TPTP Problem File: SLH0156^1.p
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- 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 : Knights_Tour/0000_KnightsTour/prob_03196_118823__6120238_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1369 ( 683 unt; 97 typ; 0 def)
% Number of atoms : 3245 (1224 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10304 ( 365 ~; 70 |; 132 &;8572 @)
% ( 0 <=>;1165 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 242 ( 242 >; 0 *; 0 +; 0 <<)
% Number of symbols : 87 ( 84 usr; 21 con; 0-3 aty)
% Number of variables : 3106 ( 187 ^;2878 !; 41 ?;3106 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:01:40.426
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
option3674462004671404037nt_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
list_l1670014477004246597nt_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
list_P5707943133018811711nt_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (84)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
uminus6221592323253981072nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Num__Onum,type,
if_num: $o > num > num > num ).
thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
if_set4441254072200386271nt_int: $o > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_KnightsTour_Oboard,type,
board: nat > nat > set_Pr958786334691620121nt_int ).
thf(sy_c_KnightsTour_Oknights__circuit,type,
knights_circuit: set_Pr958786334691620121nt_int > list_P5707943133018811711nt_int > $o ).
thf(sy_c_KnightsTour_Oknights__path,type,
knights_path: set_Pr958786334691620121nt_int > list_P5707943133018811711nt_int > $o ).
thf(sy_c_KnightsTour_Oto__path,type,
to_path: list_list_nat > option3674462004671404037nt_int ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
cons_l7309679040211256053nt_int: list_P5707943133018811711nt_int > list_l1670014477004246597nt_int > list_l1670014477004246597nt_int ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
cons_P3334398858971670639nt_int: product_prod_int_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
nil_li8670148097206105925nt_int: list_l1670014477004246597nt_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
nil_Pr2300489316682597567nt_int: list_P5707943133018811711nt_int ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
produc6783906451316923569st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc5568053154996169768nt_int: list_l1670014477004246597nt_int > list_l1670014477004246597nt_int ).
thf(sy_c_List_Orev_001t__Int__Oint,type,
rev_int: list_int > list_int ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
rev_Pr2923690841345412895nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
subseqs_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
subseq1357044202310323342nt_int: list_P5707943133018811711nt_int > list_l1670014477004246597nt_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
neg_numeral_sub_int: num > num > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Option_Ooption_Othe_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
the_li7431803565598847443nt_int: option3674462004671404037nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le7563427860532173253nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
ord_min_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Num__Onum,type,
ord_min_num: num > num > num ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_mi3891036259616950784nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_ps_092_060_094sub_0622____,type,
ps_2: list_P5707943133018811711nt_int ).
thf(sy_v_x____,type,
x: nat ).
thf(sy_v_xa____,type,
xa: nat ).
% Relevant facts (1260)
thf(fact_0_kp__5x5__ul,axiom,
knights_path @ ( board @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( the_li7431803565598847443nt_int @ ( to_path @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ one_one_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ nil_nat ) ) ) ) ) @ nil_list_nat ) ) ) ) ) ) ) ).
% kp_5x5_ul
thf(fact_1_num__elems_Ocases,axiom,
! [X: list_list_nat] :
( ! [R: list_nat,Rs: list_list_nat] :
( X
!= ( cons_list_nat @ R @ Rs ) )
=> ( X = nil_list_nat ) ) ).
% num_elems.cases
thf(fact_2_kp__5x5__lr,axiom,
knights_path @ ( board @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( the_li7431803565598847443nt_int @ ( to_path @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ one_one_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ nil_nat ) ) ) ) ) @ nil_list_nat ) ) ) ) ) ) ) ).
% kp_5x5_lr
thf(fact_3_kp__5x5__ur,axiom,
knights_path @ ( board @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( the_li7431803565598847443nt_int @ ( to_path @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ one_one_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ nil_nat ) ) ) ) ) @ nil_list_nat ) ) ) ) ) ) ) ).
% kp_5x5_ur
thf(fact_4_knights__path__board__unique,axiom,
! [B_1: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int,B_2: set_Pr958786334691620121nt_int] :
( ( knights_path @ B_1 @ Ps )
=> ( ( knights_path @ B_2 @ Ps )
=> ( B_1 = B_2 ) ) ) ).
% knights_path_board_unique
thf(fact_5_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_6_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_7_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_8_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_9_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_10_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_11_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_12_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_13_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_14_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_15_List_Otranspose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X2: list_nat,Xs: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_16_List_Otranspose_Ocases,axiom,
! [X: list_l1670014477004246597nt_int] :
( ( X != nil_li8670148097206105925nt_int )
=> ( ! [Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss ) )
=> ~ ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_17_List_Otranspose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X2: int,Xs: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_18_List_Otranspose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_19_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_20_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X3: num] :
( Y
!= ( bit1 @ X3 ) ) ) ) ).
% num.exhaust
thf(fact_21_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_22_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_23_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_24_list_Oinject,axiom,
! [X21: list_nat,X222: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X222 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_25_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_26_list_Oinject,axiom,
! [X21: product_prod_int_int,X222: list_P5707943133018811711nt_int,Y21: product_prod_int_int,Y22: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X21 @ X222 )
= ( cons_P3334398858971670639nt_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_27_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_28_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_29_knights__path__non__nil,axiom,
! [B: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps )
=> ( Ps != nil_Pr2300489316682597567nt_int ) ) ).
% knights_path_non_nil
thf(fact_30_not__Cons__self2,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( cons_list_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_31_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_32_not__Cons__self2,axiom,
! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( cons_P3334398858971670639nt_int @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_33_not__Cons__self2,axiom,
! [X: int,Xs2: list_int] :
( ( cons_int @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_34_list__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_35_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_36_list__nonempty__induct,axiom,
! [Xs2: list_P5707943133018811711nt_int,P: list_P5707943133018811711nt_int > $o] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ! [X2: product_prod_int_int] : ( P @ ( cons_P3334398858971670639nt_int @ X2 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [X2: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( P @ Xs )
=> ( P @ ( cons_P3334398858971670639nt_int @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_37_list__nonempty__induct,axiom,
! [Xs2: list_int,P: list_int > $o] :
( ( Xs2 != nil_int )
=> ( ! [X2: int] : ( P @ ( cons_int @ X2 @ nil_int ) )
=> ( ! [X2: int,Xs: list_int] :
( ( Xs != nil_int )
=> ( ( P @ Xs )
=> ( P @ ( cons_int @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_38_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_39_list__induct2_H,axiom,
! [P: list_nat > list_int > $o,Xs2: list_nat,Ys: list_int] :
( ( P @ nil_nat @ nil_int )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_nat @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: int,Ys2: list_int] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_40_list__induct2_H,axiom,
! [P: list_int > list_nat > $o,Xs2: list_int,Ys: list_nat] :
( ( P @ nil_int @ nil_nat )
=> ( ! [X2: int,Xs: list_int] : ( P @ ( cons_int @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_int @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: int,Xs: list_int,Y2: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_41_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs2: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X2: int,Xs: list_int] : ( P @ ( cons_int @ X2 @ Xs ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X2: int,Xs: list_int,Y2: int,Ys2: list_int] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_42_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs2: list_list_nat,Ys: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_43_list__induct2_H,axiom,
! [P: list_list_nat > list_int > $o,Xs2: list_list_nat,Ys: list_int] :
( ( P @ nil_list_nat @ nil_int )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_list_nat @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: int,Ys2: list_int] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_44_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs2: list_nat,Ys: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys2: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_45_list__induct2_H,axiom,
! [P: list_int > list_list_nat > $o,Xs2: list_int,Ys: list_list_nat] :
( ( P @ nil_int @ nil_list_nat )
=> ( ! [X2: int,Xs: list_int] : ( P @ ( cons_int @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys2: list_list_nat] : ( P @ nil_int @ ( cons_list_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: int,Xs: list_int,Y2: list_nat,Ys2: list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_46_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs2: list_list_nat,Ys: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys2: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_47_list__induct2_H,axiom,
! [P: list_nat > list_P5707943133018811711nt_int > $o,Xs2: list_nat,Ys: list_P5707943133018811711nt_int] :
( ( P @ nil_nat @ nil_Pr2300489316682597567nt_int )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_Pr2300489316682597567nt_int )
=> ( ! [Y2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] : ( P @ nil_nat @ ( cons_P3334398858971670639nt_int @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_48_neq__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
= ( ? [Y3: list_nat,Ys3: list_list_nat] :
( Xs2
= ( cons_list_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_49_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_50_neq__Nil__conv,axiom,
! [Xs2: list_P5707943133018811711nt_int] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
= ( ? [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( Xs2
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_51_neq__Nil__conv,axiom,
! [Xs2: list_int] :
( ( Xs2 != nil_int )
= ( ? [Y3: int,Ys3: list_int] :
( Xs2
= ( cons_int @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_52_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X2: list_nat] :
( X
!= ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ~ ! [X2: list_nat,Y2: list_nat,Xs: list_list_nat] :
( X
!= ( cons_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_53_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_54_remdups__adj_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( X != nil_Pr2300489316682597567nt_int )
=> ( ! [X2: product_prod_int_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X2: product_prod_int_int,Y2: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X2 @ ( cons_P3334398858971670639nt_int @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_55_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X2: int] :
( X
!= ( cons_int @ X2 @ nil_int ) )
=> ~ ! [X2: int,Y2: int,Xs: list_int] :
( X
!= ( cons_int @ X2 @ ( cons_int @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_56_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_57_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X2: int,Xs: list_int] :
( X
!= ( cons_int @ X2 @ Xs ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_58_list_Oexhaust,axiom,
! [Y: list_list_nat] :
( ( Y != nil_list_nat )
=> ~ ! [X212: list_nat,X223: list_list_nat] :
( Y
!= ( cons_list_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_59_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_60_list_Oexhaust,axiom,
! [Y: list_P5707943133018811711nt_int] :
( ( Y != nil_Pr2300489316682597567nt_int )
=> ~ ! [X212: product_prod_int_int,X223: list_P5707943133018811711nt_int] :
( Y
!= ( cons_P3334398858971670639nt_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_61_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X223: list_int] :
( Y
!= ( cons_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_62_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X222: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X222 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_63_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_64_list_OdiscI,axiom,
! [List: list_P5707943133018811711nt_int,X21: product_prod_int_int,X222: list_P5707943133018811711nt_int] :
( ( List
= ( cons_P3334398858971670639nt_int @ X21 @ X222 ) )
=> ( List != nil_Pr2300489316682597567nt_int ) ) ).
% list.discI
thf(fact_65_list_OdiscI,axiom,
! [List: list_int,X21: int,X222: list_int] :
( ( List
= ( cons_int @ X21 @ X222 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_66_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_67_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_68_list_Odistinct_I1_J,axiom,
! [X21: product_prod_int_int,X222: list_P5707943133018811711nt_int] :
( nil_Pr2300489316682597567nt_int
!= ( cons_P3334398858971670639nt_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_69_list_Odistinct_I1_J,axiom,
! [X21: int,X222: list_int] :
( nil_int
!= ( cons_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_70_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_71_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_72_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_73_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_74_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_75_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y23: num] :
( ( ( bit0 @ X23 )
= ( bit0 @ Y23 ) )
= ( X23 = Y23 ) ) ).
% verit_eq_simplify(8)
thf(fact_76_assms_I1_J,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n ) ).
% assms(1)
thf(fact_77_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_78_verit__eq__simplify_I14_J,axiom,
! [X23: num,X32: num] :
( ( bit0 @ X23 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_79_verit__eq__simplify_I10_J,axiom,
! [X23: num] :
( one
!= ( bit0 @ X23 ) ) ).
% verit_eq_simplify(10)
thf(fact_80_product__lists_Osimps_I1_J,axiom,
( ( produc6783906451316923569st_nat @ nil_list_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_81_product__lists_Osimps_I1_J,axiom,
( ( produc5568053154996169768nt_int @ nil_li8670148097206105925nt_int )
= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).
% product_lists.simps(1)
thf(fact_82_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_83_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_84_less_Oprems_I2_J,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ).
% less.prems(2)
thf(fact_85_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_86_test__path_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ! [S_i: product_prod_int_int,S_j: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ S_i @ ( cons_P3334398858971670639nt_int @ S_j @ Xs ) ) )
=> ( ( X != nil_Pr2300489316682597567nt_int )
=> ~ ! [V: product_prod_int_int] :
( X
!= ( cons_P3334398858971670639nt_int @ V @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% test_path.cases
thf(fact_87_KnightsTour_Otranspose_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( X != nil_Pr2300489316682597567nt_int )
=> ~ ! [S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) ) ) ).
% KnightsTour.transpose.cases
thf(fact_88_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_89_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_90_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_91_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_92_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_93_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_94_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_95_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_96_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_97_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_98_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_99_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_100_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_101_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_102_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_103_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_104_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_105__092_060open_062m_A_061_A5_092_060close_062,axiom,
( ma
= ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) ).
% \<open>m = 5\<close>
thf(fact_106_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_107_subseqs_Osimps_I1_J,axiom,
( ( subseq1357044202310323342nt_int @ nil_Pr2300489316682597567nt_int )
= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).
% subseqs.simps(1)
thf(fact_108_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_109_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_110_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_111_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_112_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_113_dvd__minus__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( dvd_dvd_int @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_114_minus__dvd__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( dvd_dvd_int @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_115_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_116_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_117_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_118_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_119_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_120_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_121_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_122_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_123_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_124_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_125_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_126_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_127_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_128_even__minus,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_129_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_130_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_131_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_132_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_133_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_134_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_135_less_Oprems_I1_J,axiom,
( xa
= ( plus_plus_nat @ na @ ma ) ) ).
% less.prems(1)
thf(fact_136__092_060open_062m_A_061_A5_A_092_060or_062_Am_A_061_A6_A_092_060or_062_Am_A_061_A7_A_092_060or_062_Am_A_061_A8_A_092_060or_062_Am_A_061_A9_A_092_060or_062_A10_A_092_060le_062_Am_092_060close_062,axiom,
( ( ma
= ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) )
| ( ma
= ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) )
| ( ma
= ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) )
| ( ma
= ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
| ( ma
= ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
| ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ma ) ) ).
% \<open>m = 5 \<or> m = 6 \<or> m = 7 \<or> m = 8 \<or> m = 9 \<or> 10 \<le> m\<close>
thf(fact_137__092_060open_0625_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ na ).
% \<open>5 \<le> n\<close>
thf(fact_138_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_139_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_140_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_141_less_Oprems_I3_J,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( ord_min_nat @ na @ ma ) ).
% less.prems(3)
thf(fact_142_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_143_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_144_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_145_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_146__092_060open_062x_____A_061_An_A_L_Am_092_060close_062,axiom,
( x
= ( plus_plus_nat @ n @ m ) ) ).
% \<open>x__ = n + m\<close>
thf(fact_147_assms_I2_J,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( ord_min_nat @ n @ m ) ).
% assms(2)
thf(fact_148_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_149_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_150_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_151_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_152_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_153_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_154_compl__le__compl__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ X ) @ ( uminus6221592323253981072nt_int @ Y ) )
= ( ord_le2843351958646193337nt_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_155_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_156_add__numeral__left,axiom,
! [V2: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_157_add__numeral__left,axiom,
! [V2: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_158_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_159_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_160_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_161_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_162_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_163_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_164_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_165_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_166_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_167_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_168_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_169_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_170_semiring__norm_I167_J,axiom,
! [V2: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_171_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_172_min__number__of_I1_J,axiom,
! [U: num,V2: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
= ( numeral_numeral_nat @ U ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
= ( numeral_numeral_nat @ V2 ) ) ) ) ).
% min_number_of(1)
thf(fact_173_min__number__of_I1_J,axiom,
! [U: num,V2: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
= ( numeral_numeral_int @ U ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
= ( numeral_numeral_int @ V2 ) ) ) ) ).
% min_number_of(1)
thf(fact_174_min__0__1_I6_J,axiom,
! [X: num] :
( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= one_one_nat ) ).
% min_0_1(6)
thf(fact_175_min__0__1_I6_J,axiom,
! [X: num] :
( ( ord_min_int @ ( numeral_numeral_int @ X ) @ one_one_int )
= one_one_int ) ).
% min_0_1(6)
thf(fact_176_min__0__1_I5_J,axiom,
! [X: num] :
( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= one_one_nat ) ).
% min_0_1(5)
thf(fact_177_min__0__1_I5_J,axiom,
! [X: num] :
( ( ord_min_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= one_one_int ) ).
% min_0_1(5)
thf(fact_178_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_179_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_180_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_181_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_182_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_183_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_184_min__number__of_I4_J,axiom,
! [U: num,V2: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) ) ) ).
% min_number_of(4)
thf(fact_185_min__number__of_I3_J,axiom,
! [U: num,V2: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
= ( numeral_numeral_int @ V2 ) ) ) ) ).
% min_number_of(3)
thf(fact_186_min__number__of_I2_J,axiom,
! [U: num,V2: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
= ( numeral_numeral_int @ U ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) ) ) ).
% min_number_of(2)
thf(fact_187_semiring__norm_I166_J,axiom,
! [V2: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V2 ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_188_semiring__norm_I165_J,axiom,
! [V2: num,W: num,Y: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V2 @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_189_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_190_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_191_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_192_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_193_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_194_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_195_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_196_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_197_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_198_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_199_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_200_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_201_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_202_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_203_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_204_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_205__092_060open_0625_A_092_060le_062_An_A_N_A5_092_060close_062,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ na @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) ).
% \<open>5 \<le> n - 5\<close>
thf(fact_206_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_207_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_208_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_209_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_210_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_211_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_212_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_213_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_214_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_215_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_216_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_217_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_218_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_219_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_220_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_221_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_222_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_223_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_224_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_225_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_226_verit__comp__simplify1_I2_J,axiom,
! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_227_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_228_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_229_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_230_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_231_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_232_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_233_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_234_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_235_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_236_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_237_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_238_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_239_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_240_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_241_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_242_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_243_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_244_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_245_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_246_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_247_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_248_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_249_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_250_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_251_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_252_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_253_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_254_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_255_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_256_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_257_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_258_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_259_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_260_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_261_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_262_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_263_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_264_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_265_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_266_min__add__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ord_min_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% min_add_distrib_left
thf(fact_267_min__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% min_add_distrib_left
thf(fact_268_min__add__distrib__right,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z ) )
= ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% min_add_distrib_right
thf(fact_269_min__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z ) )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% min_add_distrib_right
thf(fact_270_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_271_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_272_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_273_compl__le__swap2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ Y ) @ X )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_274_compl__le__swap1,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ ( uminus6221592323253981072nt_int @ X ) )
=> ( ord_le2843351958646193337nt_int @ X @ ( uminus6221592323253981072nt_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_275_compl__mono,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ Y ) @ ( uminus6221592323253981072nt_int @ X ) ) ) ).
% compl_mono
thf(fact_276_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_277_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_278_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_279_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_280_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_281_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_282_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_283_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_284_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_285_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_286_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_287_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_288_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_289_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_290_knights__path__board__m__n__geq__1,axiom,
! [N: nat,M: nat,Ps: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M ) @ Ps )
=> ( ord_less_eq_nat @ one_one_nat @ ( ord_min_nat @ N @ M ) ) ) ).
% knights_path_board_m_n_geq_1
thf(fact_291_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_292_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_293_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_294_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_295_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_296_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_297_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_298_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_299_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_300_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_301_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_302_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_303_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_304_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_305_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_306_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_307_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_308_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_309_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_310_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_311_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_312_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_313__092_060open_062n_A_061_A5_A_092_060or_062_An_A_061_A7_A_092_060or_062_An_A_061_A9_A_092_060or_062_A5_A_092_060le_062_An_A_N_A5_092_060close_062,axiom,
( ( na
= ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) )
| ( na
= ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) )
| ( na
= ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
| ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ na @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ).
% \<open>n = 5 \<or> n = 7 \<or> n = 9 \<or> 5 \<le> n - 5\<close>
thf(fact_314_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_315_min_Obounded__iff,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) )
= ( ( ord_less_eq_num @ A @ B )
& ( ord_less_eq_num @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_316_min_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_317_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_318_min_Oabsorb2,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_min_num @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_319_min_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_320_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_321_min_Oabsorb1,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_min_num @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_322_min_Oabsorb1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_323_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_324_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_325_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_326_knights__circuit__6xm__exists,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ M )
=> ? [X_1: list_P5707943133018811711nt_int] : ( knights_circuit @ ( board @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ M ) @ X_1 ) ) ).
% knights_circuit_6xm_exists
thf(fact_327_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_328_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_329_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_330_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_331_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_332_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_333_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_334_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_335_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_336_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_337_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_338_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_339_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_340_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_341_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_342_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_343_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_344__092_060open_062_092_060exists_062ps_O_Aknights__circuit_A_Iboard_Am_A_In_A_N_A5_J_J_Aps_092_060close_062,axiom,
? [X_1: list_P5707943133018811711nt_int] : ( knights_circuit @ ( board @ ma @ ( minus_minus_nat @ na @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ X_1 ) ).
% \<open>\<exists>ps. knights_circuit (board m (n - 5)) ps\<close>
thf(fact_345_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_346_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_347_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_348_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_349_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_350_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_351_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_352_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_353_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_354_diff__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_355_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_356_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_357_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_358_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_359_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_360_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_361_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_362_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_363_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_364_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_365_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_366_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_367_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_368_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_369__092_060open_062knights__circuit_A_Iboard_Am_A_In_A_N_A5_J_J_Aps_092_060_094sub_0622_092_060close_062,axiom,
knights_circuit @ ( board @ ma @ ( minus_minus_nat @ na @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ps_2 ).
% \<open>knights_circuit (board m (n - 5)) ps\<^sub>2\<close>
thf(fact_370_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_371_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_372_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_373_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_374_even__diff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% even_diff
thf(fact_375_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_376_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_377_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_378_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_379_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_380_diff__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 ) ) ).
% diff_right_commute
thf(fact_381_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_382_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_383_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_384_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_385_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_386_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_387_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_388_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_389_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_390_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_391_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_392_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_393_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_394_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_395_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_396_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_397_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_398_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_399_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_400_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_401_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_402_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_403_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_404_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_405_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_406_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_407_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_408_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_409_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_410_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_411_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_412_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_413_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_414_min__diff__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% min_diff_distrib_left
thf(fact_415_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_416_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_417_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_418_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_419_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_420_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_421_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_422_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_423_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_424_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_425_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_426_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_427_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_428_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_429_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_430_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_431_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_432_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_433_group__cancel_Osub2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_434_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_435_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_436_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_437_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_438_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_439_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_440_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_441_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K2: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_442_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_443_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_444_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_445_board__leq__subset,axiom,
! [N_1: nat,N_2: nat,M_1: nat,M_2: nat] :
( ( ( ord_less_eq_nat @ N_1 @ N_2 )
& ( ord_less_eq_nat @ M_1 @ M_2 ) )
=> ( ord_le2843351958646193337nt_int @ ( board @ N_1 @ M_1 ) @ ( board @ N_2 @ M_2 ) ) ) ).
% board_leq_subset
thf(fact_446_board__subset,axiom,
! [N_1: nat,N_2: nat,M_1: nat,M_2: nat] :
( ( ord_less_eq_nat @ N_1 @ N_2 )
=> ( ( ord_less_eq_nat @ M_1 @ M_2 )
=> ( ord_le2843351958646193337nt_int @ ( board @ N_1 @ M_1 ) @ ( board @ N_2 @ M_2 ) ) ) ) ).
% board_subset
thf(fact_447_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_448_min_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.assoc
thf(fact_449_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A3: nat,B3: nat] : ( ord_min_nat @ B3 @ A3 ) ) ) ).
% min.commute
thf(fact_450_min_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.left_commute
thf(fact_451_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% min.mono
thf(fact_452_min_Omono,axiom,
! [A: num,C: num,B: num,D: num] :
( ( ord_less_eq_num @ A @ C )
=> ( ( ord_less_eq_num @ B @ D )
=> ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ ( ord_min_num @ C @ D ) ) ) ) ).
% min.mono
thf(fact_453_min_Omono,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ ( ord_min_int @ C @ D ) ) ) ) ).
% min.mono
thf(fact_454_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_455_min_OorderE,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( A
= ( ord_min_num @ A @ B ) ) ) ).
% min.orderE
thf(fact_456_min_OorderE,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( A
= ( ord_min_int @ A @ B ) ) ) ).
% min.orderE
thf(fact_457_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_458_min_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_min_num @ A @ B ) )
=> ( ord_less_eq_num @ A @ B ) ) ).
% min.orderI
thf(fact_459_min_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_min_int @ A @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% min.orderI
thf(fact_460_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_461_min_OboundedE,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) )
=> ~ ( ( ord_less_eq_num @ A @ B )
=> ~ ( ord_less_eq_num @ A @ C ) ) ) ).
% min.boundedE
thf(fact_462_min_OboundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C ) ) ) ).
% min.boundedE
thf(fact_463_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_464_min_OboundedI,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ A @ C )
=> ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_465_min_OboundedI,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_466_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( ord_min_nat @ A3 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_467_min_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( A3
= ( ord_min_num @ A3 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_468_min_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( A3
= ( ord_min_int @ A3 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_469_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_470_min_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_471_min_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_472_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_473_min_Ocobounded2,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_474_min_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_475_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_min_nat @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb_iff1
thf(fact_476_min_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( ( ord_min_num @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb_iff1
thf(fact_477_min_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_min_int @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb_iff1
thf(fact_478_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_min_nat @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_479_min_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A3: num] :
( ( ord_min_num @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_480_min_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_min_int @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_481_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_482_min_OcoboundedI1,axiom,
! [A: num,C: num,B: num] :
( ( ord_less_eq_num @ A @ C )
=> ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_483_min_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_484_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_485_min_OcoboundedI2,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_486_min_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_487_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_488_min__le__iff__disj,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ ( ord_min_num @ X @ Y ) @ Z )
= ( ( ord_less_eq_num @ X @ Z )
| ( ord_less_eq_num @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_489_min__le__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
| ( ord_less_eq_int @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_490_rev__ps_092_060_094sub_0622__prems_I1_J,axiom,
knights_path @ ( board @ ma @ ( minus_minus_nat @ na @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( rev_Pr2923690841345412895nt_int @ ps_2 ) ).
% rev_ps\<^sub>2_prems(1)
thf(fact_491_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_492_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_493_order__refl,axiom,
! [X: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X @ X ) ).
% order_refl
thf(fact_494_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_495_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_496_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_497_dual__order_Orefl,axiom,
! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ A ) ).
% dual_order.refl
thf(fact_498_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_499_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_500_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_501_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) ) ).
% neg_one_even_power
thf(fact_502_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% neg_one_odd_power
thf(fact_503_rev__rev__ident,axiom,
! [Xs2: list_P5707943133018811711nt_int] :
( ( rev_Pr2923690841345412895nt_int @ ( rev_Pr2923690841345412895nt_int @ Xs2 ) )
= Xs2 ) ).
% rev_rev_ident
thf(fact_504_rev__is__rev__conv,axiom,
! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs2 )
= ( rev_Pr2923690841345412895nt_int @ Ys ) )
= ( Xs2 = Ys ) ) ).
% rev_is_rev_conv
thf(fact_505_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_506_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_507_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_508_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_509_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_510_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_511_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_512_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_513_min__less__iff__conj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
& ( ord_less_nat @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_514_min__less__iff__conj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_num @ Z @ ( ord_min_num @ X @ Y ) )
= ( ( ord_less_num @ Z @ X )
& ( ord_less_num @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_515_min__less__iff__conj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
& ( ord_less_int @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_516_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_517_min_Oabsorb4,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_min_num @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_518_min_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_519_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_520_min_Oabsorb3,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_min_num @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_521_min_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_522_Nil__is__rev__conv,axiom,
! [Xs2: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_523_Nil__is__rev__conv,axiom,
! [Xs2: list_list_nat] :
( ( nil_list_nat
= ( rev_list_nat @ Xs2 ) )
= ( Xs2 = nil_list_nat ) ) ).
% Nil_is_rev_conv
thf(fact_524_Nil__is__rev__conv,axiom,
! [Xs2: list_P5707943133018811711nt_int] :
( ( nil_Pr2300489316682597567nt_int
= ( rev_Pr2923690841345412895nt_int @ Xs2 ) )
= ( Xs2 = nil_Pr2300489316682597567nt_int ) ) ).
% Nil_is_rev_conv
thf(fact_525_rev__is__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( ( rev_nat @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_526_rev__is__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( ( rev_list_nat @ Xs2 )
= nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% rev_is_Nil_conv
thf(fact_527_rev__is__Nil__conv,axiom,
! [Xs2: list_P5707943133018811711nt_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs2 )
= nil_Pr2300489316682597567nt_int )
= ( Xs2 = nil_Pr2300489316682597567nt_int ) ) ).
% rev_is_Nil_conv
thf(fact_528_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_529_rev__singleton__conv,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( ( rev_list_nat @ Xs2 )
= ( cons_list_nat @ X @ nil_list_nat ) )
= ( Xs2
= ( cons_list_nat @ X @ nil_list_nat ) ) ) ).
% rev_singleton_conv
thf(fact_530_rev__singleton__conv,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( rev_nat @ Xs2 )
= ( cons_nat @ X @ nil_nat ) )
= ( Xs2
= ( cons_nat @ X @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_531_rev__singleton__conv,axiom,
! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs2 )
= ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) )
= ( Xs2
= ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ) ).
% rev_singleton_conv
thf(fact_532_rev__singleton__conv,axiom,
! [Xs2: list_int,X: int] :
( ( ( rev_int @ Xs2 )
= ( cons_int @ X @ nil_int ) )
= ( Xs2
= ( cons_int @ X @ nil_int ) ) ) ).
% rev_singleton_conv
thf(fact_533_singleton__rev__conv,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( ( cons_list_nat @ X @ nil_list_nat )
= ( rev_list_nat @ Xs2 ) )
= ( ( cons_list_nat @ X @ nil_list_nat )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_534_singleton__rev__conv,axiom,
! [X: nat,Xs2: list_nat] :
( ( ( cons_nat @ X @ nil_nat )
= ( rev_nat @ Xs2 ) )
= ( ( cons_nat @ X @ nil_nat )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_535_singleton__rev__conv,axiom,
! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int )
= ( rev_Pr2923690841345412895nt_int @ Xs2 ) )
= ( ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_536_singleton__rev__conv,axiom,
! [X: int,Xs2: list_int] :
( ( ( cons_int @ X @ nil_int )
= ( rev_int @ Xs2 ) )
= ( ( cons_int @ X @ nil_int )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_537_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_538_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_539_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_540_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_541_sub__num__simps_I4_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% sub_num_simps(4)
thf(fact_542_power__minus__odd,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_543_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_544_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_545_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_546_order__le__imp__less__or__eq,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le7563427860532173253nt_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_547_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_548_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_549_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_550_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_551_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_552_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_553_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_554_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_555_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_556_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_557_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_558_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_559_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_560_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le7563427860532173253nt_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_561_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_562_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_563_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_564_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_565_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_566_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_567_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_568_order__less__le__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_569_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_570_order__less__le__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( ord_le7563427860532173253nt_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_571_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_572_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_573_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_574_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_575_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_576_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_577_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_578_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_579_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_580_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_581_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_582_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_583_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_584_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_585_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_586_order__le__less__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_587_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_588_order__le__less__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_589_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_590_order__le__less__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( ord_le2843351958646193337nt_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le7563427860532173253nt_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_591_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_592_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_593_order__less__le__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ Z )
=> ( ord_le7563427860532173253nt_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_594_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_595_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_596_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_597_order__le__less__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le7563427860532173253nt_int @ Y @ Z )
=> ( ord_le7563427860532173253nt_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_598_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_599_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_600_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_601_order__neq__le__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( A != B )
=> ( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ord_le7563427860532173253nt_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_602_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_603_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_604_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_605_order__le__neq__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( A != B )
=> ( ord_le7563427860532173253nt_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_606_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_607_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_608_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_609_order__less__imp__le,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ord_le2843351958646193337nt_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_610_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_611_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_612_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_613_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_614_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_615_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_616_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_617_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_618_order__less__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_619_order__less__le,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_620_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_621_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_622_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_623_order__le__less,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_624_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_625_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_626_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_627_dual__order_Ostrict__implies__order,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ord_le2843351958646193337nt_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_628_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_629_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_630_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_631_order_Ostrict__implies__order,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ord_le2843351958646193337nt_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_632_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_633_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_634_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_635_dual__order_Ostrict__iff__not,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
& ~ ( ord_le2843351958646193337nt_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_636_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_637_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_638_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_639_dual__order_Ostrict__trans2,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le7563427860532173253nt_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_640_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_641_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_642_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_643_dual__order_Ostrict__trans1,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le7563427860532173253nt_int @ C @ B )
=> ( ord_le7563427860532173253nt_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_644_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_645_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_646_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_647_dual__order_Ostrict__iff__order,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_648_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_649_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_650_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_651_dual__order_Oorder__iff__strict,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_652_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_653_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_654_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_655_order_Ostrict__iff__not,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
& ~ ( ord_le2843351958646193337nt_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_656_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_657_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_658_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_659_order_Ostrict__trans2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le7563427860532173253nt_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_660_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_661_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_662_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_663_order_Ostrict__trans1,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ B @ C )
=> ( ord_le7563427860532173253nt_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_664_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_665_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_666_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_667_order_Ostrict__iff__order,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_668_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_669_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_670_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_671_order_Oorder__iff__strict,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_672_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_673_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_674_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_675_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_676_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_677_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
& ~ ( ord_less_eq_num @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_678_less__le__not__le,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X4 @ Y3 )
& ~ ( ord_le2843351958646193337nt_int @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_679_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_680_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_681_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_682_antisym__conv2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ~ ( ord_le7563427860532173253nt_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_683_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_684_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_685_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_686_antisym__conv1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ~ ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_687_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_688_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_689_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_690_nless__le,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ~ ( ord_le7563427860532173253nt_int @ A @ B ) )
= ( ~ ( ord_le2843351958646193337nt_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_691_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_692_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_693_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_694_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_695_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_696_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_697_leD,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ~ ( ord_le7563427860532173253nt_int @ X @ Y ) ) ).
% leD
thf(fact_698_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_699_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_700_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_701_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_702_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_703_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_704_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_705_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_706_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_707_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_708_rev__swap,axiom,
! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs2 )
= Ys )
= ( Xs2
= ( rev_Pr2923690841345412895nt_int @ Ys ) ) ) ).
% rev_swap
thf(fact_709_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_710_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_711_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_712_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_713_rev_Osimps_I1_J,axiom,
( ( rev_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% rev.simps(1)
thf(fact_714_rev_Osimps_I1_J,axiom,
( ( rev_Pr2923690841345412895nt_int @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% rev.simps(1)
thf(fact_715_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_716_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A4: num] :
( ( ~ ( ord_less_eq_num @ B4 @ A4 ) )
= ( ord_less_num @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_717_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
= ( ord_less_int @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_718_knights__path__rev,axiom,
! [B: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps )
=> ( knights_path @ B @ ( rev_Pr2923690841345412895nt_int @ Ps ) ) ) ).
% knights_path_rev
thf(fact_719_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_720_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_721_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_722_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_723_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_724_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_725_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_726_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_727_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_728_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_729_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_730_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_731_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_732_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_733_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_734_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_735_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_736_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_737_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_738_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_739_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_740_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_741_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_742_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_743_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_744_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_745_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_746_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_747_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_748_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_749_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_750_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_751_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_752_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_753_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_754_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_755_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_756_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_757_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_758_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_759_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_760_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_761_min_Ostrict__coboundedI2,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_num @ B @ C )
=> ( ord_less_num @ ( ord_min_num @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_762_min_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ B @ C )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_763_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_764_min_Ostrict__coboundedI1,axiom,
! [A: num,C: num,B: num] :
( ( ord_less_num @ A @ C )
=> ( ord_less_num @ ( ord_min_num @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_765_min_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ C )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_766_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( ord_min_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_767_min_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( A3
= ( ord_min_num @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_768_min_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( A3
= ( ord_min_int @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_769_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_770_min_Ostrict__boundedE,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ ( ord_min_num @ B @ C ) )
=> ~ ( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_771_min_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ ( ord_min_int @ B @ C ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_772_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
| ( ord_less_nat @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_773_min__less__iff__disj,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ ( ord_min_num @ X @ Y ) @ Z )
= ( ( ord_less_num @ X @ Z )
| ( ord_less_num @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_774_min__less__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ( ord_less_int @ X @ Z )
| ( ord_less_int @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_775_knights__circuit__rev,axiom,
! [B: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int] :
( ( knights_circuit @ B @ Ps )
=> ( knights_circuit @ B @ ( rev_Pr2923690841345412895nt_int @ Ps ) ) ) ).
% knights_circuit_rev
thf(fact_776_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_777_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_778_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_779_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_780_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_781_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_782_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_783_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_784_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_785_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_786_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_787_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_788_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_789_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_790_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_791_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_792_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_793_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_794_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_795_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_796_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_797_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_798_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_799_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_800_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_801_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_802_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_803_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_804_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_805_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_806_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_807_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_808_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_809_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_810_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_811_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_812_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_813_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_814_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_815_order__antisym__conv,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( ord_le2843351958646193337nt_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_816_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_817_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_818_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_819_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_820_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_821_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_822_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_823_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_824_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_825_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_826_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_827_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_828_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_829_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_830_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_831_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_832_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_833_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_834_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_835_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_836_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_837_ord__eq__le__subst,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_838_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_839_ord__eq__le__subst,axiom,
! [A: set_Pr958786334691620121nt_int,F: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_840_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_841_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_842_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_843_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_844_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_845_order__eq__refl,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( X = Y )
=> ( ord_le2843351958646193337nt_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_846_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_847_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_848_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_849_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_850_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_851_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_852_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_853_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_854_order__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_855_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_856_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_857_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_858_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_859_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_860_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_861_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_862_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_863_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_864_order__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_865_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_866_order__subst1,axiom,
! [A: nat,F: set_Pr958786334691620121nt_int > nat,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X2: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_867_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_868_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z2: num] : ( Y5 = Z2 ) )
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_869_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
= ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
& ( ord_le2843351958646193337nt_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_870_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_871_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_872_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_873_antisym,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_874_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_875_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_876_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_877_dual__order_Otrans,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le2843351958646193337nt_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_878_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_879_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_880_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_881_dual__order_Oantisym,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_882_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_883_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_884_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z2: num] : ( Y5 = Z2 ) )
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_885_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
= ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
& ( ord_le2843351958646193337nt_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_886_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_887_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_888_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: num,B5: num] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_889_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_890_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_891_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_892_order__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ Z )
=> ( ord_le2843351958646193337nt_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_893_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_894_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_895_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_896_order_Otrans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% order.trans
thf(fact_897_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_898_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_899_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_900_order__antisym,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_901_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_902_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_903_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_904_ord__le__eq__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( B = C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_905_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_906_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_907_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_908_ord__eq__le__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( A = B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_909_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_910_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_911_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z2: num] : ( Y5 = Z2 ) )
= ( ^ [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
& ( ord_less_eq_num @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_912_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X4 @ Y3 )
& ( ord_le2843351958646193337nt_int @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_913_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_914_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_915_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_916_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_917_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_918_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_919_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_920_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_921_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_922_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_923_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_924_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_925_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_926_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_927_uminus__power__if,axiom,
! [N: nat,A: int] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_928_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_929_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% minus_one_power_iff
thf(fact_930_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_931_min__absorb2,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_min_num @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_932_min__absorb2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( ord_mi3891036259616950784nt_int @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_933_min__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_min_int @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_934_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_935_min__absorb1,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_min_num @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_936_min__absorb1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_mi3891036259616950784nt_int @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_937_min__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_min_int @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_938_min__def,axiom,
( ord_min_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_939_min__def,axiom,
( ord_min_num
= ( ^ [A3: num,B3: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_940_min__def,axiom,
( ord_mi3891036259616950784nt_int
= ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : ( if_set4441254072200386271nt_int @ ( ord_le2843351958646193337nt_int @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_941_min__def,axiom,
( ord_min_int
= ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_942_power2__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_943_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_944_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_945_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_946_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_947_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_948_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_949_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_950_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_951_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_952_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_953_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_954_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_955_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_956_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_957_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_958_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_959_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_960_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_961_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_962_dvd__power__same,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_963_dvd__power__same,axiom,
! [X: int,Y: int,N: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_964_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_965_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_966_dvd__power__le,axiom,
! [X: nat,Y: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_967_dvd__power__le,axiom,
! [X: int,Y: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_968_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_969_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_970_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_971_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_972_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_973_power__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_974_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_975_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_976_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_977_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_978_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_979_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_980_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_981_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_982_power__minus__Bit0,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_983_power__minus__Bit1,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_984_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_985_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_986_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_987_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_988_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_989_power2__commute,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_990_power2__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_991_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_992_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_993_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_994_square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
=> ( ( ord_less_eq_int @ X @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_995_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_996_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_997_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_998_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_999_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_1000_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1001_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1002_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D2: int,T2: int] : ( dvd_dvd_int @ D2 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_1003_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D2: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D2 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_1004_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1005_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1006_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1007_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1008_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ord_less_eq_num @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1009_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1010_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_1011_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ord_less_eq_num @ X5 @ T ) ) ).
% minf(6)
thf(fact_1012_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_1013_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_1014_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% minf(8)
thf(fact_1015_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_1016_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_1017_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_1018_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_1019_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_1020_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1021_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1022_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1023_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1024_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1025_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1026_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1027_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1028_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1029_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1030_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1031_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_1032_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_1033_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_1034_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1035_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1036_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1037_push__bit__numeral__minus__1,axiom,
! [N: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_1038_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1039_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1040_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1041_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1042_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1043_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_1044_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1045_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_1046_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1047_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_1048_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1049_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_1050_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1051_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_1052_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_1053_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1054_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1055_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1056_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1057_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1058_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1059_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1060_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1061_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_1062_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1063_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1064_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1065_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_1066_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1067_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1068_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_1069_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_1070_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_1071_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_1072_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_1073_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1074_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1075_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1076_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1077_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1078_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1079_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1080_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1081_push__bit__eq__0__iff,axiom,
! [N: nat,A: nat] :
( ( ( bit_se547839408752420682it_nat @ N @ A )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% push_bit_eq_0_iff
thf(fact_1082_push__bit__eq__0__iff,axiom,
! [N: nat,A: int] :
( ( ( bit_se545348938243370406it_int @ N @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% push_bit_eq_0_iff
thf(fact_1083_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% push_bit_of_0
thf(fact_1084_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% push_bit_of_0
thf(fact_1085_push__bit__push__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% push_bit_push_bit
thf(fact_1086_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_1087_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1088_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_1089_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1090_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1091_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_1092_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1093_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_1094_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1095_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_1096_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1097_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_1098_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1099_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1100_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_1101_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1102_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_1103_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1104_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_1105_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_1106_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_1107_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_1108_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1109_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1110_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_1111_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_1112_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_1113_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_1114_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1115_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1116_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_1117_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1118_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1119_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_1120_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1121_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1122_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1123_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1124_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1125_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_1126_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_1127_min__0__1_I4_J,axiom,
! [X: num] :
( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(4)
thf(fact_1128_min__0__1_I4_J,axiom,
! [X: num] :
( ( ord_min_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
= zero_zero_int ) ).
% min_0_1(4)
thf(fact_1129_min__0__1_I3_J,axiom,
! [X: num] :
( ( ord_min_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
= zero_zero_nat ) ).
% min_0_1(3)
thf(fact_1130_min__0__1_I3_J,axiom,
! [X: num] :
( ( ord_min_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
= zero_zero_int ) ).
% min_0_1(3)
thf(fact_1131_min__0__1_I2_J,axiom,
( ( ord_min_int @ one_one_int @ zero_zero_int )
= zero_zero_int ) ).
% min_0_1(2)
thf(fact_1132_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_1133_min__0__1_I1_J,axiom,
( ( ord_min_int @ zero_zero_int @ one_one_int )
= zero_zero_int ) ).
% min_0_1(1)
thf(fact_1134_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_1135_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1136_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1137_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1138_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1139_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1140_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_1141_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1142_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1143_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1144_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1145_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1146_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1147_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1148_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1149_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1150_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_1151_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_1152_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1153_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1154_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1155_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1156_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_1157_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_1158_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_1159_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ one_one_int )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_1160_even__push__bit__iff,axiom,
! [N: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
= ( ( N != zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_1161_even__push__bit__iff,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( ( N != zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_1162_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_1163_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_1164_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1165_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1166_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_1167_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1168_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1169_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1170_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1171_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1172_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_1173_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_1174_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1175_push__bit__int__code_I1_J,axiom,
! [I: int] :
( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
= I ) ).
% push_bit_int_code(1)
thf(fact_1176_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1177_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1178_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1179_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1180_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1181_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1182_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1183_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1184_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1185_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1186_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1187_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1188_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1189_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1190_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1191_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1192_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 ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1193_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 ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1194_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1195_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_1196_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_1197_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1198_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_1199_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% push_bit_negative_int_iff
thf(fact_1200_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1201_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1202_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1203_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1204_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1205_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1206_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1207_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1208_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1209_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1210_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1211_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1212_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1213_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1214_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1215_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1216_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B3: nat] :
( ( ( dvd_dvd_nat @ A3 @ B3 )
& ( A3 != B3 ) )
| ( A3 = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1217_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1218_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1219_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1220_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1221_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_1222_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_1223_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
& ( dvd_dvd_nat @ B3 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1224_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_1225_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_1226_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_1227_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1228_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_1229_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1230_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1231_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1232_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1233_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1234_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1235_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1236_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1237_verit__less__mono__div__int2,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1238_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_1239_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1240_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1241_zdiv__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1242_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_1243_zdiv__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_1244_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1245_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1246_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1247_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1248_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1249_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1250_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1251_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1252_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1253_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1254_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1255_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1256_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1257_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1258_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1259_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
% Helper facts (11)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
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 ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( if_set4441254072200386271nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( if_set4441254072200386271nt_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
knights_path @ ( board @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( the_li7431803565598847443nt_int @ ( to_path @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ nil_nat ) ) ) ) ) @ ( cons_list_nat @ ( cons_nat @ one_one_nat @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( cons_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ nil_nat ) ) ) ) ) @ nil_list_nat ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------