 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_970_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A2: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A2 )
=> ~ ! [L2: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ L2 )
=> ( ( ( set_Extended_enat2 @ L2 )
= A2 )
=> ( ( size_s3941691890525107288d_enat @ L2 )
!= ( finite121521170596916366d_enat @ A2 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_971_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A2: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B9: a] :
( ( member_a @ B9 @ B5 )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_972_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A2: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B9: a] :
( ( member_a @ B9 @ B5 )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_973_card__le__if__inj__on__rel,axiom,
! [B5: set_nat,A2: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B5 )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_974_card__le__if__inj__on__rel,axiom,
! [B5: set_nat,A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B5 )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_975_ex__card,axiom,
! [N: nat,A2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A2 ) )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ A2 )
& ( ( finite_card_nat @ S3 )
= N ) ) ) ).
% ex_card
thf(fact_976_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_977_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_978_not__one__less__zero,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_less_zero
thf(fact_979_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_980_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_981_zero__less__one,axiom,
ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one
thf(fact_982_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B2 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_983_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_984_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_985_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_986_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_987_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ N6 )
=> ( ord_less_nat @ X6 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_988_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_989_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_990_zero__neq__one,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_neq_one
thf(fact_991_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_992_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_993_not__one__le__zero,axiom,
~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_le_zero
thf(fact_994_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_995_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_996_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_997_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_998_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_999_zero__less__one__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one_class.zero_le_one
thf(fact_1000_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X6: nat > real] :
( ( P @ X6 )
=> ( P @ ( F @ X6 ) ) )
=> ( ! [X6: nat > real] :
( ( P @ X6 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X6 @ I3 ) )
& ( ord_less_eq_real @ ( X6 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I7: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I7 ) @ one_one_nat )
& ! [X5: nat > real,I7: nat] :
( ( ( P @ X5 )
& ( Q @ I7 )
& ( ( X5 @ I7 )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I7 )
= zero_zero_nat ) )
& ! [X5: nat > real,I7: nat] :
( ( ( P @ X5 )
& ( Q @ I7 )
& ( ( X5 @ I7 )
= one_one_real ) )
=> ( ( L2 @ X5 @ I7 )
= one_one_nat ) )
& ! [X5: nat > real,I7: nat] :
( ( ( P @ X5 )
& ( Q @ I7 )
& ( ( L2 @ X5 @ I7 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I7 ) @ ( F @ X5 @ I7 ) ) )
& ! [X5: nat > real,I7: nat] :
( ( ( P @ X5 )
& ( Q @ I7 )
& ( ( L2 @ X5 @ I7 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I7 ) @ ( X5 @ I7 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1001_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1002_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1003_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_1004_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N7: nat] :
( ( ord_less_nat @ M5 @ N7 )
& ( member_nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1005_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_nat @ M @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1006_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1007_seq__mono__lemma,axiom,
! [M2: nat,D2: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M2 ) ) )
=> ! [N7: nat] :
( ( ord_less_eq_nat @ M2 @ N7 )
=> ( ord_less_real @ ( D2 @ N7 ) @ ( E2 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1008_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B3: real,C2: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C2 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( P @ A4 @ C2 ) ) ) ) )
=> ( ! [X6: real] :
( ( ord_less_eq_real @ A @ X6 )
=> ( ( ord_less_eq_real @ X6 @ B )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X6 )
& ( ord_less_eq_real @ X6 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1009_bgauge__existence__lemma,axiom,
! [S: set_a,Q3: real > a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ S )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q3 @ D3 @ X2 ) ) ) )
= ( ! [X2: a] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_a @ X2 @ S )
=> ( Q3 @ D3 @ X2 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_1010_bgauge__existence__lemma,axiom,
! [S: set_nat,Q3: real > nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q3 @ D3 @ X2 ) ) ) )
= ( ! [X2: nat] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_nat @ X2 @ S )
=> ( Q3 @ D3 @ X2 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_1011_nth__Cons__pos,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1012_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1013_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1014_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1015_finite__enumerate__mono__iff,axiom,
! [S2: set_Extended_enat,M2: nat,N: nat] :
( ( finite4001608067531595151d_enat @ S2 )
=> ( ( ord_less_nat @ M2 @ ( finite121521170596916366d_enat @ S2 ) )
=> ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S2 ) )
=> ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_1016_finite__enumerate__mono__iff,axiom,
! [S2: set_nat,M2: nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ M2 @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_1017_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1018_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1019_insertCI,axiom,
! [A: a,B5: set_a,B: a] :
( ( ~ ( member_a @ A @ B5 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_1020_insertCI,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_1021_insert__subset,axiom,
! [X: a,A2: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B5 )
= ( ( member_a @ X @ B5 )
& ( ord_less_eq_set_a @ A2 @ B5 ) ) ) ).
% insert_subset
thf(fact_1022_insert__subset,axiom,
! [X: nat,A2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
= ( ( member_nat @ X @ B5 )
& ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).
% insert_subset
thf(fact_1023_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_1024_Diff__insert0,axiom,
! [X: a,A2: set_a,B5: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B5 ) )
= ( minus_minus_set_a @ A2 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1025_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B5: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
= ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1026_insert__Diff1,axiom,
! [X: a,B5: set_a,A2: set_a] :
( ( member_a @ X @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B5 )
= ( minus_minus_set_a @ A2 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1027_insert__Diff1,axiom,
! [X: nat,B5: set_nat,A2: set_nat] :
( ( member_nat @ X @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
= ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1028_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B5: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_1029_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1030_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1031_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1032_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1033_card__insert__disjoint,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1034_card__insert__disjoint,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1035_enumerate__mono__le__iff,axiom,
! [S2: set_Extended_enat,M2: nat,N: nat] :
( ~ ( finite4001608067531595151d_enat @ S2 )
=> ( ( ord_le2932123472753598470d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% enumerate_mono_le_iff
thf(fact_1036_enumerate__mono__le__iff,axiom,
! [S2: set_nat,M2: nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% enumerate_mono_le_iff
thf(fact_1037_enumerate__mono__iff,axiom,
! [S2: set_Extended_enat,M2: nat,N: nat] :
( ~ ( finite4001608067531595151d_enat @ S2 )
=> ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% enumerate_mono_iff
thf(fact_1038_enumerate__mono__iff,axiom,
! [S2: set_nat,M2: nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% enumerate_mono_iff
thf(fact_1039_card__Diff__insert,axiom,
! [A: a,A2: set_a,B5: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B5 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B5 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1040_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B5 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1041_insert__Diff__if,axiom,
! [X: a,B5: set_a,A2: set_a] :
( ( ( member_a @ X @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B5 )
= ( minus_minus_set_a @ A2 @ B5 ) ) )
& ( ~ ( member_a @ X @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B5 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1042_insert__Diff__if,axiom,
! [X: nat,B5: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
= ( minus_minus_set_nat @ A2 @ B5 ) ) )
& ( ~ ( member_nat @ X @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1043_subset__insert,axiom,
! [X: a,A2: set_a,B5: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B5 ) )
= ( ord_less_eq_set_a @ A2 @ B5 ) ) ) ).
% subset_insert
thf(fact_1044_subset__insert,axiom,
! [X: nat,A2: set_nat,B5: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
= ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).
% subset_insert
thf(fact_1045_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1046_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B8: set_a] :
( ( A2
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_1047_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B8: set_nat] :
( ( A2
= ( insert_nat @ A @ B8 ) )
& ~ ( member_nat @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_1048_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B5: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B5 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B5 ) )
= ( ( ( A = B )
=> ( A2 = B5 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B5
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1049_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B5: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B5 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A2 = B5 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B5
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1050_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1051_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1052_insert__ident,axiom,
! [X: a,A2: set_a,B5: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B5 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B5 ) )
= ( A2 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1053_insert__ident,axiom,
! [X: nat,A2: set_nat,B5: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B5 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B5 ) )
= ( A2 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1054_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B8: set_a] :
( ( A2
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_1055_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B8: set_nat] :
( ( A2
= ( insert_nat @ X @ B8 ) )
=> ( member_nat @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_1056_insertI2,axiom,
! [A: a,B5: set_a,B: a] :
( ( member_a @ A @ B5 )
=> ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).
% insertI2
thf(fact_1057_insertI2,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( member_nat @ A @ B5 )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_1058_insertI1,axiom,
! [A: a,B5: set_a] : ( member_a @ A @ ( insert_a @ A @ B5 ) ) ).
% insertI1
thf(fact_1059_insertI1,axiom,
! [A: nat,B5: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_1060_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_1061_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_1062_list_Oset__intros_I2_J,axiom,
! [Y: a,X222: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X222 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1063_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1064_list_Oset__intros_I1_J,axiom,
! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1065_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1066_list_Oset__cases,axiom,
! [E2: a,A: list_a] :
( ( member_a @ E2 @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E2 @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1067_list_Oset__cases,axiom,
! [E2: nat,A: list_nat] :
( ( member_nat @ E2 @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1068_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1069_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1070_enumerate__Ex,axiom,
! [S2: set_nat,S: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( member_nat @ S @ S2 )
=> ? [N3: nat] :
( ( infini8530281810654367211te_nat @ S2 @ N3 )
= S ) ) ) ).
% enumerate_Ex
thf(fact_1071_enumerate__in__set,axiom,
! [S2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ S2 ) ) ).
% enumerate_in_set
thf(fact_1072_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y2: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1073_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1074_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y2: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1075_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1076_impossible__Cons,axiom,
! [Xs: list_a,Ys2: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys2 ) )
=> ( Xs
!= ( cons_a @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1077_impossible__Cons,axiom,
! [Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs
!= ( cons_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1078_subset__Diff__insert,axiom,
! [A2: set_a,B5: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B5 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B5 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1079_subset__Diff__insert,axiom,
! [A2: set_nat,B5: set_nat,X: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B5 @ C3 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1080_sorted2,axiom,
! [X: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1081_sorted2,axiom,
! [X: a,Y: a,Zs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( cons_a @ X @ ( cons_a @ Y @ Zs ) ) )
= ( ( ord_less_eq_a @ X @ Y )
& ( sorted_wrt_a @ ord_less_eq_a @ ( cons_a @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1082_sorted2,axiom,
! [X: real,Y: real,Zs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( cons_real @ X @ ( cons_real @ Y @ Zs ) ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( sorted_wrt_real @ ord_less_eq_real @ ( cons_real @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1083_sorted2,axiom,
! [X: extended_enat,Y: extended_enat,Zs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( cons_Extended_enat @ X @ ( cons_Extended_enat @ Y @ Zs ) ) )
= ( ( ord_le2932123472753598470d_enat @ X @ Y )
& ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( cons_Extended_enat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1084_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1085_distinct_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X @ Xs ) )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1086_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1087_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1088_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1089_list__update__code_I3_J,axiom,
! [X: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1090_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1091_le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_1092_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X2: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1093_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1094_card__Suc__eq__finite,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B2: a,B6: set_a] :
( ( A2
= ( insert_a @ B2 @ B6 ) )
& ~ ( member_a @ B2 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1095_card__Suc__eq__finite,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
= ( ? [B2: nat,B6: set_nat] :
( ( A2
= ( insert_nat @ B2 @ B6 ) )
& ~ ( member_nat @ B2 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1096_card__insert__if,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( finite_card_a @ A2 ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1097_card__insert__if,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( finite_card_nat @ A2 ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1098_sorted__simps_I2_J,axiom,
! [X: nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) )
=> ( ord_less_eq_nat @ X @ X2 ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys2 ) ) ) ).
% sorted_simps(2)
thf(fact_1099_sorted__simps_I2_J,axiom,
! [X: a,Ys2: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( cons_a @ X @ Ys2 ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys2 ) )
=> ( ord_less_eq_a @ X @ X2 ) )
& ( sorted_wrt_a @ ord_less_eq_a @ Ys2 ) ) ) ).
% sorted_simps(2)
thf(fact_1100_sorted__simps_I2_J,axiom,
! [X: real,Ys2: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( cons_real @ X @ Ys2 ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ ( set_real2 @ Ys2 ) )
=> ( ord_less_eq_real @ X @ X2 ) )
& ( sorted_wrt_real @ ord_less_eq_real @ Ys2 ) ) ) ).
% sorted_simps(2)
thf(fact_1101_sorted__simps_I2_J,axiom,
! [X: extended_enat,Ys2: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( cons_Extended_enat @ X @ Ys2 ) )
= ( ! [X2: extended_enat] :
( ( member_Extended_enat @ X2 @ ( set_Extended_enat2 @ Ys2 ) )
=> ( ord_le2932123472753598470d_enat @ X @ X2 ) )
& ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Ys2 ) ) ) ).
% sorted_simps(2)
thf(fact_1102_strict__sorted__simps_I2_J,axiom,
! [X: a,Ys2: list_a] :
( ( sorted_wrt_a @ ord_less_a @ ( cons_a @ X @ Ys2 ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys2 ) )
=> ( ord_less_a @ X @ X2 ) )
& ( sorted_wrt_a @ ord_less_a @ Ys2 ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1103_strict__sorted__simps_I2_J,axiom,
! [X: nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X @ Ys2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) )
=> ( ord_less_nat @ X @ X2 ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys2 ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1104_strict__sorted__simps_I2_J,axiom,
! [X: real,Ys2: list_real] :
( ( sorted_wrt_real @ ord_less_real @ ( cons_real @ X @ Ys2 ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ ( set_real2 @ Ys2 ) )
=> ( ord_less_real @ X @ X2 ) )
& ( sorted_wrt_real @ ord_less_real @ Ys2 ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1105_strict__sorted__simps_I2_J,axiom,
! [X: extended_enat,Ys2: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ ( cons_Extended_enat @ X @ Ys2 ) )
= ( ! [X2: extended_enat] :
( ( member_Extended_enat @ X2 @ ( set_Extended_enat2 @ Ys2 ) )
=> ( ord_le72135733267957522d_enat @ X @ X2 ) )
& ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ Ys2 ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1106_set__update__subset__insert,axiom,
! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1107_set__update__subset__insert,axiom,
! [Xs: list_a,I: nat,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X ) ) @ ( insert_a @ X @ ( set_a2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1108_enumerate__step,axiom,
! [S2: set_Extended_enat,N: nat] :
( ~ ( finite4001608067531595151d_enat @ S2 )
=> ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ N ) @ ( infini7641415182203889163d_enat @ S2 @ ( suc @ N ) ) ) ) ).
% enumerate_step
thf(fact_1109_enumerate__step,axiom,
! [S2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N ) ) ) ) ).
% enumerate_step
thf(fact_1110_enumerate__mono,axiom,
! [M2: nat,N: nat,S2: set_Extended_enat] :
( ( ord_less_nat @ M2 @ N )
=> ( ~ ( finite4001608067531595151d_enat @ S2 )
=> ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) ) ) ) ).
% enumerate_mono
thf(fact_1111_enumerate__mono,axiom,
! [M2: nat,N: nat,S2: set_nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% enumerate_mono
thf(fact_1112_finite__le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_1113_finite__enum__ext,axiom,
! [X8: set_nat,Y6: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X8 ) )
=> ( ( infini8530281810654367211te_nat @ X8 @ I3 )
= ( infini8530281810654367211te_nat @ Y6 @ I3 ) ) )
=> ( ( finite_finite_nat @ X8 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ( finite_card_nat @ X8 )
= ( finite_card_nat @ Y6 ) )
=> ( X8 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_1114_finite__enumerate__Ex,axiom,
! [S2: set_nat,S: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( member_nat @ S @ S2 )
=> ? [N3: nat] :
( ( ord_less_nat @ N3 @ ( finite_card_nat @ S2 ) )
& ( ( infini8530281810654367211te_nat @ S2 @ N3 )
= S ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_1115_finite__enumerate__in__set,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ S2 ) ) ) ).
% finite_enumerate_in_set
thf(fact_1116_card__le__Suc__iff,axiom,
! [N: nat,A2: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A2 ) )
= ( ? [A3: a,B6: set_a] :
( ( A2
= ( insert_a @ A3 @ B6 ) )
& ~ ( member_a @ A3 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B6 ) )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1117_card__le__Suc__iff,axiom,
! [N: nat,A2: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A2 ) )
= ( ? [A3: nat,B6: set_nat] :
( ( A2
= ( insert_nat @ A3 @ B6 ) )
& ~ ( member_nat @ A3 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1118_nth__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1119_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1120_finite__enumerate__mono,axiom,
! [M2: nat,N: nat,S2: set_Extended_enat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( finite4001608067531595151d_enat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S2 ) )
=> ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_1121_finite__enumerate__mono,axiom,
! [M2: nat,N: nat,S2: set_nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_1122_nth__equal__first__eq,axiom,
! [X: a,Xs: list_a,N: nat] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1123_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1124_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs: list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1125_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1126_finite__enumerate__step,axiom,
! [S2: set_Extended_enat,N: nat] :
( ( finite4001608067531595151d_enat @ S2 )
=> ( ( ord_less_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ S2 ) )
=> ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ N ) @ ( infini7641415182203889163d_enat @ S2 @ ( suc @ N ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_1127_finite__enumerate__step,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S2 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_1128_finite__enum__subset,axiom,
! [X8: set_nat,Y6: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X8 ) )
=> ( ( infini8530281810654367211te_nat @ X8 @ I3 )
= ( infini8530281810654367211te_nat @ Y6 @ I3 ) ) )
=> ( ( finite_finite_nat @ X8 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X8 ) @ ( finite_card_nat @ Y6 ) )
=> ( ord_less_eq_set_nat @ X8 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_1129_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_1130_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_1131_set__update__distinct,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ Xs @ N @ X ) )
= ( insert_a @ X @ ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a @ ( nth_a @ Xs @ N ) @ bot_bot_set_a ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1132_set__update__distinct,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) )
= ( insert_nat @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1133_distinct__list__update,axiom,
! [Xs: list_nat,A: nat,I: nat] :
( ( distinct_nat @ Xs )
=> ( ~ ( member_nat @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ I ) @ bot_bot_set_nat ) ) )
=> ( distinct_nat @ ( list_update_nat @ Xs @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1134_distinct__list__update,axiom,
! [Xs: list_a,A: a,I: nat] :
( ( distinct_a @ Xs )
=> ( ~ ( member_a @ A @ ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a @ ( nth_a @ Xs @ I ) @ bot_bot_set_a ) ) )
=> ( distinct_a @ ( list_update_a @ Xs @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1135_insert__subsetI,axiom,
! [X: a,A2: set_a,X8: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X8 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X8 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1136_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X8: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X8 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X8 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1137_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_1138_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1139_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_1140_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1141_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1142_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1143_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1144_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1145_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1146_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1147_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1148_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1149_finite__transitivity__chain,axiom,
! [A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X6: a] :
~ ( R @ X6 @ X6 )
=> ( ! [X6: a,Y4: a,Z3: a] :
( ( R @ X6 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X6 @ Z3 ) ) )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ? [Y5: a] :
( ( member_a @ Y5 @ A2 )
& ( R @ X6 @ Y5 ) ) )
=> ( A2 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1150_finite__transitivity__chain,axiom,
! [A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X6: nat] :
~ ( R @ X6 @ X6 )
=> ( ! [X6: nat,Y4: nat,Z3: nat] :
( ( R @ X6 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X6 @ Z3 ) ) )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A2 )
& ( R @ X6 @ Y5 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1151_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1152_bot_Onot__eq__extremum,axiom,
! [A: extended_enat] :
( ( A != bot_bo4199563552545308370d_enat )
= ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1153_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1154_bot_Oextremum__strict,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% bot.extremum_strict
thf(fact_1155_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_1156_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1157_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_1158_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1159_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_1160_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1161_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_1162_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1163_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1164_bot_Oextremum__uniqueI,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
=> ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_uniqueI
thf(fact_1165_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1166_bot_Oextremum__unique,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
= ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_unique
thf(fact_1167_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1168_bot_Oextremum,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% bot.extremum
thf(fact_1169_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X6: a] :
~ ( member_a @ X6 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1170_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X6: nat] :
~ ( member_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1171_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1172_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1173_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1174_finite__has__maximal,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A2 )
=> ( ( ord_less_eq_a @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1175_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X6: real] :
( ( member_real @ X6 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1176_finite__has__maximal,axiom,
! [A2: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A2 )
=> ( ( A2 != bot_bo7653980558646680370d_enat )
=> ? [X6: extended_enat] :
( ( member_Extended_enat @ X6 @ A2 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1177_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1178_finite__has__minimal,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A2 )
=> ( ( ord_less_eq_a @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1179_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X6: real] :
( ( member_real @ X6 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1180_finite__has__minimal,axiom,
! [A2: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A2 )
=> ( ( A2 != bot_bo7653980558646680370d_enat )
=> ? [X6: extended_enat] :
( ( member_Extended_enat @ X6 @ A2 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1181_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X6 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1182_ex__min__if__finite,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ? [X6: real] :
( ( member_real @ X6 @ S2 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S2 )
& ( ord_less_real @ Xa @ X6 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1183_ex__min__if__finite,axiom,
! [S2: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ S2 )
=> ( ( S2 != bot_bo7653980558646680370d_enat )
=> ? [X6: extended_enat] :
( ( member_Extended_enat @ X6 @ S2 )
& ~ ? [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ S2 )
& ( ord_le72135733267957522d_enat @ Xa @ X6 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1184_infinite__growing,axiom,
! [X8: set_a] :
( ( X8 != bot_bot_set_a )
=> ( ! [X6: a] :
( ( member_a @ X6 @ X8 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X8 )
& ( ord_less_a @ X6 @ Xa ) ) )
=> ~ ( finite_finite_a @ X8 ) ) ) ).
% infinite_growing
thf(fact_1185_infinite__growing,axiom,
! [X8: set_nat] :
( ( X8 != bot_bot_set_nat )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ X8 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X8 )
& ( ord_less_nat @ X6 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_1186_infinite__growing,axiom,
! [X8: set_real] :
( ( X8 != bot_bot_set_real )
=> ( ! [X6: real] :
( ( member_real @ X6 @ X8 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X8 )
& ( ord_less_real @ X6 @ Xa ) ) )
=> ~ ( finite_finite_real @ X8 ) ) ) ).
% infinite_growing
thf(fact_1187_infinite__growing,axiom,
! [X8: set_Extended_enat] :
( ( X8 != bot_bo7653980558646680370d_enat )
=> ( ! [X6: extended_enat] :
( ( member_Extended_enat @ X6 @ X8 )
=> ? [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ X8 )
& ( ord_le72135733267957522d_enat @ X6 @ Xa ) ) )
=> ~ ( finite4001608067531595151d_enat @ X8 ) ) ) ).
% infinite_growing
thf(fact_1188_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A4: nat] :
( A
= ( insert_nat @ A4 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1189_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A6: set_nat,B2: nat] :
( ( A3
= ( insert_nat @ B2 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_1190_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X6 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1191_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X6 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1192_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X6: a] : ( P @ ( insert_a @ X6 @ bot_bot_set_a ) )
=> ( ! [X6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1193_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X6: nat] : ( P @ ( insert_nat @ X6 @ bot_bot_set_nat ) )
=> ( ! [X6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1194_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X6 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1195_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X6 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1196_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1197_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1198_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1199_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1200_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1201_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1202_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1203_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > a] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1204_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > real] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1205_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1206_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > extended_enat] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1207_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > extended_enat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ Y5 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X6 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1208_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( ord_less_nat @ X5 @ B3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1209_finite__linorder__max__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B3: real,A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A7 )
=> ( ord_less_real @ X5 @ B3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_real @ B3 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1210_finite__linorder__max__induct,axiom,
! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
( ( finite4001608067531595151d_enat @ A2 )
=> ( ( P @ bot_bo7653980558646680370d_enat )
=> ( ! [B3: extended_enat,A7: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A7 )
=> ( ! [X5: extended_enat] :
( ( member_Extended_enat @ X5 @ A7 )
=> ( ord_le72135733267957522d_enat @ X5 @ B3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_Extended_enat @ B3 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1211_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( ord_less_nat @ B3 @ X5 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1212_finite__linorder__min__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B3: real,A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A7 )
=> ( ord_less_real @ B3 @ X5 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_real @ B3 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1213_finite__linorder__min__induct,axiom,
! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
( ( finite4001608067531595151d_enat @ A2 )
=> ( ( P @ bot_bo7653980558646680370d_enat )
=> ( ! [B3: extended_enat,A7: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A7 )
=> ( ! [X5: extended_enat] :
( ( member_Extended_enat @ X5 @ A7 )
=> ( ord_le72135733267957522d_enat @ B3 @ X5 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_Extended_enat @ B3 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1214_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1215_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1216_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A2 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1217_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1218_subset__insert__iff,axiom,
! [A2: set_a,X: a,B5: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B5 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1219_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1220_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1221_infinite__remove,axiom,
! [S2: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1222_infinite__coinduct,axiom,
! [X8: set_nat > $o,A2: set_nat] :
( ( X8 @ A2 )
=> ( ! [A7: set_nat] :
( ( X8 @ A7 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A7 )
& ( ( X8 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1223_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1224_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1225_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X6: nat] :
( A2
!= ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1226_finite__remove__induct,axiom,
! [B5: set_a,P: set_a > $o] :
( ( finite_finite_a @ B5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B5 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_1227_finite__remove__induct,axiom,
! [B5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_1228_remove__induct,axiom,
! [P: set_a > $o,B5: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B5 )
=> ( P @ B5 ) )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B5 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_1229_remove__induct,axiom,
! [P: set_nat > $o,B5: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B5 )
=> ( P @ B5 ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_1230_card__1__singleton__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1231_card__eq__SucD,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
=> ? [B3: a,B8: set_a] :
( ( A2
= ( insert_a @ B3 @ B8 ) )
& ~ ( member_a @ B3 @ B8 )
& ( ( finite_card_a @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1232_card__eq__SucD,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
=> ? [B3: nat,B8: set_nat] :
( ( A2
= ( insert_nat @ B3 @ B8 ) )
& ~ ( member_nat @ B3 @ B8 )
& ( ( finite_card_nat @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1233_card__Suc__eq,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B2: a,B6: set_a] :
( ( A2
= ( insert_a @ B2 @ B6 ) )
& ~ ( member_a @ B2 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1234_card__Suc__eq,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
= ( ? [B2: nat,B6: set_nat] :
( ( A2
= ( insert_nat @ B2 @ B6 ) )
& ~ ( member_nat @ B2 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1235_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1236_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1237_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B5: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B5 ) )
= ( ( ( member_a @ X @ B5 )
=> ( ord_less_set_a @ A2 @ B5 ) )
& ( ~ ( member_a @ X @ B5 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1238_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B5: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
= ( ( ( member_nat @ X @ B5 )
=> ( ord_less_set_nat @ A2 @ B5 ) )
& ( ~ ( member_nat @ X @ B5 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1239_finite__induct__select,axiom,
! [S2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S2 @ T4 ) )
& ( P @ ( insert_nat @ X5 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1240_card_Oremove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ A2 )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1241_card_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ A2 )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1242_card_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1243_card__Suc__Diff1,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1244_card__Suc__Diff1,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1245_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1246_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1247_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1248_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1249_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1250_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1251_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1252_card__Diff__singleton__if,axiom,
! [X: nat,A2: set_nat] :
( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1253_card__Diff__singleton,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1254_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1255_enat__ord__number_I2_J,axiom,
! [M2: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_1256_enat__ord__number_I1_J,axiom,
! [M2: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_1257_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_1258_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1259_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_le72135733267957522d_enat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_1260_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_1261_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_1262_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
% Conjectures (1)
thf(conj_0,conjecture,
member_a @ ( nth_a @ xs @ j ) @ i ).
%------------------------------------------------------------------------------