TPTP Problem File: SLH0281^1.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 : Frequency_Moments/0082_K_Smallest/prob_00357_012910__19829584_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1393 ( 501 unt; 124 typ; 0 def)
% Number of atoms : 3519 (1128 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10812 ( 362 ~; 71 |; 144 &;8588 @)
% ( 0 <=>;1647 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 423 ( 423 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 108 usr; 12 con; 0-4 aty)
% Number of variables : 3526 ( 227 ^;3232 !; 67 ?;3526 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:13:42.072
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__Multiset__Omultiset_It__Pratt____Certificate__Opratt_J,type,
multiset_Pratt_pratt: $tType ).
thf(ty_n_t__List__Olist_It__Pratt____Certificate__Opratt_J,type,
list_Pratt_pratt: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Pratt____Certificate__Opratt_J,type,
set_Pratt_pratt: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
set_multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Pratt____Certificate__Opratt,type,
pratt_pratt: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $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__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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (108)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Pratt____Certificate__Opratt_J,type,
minus_5321434743015354872_pratt: multiset_Pratt_pratt > multiset_Pratt_pratt > multiset_Pratt_pratt ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
plus_p5152935555875550043iset_a: set_multiset_a > set_multiset_a > set_multiset_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
plus_p4817606893110106565et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Pratt____Certificate__Opratt_J,type,
zero_z502583456159058376_pratt: multiset_Pratt_pratt ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_K__Smallest_Ocount__le_001t__Nat__Onat,type,
k_count_le_nat: nat > multiset_nat > nat ).
thf(sy_c_K__Smallest_Ocount__le_001tf__a,type,
k_count_le_a: a > multiset_a > nat ).
thf(sy_c_K__Smallest_Ocount__less_001t__Nat__Onat,type,
k_count_less_nat: nat > multiset_nat > nat ).
thf(sy_c_K__Smallest_Ocount__less_001tf__a,type,
k_count_less_a: a > multiset_a > nat ).
thf(sy_c_K__Smallest_Oleast_001tf__a,type,
k_least_a: nat > set_a > set_a ).
thf(sy_c_K__Smallest_Onth__mset_001t__Nat__Onat,type,
k_nth_mset_nat: nat > multiset_nat > nat ).
thf(sy_c_K__Smallest_Onth__mset_001tf__a,type,
k_nth_mset_a: nat > multiset_a > a ).
thf(sy_c_K__Smallest_Orank__of_001tf__a,type,
k_rank_of_a: a > set_a > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Pratt____Certificate__Opratt,type,
count_4999389613427621831_pratt: list_Pratt_pratt > pratt_pratt > nat ).
thf(sy_c_List_Ocount__list_001tf__a,type,
count_list_a: list_a > a > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Pratt____Certificate__Opratt,type,
gen_le3130621340910794462_pratt: nat > list_Pratt_pratt > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Pratt____Certificate__Opratt,type,
cons_Pratt_pratt: pratt_pratt > list_Pratt_pratt > list_Pratt_pratt ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Pratt____Certificate__Opratt,type,
set_Pratt_pratt2: list_Pratt_pratt > set_Pratt_pratt ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Pratt____Certificate__Opratt,type,
list_u6220086826248601529_pratt: list_Pratt_pratt > nat > pratt_pratt > list_Pratt_pratt ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Pratt____Certificate__Opratt,type,
nth_Pratt_pratt: list_Pratt_pratt > nat > pratt_pratt ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Pratt____Certificate__Opratt,type,
rev_Pratt_pratt: list_Pratt_pratt > list_Pratt_pratt ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Pratt____Certificate__Opratt,type,
rotate1_Pratt_pratt: list_Pratt_pratt > list_Pratt_pratt ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Pratt____Certificate__Opratt,type,
rotate_Pratt_pratt: nat > list_Pratt_pratt > list_Pratt_pratt ).
thf(sy_c_List_Orotate_001tf__a,type,
rotate_a: nat > list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Nat__Onat_J,type,
sorted_wrt_list_nat: ( list_nat > list_nat > $o ) > list_list_nat > $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__Pratt____Certificate__Opratt,type,
sorted3866451260844240568_pratt: ( pratt_pratt > pratt_pratt > $o ) > list_Pratt_pratt > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
add_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Oadd__mset_001t__Pratt____Certificate__Opratt,type,
add_mset_Pratt_pratt: pratt_pratt > multiset_Pratt_pratt > multiset_Pratt_pratt ).
thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
linord3047872887403683810et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001tf__a,type,
linord814965612141868908iset_a: multiset_a > list_a ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Pratt____Certificate__Opratt,type,
mset_Pratt_pratt: list_Pratt_pratt > multiset_Pratt_pratt ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Pratt____Certificate__Opratt,type,
set_mset_Pratt_pratt: multiset_Pratt_pratt > set_Pratt_pratt ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Pratt____Certificate__Opratt_J,type,
size_s4929225773697354557_pratt: list_Pratt_pratt > 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_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
size_s5917832649809541300et_nat: multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Pratt____Certificate__Opratt_J,type,
size_s6011366409283536893_pratt: multiset_Pratt_pratt > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_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__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_Itf__a_J,type,
ord_le7200615133682826212iset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Pratt____Certificate__Opratt_J,type,
ord_le3428837165542985291_pratt: set_Pratt_pratt > set_Pratt_pratt > $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_Pratt__Certificate_Obuild__fpc,type,
pratt_build_fpc: nat > nat > nat > list_nat > list_Pratt_pratt ).
thf(sy_c_Pratt__Certificate_Omod__exp__nat,type,
pratt_mod_exp_nat: nat > nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Pratt____Certificate__Opratt,type,
collect_Pratt_pratt: ( pratt_pratt > $o ) > set_Pratt_pratt ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Pratt____Certificate__Opratt,type,
member_Pratt_pratt: pratt_pratt > set_Pratt_pratt > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: multiset_a ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_xs____,type,
xs: list_a ).
thf(sy_v_y____,type,
y: a ).
% Relevant facts (1265)
thf(fact_0_s__xs,axiom,
sorted_wrt_a @ ord_less_eq_a @ xs ).
% s_xs
thf(fact_1_sorted__nth__mono,axiom,
! [Xs: list_list_nat,I: nat,J: nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ord_less_eq_list_nat @ ( nth_list_nat @ Xs @ I ) @ ( nth_list_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_2_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_3_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_4__092_060open_062i_A_092_060le_062_Ak_092_060close_062,axiom,
ord_less_eq_nat @ i @ k ).
% \<open>i \<le> k\<close>
thf(fact_5_k__bound,axiom,
ord_less_nat @ k @ ( size_size_list_a @ xs ) ).
% k_bound
thf(fact_6_i__bound,axiom,
ord_less_nat @ i @ ( size_size_list_a @ xs ) ).
% i_bound
thf(fact_7_y__def,axiom,
( y
= ( nth_a @ xs @ i ) ) ).
% y_def
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062i_O_A_092_060lbrakk_062i_A_060_Alength_Axs_059_Ay_A_061_Axs_A_B_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ xs ) )
=> ( y
!= ( nth_a @ xs @ I2 ) ) ) ).
% \<open>\<And>thesis. (\<And>i. \<lbrakk>i < length xs; y = xs ! i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_10_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_11_order__refl,axiom,
! [X: list_nat] : ( ord_less_eq_list_nat @ X @ X ) ).
% order_refl
thf(fact_12_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_14_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_15_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_16_dual__order_Orefl,axiom,
! [A: list_nat] : ( ord_less_eq_list_nat @ A @ A ) ).
% dual_order.refl
thf(fact_17_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_18_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_19_no__col,axiom,
! [X: a] :
( ( ord_less_eq_a @ X @ ( nth_a @ xs @ k ) )
=> ( ord_less_eq_nat @ ( count_list_a @ xs @ X ) @ one_one_nat ) ) ).
% no_col
thf(fact_20_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_21_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_22_verit__comp__simplify1_I2_J,axiom,
! [A: list_nat] : ( ord_less_eq_list_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_23_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_24_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_25_nle__le,axiom,
! [A: list_nat,B: list_nat] :
( ( ~ ( ord_less_eq_list_nat @ A @ B ) )
= ( ( ord_less_eq_list_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_26_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_27_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_28_le__cases3,axiom,
! [X: list_nat,Y: list_nat,Z: list_nat] :
( ( ( ord_less_eq_list_nat @ X @ Y )
=> ~ ( ord_less_eq_list_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_list_nat @ Y @ X )
=> ~ ( ord_less_eq_list_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_list_nat @ X @ Z )
=> ~ ( ord_less_eq_list_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_list_nat @ Z @ Y )
=> ~ ( ord_less_eq_list_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_list_nat @ Y @ Z )
=> ~ ( ord_less_eq_list_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_list_nat @ Z @ X )
=> ~ ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_29_le__cases3,axiom,
! [X: a,Y: a,Z: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z ) )
=> ( ( ( ord_less_eq_a @ X @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y ) )
=> ( ( ( ord_less_eq_a @ Z @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z )
=> ~ ( ord_less_eq_a @ Z @ X ) )
=> ~ ( ( ord_less_eq_a @ Z @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_30_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_31_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_32_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z2: set_nat] : ( Y2 = Z2 ) )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_33_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ Y3 )
& ( ord_less_eq_list_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_34_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ( ord_less_eq_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_35_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_36_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_37_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_38_ord__eq__le__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ord_less_eq_list_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_39_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_40_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_41_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_42_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_43_ord__le__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_list_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_44_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_45_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_46_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_47_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_48_order__antisym,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
=> ( ( ord_less_eq_list_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_49_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_50_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_51_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_52_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_53_order_Otrans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ord_less_eq_list_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_54_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_55_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_56_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_57_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_58_verit__comp__simplify1_I1_J,axiom,
! [A: list_nat] :
~ ( ord_less_list_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_59_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_60_sorted01,axiom,
! [Xs: list_list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs ) ) ).
% sorted01
thf(fact_61_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_62_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_63_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_64_less__imp__neq,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_65_less__imp__neq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_66_less__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_67_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_68_order_Oasym,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order.asym
thf(fact_69_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_70_order_Oasym,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ~ ( ord_less_list_nat @ B @ A ) ) ).
% order.asym
thf(fact_71_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_72_ord__eq__less__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_73_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_74_ord__eq__less__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_75_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_76_ord__less__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_77_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_78_ord__less__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_79_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_80_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_81_antisym__conv3,axiom,
! [Y: a,X: a] :
( ~ ( ord_less_a @ Y @ X )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_82_antisym__conv3,axiom,
! [Y: list_nat,X: list_nat] :
( ~ ( ord_less_list_nat @ Y @ X )
=> ( ( ~ ( ord_less_list_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_83_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_84_linorder__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_a @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_85_linorder__cases,axiom,
! [X: list_nat,Y: list_nat] :
( ~ ( ord_less_list_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_list_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_86_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_87_dual__order_Oasym,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ~ ( ord_less_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_88_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_89_dual__order_Oasym,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ~ ( ord_less_list_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_90_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_91_dual__order_Oirrefl,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_92_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_93_dual__order_Oirrefl,axiom,
! [A: list_nat] :
~ ( ord_less_list_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_94_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_95_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_96_linorder__less__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: a] : ( P @ A2 @ A2 )
=> ( ! [A2: a,B2: a] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_97_linorder__less__wlog,axiom,
! [P: list_nat > list_nat > $o,A: list_nat,B: list_nat] :
( ! [A2: list_nat,B2: list_nat] :
( ( ord_less_list_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: list_nat] : ( P @ A2 @ A2 )
=> ( ! [A2: list_nat,B2: list_nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_98_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_99_order_Ostrict__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_100_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_101_order_Ostrict__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_102_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_103_not__less__iff__gr__or__eq,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ( ord_less_a @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_104_not__less__iff__gr__or__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ~ ( ord_less_list_nat @ X @ Y ) )
= ( ( ord_less_list_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_105_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_106_dual__order_Ostrict__trans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_107_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_108_dual__order_Ostrict__trans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ( ( ord_less_list_nat @ C @ B )
=> ( ord_less_list_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_109_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_110_order_Ostrict__implies__not__eq,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_111_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_112_order_Ostrict__implies__not__eq,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_113_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_114_dual__order_Ostrict__implies__not__eq,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_115_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_116_dual__order_Ostrict__implies__not__eq,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_117_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_118_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_119_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_120_sorted__wrt01,axiom,
! [Xs: list_Pratt_pratt,P: pratt_pratt > pratt_pratt > $o] :
( ( ord_less_eq_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ one_one_nat )
=> ( sorted3866451260844240568_pratt @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_121_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_122_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_123_length__induct,axiom,
! [P: list_Pratt_pratt > $o,Xs: list_Pratt_pratt] :
( ! [Xs2: list_Pratt_pratt] :
( ! [Ys: list_Pratt_pratt] :
( ( ord_less_nat @ ( size_s4929225773697354557_pratt @ Ys ) @ ( size_s4929225773697354557_pratt @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_124_count__le__length,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( size_size_list_a @ Xs ) ) ).
% count_le_length
thf(fact_125_count__le__length,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_126_count__le__length,axiom,
! [Xs: list_Pratt_pratt,X: pratt_pratt] : ( ord_less_eq_nat @ ( count_4999389613427621831_pratt @ Xs @ X ) @ ( size_s4929225773697354557_pratt @ Xs ) ) ).
% count_le_length
thf(fact_127_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_128_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_129_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_Pratt_pratt] :
( ( size_s4929225773697354557_pratt @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_130_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_131_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_132_neq__if__length__neq,axiom,
! [Xs: list_Pratt_pratt,Ys2: list_Pratt_pratt] :
( ( ( size_s4929225773697354557_pratt @ Xs )
!= ( size_s4929225773697354557_pratt @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_133_linorder__neqE,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_a @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_134_linorder__neqE,axiom,
! [X: list_nat,Y: list_nat] :
( ( X != Y )
=> ( ~ ( ord_less_list_nat @ X @ Y )
=> ( ord_less_list_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_135_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_136_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A: pratt_pratt,P: pratt_pratt > $o] :
( ( member_Pratt_pratt @ A @ ( collect_Pratt_pratt @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A3: set_Pratt_pratt] :
( ( collect_Pratt_pratt
@ ^ [X2: pratt_pratt] : ( member_Pratt_pratt @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_142_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_143_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_144_order__less__asym,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ~ ( ord_less_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_145_order__less__asym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_146_order__less__asym,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ~ ( ord_less_list_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_147_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_148_linorder__neq__iff,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ( ord_less_a @ X @ Y )
| ( ord_less_a @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_149_linorder__neq__iff,axiom,
! [X: list_nat,Y: list_nat] :
( ( X != Y )
= ( ( ord_less_list_nat @ X @ Y )
| ( ord_less_list_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_150_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_151_order__less__asym_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_152_order__less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_153_order__less__asym_H,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ~ ( ord_less_list_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_154_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_155_order__less__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_156_order__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_157_order__less__trans,axiom,
! [X: list_nat,Y: list_nat,Z: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( ( ord_less_list_nat @ Y @ Z )
=> ( ord_less_list_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_158_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_159_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_160_ord__eq__less__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_161_ord__eq__less__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_162_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_163_ord__eq__less__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_164_ord__eq__less__subst,axiom,
! [A: list_nat,F: nat > list_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_165_ord__eq__less__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_166_ord__eq__less__subst,axiom,
! [A: list_nat,F: a > list_nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_167_ord__eq__less__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_168_ord__eq__less__subst,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_169_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_170_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_171_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_172_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_173_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_174_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_175_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_176_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > list_nat,C: list_nat] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_177_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_178_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_179_order__less__irrefl,axiom,
! [X: a] :
~ ( ord_less_a @ X @ X ) ).
% order_less_irrefl
thf(fact_180_order__less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_181_order__less__irrefl,axiom,
! [X: list_nat] :
~ ( ord_less_list_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_182_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_183_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_184_order__less__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_185_order__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_186_order__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_187_order__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_188_order__less__subst1,axiom,
! [A: nat,F: list_nat > nat,B: list_nat,C: list_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_list_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_189_order__less__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_190_order__less__subst1,axiom,
! [A: a,F: list_nat > a,B: list_nat,C: list_nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_list_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_191_order__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_192_order__less__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_193_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_194_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_195_order__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_196_order__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_197_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_198_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_199_order__less__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_200_order__less__subst2,axiom,
! [A: a,B: a,F: a > list_nat,C: list_nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_201_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_202_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_203_order__less__not__sym,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ~ ( ord_less_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_204_order__less__not__sym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_205_order__less__not__sym,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ~ ( ord_less_list_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_206_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_207_order__less__imp__triv,axiom,
! [X: a,Y: a,P: $o] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_a @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_208_order__less__imp__triv,axiom,
! [X: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_209_order__less__imp__triv,axiom,
! [X: list_nat,Y: list_nat,P: $o] :
( ( ord_less_list_nat @ X @ Y )
=> ( ( ord_less_list_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_210_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_211_linorder__less__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y )
| ( ord_less_a @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_212_linorder__less__linear,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
| ( X = Y )
| ( ord_less_list_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_213_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_214_order__less__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_215_order__less__imp__not__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_216_order__less__imp__not__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_217_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_218_order__less__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_219_order__less__imp__not__eq2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_220_order__less__imp__not__eq2,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_221_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_222_order__less__imp__not__less,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ~ ( ord_less_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_223_order__less__imp__not__less,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_224_order__less__imp__not__less,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ~ ( ord_less_list_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_225_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_226_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_a
= ( ^ [P3: a > a > $o,Xs3: list_a] :
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs3 ) )
=> ( P3 @ ( nth_a @ Xs3 @ I3 ) @ ( nth_a @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_227_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs3: list_nat] :
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
=> ( P3 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_228_sorted__wrt__iff__nth__less,axiom,
( sorted3866451260844240568_pratt
= ( ^ [P3: pratt_pratt > pratt_pratt > $o,Xs3: list_Pratt_pratt] :
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s4929225773697354557_pratt @ Xs3 ) )
=> ( P3 @ ( nth_Pratt_pratt @ Xs3 @ I3 ) @ ( nth_Pratt_pratt @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_229_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_230_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_231_sorted__wrt__nth__less,axiom,
! [P: pratt_pratt > pratt_pratt > $o,Xs: list_Pratt_pratt,I: nat,J: nat] :
( ( sorted3866451260844240568_pratt @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( P @ ( nth_Pratt_pratt @ Xs @ I ) @ ( nth_Pratt_pratt @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_232_strict__sorted__imp__sorted,axiom,
! [Xs: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_list_nat @ Xs )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_233_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_234_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_235_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_a,Z2: list_a] : ( Y2 = Z2 ) )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I3 )
= ( nth_a @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_236_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_237_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_Pratt_pratt,Z2: list_Pratt_pratt] : ( Y2 = Z2 ) )
= ( ^ [Xs3: list_Pratt_pratt,Ys3: list_Pratt_pratt] :
( ( ( size_s4929225773697354557_pratt @ Xs3 )
= ( size_s4929225773697354557_pratt @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s4929225773697354557_pratt @ Xs3 ) )
=> ( ( nth_Pratt_pratt @ Xs3 @ I3 )
= ( nth_Pratt_pratt @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_238_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: a] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_a @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_239_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: nat] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_240_Skolem__list__nth,axiom,
! [K: nat,P: nat > pratt_pratt > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: pratt_pratt] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_Pratt_pratt] :
( ( ( size_s4929225773697354557_pratt @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_Pratt_pratt @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_241_nth__equalityI,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_242_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_243_nth__equalityI,axiom,
! [Xs: list_Pratt_pratt,Ys2: list_Pratt_pratt] :
( ( ( size_s4929225773697354557_pratt @ Xs )
= ( size_s4929225773697354557_pratt @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( nth_Pratt_pratt @ Xs @ I2 )
= ( nth_Pratt_pratt @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_244_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_245_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_246_order__le__imp__less__or__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
=> ( ( ord_less_list_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_247_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_248_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_249_linorder__le__less__linear,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
| ( ord_less_list_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_250_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_251_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_252_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_253_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_254_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_255_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_256_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_257_order__less__le__subst2,axiom,
! [A: list_nat,B: list_nat,F: list_nat > a,C: a] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_list_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_258_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_259_order__less__le__subst2,axiom,
! [A: list_nat,B: list_nat,F: list_nat > nat,C: nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_list_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_260_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_261_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_262_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_263_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_264_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_265_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_266_order__less__le__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_267_order__less__le__subst1,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_268_order__less__le__subst1,axiom,
! [A: list_nat,F: a > list_nat,B: a,C: a] :
( ( ord_less_list_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_269_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_270_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_271_order__less__le__subst1,axiom,
! [A: list_nat,F: nat > list_nat,B: nat,C: nat] :
( ( ord_less_list_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_272_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_273_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_274_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_275_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_276_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_277_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_278_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > list_nat,C: list_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_279_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_280_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_281_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_282_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_283_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_284_order__le__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_285_order__le__less__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_286_order__le__less__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_287_order__le__less__subst1,axiom,
! [A: a,F: list_nat > a,B: list_nat,C: list_nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_list_nat @ X3 @ Y5 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_288_order__le__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_289_order__le__less__subst1,axiom,
! [A: nat,F: list_nat > nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_list_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_290_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_291_order__le__less__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_a @ X3 @ Y5 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_292_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_293_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_294_order__less__le__trans,axiom,
! [X: list_nat,Y: list_nat,Z: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( ( ord_less_eq_list_nat @ Y @ Z )
=> ( ord_less_list_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_295_order__less__le__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_296_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_297_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_298_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_299_order__le__less__trans,axiom,
! [X: list_nat,Y: list_nat,Z: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
=> ( ( ord_less_list_nat @ Y @ Z )
=> ( ord_less_list_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_300_order__le__less__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_301_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_302_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_303_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_304_order__neq__le__trans,axiom,
! [A: list_nat,B: list_nat] :
( ( A != B )
=> ( ( ord_less_eq_list_nat @ A @ B )
=> ( ord_less_list_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_305_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_306_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_307_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_308_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_309_order__le__neq__trans,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_list_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_310_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_311_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_312_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_313_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_314_order__less__imp__le,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( ord_less_eq_list_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_315_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_316_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_317_linorder__not__less,axiom,
! [X: list_nat,Y: list_nat] :
( ( ~ ( ord_less_list_nat @ X @ Y ) )
= ( ord_less_eq_list_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_318_linorder__not__less,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_not_less
thf(fact_319_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_320_linorder__not__le,axiom,
! [X: list_nat,Y: list_nat] :
( ( ~ ( ord_less_eq_list_nat @ X @ Y ) )
= ( ord_less_list_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_321_linorder__not__le,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X @ Y ) )
= ( ord_less_a @ Y @ X ) ) ).
% linorder_not_le
thf(fact_322_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_323_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_324_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_325_order__less__le,axiom,
( ord_less_list_nat
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_326_order__less__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_327_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_328_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_329_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_330_order__le__less,axiom,
( ord_less_eq_list_nat
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_331_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_332_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_333_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_334_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_335_dual__order_Ostrict__implies__order,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ( ord_less_eq_list_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_336_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_337_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_338_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_339_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_340_order_Ostrict__implies__order,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ord_less_eq_list_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_341_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_342_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_343_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_344_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_345_dual__order_Ostrict__iff__not,axiom,
( ord_less_list_nat
= ( ^ [B3: list_nat,A4: list_nat] :
( ( ord_less_eq_list_nat @ B3 @ A4 )
& ~ ( ord_less_eq_list_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_346_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_347_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_348_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_349_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_350_dual__order_Ostrict__trans2,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ( ( ord_less_eq_list_nat @ C @ B )
=> ( ord_less_list_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_351_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_352_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_353_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_354_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_355_dual__order_Ostrict__trans1,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ B @ A )
=> ( ( ord_less_list_nat @ C @ B )
=> ( ord_less_list_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_356_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_357_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_358_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_359_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_360_dual__order_Ostrict__iff__order,axiom,
( ord_less_list_nat
= ( ^ [B3: list_nat,A4: list_nat] :
( ( ord_less_eq_list_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_361_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_362_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_363_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_364_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_365_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_list_nat
= ( ^ [B3: list_nat,A4: list_nat] :
( ( ord_less_list_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_366_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_367_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_368_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_369_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_370_order_Ostrict__iff__not,axiom,
( ord_less_list_nat
= ( ^ [A4: list_nat,B3: list_nat] :
( ( ord_less_eq_list_nat @ A4 @ B3 )
& ~ ( ord_less_eq_list_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_371_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ~ ( ord_less_eq_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_372_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_373_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_374_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_375_order_Ostrict__trans2,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_376_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_377_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_378_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_379_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_380_order_Ostrict__trans1,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_381_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_382_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_383_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_384_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_385_order_Ostrict__iff__order,axiom,
( ord_less_list_nat
= ( ^ [A4: list_nat,B3: list_nat] :
( ( ord_less_eq_list_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_386_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_387_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_388_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_389_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_390_order_Oorder__iff__strict,axiom,
( ord_less_eq_list_nat
= ( ^ [A4: list_nat,B3: list_nat] :
( ( ord_less_list_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_391_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_392_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_393_not__le__imp__less,axiom,
! [Y: list_nat,X: list_nat] :
( ~ ( ord_less_eq_list_nat @ Y @ X )
=> ( ord_less_list_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_394_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_395_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_396_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ~ ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_397_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_398_less__le__not__le,axiom,
( ord_less_list_nat
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_list_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_399_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ~ ( ord_less_eq_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_400_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_401_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_402_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_403_antisym__conv2,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
=> ( ( ~ ( ord_less_list_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_404_antisym__conv2,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_405_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_406_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_407_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_408_antisym__conv1,axiom,
! [X: list_nat,Y: list_nat] :
( ~ ( ord_less_list_nat @ X @ Y )
=> ( ( ord_less_eq_list_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_409_antisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_410_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_411_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_412_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_413_nless__le,axiom,
! [A: list_nat,B: list_nat] :
( ( ~ ( ord_less_list_nat @ A @ B ) )
= ( ~ ( ord_less_eq_list_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_414_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_415_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_416_leI,axiom,
! [X: list_nat,Y: list_nat] :
( ~ ( ord_less_list_nat @ X @ Y )
=> ( ord_less_eq_list_nat @ Y @ X ) ) ).
% leI
thf(fact_417_leI,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% leI
thf(fact_418_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_419_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_420_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_421_leD,axiom,
! [Y: list_nat,X: list_nat] :
( ( ord_less_eq_list_nat @ Y @ X )
=> ~ ( ord_less_list_nat @ X @ Y ) ) ).
% leD
thf(fact_422_leD,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_a @ X @ Y ) ) ).
% leD
thf(fact_423_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_424_verit__comp__simplify1_I3_J,axiom,
! [B4: list_nat,A5: list_nat] :
( ( ~ ( ord_less_eq_list_nat @ B4 @ A5 ) )
= ( ord_less_list_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_425_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_426_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_427_sorted__iff__nth__mono__less,axiom,
! [Xs: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ord_less_eq_list_nat @ ( nth_list_nat @ Xs @ I3 ) @ ( nth_list_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_428_sorted__iff__nth__mono__less,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_429_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_430_sorted__iff__nth__mono,axiom,
! [Xs: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ord_less_eq_list_nat @ ( nth_list_nat @ Xs @ I3 ) @ ( nth_list_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_431_sorted__iff__nth__mono,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_432_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_433_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_434_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_435_order__antisym__conv,axiom,
! [Y: list_nat,X: list_nat] :
( ( ord_less_eq_list_nat @ Y @ X )
=> ( ( ord_less_eq_list_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_436_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_437_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_438_linorder__le__cases,axiom,
! [X: list_nat,Y: list_nat] :
( ~ ( ord_less_eq_list_nat @ X @ Y )
=> ( ord_less_eq_list_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_439_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_440_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_441_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_442_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_443_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_444_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_445_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_446_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_447_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > list_nat,C: list_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_450_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_451_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_452_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_453_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_454_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_455_ord__eq__le__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_456_ord__eq__le__subst,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_457_ord__eq__le__subst,axiom,
! [A: list_nat,F: a > list_nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_458_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_459_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_460_ord__eq__le__subst,axiom,
! [A: list_nat,F: nat > list_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_461_linorder__linear,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
| ( ord_less_eq_list_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_462_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_463_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_464_verit__la__disequality,axiom,
! [A: list_nat,B: list_nat] :
( ( A = B )
| ~ ( ord_less_eq_list_nat @ A @ B )
| ~ ( ord_less_eq_list_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_465_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_466_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_467_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_468_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_469_order__eq__refl,axiom,
! [X: list_nat,Y: list_nat] :
( ( X = Y )
=> ( ord_less_eq_list_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_470_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_471_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_472_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_473_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_474_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_475_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_476_order__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_477_order__subst2,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_478_order__subst2,axiom,
! [A: a,B: a,F: a > list_nat,C: list_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_479_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_480_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_481_order__subst2,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_482_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_483_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_484_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 )
=> ( ! [X3: a,Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_485_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 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_486_order__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_487_order__subst1,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_488_order__subst1,axiom,
! [A: a,F: list_nat > a,B: list_nat,C: list_nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_489_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_490_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_491_order__subst1,axiom,
! [A: nat,F: list_nat > nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_492_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_493_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z2: set_nat] : ( Y2 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_494_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [A4: list_nat,B3: list_nat] :
( ( ord_less_eq_list_nat @ A4 @ B3 )
& ( ord_less_eq_list_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_495_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ( ord_less_eq_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_496_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_497_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_498_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_499_antisym,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_500_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_501_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_502_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_503_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_504_dual__order_Otrans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ B @ A )
=> ( ( ord_less_eq_list_nat @ C @ B )
=> ( ord_less_eq_list_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_505_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_506_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_507_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_508_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_509_dual__order_Oantisym,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_eq_list_nat @ B @ A )
=> ( ( ord_less_eq_list_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_510_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_511_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_512_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_513_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z2: set_nat] : ( Y2 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_514_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
= ( ^ [A4: list_nat,B3: list_nat] :
( ( ord_less_eq_list_nat @ B3 @ A4 )
& ( ord_less_eq_list_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_515_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ( ord_less_eq_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_516_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_517_linorder__wlog,axiom,
! [P: list_nat > list_nat > $o,A: list_nat,B: list_nat] :
( ! [A2: list_nat,B2: list_nat] :
( ( ord_less_eq_list_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: list_nat,B2: list_nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_518_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: a,B2: a] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_519_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_520_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_521_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_522_order__trans,axiom,
! [X: list_nat,Y: list_nat,Z: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
=> ( ( ord_less_eq_list_nat @ Y @ Z )
=> ( ord_less_eq_list_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_523_order__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( ord_less_eq_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_524_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_525_count__list__ge__2__iff,axiom,
! [Y: nat,Z: nat,Xs: list_a] :
( ( ord_less_nat @ Y @ Z )
=> ( ( ord_less_nat @ Z @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ Xs @ Y )
= ( nth_a @ Xs @ Z ) )
=> ( ord_less_nat @ one_one_nat @ ( count_list_a @ Xs @ ( nth_a @ Xs @ Y ) ) ) ) ) ) ).
% count_list_ge_2_iff
thf(fact_526_count__list__ge__2__iff,axiom,
! [Y: nat,Z: nat,Xs: list_nat] :
( ( ord_less_nat @ Y @ Z )
=> ( ( ord_less_nat @ Z @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ Y )
= ( nth_nat @ Xs @ Z ) )
=> ( ord_less_nat @ one_one_nat @ ( count_list_nat @ Xs @ ( nth_nat @ Xs @ Y ) ) ) ) ) ) ).
% count_list_ge_2_iff
thf(fact_527_count__list__ge__2__iff,axiom,
! [Y: nat,Z: nat,Xs: list_Pratt_pratt] :
( ( ord_less_nat @ Y @ Z )
=> ( ( ord_less_nat @ Z @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( ( nth_Pratt_pratt @ Xs @ Y )
= ( nth_Pratt_pratt @ Xs @ Z ) )
=> ( ord_less_nat @ one_one_nat @ ( count_4999389613427621831_pratt @ Xs @ ( nth_Pratt_pratt @ Xs @ Y ) ) ) ) ) ) ).
% count_list_ge_2_iff
thf(fact_528__092_060open_062i_A_060_Ak_A_L_A1_092_060close_062,axiom,
ord_less_nat @ i @ ( plus_plus_nat @ k @ one_one_nat ) ).
% \<open>i < k + 1\<close>
thf(fact_529__092_060open_062y_A_092_060in_062_Aset_Axs_092_060close_062,axiom,
member_a @ y @ ( set_a2 @ xs ) ).
% \<open>y \<in> set xs\<close>
thf(fact_530_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_531_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_532_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_533_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_534_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_535_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_536_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_537_add__right__cancel,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= ( plus_plus_multiset_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_538_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_539_add__left__cancel,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= ( plus_plus_multiset_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_540_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_541_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_542_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_543_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_544_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_545_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_546_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_547_that,axiom,
member_a @ y @ ( k_least_a @ ( plus_plus_nat @ k @ one_one_nat ) @ ( set_a2 @ xs ) ) ).
% that
thf(fact_548_subset__code_I1_J,axiom,
! [Xs: list_Pratt_pratt,B5: set_Pratt_pratt] :
( ( ord_le3428837165542985291_pratt @ ( set_Pratt_pratt2 @ Xs ) @ B5 )
= ( ! [X2: pratt_pratt] :
( ( member_Pratt_pratt @ X2 @ ( set_Pratt_pratt2 @ Xs ) )
=> ( member_Pratt_pratt @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_549_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_550_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_551_add__right__imp__eq,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= ( plus_plus_multiset_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_552_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_553_add__left__imp__eq,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= ( plus_plus_multiset_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_554_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_555_add_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ B @ ( plus_plus_set_nat @ A @ C ) )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_556_add_Oleft__commute,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_557_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_558_add_Ocommute,axiom,
( plus_plus_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( plus_plus_set_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_559_add_Ocommute,axiom,
( plus_plus_multiset_a
= ( ^ [A4: multiset_a,B3: multiset_a] : ( plus_plus_multiset_a @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_560_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_561_add_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_562_add_Oassoc,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_563_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_564_group__cancel_Oadd2,axiom,
! [B5: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B5
= ( plus_plus_set_nat @ K @ B ) )
=> ( ( plus_plus_set_nat @ A @ B5 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_565_group__cancel_Oadd2,axiom,
! [B5: multiset_a,K: multiset_a,B: multiset_a,A: multiset_a] :
( ( B5
= ( plus_plus_multiset_a @ K @ B ) )
=> ( ( plus_plus_multiset_a @ A @ B5 )
= ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_566_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_567_group__cancel_Oadd1,axiom,
! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A3
= ( plus_plus_set_nat @ K @ A ) )
=> ( ( plus_plus_set_nat @ A3 @ B )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_568_group__cancel_Oadd1,axiom,
! [A3: multiset_a,K: multiset_a,A: multiset_a,B: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ K @ A ) )
=> ( ( plus_plus_multiset_a @ A3 @ B )
= ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_569_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_570_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_multiset_a @ I @ K )
= ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_571_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_572_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_573_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_574_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_575_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
( ( ( ord_le7200615133682826212iset_a @ I @ J )
& ( K = L ) )
=> ( ord_le7200615133682826212iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_576_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_577_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
( ( ( I = J )
& ( ord_le7200615133682826212iset_a @ K @ L ) )
=> ( ord_le7200615133682826212iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_578_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_579_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
( ( ( ord_le7200615133682826212iset_a @ I @ J )
& ( ord_le7200615133682826212iset_a @ K @ L ) )
=> ( ord_le7200615133682826212iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_580_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_581_add__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( ord_le7200615133682826212iset_a @ A @ B )
=> ( ( ord_le7200615133682826212iset_a @ C @ D )
=> ( ord_le7200615133682826212iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% add_mono
thf(fact_582_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_583_add__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ord_le7200615133682826212iset_a @ A @ B )
=> ( ord_le7200615133682826212iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_584_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_585_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_586_add__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ord_le7200615133682826212iset_a @ A @ B )
=> ( ord_le7200615133682826212iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_587_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_588_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_589_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_590_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_591_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_592_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_593_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_594_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_595_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_596_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_597_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_598_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_599_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_600_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_601_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_602_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_603_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_604_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_605_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_606_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_607_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_608_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_609_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_610_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_611_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_612_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_613_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_614_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_615_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_616_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_617_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_618_sorted__wrt__mono__rel,axiom,
! [Xs: list_Pratt_pratt,P: pratt_pratt > pratt_pratt > $o,Q: pratt_pratt > pratt_pratt > $o] :
( ! [X3: pratt_pratt,Y5: pratt_pratt] :
( ( member_Pratt_pratt @ X3 @ ( set_Pratt_pratt2 @ Xs ) )
=> ( ( member_Pratt_pratt @ Y5 @ ( set_Pratt_pratt2 @ Xs ) )
=> ( ( P @ X3 @ Y5 )
=> ( Q @ X3 @ Y5 ) ) ) )
=> ( ( sorted3866451260844240568_pratt @ P @ Xs )
=> ( sorted3866451260844240568_pratt @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_619_sorted__wrt__mono__rel,axiom,
! [Xs: list_a,P: a > a > $o,Q: a > a > $o] :
( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y5 @ ( set_a2 @ Xs ) )
=> ( ( P @ X3 @ Y5 )
=> ( Q @ X3 @ Y5 ) ) ) )
=> ( ( sorted_wrt_a @ P @ Xs )
=> ( sorted_wrt_a @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_620_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y5: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y5 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y5 )
=> ( Q @ X3 @ Y5 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_621_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_622_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_623_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_624_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_625_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_626_strict__sorted__equal,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_list_nat @ Xs )
=> ( ( sorted_wrt_list_nat @ ord_less_list_nat @ Ys2 )
=> ( ( ( set_list_nat2 @ Ys2 )
= ( set_list_nat2 @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_627_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_628_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_629_count__list__gr__1,axiom,
! [X: pratt_pratt,Xs: list_Pratt_pratt] :
( ( member_Pratt_pratt @ X @ ( set_Pratt_pratt2 @ Xs ) )
= ( ord_less_eq_nat @ one_one_nat @ ( count_4999389613427621831_pratt @ Xs @ X ) ) ) ).
% count_list_gr_1
thf(fact_630_count__list__gr__1,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ord_less_eq_nat @ one_one_nat @ ( count_list_nat @ Xs @ X ) ) ) ).
% count_list_gr_1
thf(fact_631_count__list__gr__1,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ord_less_eq_nat @ one_one_nat @ ( count_list_a @ Xs @ X ) ) ) ).
% count_list_gr_1
thf(fact_632_nth__mem,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N2 ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_633_nth__mem,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_634_nth__mem,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( ord_less_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( member_Pratt_pratt @ ( nth_Pratt_pratt @ Xs @ N2 ) @ ( set_Pratt_pratt2 @ Xs ) ) ) ).
% nth_mem
thf(fact_635_list__ball__nth,axiom,
! [N2: nat,Xs: list_a,P: a > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_a @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_636_list__ball__nth,axiom,
! [N2: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_637_list__ball__nth,axiom,
! [N2: nat,Xs: list_Pratt_pratt,P: pratt_pratt > $o] :
( ( ord_less_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ! [X3: pratt_pratt] :
( ( member_Pratt_pratt @ X3 @ ( set_Pratt_pratt2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_Pratt_pratt @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_638_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_639_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_640_in__set__conv__nth,axiom,
! [X: pratt_pratt,Xs: list_Pratt_pratt] :
( ( member_Pratt_pratt @ X @ ( set_Pratt_pratt2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s4929225773697354557_pratt @ Xs ) )
& ( ( nth_Pratt_pratt @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_641_all__nth__imp__all__set,axiom,
! [Xs: list_a,P: a > $o,X: a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I2 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_642_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_643_all__nth__imp__all__set,axiom,
! [Xs: list_Pratt_pratt,P: pratt_pratt > $o,X: pratt_pratt] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( P @ ( nth_Pratt_pratt @ Xs @ I2 ) ) )
=> ( ( member_Pratt_pratt @ X @ ( set_Pratt_pratt2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_644_all__set__conv__all__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_645_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_646_all__set__conv__all__nth,axiom,
! [Xs: list_Pratt_pratt,P: pratt_pratt > $o] :
( ( ! [X2: pratt_pratt] :
( ( member_Pratt_pratt @ X2 @ ( set_Pratt_pratt2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( P @ ( nth_Pratt_pratt @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_647_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_648_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_649_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_650_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_651_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_652_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_653_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_654_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_655_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_656_size__neq__size__imp__neq,axiom,
! [X: multiset_nat,Y: multiset_nat] :
( ( ( size_s5917832649809541300et_nat @ X )
!= ( size_s5917832649809541300et_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_657_size__neq__size__imp__neq,axiom,
! [X: multiset_Pratt_pratt,Y: multiset_Pratt_pratt] :
( ( ( size_s6011366409283536893_pratt @ X )
!= ( size_s6011366409283536893_pratt @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_658_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_659_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_660_size__neq__size__imp__neq,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( size_size_multiset_a @ X )
!= ( size_size_multiset_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_661_size__neq__size__imp__neq,axiom,
! [X: list_Pratt_pratt,Y: list_Pratt_pratt] :
( ( ( size_s4929225773697354557_pratt @ X )
!= ( size_s4929225773697354557_pratt @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_662_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_663_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_664_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_665_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_666_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_667_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_668__092_060open_062rank__of_A_Ixs_A_B_Ai_J_A_Iset_Axs_J_A_060_Ak_A_L_A1_092_060close_062,axiom,
ord_less_nat @ ( k_rank_of_a @ ( nth_a @ xs @ i ) @ ( set_a2 @ xs ) ) @ ( plus_plus_nat @ k @ one_one_nat ) ).
% \<open>rank_of (xs ! i) (set xs) < k + 1\<close>
thf(fact_669__092_060open_062_092_060And_062y_O_A_092_060lbrakk_062y_A_060_Alength_Axs_059_Arank__of_A_Ixs_A_B_Ay_J_A_Iset_Axs_J_A_060_Ak_A_L_A1_092_060rbrakk_062_A_092_060Longrightarrow_062_Ay_A_060_Ak_A_L_A1_092_060close_062,axiom,
! [Y: nat] :
( ( ord_less_nat @ Y @ ( size_size_list_a @ xs ) )
=> ( ( ord_less_nat @ ( k_rank_of_a @ ( nth_a @ xs @ Y ) @ ( set_a2 @ xs ) ) @ ( plus_plus_nat @ k @ one_one_nat ) )
=> ( ord_less_nat @ Y @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ) ).
% \<open>\<And>y. \<lbrakk>y < length xs; rank_of (xs ! y) (set xs) < k + 1\<rbrakk> \<Longrightarrow> y < k + 1\<close>
thf(fact_670_rank__conv,axiom,
! [Y: nat] :
( ( ord_less_nat @ Y @ ( size_size_list_a @ xs ) )
=> ( ( ord_less_nat @ ( k_rank_of_a @ ( nth_a @ xs @ Y ) @ ( set_a2 @ xs ) ) @ ( plus_plus_nat @ k @ one_one_nat ) )
= ( ord_less_nat @ Y @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ) ).
% rank_conv
thf(fact_671__092_060open_062_092_060And_062y_O_Ay_A_060_Ak_A_L_A1_A_092_060Longrightarrow_062_Arank__of_A_Ixs_A_B_Ay_J_A_Iset_Axs_J_A_060_Ak_A_L_A1_092_060close_062,axiom,
! [Y: nat] :
( ( ord_less_nat @ Y @ ( plus_plus_nat @ k @ one_one_nat ) )
=> ( ord_less_nat @ ( k_rank_of_a @ ( nth_a @ xs @ Y ) @ ( set_a2 @ xs ) ) @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ).
% \<open>\<And>y. y < k + 1 \<Longrightarrow> rank_of (xs ! y) (set xs) < k + 1\<close>
thf(fact_672_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_673_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_674_set__plus__intro,axiom,
! [A: set_nat,C4: set_set_nat,B: set_nat,D2: set_set_nat] :
( ( member_set_nat @ A @ C4 )
=> ( ( member_set_nat @ B @ D2 )
=> ( member_set_nat @ ( plus_plus_set_nat @ A @ B ) @ ( plus_p4817606893110106565et_nat @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_675_set__plus__intro,axiom,
! [A: multiset_a,C4: set_multiset_a,B: multiset_a,D2: set_multiset_a] :
( ( member_multiset_a @ A @ C4 )
=> ( ( member_multiset_a @ B @ D2 )
=> ( member_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ ( plus_p5152935555875550043iset_a @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_676_set__plus__intro,axiom,
! [A: nat,C4: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C4 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_677_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_678_assms_I1_J,axiom,
ord_less_nat @ k @ ( size_size_multiset_a @ a2 ) ).
% assms(1)
thf(fact_679_set__plus__mono2,axiom,
! [C4: set_nat,D2: set_nat,E: set_nat,F2: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ D2 )
=> ( ( ord_less_eq_set_nat @ E @ F2 )
=> ( ord_less_eq_set_nat @ ( plus_plus_set_nat @ C4 @ E ) @ ( plus_plus_set_nat @ D2 @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_680_ex__subset,axiom,
! [A3: set_nat,P: nat > $o,B5: set_nat] :
( ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ( P @ X6 ) )
=> ( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B5 )
& ( P @ X3 ) ) ) ) ).
% ex_subset
thf(fact_681_least__subset,axiom,
! [K: nat,S2: set_a] : ( ord_less_eq_set_a @ ( k_least_a @ K @ S2 ) @ S2 ) ).
% least_subset
thf(fact_682_set__plus__elim,axiom,
! [X: nat,A3: set_nat,B5: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A3 @ B5 ) )
=> ~ ! [A2: nat,B2: nat] :
( ( X
= ( plus_plus_nat @ A2 @ B2 ) )
=> ( ( member_nat @ A2 @ A3 )
=> ~ ( member_nat @ B2 @ B5 ) ) ) ) ).
% set_plus_elim
thf(fact_683_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_684_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_685_size__union,axiom,
! [M5: multiset_a,N4: multiset_a] :
( ( size_size_multiset_a @ ( plus_plus_multiset_a @ M5 @ N4 ) )
= ( plus_plus_nat @ ( size_size_multiset_a @ M5 ) @ ( size_size_multiset_a @ N4 ) ) ) ).
% size_union
thf(fact_686_subsetI,axiom,
! [A3: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B5 ) )
=> ( ord_less_eq_set_a @ A3 @ B5 ) ) ).
% subsetI
thf(fact_687_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 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_688_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ J2 ) @ ( nth_a @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_689_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_690_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_691_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_692_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_693_length__rev,axiom,
! [Xs: list_Pratt_pratt] :
( ( size_s4929225773697354557_pratt @ ( rev_Pratt_pratt @ Xs ) )
= ( size_s4929225773697354557_pratt @ Xs ) ) ).
% length_rev
thf(fact_694_count__list__rev,axiom,
! [Xs: list_a,X: a] :
( ( count_list_a @ ( rev_a @ Xs ) @ X )
= ( count_list_a @ Xs @ X ) ) ).
% count_list_rev
thf(fact_695_psubsetD,axiom,
! [A3: set_a,B5: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B5 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_696_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_697_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_698_subsetD,axiom,
! [A3: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_699_in__mono,axiom,
! [A3: set_a,B5: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_700_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_701_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_702_xs__def,axiom,
( xs
= ( linord814965612141868908iset_a @ a2 ) ) ).
% xs_def
thf(fact_703_A__def,axiom,
( a2
= ( mset_a @ xs ) ) ).
% A_def
thf(fact_704_assms_I2_J,axiom,
! [X: a] :
( ( ord_less_eq_a @ X @ ( k_nth_mset_a @ k @ a2 ) )
=> ( ord_less_eq_nat @ ( count_a @ a2 @ X ) @ one_one_nat ) ) ).
% assms(2)
thf(fact_705_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ ( rev_a @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ ( suc @ I3 ) ) @ ( nth_a @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_706_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I3 ) ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_707_nth__mset__bound__left,axiom,
! [K: nat,M5: multiset_a,X: a] :
( ( ord_less_nat @ K @ ( size_size_multiset_a @ M5 ) )
=> ( ( ord_less_eq_nat @ ( k_count_less_a @ X @ M5 ) @ K )
=> ( ord_less_eq_a @ X @ ( k_nth_mset_a @ K @ M5 ) ) ) ) ).
% nth_mset_bound_left
thf(fact_708_nth__mset__bound__left,axiom,
! [K: nat,M5: multiset_nat,X: nat] :
( ( ord_less_nat @ K @ ( size_s5917832649809541300et_nat @ M5 ) )
=> ( ( ord_less_eq_nat @ ( k_count_less_nat @ X @ M5 ) @ K )
=> ( ord_less_eq_nat @ X @ ( k_nth_mset_nat @ K @ M5 ) ) ) ) ).
% nth_mset_bound_left
thf(fact_709_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_710_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_711_rotate1__length01,axiom,
! [Xs: list_Pratt_pratt] :
( ( ord_less_eq_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ one_one_nat )
=> ( ( rotate1_Pratt_pratt @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_712_sorted__iff__nth__Suc,axiom,
! [Xs: list_a] :
( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_713_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_714_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_715_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_716_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_717_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_718_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_719_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_720_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_721_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_722_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_723_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_724_length__rotate1,axiom,
! [Xs: list_Pratt_pratt] :
( ( size_s4929225773697354557_pratt @ ( rotate1_Pratt_pratt @ Xs ) )
= ( size_s4929225773697354557_pratt @ Xs ) ) ).
% length_rotate1
thf(fact_725_mset__sorted__list__of__multiset,axiom,
! [M5: multiset_a] :
( ( mset_a @ ( linord814965612141868908iset_a @ M5 ) )
= M5 ) ).
% mset_sorted_list_of_multiset
thf(fact_726_count__union,axiom,
! [M5: multiset_a,N4: multiset_a,A: a] :
( ( count_a @ ( plus_plus_multiset_a @ M5 @ N4 ) @ A )
= ( plus_plus_nat @ ( count_a @ M5 @ A ) @ ( count_a @ N4 @ A ) ) ) ).
% count_union
thf(fact_727_size__mset,axiom,
! [Xs: list_nat] :
( ( size_s5917832649809541300et_nat @ ( mset_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% size_mset
thf(fact_728_size__mset,axiom,
! [Xs: list_Pratt_pratt] :
( ( size_s6011366409283536893_pratt @ ( mset_Pratt_pratt @ Xs ) )
= ( size_s4929225773697354557_pratt @ Xs ) ) ).
% size_mset
thf(fact_729_size__mset,axiom,
! [Xs: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% size_mset
thf(fact_730_multiset__eq__iff,axiom,
( ( ^ [Y2: multiset_a,Z2: multiset_a] : ( Y2 = Z2 ) )
= ( ^ [M6: multiset_a,N5: multiset_a] :
! [A4: a] :
( ( count_a @ M6 @ A4 )
= ( count_a @ N5 @ A4 ) ) ) ) ).
% multiset_eq_iff
thf(fact_731_multiset__eqI,axiom,
! [A3: multiset_a,B5: multiset_a] :
( ! [X3: a] :
( ( count_a @ A3 @ X3 )
= ( count_a @ B5 @ X3 ) )
=> ( A3 = B5 ) ) ).
% multiset_eqI
thf(fact_732_ex__mset,axiom,
! [X7: multiset_a] :
? [Xs2: list_a] :
( ( mset_a @ Xs2 )
= X7 ) ).
% ex_mset
thf(fact_733_count__inject,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( count_a @ X )
= ( count_a @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_734_Frequency__Moments__Preliminary__Results_Ocount__mset,axiom,
! [Xs: list_a,A: a] :
( ( count_a @ ( mset_a @ Xs ) @ A )
= ( count_list_a @ Xs @ A ) ) ).
% Frequency_Moments_Preliminary_Results.count_mset
thf(fact_735_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_736_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_737_nth__mset__def,axiom,
( k_nth_mset_nat
= ( ^ [K3: nat,M6: multiset_nat] : ( nth_nat @ ( linord3047872887403683810et_nat @ M6 ) @ K3 ) ) ) ).
% nth_mset_def
thf(fact_738_nth__mset__def,axiom,
( k_nth_mset_a
= ( ^ [K3: nat,M6: multiset_a] : ( nth_a @ ( linord814965612141868908iset_a @ M6 ) @ K3 ) ) ) ).
% nth_mset_def
thf(fact_739_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_740_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_741_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_742_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_743_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_744_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_745_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_746_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_747_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_748_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_749_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_750_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_751_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_752_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_753_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_754_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_755_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_756_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_757_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_758_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_759_Suc__le__D,axiom,
! [N2: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M8 )
=> ? [M3: nat] :
( M8
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_760_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_761_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_762_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_763_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_764_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_765_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y5: nat,Z3: nat] :
( ( R @ X3 @ Y5 )
=> ( ( R @ Y5 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_766_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_767_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_768_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_769_plus__multiset_Orep__eq,axiom,
! [X: multiset_a,Xa: multiset_a] :
( ( count_a @ ( plus_plus_multiset_a @ X @ Xa ) )
= ( ^ [A4: a] : ( plus_plus_nat @ ( count_a @ X @ A4 ) @ ( count_a @ Xa @ A4 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_770_list__eq__iff,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys2 ) )
=> ( ( sorted_wrt_a @ ord_less_eq_a @ Xs )
=> ( ( sorted_wrt_a @ ord_less_eq_a @ Ys2 )
=> ( Xs = Ys2 ) ) ) ) ).
% list_eq_iff
thf(fact_771_list__eq__iff,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys2 )
=> ( Xs = Ys2 ) ) ) ) ).
% list_eq_iff
thf(fact_772_lift__Suc__antimono__le,axiom,
! [F: nat > a,N2: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_a @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_773_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_774_lift__Suc__mono__le,axiom,
! [F: nat > a,N2: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_eq_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_a @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_775_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_776_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_777_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_778_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_779_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_780_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_781_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_782_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_783_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_784_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_785_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_786_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_787_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_788_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_789_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_790_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_791_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_792_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_793_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_794_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_795_sorted__sorted__list__of__multiset,axiom,
! [M5: multiset_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord3047872887403683810et_nat @ M5 ) ) ).
% sorted_sorted_list_of_multiset
thf(fact_796_sorted__sorted__list__of__multiset,axiom,
! [M5: multiset_a] : ( sorted_wrt_a @ ord_less_eq_a @ ( linord814965612141868908iset_a @ M5 ) ) ).
% sorted_sorted_list_of_multiset
thf(fact_797__092_060open_062_092_060And_062xa_O_Axa_A_092_060le_062_Axs_A_B_Ak_A_092_060Longrightarrow_062_Acount_AA_Axa_A_092_060le_062_ASuc_A0_092_060close_062,axiom,
! [X: a] :
( ( ord_less_eq_a @ X @ ( nth_a @ xs @ k ) )
=> ( ord_less_eq_nat @ ( count_a @ a2 @ X ) @ ( suc @ zero_zero_nat ) ) ) ).
% \<open>\<And>xa. xa \<le> xs ! k \<Longrightarrow> count A xa \<le> Suc 0\<close>
thf(fact_798_perm__rev,axiom,
! [Xs: list_a] :
( ( mset_a @ ( rev_a @ Xs ) )
= ( mset_a @ Xs ) ) ).
% perm_rev
thf(fact_799_nth__mset__bound__left__excl,axiom,
! [K: nat,M5: multiset_a,X: a] :
( ( ord_less_nat @ K @ ( size_size_multiset_a @ M5 ) )
=> ( ( ord_less_eq_nat @ ( k_count_le_a @ X @ M5 ) @ K )
=> ( ord_less_a @ X @ ( k_nth_mset_a @ K @ M5 ) ) ) ) ).
% nth_mset_bound_left_excl
thf(fact_800_nth__mset__bound__left__excl,axiom,
! [K: nat,M5: multiset_nat,X: nat] :
( ( ord_less_nat @ K @ ( size_s5917832649809541300et_nat @ M5 ) )
=> ( ( ord_less_eq_nat @ ( k_count_le_nat @ X @ M5 ) @ K )
=> ( ord_less_nat @ X @ ( k_nth_mset_nat @ K @ M5 ) ) ) ) ).
% nth_mset_bound_left_excl
thf(fact_801_nth__mset__bound__right,axiom,
! [K: nat,M5: multiset_a,X: a] :
( ( ord_less_nat @ K @ ( size_size_multiset_a @ M5 ) )
=> ( ( ord_less_nat @ K @ ( k_count_le_a @ X @ M5 ) )
=> ( ord_less_eq_a @ ( k_nth_mset_a @ K @ M5 ) @ X ) ) ) ).
% nth_mset_bound_right
thf(fact_802_nth__mset__bound__right,axiom,
! [K: nat,M5: multiset_nat,X: nat] :
( ( ord_less_nat @ K @ ( size_s5917832649809541300et_nat @ M5 ) )
=> ( ( ord_less_nat @ K @ ( k_count_le_nat @ X @ M5 ) )
=> ( ord_less_eq_nat @ ( k_nth_mset_nat @ K @ M5 ) @ X ) ) ) ).
% nth_mset_bound_right
thf(fact_803_perm__length,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) ) ) ).
% perm_length
thf(fact_804_perm__length,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% perm_length
thf(fact_805_perm__length,axiom,
! [Xs: list_Pratt_pratt,Ys2: list_Pratt_pratt] :
( ( ( mset_Pratt_pratt @ Xs )
= ( mset_Pratt_pratt @ Ys2 ) )
=> ( ( size_s4929225773697354557_pratt @ Xs )
= ( size_s4929225773697354557_pratt @ Ys2 ) ) ) ).
% perm_length
thf(fact_806_perm__setP,axiom,
! [As: list_a,Bs: list_a,P: set_a > $o] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( P @ ( set_a2 @ As ) )
=> ( P @ ( set_a2 @ Bs ) ) ) ) ).
% perm_setP
thf(fact_807_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_808_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_809_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_810_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_811_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_812_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_813_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_814_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_815_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_816_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_817_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_818_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_819_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_820_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_821_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_822_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_823_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_824_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_825_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_826_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_827_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_828_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_829_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_830_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_831_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_832_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_833_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_834_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_835_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_836_count__notin,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( count_list_a @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_837_count__mset__0__iff,axiom,
! [Xs: list_a,X: a] :
( ( ( count_a @ ( mset_a @ Xs ) @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_838_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_839_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_840_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_841_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_842_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_843_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_844_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_845_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_846_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_847_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_848_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_849_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_850_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_851_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_852_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_853_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_854_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_855_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_856_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] :
( ( P @ X3 @ Y5 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_857_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_858_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_859_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_860_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_861_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_862_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_863_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_864_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_865_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_866_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_867_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_868_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_869_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_870_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_871_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_872_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_873_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_874_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_875_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_876_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_877_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_878_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_879_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_880_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_881_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_882_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_883_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_884_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_885_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_886_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_887_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_888_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_889_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_890_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_891_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_892_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_893_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_894_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_895_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_896_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_897_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_898_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_899_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_900_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_901_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_902_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_903_set__zero__plus2,axiom,
! [A3: set_nat,B5: set_nat] :
( ( member_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_set_nat @ B5 @ ( plus_plus_set_nat @ A3 @ B5 ) ) ) ).
% set_zero_plus2
thf(fact_904_count__list__0__iff,axiom,
! [Xs: list_a,X: a] :
( ( ( count_list_a @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_905_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_906_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_907_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_908_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_909_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_910_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_911_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_912_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_913_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_914_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_915_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_916_length__pos__if__in__set,axiom,
! [X: pratt_pratt,Xs: list_Pratt_pratt] :
( ( member_Pratt_pratt @ X @ ( set_Pratt_pratt2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s4929225773697354557_pratt @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_917_perm__sym,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys2 ) )
=> ( ( mset_a @ Ys2 )
= ( mset_a @ Xs ) ) ) ).
% perm_sym
thf(fact_918_count__mset__gt__0,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_a @ ( mset_a @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_919_perm__set__eq,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys2 ) )
=> ( ( set_a2 @ Xs )
= ( set_a2 @ Ys2 ) ) ) ).
% perm_set_eq
thf(fact_920_count__empty,axiom,
! [A: a] :
( ( count_a @ zero_zero_multiset_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_921_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_922_size__eq__0__iff__empty,axiom,
! [M5: multiset_a] :
( ( ( size_size_multiset_a @ M5 )
= zero_zero_nat )
= ( M5 = zero_zero_multiset_a ) ) ).
% size_eq_0_iff_empty
thf(fact_923_zero__multiset_Orep__eq,axiom,
( ( count_a @ zero_zero_multiset_a )
= ( ^ [A4: a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_924_nonempty__has__size,axiom,
! [S2: multiset_a] :
( ( S2 != zero_zero_multiset_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_925_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
= ( P @ B2 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_926_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_927_fib_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ ( suc @ N3 ) ) ) ) ) ).
% fib.cases
thf(fact_928_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_929_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_930_encode__unary__nat_Ocases,axiom,
! [X: nat] :
( ! [L2: nat] :
( X
!= ( suc @ L2 ) )
=> ( X = zero_zero_nat ) ) ).
% encode_unary_nat.cases
thf(fact_931_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_932_rev__nth,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rev_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_933_rev__nth,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_934_rev__nth,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( ord_less_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( nth_Pratt_pratt @ ( rev_Pratt_pratt @ Xs ) @ N2 )
= ( nth_Pratt_pratt @ Xs @ ( minus_minus_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_935_nth__rotate1,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate1_a @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_936_nth__rotate1,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_937_nth__rotate1,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( ord_less_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( nth_Pratt_pratt @ ( rotate1_Pratt_pratt @ Xs ) @ N2 )
= ( nth_Pratt_pratt @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_s4929225773697354557_pratt @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_938_gen__length__def,axiom,
( gen_length_a
= ( ^ [N: nat,Xs3: list_a] : ( plus_plus_nat @ N @ ( size_size_list_a @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_939_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N: nat,Xs3: list_nat] : ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_940_gen__length__def,axiom,
( gen_le3130621340910794462_pratt
= ( ^ [N: nat,Xs3: list_Pratt_pratt] : ( plus_plus_nat @ N @ ( size_s4929225773697354557_pratt @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_941_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_942_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_943_length__code,axiom,
( size_s4929225773697354557_pratt
= ( gen_le3130621340910794462_pratt @ zero_zero_nat ) ) ).
% length_code
thf(fact_944_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_945_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_946_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_947_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_948_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_949_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_950_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_951_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_952_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_953_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_954_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_955_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_956_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_957_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_958_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_959_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_960_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_961_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_962_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_963_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_964_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_965_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_966_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_967_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_968_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_969_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_970_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_971_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_972_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_973_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_974_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_975_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_976_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_977_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_978_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_979_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_980_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_981_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_982_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_983_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_984_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_985_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_986_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_987_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_988_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_989_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_990_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_991_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_992_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_993_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_994_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_995_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_996_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_997_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_998_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_999_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1000_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1001_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1002_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1003_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1004_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1005_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1006_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1007_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1008_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
=> ( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M3 @ N3 ) )
=> ( P @ M3 @ N3 ) ) )
=> ( P @ M2 @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_1009_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1010_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1011_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1012_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1013_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1014_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1015_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_1016_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1017_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1018_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1019_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1020_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1021_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1022_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1023_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1024_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1025_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1026_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1027_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1028_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1029_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1030_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1031_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1032_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_1033_mod__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( modulo_modulo_nat @ M2 @ N2 )
= M2 ) ) ).
% mod_less
thf(fact_1034_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_1035_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_1036_count__diff,axiom,
! [M5: multiset_a,N4: multiset_a,A: a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ M5 @ N4 ) @ A )
= ( minus_minus_nat @ ( count_a @ M5 @ A ) @ ( count_a @ N4 @ A ) ) ) ).
% count_diff
thf(fact_1037_psubset__imp__ex__mem,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_set_a @ A3 @ B5 )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1038_minus__multiset_Orep__eq,axiom,
! [X: multiset_a,Xa: multiset_a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ X @ Xa ) )
= ( ^ [A4: a] : ( minus_minus_nat @ ( count_a @ X @ A4 ) @ ( count_a @ Xa @ A4 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_1039_diff__size__le__size__Diff,axiom,
! [M5: multiset_a,M9: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M5 ) @ ( size_size_multiset_a @ M9 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ M9 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1040_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1041_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1042_mod__add__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B4: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B4 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1043_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1044_mod__Suc__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N2 ) ) ).
% mod_Suc_Suc_eq
thf(fact_1045_mod__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).
% mod_Suc_eq
thf(fact_1046_mod__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_1047_mod__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
!= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ) ).
% mod_Suc
thf(fact_1048_mod__less__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_1049_mod__induct,axiom,
! [P: nat > $o,N2: nat,P4: nat,M2: nat] :
( ( P @ N2 )
=> ( ( ord_less_nat @ N2 @ P4 )
=> ( ( ord_less_nat @ M2 @ P4 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P4 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_1050_mod__Suc__le__divisor,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_1051_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M @ N ) @ M @ ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ).
% mod_if
thf(fact_1052_le__mod__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( modulo_modulo_nat @ M2 @ N2 )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_1053_mod__le__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_1054_nth__rotate,axiom,
! [N2: nat,Xs: list_a,M2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate_a @ M2 @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1055_nth__rotate,axiom,
! [N2: nat,Xs: list_nat,M2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M2 @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1056_nth__rotate,axiom,
! [N2: nat,Xs: list_Pratt_pratt,M2: nat] :
( ( ord_less_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( nth_Pratt_pratt @ ( rotate_Pratt_pratt @ M2 @ Xs ) @ N2 )
= ( nth_Pratt_pratt @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( size_s4929225773697354557_pratt @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1057_nth__Cons__pos,axiom,
! [N2: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1058_nth__Cons__pos,axiom,
! [N2: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1059_DiffI,axiom,
! [C: a,A3: set_a,B5: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1060_Diff__iff,axiom,
! [C: a,A3: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B5 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_1061_cons__perm__eq,axiom,
! [Z: a,Xs: list_a,Ys2: list_a] :
( ( ( mset_a @ ( cons_a @ Z @ Xs ) )
= ( mset_a @ ( cons_a @ Z @ Ys2 ) ) )
= ( ( mset_a @ Xs )
= ( mset_a @ Ys2 ) ) ) ).
% cons_perm_eq
thf(fact_1062_set__rotate,axiom,
! [N2: nat,Xs: list_a] :
( ( set_a2 @ ( rotate_a @ N2 @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate
thf(fact_1063_length__rotate,axiom,
! [N2: nat,Xs: list_a] :
( ( size_size_list_a @ ( rotate_a @ N2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate
thf(fact_1064_length__rotate,axiom,
! [N2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_1065_length__rotate,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( size_s4929225773697354557_pratt @ ( rotate_Pratt_pratt @ N2 @ Xs ) )
= ( size_s4929225773697354557_pratt @ Xs ) ) ).
% length_rotate
thf(fact_1066_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1067_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1068_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N2: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N2 ) )
= ( nth_a @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_1069_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N2: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N2 ) )
= ( nth_nat @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_1070_rotate__length01,axiom,
! [Xs: list_a,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate_a @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1071_rotate__length01,axiom,
! [Xs: list_nat,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1072_rotate__length01,axiom,
! [Xs: list_Pratt_pratt,N2: nat] :
( ( ord_less_eq_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ one_one_nat )
=> ( ( rotate_Pratt_pratt @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1073_rotate__id,axiom,
! [N2: nat,Xs: list_a] :
( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_a @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1074_rotate__id,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1075_rotate__id,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( ( modulo_modulo_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_Pratt_pratt @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1076_DiffE,axiom,
! [C: a,A3: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B5 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B5 ) ) ) ).
% DiffE
thf(fact_1077_DiffD1,axiom,
! [C: a,A3: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B5 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1078_DiffD2,axiom,
! [C: a,A3: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B5 ) )
=> ~ ( member_a @ C @ B5 ) ) ).
% DiffD2
thf(fact_1079_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_1080_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_1081_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_1082_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_1083_cons__perm__imp__perm,axiom,
! [Z: a,Xs: list_a,Ys2: list_a] :
( ( ( mset_a @ ( cons_a @ Z @ Xs ) )
= ( mset_a @ ( cons_a @ Z @ Ys2 ) ) )
=> ( ( mset_a @ Xs )
= ( mset_a @ Ys2 ) ) ) ).
% cons_perm_imp_perm
thf(fact_1084_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1085_Suc__length__conv,axiom,
! [N2: nat,Xs: list_a] :
( ( ( suc @ N2 )
= ( size_size_list_a @ Xs ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_1086_Suc__length__conv,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( suc @ N2 )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_1087_Suc__length__conv,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( ( suc @ N2 )
= ( size_s4929225773697354557_pratt @ Xs ) )
= ( ? [Y3: pratt_pratt,Ys3: list_Pratt_pratt] :
( ( Xs
= ( cons_Pratt_pratt @ Y3 @ Ys3 ) )
& ( ( size_s4929225773697354557_pratt @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_1088_length__Suc__conv,axiom,
! [Xs: list_a,N2: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N2 ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_1089_length__Suc__conv,axiom,
! [Xs: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N2 ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_1090_length__Suc__conv,axiom,
! [Xs: list_Pratt_pratt,N2: nat] :
( ( ( size_s4929225773697354557_pratt @ Xs )
= ( suc @ N2 ) )
= ( ? [Y3: pratt_pratt,Ys3: list_Pratt_pratt] :
( ( Xs
= ( cons_Pratt_pratt @ Y3 @ Ys3 ) )
& ( ( size_s4929225773697354557_pratt @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_1091_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_1092_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_1093_impossible__Cons,axiom,
! [Xs: list_Pratt_pratt,Ys2: list_Pratt_pratt,X: pratt_pratt] :
( ( ord_less_eq_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ ( size_s4929225773697354557_pratt @ Ys2 ) )
=> ( Xs
!= ( cons_Pratt_pratt @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1094_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_1095_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_1096_rotate__conv__mod,axiom,
( rotate_a
= ( ^ [N: nat,Xs3: list_a] : ( rotate_a @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_1097_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N: nat,Xs3: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_1098_rotate__conv__mod,axiom,
( rotate_Pratt_pratt
= ( ^ [N: nat,Xs3: list_Pratt_pratt] : ( rotate_Pratt_pratt @ ( modulo_modulo_nat @ N @ ( size_s4929225773697354557_pratt @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_1099_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_a @ Xs ) )
= ( ? [X2: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N2 @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1100_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1101_Suc__le__length__iff,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_s4929225773697354557_pratt @ Xs ) )
= ( ? [X2: pratt_pratt,Ys3: list_Pratt_pratt] :
( ( Xs
= ( cons_Pratt_pratt @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N2 @ ( size_s4929225773697354557_pratt @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1102_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_1103_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_1104_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_1105_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_1106_count__list_Osimps_I2_J,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( ( X = Y )
=> ( ( count_list_a @ ( cons_a @ X @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_a @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_a @ ( cons_a @ X @ Xs ) @ Y )
= ( count_list_a @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_1107_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1108_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1109_list_Osize_I4_J,axiom,
! [X21: pratt_pratt,X222: list_Pratt_pratt] :
( ( size_s4929225773697354557_pratt @ ( cons_Pratt_pratt @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_s4929225773697354557_pratt @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1110_nth__Cons_H,axiom,
! [N2: nat,X: a,Xs: list_a] :
( ( ( N2 = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N2 )
= X ) )
& ( ( N2 != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N2 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1111_nth__Cons_H,axiom,
! [N2: nat,X: nat,Xs: list_nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
= X ) )
& ( ( N2 != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1112_nth__equal__first__eq,axiom,
! [X: a,Xs: list_a,N2: nat] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N2 )
= X )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1113_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N2: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
= X )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1114_nth__equal__first__eq,axiom,
! [X: pratt_pratt,Xs: list_Pratt_pratt,N2: nat] :
( ~ ( member_Pratt_pratt @ X @ ( set_Pratt_pratt2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( ( nth_Pratt_pratt @ ( cons_Pratt_pratt @ X @ Xs ) @ N2 )
= X )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1115_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs: list_a,N2: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N2 )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1116_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N2: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1117_rotate__rev,axiom,
! [N2: nat,Xs: list_a] :
( ( rotate_a @ N2 @ ( rev_a @ Xs ) )
= ( rev_a @ ( rotate_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( modulo_modulo_nat @ N2 @ ( size_size_list_a @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1118_rotate__rev,axiom,
! [N2: nat,Xs: list_nat] :
( ( rotate_nat @ N2 @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1119_rotate__rev,axiom,
! [N2: nat,Xs: list_Pratt_pratt] :
( ( rotate_Pratt_pratt @ N2 @ ( rev_Pratt_pratt @ Xs ) )
= ( rev_Pratt_pratt @ ( rotate_Pratt_pratt @ ( minus_minus_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ ( modulo_modulo_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1120_Cons__less__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X @ Y ) ) ) ) ).
% Cons_less_Cons
thf(fact_1121_Cons__le__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ).
% Cons_le_Cons
thf(fact_1122_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_1123_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_1124_length__Cons,axiom,
! [X: pratt_pratt,Xs: list_Pratt_pratt] :
( ( size_s4929225773697354557_pratt @ ( cons_Pratt_pratt @ X @ Xs ) )
= ( suc @ ( size_s4929225773697354557_pratt @ Xs ) ) ) ).
% length_Cons
thf(fact_1125_less__eq__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_eq_list_nat @ Xs @ Ys2 ) ) ) ) ).
% less_eq_list_code(3)
thf(fact_1126_less__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_list_nat @ Xs @ Ys2 ) ) ) ) ).
% less_list_code(3)
thf(fact_1127_verit__le__mono__div,axiom,
! [A3: nat,B5: nat,N2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N2 )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N2 )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1128_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1129_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1130_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1131_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1132_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1133_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1134_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1135_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1136_div__le__mono,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1137_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_1138_Suc__div__le__mono,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1139_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_1140_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1141_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1142_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1143_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1144_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1145_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1146_div__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( divide_divide_nat @ M2 @ N2 ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M2 ) @ N2 )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% div_Suc
thf(fact_1147_div__less__mono,axiom,
! [A3: nat,B5: nat,N2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( modulo_modulo_nat @ A3 @ N2 )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N2 )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A3 @ N2 ) @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1148_div__if,axiom,
( divide_divide_nat
= ( ^ [M: nat,N: nat] :
( if_nat
@ ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_1149_le__div__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% le_div_geq
thf(fact_1150_div__gt__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ M2 ) ) ) ) ).
% div_gt_0
thf(fact_1151_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_1152_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_1153_length__list__update,axiom,
! [Xs: list_Pratt_pratt,I: nat,X: pratt_pratt] :
( ( size_s4929225773697354557_pratt @ ( list_u6220086826248601529_pratt @ Xs @ I @ X ) )
= ( size_s4929225773697354557_pratt @ Xs ) ) ).
% length_list_update
thf(fact_1154_list__update__id,axiom,
! [Xs: list_a,I: nat] :
( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_1155_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_1156_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_1157_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_1158_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_1159_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_1160_list__update__beyond,axiom,
! [Xs: list_Pratt_pratt,I: nat,X: pratt_pratt] :
( ( ord_less_eq_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ I )
=> ( ( list_u6220086826248601529_pratt @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1161_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_1162_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_1163_nth__list__update__eq,axiom,
! [I: nat,Xs: list_Pratt_pratt,X: pratt_pratt] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( nth_Pratt_pratt @ ( list_u6220086826248601529_pratt @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_1164_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_1165_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_1166_set__swap,axiom,
! [I: nat,Xs: list_Pratt_pratt,J: nat] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( set_Pratt_pratt2 @ ( list_u6220086826248601529_pratt @ ( list_u6220086826248601529_pratt @ Xs @ I @ ( nth_Pratt_pratt @ Xs @ J ) ) @ J @ ( nth_Pratt_pratt @ Xs @ I ) ) )
= ( set_Pratt_pratt2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1167_set__update__subsetI,axiom,
! [Xs: list_a,A3: set_a,X: a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A3 )
=> ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_1168_set__update__memI,axiom,
! [N2: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1169_set__update__memI,axiom,
! [N2: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1170_set__update__memI,axiom,
! [N2: nat,Xs: list_Pratt_pratt,X: pratt_pratt] :
( ( ord_less_nat @ N2 @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( member_Pratt_pratt @ X @ ( set_Pratt_pratt2 @ ( list_u6220086826248601529_pratt @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1171_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_1172_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_1173_list__update__same__conv,axiom,
! [I: nat,Xs: list_Pratt_pratt,X: pratt_pratt] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( ( list_u6220086826248601529_pratt @ Xs @ I @ X )
= Xs )
= ( ( nth_Pratt_pratt @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1174_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_1175_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_1176_nth__list__update,axiom,
! [I: nat,Xs: list_Pratt_pratt,J: nat,X: pratt_pratt] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_Pratt_pratt @ ( list_u6220086826248601529_pratt @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_Pratt_pratt @ ( list_u6220086826248601529_pratt @ Xs @ I @ X ) @ J )
= ( nth_Pratt_pratt @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1177_mset__swap,axiom,
! [I: nat,Ls: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls @ J @ ( nth_a @ Ls @ I ) ) @ I @ ( nth_a @ Ls @ J ) ) )
= ( mset_a @ Ls ) ) ) ) ).
% mset_swap
thf(fact_1178_mset__swap,axiom,
! [I: nat,Ls: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ls ) )
=> ( ( mset_nat @ ( list_update_nat @ ( list_update_nat @ Ls @ J @ ( nth_nat @ Ls @ I ) ) @ I @ ( nth_nat @ Ls @ J ) ) )
= ( mset_nat @ Ls ) ) ) ) ).
% mset_swap
thf(fact_1179_mset__swap,axiom,
! [I: nat,Ls: list_Pratt_pratt,J: nat] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_s4929225773697354557_pratt @ Ls ) )
=> ( ( mset_Pratt_pratt @ ( list_u6220086826248601529_pratt @ ( list_u6220086826248601529_pratt @ Ls @ J @ ( nth_Pratt_pratt @ Ls @ I ) ) @ I @ ( nth_Pratt_pratt @ Ls @ J ) ) )
= ( mset_Pratt_pratt @ Ls ) ) ) ) ).
% mset_swap
thf(fact_1180_perm__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 ) )
=> ( ( mset_a @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
= ( mset_a @ Xs ) ) ) ) ).
% perm_swap
thf(fact_1181_perm__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 ) )
=> ( ( mset_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( mset_nat @ Xs ) ) ) ) ).
% perm_swap
thf(fact_1182_perm__swap,axiom,
! [I: nat,Xs: list_Pratt_pratt,J: nat] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( mset_Pratt_pratt @ ( list_u6220086826248601529_pratt @ ( list_u6220086826248601529_pratt @ Xs @ I @ ( nth_Pratt_pratt @ Xs @ J ) ) @ J @ ( nth_Pratt_pratt @ Xs @ I ) ) )
= ( mset_Pratt_pratt @ Xs ) ) ) ) ).
% perm_swap
thf(fact_1183_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_1184_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_1185_rev__update,axiom,
! [K: nat,Xs: list_Pratt_pratt,Y: pratt_pratt] :
( ( ord_less_nat @ K @ ( size_s4929225773697354557_pratt @ Xs ) )
=> ( ( rev_Pratt_pratt @ ( list_u6220086826248601529_pratt @ Xs @ K @ Y ) )
= ( list_u6220086826248601529_pratt @ ( rev_Pratt_pratt @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_s4929225773697354557_pratt @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1186_Pratt__Certificate_Oeval__mod__exp_I11_J,axiom,
! [E3: nat,M8: nat] :
( ( pratt_mod_exp_nat @ ( suc @ zero_zero_nat ) @ E3 @ M8 )
= ( modulo_modulo_nat @ one_one_nat @ M8 ) ) ).
% Pratt_Certificate.eval_mod_exp(11)
thf(fact_1187_Pratt__Certificate_Oeval__mod__exp_I6_J,axiom,
! [B4: nat,E3: nat] :
( ( pratt_mod_exp_nat @ B4 @ E3 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% Pratt_Certificate.eval_mod_exp(6)
thf(fact_1188_Pratt__Certificate_Oeval__mod__exp_I8_J,axiom,
! [M8: nat] :
( ( pratt_mod_exp_nat @ zero_zero_nat @ ( suc @ zero_zero_nat ) @ M8 )
= zero_zero_nat ) ).
% Pratt_Certificate.eval_mod_exp(8)
thf(fact_1189_Pratt__Certificate_Oeval__mod__exp_I7_J,axiom,
! [M8: nat] :
( ( pratt_mod_exp_nat @ zero_zero_nat @ one_one_nat @ M8 )
= zero_zero_nat ) ).
% Pratt_Certificate.eval_mod_exp(7)
thf(fact_1190_Pratt__Certificate_Oeval__mod__exp_I5_J,axiom,
! [B4: nat,E3: nat] :
( ( pratt_mod_exp_nat @ B4 @ E3 @ one_one_nat )
= zero_zero_nat ) ).
% Pratt_Certificate.eval_mod_exp(5)
thf(fact_1191_Pratt__Certificate_Oeval__mod__exp_I10_J,axiom,
! [E3: nat,M8: nat] :
( ( pratt_mod_exp_nat @ one_one_nat @ E3 @ M8 )
= ( modulo_modulo_nat @ one_one_nat @ M8 ) ) ).
% Pratt_Certificate.eval_mod_exp(10)
thf(fact_1192_Pratt__Certificate_Oeval__mod__exp_I2_J,axiom,
! [B4: nat,M8: nat] :
( ( pratt_mod_exp_nat @ B4 @ one_one_nat @ M8 )
= ( modulo_modulo_nat @ B4 @ M8 ) ) ).
% Pratt_Certificate.eval_mod_exp(2)
thf(fact_1193_Pratt__Certificate_Oeval__mod__exp_I3_J,axiom,
! [B4: nat,M8: nat] :
( ( pratt_mod_exp_nat @ B4 @ ( suc @ zero_zero_nat ) @ M8 )
= ( modulo_modulo_nat @ B4 @ M8 ) ) ).
% Pratt_Certificate.eval_mod_exp(3)
thf(fact_1194_Pratt__Certificate_Oeval__mod__exp_I1_J,axiom,
! [B4: nat,M8: nat] :
( ( pratt_mod_exp_nat @ B4 @ zero_zero_nat @ M8 )
= ( modulo_modulo_nat @ one_one_nat @ M8 ) ) ).
% Pratt_Certificate.eval_mod_exp(1)
thf(fact_1195_size__mset__mono,axiom,
! [A3: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ A3 @ B5 )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ A3 ) @ ( size_size_multiset_a @ B5 ) ) ) ).
% size_mset_mono
thf(fact_1196_mset__subset__eqI,axiom,
! [A3: multiset_a,B5: multiset_a] :
( ! [A2: a] : ( ord_less_eq_nat @ ( count_a @ A3 @ A2 ) @ ( count_a @ B5 @ A2 ) )
=> ( subseteq_mset_a @ A3 @ B5 ) ) ).
% mset_subset_eqI
thf(fact_1197_subseteq__mset__def,axiom,
( subseteq_mset_a
= ( ^ [A6: multiset_a,B6: multiset_a] :
! [A4: a] : ( ord_less_eq_nat @ ( count_a @ A6 @ A4 ) @ ( count_a @ B6 @ A4 ) ) ) ) ).
% subseteq_mset_def
thf(fact_1198_mset__subset__eq__count,axiom,
! [A3: multiset_a,B5: multiset_a,A: a] :
( ( subseteq_mset_a @ A3 @ B5 )
=> ( ord_less_eq_nat @ ( count_a @ A3 @ A ) @ ( count_a @ B5 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_1199_size__Diff__submset,axiom,
! [M5: multiset_a,M9: multiset_a] :
( ( subseteq_mset_a @ M5 @ M9 )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M9 @ M5 ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M9 ) @ ( size_size_multiset_a @ M5 ) ) ) ) ).
% size_Diff_submset
thf(fact_1200_mset__update,axiom,
! [I: nat,Ls: list_a,V: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ Ls @ I @ V ) )
= ( add_mset_a @ V @ ( minus_3765977307040488491iset_a @ ( mset_a @ Ls ) @ ( add_mset_a @ ( nth_a @ Ls @ I ) @ zero_zero_multiset_a ) ) ) ) ) ).
% mset_update
thf(fact_1201_mset__update,axiom,
! [I: nat,Ls: list_nat,V: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ls ) )
=> ( ( mset_nat @ ( list_update_nat @ Ls @ I @ V ) )
= ( add_mset_nat @ V @ ( minus_8522176038001411705et_nat @ ( mset_nat @ Ls ) @ ( add_mset_nat @ ( nth_nat @ Ls @ I ) @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% mset_update
thf(fact_1202_mset__update,axiom,
! [I: nat,Ls: list_Pratt_pratt,V: pratt_pratt] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Ls ) )
=> ( ( mset_Pratt_pratt @ ( list_u6220086826248601529_pratt @ Ls @ I @ V ) )
= ( add_mset_Pratt_pratt @ V @ ( minus_5321434743015354872_pratt @ ( mset_Pratt_pratt @ Ls ) @ ( add_mset_Pratt_pratt @ ( nth_Pratt_pratt @ Ls @ I ) @ zero_z502583456159058376_pratt ) ) ) ) ) ).
% mset_update
thf(fact_1203_length__fpc,axiom,
! [P4: nat,A: nat,R2: nat,Qs: list_nat] :
( ( size_s4929225773697354557_pratt @ ( pratt_build_fpc @ P4 @ A @ R2 @ Qs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Qs ) @ one_one_nat ) ) ).
% length_fpc
thf(fact_1204_count__add__mset,axiom,
! [B: a,A: a,A3: multiset_a] :
( ( ( B = A )
=> ( ( count_a @ ( add_mset_a @ B @ A3 ) @ A )
= ( suc @ ( count_a @ A3 @ A ) ) ) )
& ( ( B != A )
=> ( ( count_a @ ( add_mset_a @ B @ A3 ) @ A )
= ( count_a @ A3 @ A ) ) ) ) ).
% count_add_mset
thf(fact_1205_size__add__mset,axiom,
! [A: a,A3: multiset_a] :
( ( size_size_multiset_a @ ( add_mset_a @ A @ A3 ) )
= ( suc @ ( size_size_multiset_a @ A3 ) ) ) ).
% size_add_mset
thf(fact_1206_mset_Osimps_I2_J,axiom,
! [A: a,X: list_a] :
( ( mset_a @ ( cons_a @ A @ X ) )
= ( add_mset_a @ A @ ( mset_a @ X ) ) ) ).
% mset.simps(2)
thf(fact_1207_add__mset_Orep__eq,axiom,
! [X: a,Xa: multiset_a] :
( ( count_a @ ( add_mset_a @ X @ Xa ) )
= ( ^ [B3: a] : ( if_nat @ ( B3 = X ) @ ( suc @ ( count_a @ Xa @ B3 ) ) @ ( count_a @ Xa @ B3 ) ) ) ) ).
% add_mset.rep_eq
thf(fact_1208_size__eq__Suc__imp__eq__union,axiom,
! [M5: multiset_a,N2: nat] :
( ( ( size_size_multiset_a @ M5 )
= ( suc @ N2 ) )
=> ? [A2: a,N7: multiset_a] :
( M5
= ( add_mset_a @ A2 @ N7 ) ) ) ).
% size_eq_Suc_imp_eq_union
thf(fact_1209_size__single,axiom,
! [B: a] :
( ( size_size_multiset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= one_one_nat ) ).
% size_single
thf(fact_1210_size__1__singleton__mset,axiom,
! [M5: multiset_a] :
( ( ( size_size_multiset_a @ M5 )
= one_one_nat )
=> ? [A2: a] :
( M5
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ).
% size_1_singleton_mset
thf(fact_1211_count__single,axiom,
! [B: a,A: a] :
( ( ( B = A )
=> ( ( count_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) @ A )
= one_one_nat ) )
& ( ( B != A )
=> ( ( count_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) @ A )
= zero_zero_nat ) ) ) ).
% count_single
thf(fact_1212_size__mset__SucE,axiom,
! [A3: multiset_a,N2: nat] :
( ( ( size_size_multiset_a @ A3 )
= ( suc @ N2 ) )
=> ~ ! [A2: a,B7: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ B7 ) )
=> ( ( size_size_multiset_a @ B7 )
!= N2 ) ) ) ).
% size_mset_SucE
thf(fact_1213_size__Diff1__le,axiom,
! [M5: multiset_a,X: a] : ( ord_less_eq_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M5 ) ) ).
% size_Diff1_le
thf(fact_1214_size__Diff__singleton__if,axiom,
! [X: a,A3: multiset_a] :
( ( ( member_a @ X @ ( set_mset_a @ A3 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ A3 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ ( set_mset_a @ A3 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( size_size_multiset_a @ A3 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_1215_size__Diff__singleton,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M5 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_1216_set__mset__mset,axiom,
! [Xs: list_a] :
( ( set_mset_a @ ( mset_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_1217_diff__add__mset__swap,axiom,
! [B: a,A3: multiset_a,M5: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A3 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M5 ) @ A3 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M5 @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_1218_set__sorted__list__of__multiset,axiom,
! [M5: multiset_a] :
( ( set_a2 @ ( linord814965612141868908iset_a @ M5 ) )
= ( set_mset_a @ M5 ) ) ).
% set_sorted_list_of_multiset
thf(fact_1219_count__greater__zero__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M5 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% count_greater_zero_iff
thf(fact_1220_count__greater__eq__one__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M5 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_1221_single__subset__iff,axiom,
! [A: a,M5: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M5 )
= ( member_a @ A @ ( set_mset_a @ M5 ) ) ) ).
% single_subset_iff
thf(fact_1222_diff__union__swap2,axiom,
! [Y: a,M5: multiset_a,X: a] :
( ( member_a @ Y @ ( set_mset_a @ M5 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M5 ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_1223_insert__DiffM,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M5 ) ) ).
% insert_DiffM
thf(fact_1224_count__greater__eq__Suc__zero__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( count_a @ M5 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_1225_in__diff__count,axiom,
! [A: a,M5: multiset_a,N4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N4 ) ) )
= ( ord_less_nat @ ( count_a @ N4 @ A ) @ ( count_a @ M5 @ A ) ) ) ).
% in_diff_count
thf(fact_1226_diff__single__trivial,axiom,
! [X: a,M5: multiset_a] :
( ~ ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M5 ) ) ).
% diff_single_trivial
thf(fact_1227_diff__single__eq__union,axiom,
! [X: a,M5: multiset_a,N4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= N4 )
= ( M5
= ( add_mset_a @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_1228_multi__drop__mem__not__eq,axiom,
! [C: a,B5: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B5 ) )
=> ( ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_1229_add__mset__remove__trivial__If,axiom,
! [A: a,N4: multiset_a] :
( ( ( member_a @ A @ ( set_mset_a @ N4 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N4 ) )
& ( ~ ( member_a @ A @ ( set_mset_a @ N4 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_1230_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_a,A: a] :
( ( N4
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_1231_multiset__add__sub__el__shuffle,axiom,
! [C: a,B5: multiset_a,B: a] :
( ( member_a @ C @ ( set_mset_a @ B5 ) )
=> ( ( B != C )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B5 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_1232_more__than__one__mset__mset__diff,axiom,
! [A: a,M5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M5 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_1233_count__in__diffI,axiom,
! [N4: multiset_a,X: a,M5: multiset_a] :
( ! [N3: nat] :
( ( count_a @ N4 @ X )
!= ( plus_plus_nat @ N3 @ ( count_a @ M5 @ X ) ) )
=> ( member_a @ X @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N4 ) ) ) ) ).
% count_in_diffI
thf(fact_1234_mset__subset__eqD,axiom,
! [A3: multiset_a,B5: multiset_a,X: a] :
( ( subseteq_mset_a @ A3 @ B5 )
=> ( ( member_a @ X @ ( set_mset_a @ A3 ) )
=> ( member_a @ X @ ( set_mset_a @ B5 ) ) ) ) ).
% mset_subset_eqD
thf(fact_1235_in__diffD,axiom,
! [A: a,M5: multiset_a,N4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N4 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M5 ) ) ) ).
% in_diffD
thf(fact_1236_in__multiset__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
= ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_1237_count__eq__zero__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ( count_a @ M5 @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1238_count__inI,axiom,
! [M5: multiset_a,X: a] :
( ( ( count_a @ M5 @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% count_inI
thf(fact_1239_in__countE,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ~ ! [N3: nat] :
( ( count_a @ M5 @ X )
!= ( suc @ N3 ) ) ) ).
% in_countE
thf(fact_1240_multiset__nonemptyE,axiom,
! [A3: multiset_a] :
( ( A3 != zero_zero_multiset_a )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ ( set_mset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_1241_union__iff,axiom,
! [A: a,A3: multiset_a,B5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A3 @ B5 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A3 ) )
| ( member_a @ A @ ( set_mset_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_1242_size__eq__Suc__imp__elem,axiom,
! [M5: multiset_a,N2: nat] :
( ( ( size_size_multiset_a @ M5 )
= ( suc @ N2 ) )
=> ? [A2: a] : ( member_a @ A2 @ ( set_mset_a @ M5 ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_1243_multiset__induct__min,axiom,
! [P: multiset_a > $o,M5: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X3: a,M10: multiset_a] :
( ( P @ M10 )
=> ( ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ M10 ) )
=> ( ord_less_eq_a @ X3 @ Xa2 ) )
=> ( P @ ( add_mset_a @ X3 @ M10 ) ) ) )
=> ( P @ M5 ) ) ) ).
% multiset_induct_min
thf(fact_1244_multiset__induct__min,axiom,
! [P: multiset_nat > $o,M5: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M10: multiset_nat] :
( ( P @ M10 )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_mset_nat @ M10 ) )
=> ( ord_less_eq_nat @ X3 @ Xa2 ) )
=> ( P @ ( add_mset_nat @ X3 @ M10 ) ) ) )
=> ( P @ M5 ) ) ) ).
% multiset_induct_min
thf(fact_1245_multiset__induct__max,axiom,
! [P: multiset_a > $o,M5: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X3: a,M10: multiset_a] :
( ( P @ M10 )
=> ( ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ M10 ) )
=> ( ord_less_eq_a @ Xa2 @ X3 ) )
=> ( P @ ( add_mset_a @ X3 @ M10 ) ) ) )
=> ( P @ M5 ) ) ) ).
% multiset_induct_max
thf(fact_1246_multiset__induct__max,axiom,
! [P: multiset_nat > $o,M5: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M10: multiset_nat] :
( ( P @ M10 )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_mset_nat @ M10 ) )
=> ( ord_less_eq_nat @ Xa2 @ X3 ) )
=> ( P @ ( add_mset_nat @ X3 @ M10 ) ) ) )
=> ( P @ M5 ) ) ) ).
% multiset_induct_max
thf(fact_1247_mset__add,axiom,
! [A: a,A3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A3 ) )
=> ~ ! [B7: multiset_a] :
( A3
!= ( add_mset_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_1248_multi__member__split,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ? [A7: multiset_a] :
( M5
= ( add_mset_a @ X @ A7 ) ) ) ).
% multi_member_split
thf(fact_1249_insert__noteq__member,axiom,
! [B: a,B5: multiset_a,C: a,C4: multiset_a] :
( ( ( add_mset_a @ B @ B5 )
= ( add_mset_a @ C @ C4 ) )
=> ( ( B != C )
=> ( member_a @ C @ ( set_mset_a @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_1250_union__single__eq__member,axiom,
! [X: a,M5: multiset_a,N4: multiset_a] :
( ( ( add_mset_a @ X @ M5 )
= N4 )
=> ( member_a @ X @ ( set_mset_a @ N4 ) ) ) ).
% union_single_eq_member
thf(fact_1251_multi__member__this,axiom,
! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_1252_multi__member__skip,axiom,
! [X: a,XS: multiset_a,Y: a] :
( ( member_a @ X @ ( set_mset_a @ XS ) )
=> ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_1253_multi__member__last,axiom,
! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_1254_multi__subset__induct,axiom,
! [F2: multiset_a,A3: multiset_a,P: multiset_a > $o] :
( ( subseteq_mset_a @ F2 @ A3 )
=> ( ( P @ zero_zero_multiset_a )
=> ( ! [A2: a,F3: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ A3 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_mset_a @ A2 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_1255_mset__subset__eq__single,axiom,
! [A: a,B5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ B5 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B5 ) ) ).
% mset_subset_eq_single
thf(fact_1256_nth__mem__mset,axiom,
! [I: nat,Ls: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_1257_nth__mem__mset,axiom,
! [I: nat,Ls: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ls ) )
=> ( member_nat @ ( nth_nat @ Ls @ I ) @ ( set_mset_nat @ ( mset_nat @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_1258_nth__mem__mset,axiom,
! [I: nat,Ls: list_Pratt_pratt] :
( ( ord_less_nat @ I @ ( size_s4929225773697354557_pratt @ Ls ) )
=> ( member_Pratt_pratt @ ( nth_Pratt_pratt @ Ls @ I ) @ ( set_mset_Pratt_pratt @ ( mset_Pratt_pratt @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_1259_in__diff__countE,axiom,
! [X: a,M5: multiset_a,N4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N4 ) ) )
=> ~ ! [N3: nat] :
( ( count_a @ M5 @ X )
!= ( plus_plus_nat @ ( suc @ N3 ) @ ( count_a @ N4 @ X ) ) ) ) ).
% in_diff_countE
thf(fact_1260_insert__subset__eq__iff,axiom,
! [A: a,A3: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ A3 ) @ B5 )
= ( ( member_a @ A @ ( set_mset_a @ B5 ) )
& ( subseteq_mset_a @ A3 @ ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_1261_diff__union__single__conv,axiom,
! [A: a,J4: multiset_a,I5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ J4 ) )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I5 @ J4 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I5 @ ( minus_3765977307040488491iset_a @ J4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_1262_insert__DiffM2,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M5 ) ) ).
% insert_DiffM2
thf(fact_1263_size__Suc__Diff1,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( suc @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) )
= ( size_size_multiset_a @ M5 ) ) ) ).
% size_Suc_Diff1
thf(fact_1264_size__Diff2__less,axiom,
! [X: a,M5: multiset_a,Y: a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( member_a @ Y @ ( set_mset_a @ M5 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M5 ) ) ) ) ).
% size_Diff2_less
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_a @ ( nth_a @ xs @ i ) @ ( nth_a @ xs @ k ) ).
%------------------------------------------------------------------------------