TPTP Problem File: SLH0059^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Combinable_Wands/0001_Mask/prob_00208_005966__7106346_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1243 ( 553 unt; 180 typ; 0 def)
% Number of atoms : 2781 (1216 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 7748 ( 277 ~; 57 |; 170 &;6261 @)
% ( 0 <=>; 983 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 616 ( 616 >; 0 *; 0 +; 0 <<)
% Number of symbols : 160 ( 157 usr; 11 con; 0-8 aty)
% Number of variables : 2301 ( 102 ^;2151 !; 48 ?;2301 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:05:10.946
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__PosRat__Oprat,type,
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thf(ty_n_t__Mask__Oval,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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thf(ty_n_tf__b,type,
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% Explicit typings (157)
thf(sy_c_Fun_Ofun__upd_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__PosRat__Oprat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_If_001t__Int__Oint,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_Int_Onat,type,
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thf(sy_c_Mask_Omultiply__mask,type,
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thf(sy_c_Mask_Onull_001t__Int__Oint,type,
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thf(sy_c_Mask_Onull_001t__Nat__Onat,type,
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thf(sy_c_Mask_Opre__get__h_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Mask_Oval_OBool,type,
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thf(sy_c_Mask_Oval_Ois__Address,type,
is_Address: val > $o ).
thf(sy_c_Mask_Oval_Ois__Bool,type,
is_Bool: val > $o ).
thf(sy_c_Mask_Oval_Ois__Rat,type,
is_Rat: val > $o ).
thf(sy_c_Mask_Oval_Osize__val,type,
size_val: val > nat ).
thf(sy_c_Mask_Oval_Othe__address,type,
the_address: val > nat ).
thf(sy_c_Mask_Oval_Othe__bool,type,
the_bool: val > $o ).
thf(sy_c_Mask_Oval_Othe__rat,type,
the_rat: val > prat ).
thf(sy_c_Mask_Ovalid__mask,type,
valid_mask: ( produc1457211279475724562t_char > prat ) > $o ).
thf(sy_c_Mask_Ovalid__mask__rel,type,
valid_mask_rel: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ).
thf(sy_c_Mask_Ovalid__null,type,
valid_null: ( produc1457211279475724562t_char > prat ) > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Mask__Oval,type,
size_size_val: val > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
size_size_typerep: typerep > nat ).
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_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__eq_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_Eo_J_J,type,
ord_le6723339807950303054prat_o: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_Eo_J,type,
ord_le2830794348966088778prat_o: ( ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J_M_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J_M_Eo_J_J,type,
ord_le6348261579570775310char_o: ( produc5803078220529002682t_char > produc5803078220529002682t_char > $o ) > ( produc5803078220529002682t_char > produc5803078220529002682t_char > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J_M_Eo_J,type,
ord_le7815076450745241763char_o: ( produc5803078220529002682t_char > $o ) > ( produc5803078220529002682t_char > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_M_Eo_J_J,type,
ord_le2827606955515295502char_o: ( produc1457211279475724562t_char > produc1457211279475724562t_char > $o ) > ( produc1457211279475724562t_char > produc1457211279475724562t_char > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_M_Eo_J,type,
ord_le5722155653058504523char_o: ( produc1457211279475724562t_char > $o ) > ( produc1457211279475724562t_char > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_Eo_J_J,type,
ord_le5632367588805359236prat_o: ( produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o ) > ( produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_Eo_J,type,
ord_le1173062544651188894prat_o: ( produc3933091914578511633r_prat > $o ) > ( produc3933091914578511633r_prat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J_J,type,
ord_le5742415681470231226t_char: set_Pr1935176096852982554t_char > set_Pr1935176096852982554t_char > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
ord_le3803256517986266150t_char: set_Pr7632435056502277254t_char > set_Pr7632435056502277254t_char > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
ord_le6315511645215477266t_char: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
ord_le9052479809666887335r_prat: set_Pr2722754889679711815r_prat > set_Pr2722754889679711815r_prat > $o ).
thf(sy_c_Parity_Oadjust__div,type,
adjust_div: product_prod_int_int > int ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
unique5329631941980267465ux_int: product_prod_int_int > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_PosRat_Ocomp__one,type,
comp_one: prat > prat ).
thf(sy_c_PosRat_Ohalf,type,
half: prat ).
thf(sy_c_PosRat_Opadd,type,
padd: prat > prat > prat ).
thf(sy_c_PosRat_Opdiv,type,
pdiv: prat > prat > prat ).
thf(sy_c_PosRat_Opgt,type,
pgt: prat > prat > $o ).
thf(sy_c_PosRat_Opgte,type,
pgte: prat > prat > $o ).
thf(sy_c_PosRat_Opinv,type,
pinv: prat > prat ).
thf(sy_c_PosRat_Opmax,type,
pmax: prat > prat > prat ).
thf(sy_c_PosRat_Opmin,type,
pmin: prat > prat > prat ).
thf(sy_c_PosRat_Opmult,type,
pmult: prat > prat > prat ).
thf(sy_c_PosRat_Opnone,type,
pnone: prat ).
thf(sy_c_PosRat_Oppos,type,
ppos: prat > $o ).
thf(sy_c_PosRat_Opwrite,type,
pwrite: prat ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__PosRat__Oprat,type,
produc5508603645230141260t_prat: ( produc1457211279475724562t_char > prat ) > prat > produc4561108331071084498t_prat ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
produc2957999048406202538t_char: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > produc5803078220529002682t_char ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
produc2591414526589101846t_char: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > produc2489117125269924006t_char ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
produc5928798746291773453r_prat: ( produc1457211279475724562t_char > prat ) > produc3933091914578511633r_prat > produc6235724684372434707r_prat ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
produc120197956887798346t_char: nat > list_char > produc1457211279475724562t_char ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__PosRat__Oprat,type,
produc2920179824973321483r_prat: produc1457211279475724562t_char > prat > produc3933091914578511633r_prat ).
thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
produc5882964721085011416t_char: produc5803078220529002682t_char > produc2489117125269924006t_char ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
product_snd_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
produc1900778367047502200t_char: produc1457211279475724562t_char > list_char ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__PosRat__Oprat,type,
produc2879476908335557213r_prat: produc3933091914578511633r_prat > prat ).
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_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
collec2458894688891239037t_char: ( produc1457211279475724562t_char > $o ) > set_Pr2507339680178222706t_char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_Oaccp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
accp_P4765339447491148764r_prat: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( produc1457211279475724562t_char > prat ) > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__PosRat__Oprat_J,type,
accp_P6822921472154139145t_prat: ( produc4561108331071084498t_prat > produc4561108331071084498t_prat > $o ) > produc4561108331071084498t_prat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
accp_P224314924008452035t_char: ( produc5803078220529002682t_char > produc5803078220529002682t_char > $o ) > produc5803078220529002682t_char > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
accp_P5529878251199943754r_prat: ( produc6235724684372434707r_prat > produc6235724684372434707r_prat > $o ) > produc6235724684372434707r_prat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
accp_P8956218777217113800r_prat: ( produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o ) > produc3933091914578511633r_prat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
member5171123587969721059t_char: produc5803078220529002682t_char > set_Pr1935176096852982554t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
member484235747325421115t_char: produc1457211279475724562t_char > set_Pr2507339680178222706t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
member7501365625282528168r_prat: produc3933091914578511633r_prat > set_Pr2722754889679711815r_prat > $o ).
thf(sy_v__092_060pi_062,type,
pi: a > $o ).
thf(sy_v_h,type,
h: a > b ).
thf(sy_v_h_H,type,
h2: a > b ).
% Relevant facts (1057)
thf(fact_0_assms,axiom,
! [Hl: a] :
( ( pi @ Hl )
=> ( ( h @ Hl )
= ( h2 @ Hl ) ) ) ).
% assms
thf(fact_1_equal__on__bmask__def,axiom,
( equal_on_bmask_a_b
= ( ^ [Pi: a > $o,H: a > b,H2: a > b] :
! [Hl2: a] :
( ( Pi @ Hl2 )
=> ( ( H @ Hl2 )
= ( H2 @ Hl2 ) ) ) ) ) ).
% equal_on_bmask_def
thf(fact_2_empty__bmask_Oelims_I2_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
~ ( empty_7634413185502380953r_prat @ X ) ).
% empty_bmask.elims(2)
thf(fact_3_empty__bmask_Oelims_I1_J,axiom,
! [X: produc1457211279475724562t_char > prat,Y: $o] :
( ( ( empty_7634413185502380953r_prat @ X )
= Y )
=> ~ Y ) ).
% empty_bmask.elims(1)
thf(fact_4_empty__bmask_Osimps,axiom,
( empty_7634413185502380953r_prat
= ( ^ [Hl2: produc1457211279475724562t_char > prat] : $false ) ) ).
% empty_bmask.simps
thf(fact_5_val_Oinject_I3_J,axiom,
! [X3: prat,Y3: prat] :
( ( ( rat @ X3 )
= ( rat @ Y3 ) )
= ( X3 = Y3 ) ) ).
% val.inject(3)
thf(fact_6_val_Oinject_I2_J,axiom,
! [X2: nat,Y2: nat] :
( ( ( address @ X2 )
= ( address @ Y2 ) )
= ( X2 = Y2 ) ) ).
% val.inject(2)
thf(fact_7_val_Oinject_I1_J,axiom,
! [X1: $o,Y1: $o] :
( ( ( bool @ X1 )
= ( bool @ Y1 ) )
= ( X1 = Y1 ) ) ).
% val.inject(1)
thf(fact_8_greater__mask__properties_I3_J,axiom,
! [Pi2: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat] :
( ( ( greate7680731506090375811t_char @ Pi2 @ Pi3 )
& ( greate7680731506090375811t_char @ Pi3 @ Pi2 ) )
=> ( Pi3 = Pi2 ) ) ).
% greater_mask_properties(3)
thf(fact_9_greater__mask__properties_I2_J,axiom,
! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( ( greate7680731506090375811t_char @ A @ B )
& ( greate7680731506090375811t_char @ B @ C ) )
=> ( greate7680731506090375811t_char @ A @ C ) ) ).
% greater_mask_properties(2)
thf(fact_10_greater__mask__properties_I1_J,axiom,
! [Pi3: produc1457211279475724562t_char > prat] : ( greate7680731506090375811t_char @ Pi3 @ Pi3 ) ).
% greater_mask_properties(1)
thf(fact_11_val_Odistinct_I5_J,axiom,
! [X2: nat,X3: prat] :
( ( address @ X2 )
!= ( rat @ X3 ) ) ).
% val.distinct(5)
thf(fact_12_val_Odistinct_I3_J,axiom,
! [X1: $o,X3: prat] :
( ( bool @ X1 )
!= ( rat @ X3 ) ) ).
% val.distinct(3)
thf(fact_13_val_Odistinct_I1_J,axiom,
! [X1: $o,X2: nat] :
( ( bool @ X1 )
!= ( address @ X2 ) ) ).
% val.distinct(1)
thf(fact_14_val_Oexhaust,axiom,
! [Y: val] :
( ! [X12: $o] :
( Y
!= ( bool @ X12 ) )
=> ( ! [X22: nat] :
( Y
!= ( address @ X22 ) )
=> ~ ! [X32: prat] :
( Y
!= ( rat @ X32 ) ) ) ) ).
% val.exhaust
thf(fact_15_val_Osel_I3_J,axiom,
! [X3: prat] :
( ( the_rat @ ( rat @ X3 ) )
= X3 ) ).
% val.sel(3)
thf(fact_16_val_Osel_I2_J,axiom,
! [X2: nat] :
( ( the_address @ ( address @ X2 ) )
= X2 ) ).
% val.sel(2)
thf(fact_17_val_Osel_I1_J,axiom,
! [X1: $o] :
( ( the_bool @ ( bool @ X1 ) )
= X1 ) ).
% val.sel(1)
thf(fact_18_val_Odisc_I9_J,axiom,
! [X3: prat] : ( is_Rat @ ( rat @ X3 ) ) ).
% val.disc(9)
thf(fact_19_val_Odisc_I8_J,axiom,
! [X2: nat] :
~ ( is_Rat @ ( address @ X2 ) ) ).
% val.disc(8)
thf(fact_20_val_Ocollapse_I3_J,axiom,
! [Val: val] :
( ( is_Rat @ Val )
=> ( ( rat @ ( the_rat @ Val ) )
= Val ) ) ).
% val.collapse(3)
thf(fact_21_val_Oexhaust__sel,axiom,
! [Val: val] :
( ( Val
!= ( bool @ ( the_bool @ Val ) ) )
=> ( ( Val
!= ( address @ ( the_address @ Val ) ) )
=> ( Val
= ( rat @ ( the_rat @ Val ) ) ) ) ) ).
% val.exhaust_sel
thf(fact_22_is__Rat__def,axiom,
( is_Rat
= ( ^ [Val2: val] :
? [X33: prat] :
( Val2
= ( rat @ X33 ) ) ) ) ).
% is_Rat_def
thf(fact_23_val_OdiscI_I3_J,axiom,
! [Val: val,X3: prat] :
( ( Val
= ( rat @ X3 ) )
=> ( is_Rat @ Val ) ) ).
% val.discI(3)
thf(fact_24_val_Odisc_I7_J,axiom,
! [X1: $o] :
~ ( is_Rat @ ( bool @ X1 ) ) ).
% val.disc(7)
thf(fact_25_val_Ocollapse_I1_J,axiom,
! [Val: val] :
( ( is_Bool @ Val )
=> ( ( bool @ ( the_bool @ Val ) )
= Val ) ) ).
% val.collapse(1)
thf(fact_26_val_Ocollapse_I2_J,axiom,
! [Val: val] :
( ( is_Address @ Val )
=> ( ( address @ ( the_address @ Val ) )
= Val ) ) ).
% val.collapse(2)
thf(fact_27_val_Oexpand,axiom,
! [Val: val,Val3: val] :
( ( ( is_Bool @ Val )
= ( is_Bool @ Val3 ) )
=> ( ( ( is_Bool @ Val )
=> ( ( is_Bool @ Val3 )
=> ( ( the_bool @ Val )
= ( the_bool @ Val3 ) ) ) )
=> ( ( ( is_Address @ Val )
= ( is_Address @ Val3 ) )
=> ( ( ( is_Address @ Val )
=> ( ( is_Address @ Val3 )
=> ( ( the_address @ Val )
= ( the_address @ Val3 ) ) ) )
=> ( ( ( is_Rat @ Val )
=> ( ( is_Rat @ Val3 )
=> ( ( the_rat @ Val )
= ( the_rat @ Val3 ) ) ) )
=> ( Val = Val3 ) ) ) ) ) ) ).
% val.expand
thf(fact_28_greater__mask__decomp,axiom,
! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( greate7680731506090375811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
=> ? [A1: produc1457211279475724562t_char > prat,A2: produc1457211279475724562t_char > prat] :
( ( A
= ( add_ma6586698345353345811t_char @ A1 @ A2 ) )
& ( greate7680731506090375811t_char @ A1 @ B )
& ( greate7680731506090375811t_char @ A2 @ C ) ) ) ).
% greater_mask_decomp
thf(fact_29_greater__mask__def,axiom,
( greate7680731506090375811t_char
= ( ^ [Pi4: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
? [R: produc1457211279475724562t_char > prat] :
( Pi4
= ( add_ma6586698345353345811t_char @ Pi @ R ) ) ) ) ).
% greater_mask_def
thf(fact_30_val_Odistinct__disc_I3_J,axiom,
! [Val: val] :
( ( is_Bool @ Val )
=> ~ ( is_Rat @ Val ) ) ).
% val.distinct_disc(3)
thf(fact_31_val_Odistinct__disc_I4_J,axiom,
! [Val: val] :
( ( is_Rat @ Val )
=> ~ ( is_Bool @ Val ) ) ).
% val.distinct_disc(4)
thf(fact_32_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_33_mem__Collect__eq,axiom,
! [A: produc1457211279475724562t_char,P: produc1457211279475724562t_char > $o] :
( ( member484235747325421115t_char @ A @ ( collec2458894688891239037t_char @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_34_Collect__mem__eq,axiom,
! [A3: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_35_Collect__mem__eq,axiom,
! [A3: set_Pr2507339680178222706t_char] :
( ( collec2458894688891239037t_char
@ ^ [X4: produc1457211279475724562t_char] : ( member484235747325421115t_char @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_36_val_Odistinct__disc_I5_J,axiom,
! [Val: val] :
( ( is_Address @ Val )
=> ~ ( is_Rat @ Val ) ) ).
% val.distinct_disc(5)
thf(fact_37_val_Odistinct__disc_I6_J,axiom,
! [Val: val] :
( ( is_Rat @ Val )
=> ~ ( is_Address @ Val ) ) ).
% val.distinct_disc(6)
thf(fact_38_is__Address__def,axiom,
( is_Address
= ( ^ [Val2: val] :
? [X23: nat] :
( Val2
= ( address @ X23 ) ) ) ) ).
% is_Address_def
thf(fact_39_val_Odistinct__disc_I2_J,axiom,
! [Val: val] :
( ( is_Address @ Val )
=> ~ ( is_Bool @ Val ) ) ).
% val.distinct_disc(2)
thf(fact_40_val_Odistinct__disc_I1_J,axiom,
! [Val: val] :
( ( is_Bool @ Val )
=> ~ ( is_Address @ Val ) ) ).
% val.distinct_disc(1)
thf(fact_41_add__masks__asso,axiom,
! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( add_ma6586698345353345811t_char @ ( add_ma6586698345353345811t_char @ A @ B ) @ C )
= ( add_ma6586698345353345811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) ) ) ).
% add_masks_asso
thf(fact_42_add__masks__comm,axiom,
( add_ma6586698345353345811t_char
= ( ^ [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B2 @ A4 ) ) ) ).
% add_masks_comm
thf(fact_43_val_Oexhaust__disc,axiom,
! [Val: val] :
( ~ ( is_Bool @ Val )
=> ( ~ ( is_Address @ Val )
=> ( is_Rat @ Val ) ) ) ).
% val.exhaust_disc
thf(fact_44_val_Odisc_I6_J,axiom,
! [X3: prat] :
~ ( is_Address @ ( rat @ X3 ) ) ).
% val.disc(6)
thf(fact_45_val_Odisc_I5_J,axiom,
! [X2: nat] : ( is_Address @ ( address @ X2 ) ) ).
% val.disc(5)
thf(fact_46_val_Odisc_I4_J,axiom,
! [X1: $o] :
~ ( is_Address @ ( bool @ X1 ) ) ).
% val.disc(4)
thf(fact_47_val_Odisc_I3_J,axiom,
! [X3: prat] :
~ ( is_Bool @ ( rat @ X3 ) ) ).
% val.disc(3)
thf(fact_48_val_Odisc_I2_J,axiom,
! [X2: nat] :
~ ( is_Bool @ ( address @ X2 ) ) ).
% val.disc(2)
thf(fact_49_val_Odisc_I1_J,axiom,
! [X1: $o] : ( is_Bool @ ( bool @ X1 ) ) ).
% val.disc(1)
thf(fact_50_val_OdiscI_I2_J,axiom,
! [Val: val,X2: nat] :
( ( Val
= ( address @ X2 ) )
=> ( is_Address @ Val ) ) ).
% val.discI(2)
thf(fact_51_val_OdiscI_I1_J,axiom,
! [Val: val,X1: $o] :
( ( Val
= ( bool @ X1 ) )
=> ( is_Bool @ Val ) ) ).
% val.discI(1)
thf(fact_52_is__Bool__def,axiom,
( is_Bool
= ( ^ [Val2: val] :
? [X13: $o] :
( Val2
= ( bool @ X13 ) ) ) ) ).
% is_Bool_def
thf(fact_53_minus__empty,axiom,
! [Pi3: ( produc1457211279475724562t_char > prat ) > prat] :
( Pi3
= ( add_ma5081279659257173972r_prat @ Pi3 @ empty_6616431780912394577r_prat ) ) ).
% minus_empty
thf(fact_54_minus__empty,axiom,
! [Pi3: produc1457211279475724562t_char > prat] :
( Pi3
= ( add_ma6586698345353345811t_char @ Pi3 @ empty_3446695950879338768t_char ) ) ).
% minus_empty
thf(fact_55_add__acc__uni__mask,axiom,
( add_ac6043276672051819257t_char
= ( ^ [Pi: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char,P2: prat] : ( add_ma6586698345353345811t_char @ Pi @ ( uni_ma6578582744724537101t_char @ Hl2 @ P2 ) ) ) ) ).
% add_acc_uni_mask
thf(fact_56_greater__maskI,axiom,
! [Pi2: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat] :
( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ ( Pi2 @ Hl3 ) @ ( Pi3 @ Hl3 ) )
=> ( greate7680731506090375811t_char @ Pi2 @ Pi3 ) ) ).
% greater_maskI
thf(fact_57_greater__mask__equiv__def,axiom,
( greate7680731506090375811t_char
= ( ^ [Pi4: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
! [Hl2: produc1457211279475724562t_char] : ( pgte @ ( Pi4 @ Hl2 ) @ ( Pi @ Hl2 ) ) ) ) ).
% greater_mask_equiv_def
thf(fact_58_add__masks_Oelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) ) ) ).
% add_masks.elims
thf(fact_59_add__masks_Osimps,axiom,
( add_ma6586698345353345811t_char
= ( ^ [Pi4: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( padd @ ( Pi4 @ Hl2 ) @ ( Pi @ Hl2 ) ) ) ) ).
% add_masks.simps
thf(fact_60_empty__bmask_Opelims_I2_J,axiom,
! [X: produc3933091914578511633r_prat] :
( ( empty_4818326827210037067r_prat @ X )
=> ~ ( accp_P8956218777217113800r_prat @ empty_8629834306080346194r_prat @ X ) ) ).
% empty_bmask.pelims(2)
thf(fact_61_empty__bmask_Opelims_I2_J,axiom,
! [X: produc1457211279475724562t_char] :
( ( empty_7833442678171572056t_char @ X )
=> ~ ( accp_P8892882183480857371t_char @ empty_5910826708504235025t_char @ X ) ) ).
% empty_bmask.pelims(2)
thf(fact_62_empty__bmask_Opelims_I2_J,axiom,
! [X: produc5803078220529002682t_char] :
( ( empty_5040027858169863424t_char @ X )
=> ~ ( accp_P224314924008452035t_char @ empty_2805327505229121209t_char @ X ) ) ).
% empty_bmask.pelims(2)
thf(fact_63_empty__bmask_Opelims_I2_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ( empty_7634413185502380953r_prat @ X )
=> ~ ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X ) ) ).
% empty_bmask.pelims(2)
thf(fact_64_empty__bmask_Opelims_I1_J,axiom,
! [X: produc3933091914578511633r_prat,Y: $o] :
( ( ( empty_4818326827210037067r_prat @ X )
= Y )
=> ( ( accp_P8956218777217113800r_prat @ empty_8629834306080346194r_prat @ X )
=> ~ ( ~ Y
=> ~ ( accp_P8956218777217113800r_prat @ empty_8629834306080346194r_prat @ X ) ) ) ) ).
% empty_bmask.pelims(1)
thf(fact_65_empty__bmask_Opelims_I1_J,axiom,
! [X: produc1457211279475724562t_char,Y: $o] :
( ( ( empty_7833442678171572056t_char @ X )
= Y )
=> ( ( accp_P8892882183480857371t_char @ empty_5910826708504235025t_char @ X )
=> ~ ( ~ Y
=> ~ ( accp_P8892882183480857371t_char @ empty_5910826708504235025t_char @ X ) ) ) ) ).
% empty_bmask.pelims(1)
thf(fact_66_empty__bmask_Opelims_I1_J,axiom,
! [X: produc5803078220529002682t_char,Y: $o] :
( ( ( empty_5040027858169863424t_char @ X )
= Y )
=> ( ( accp_P224314924008452035t_char @ empty_2805327505229121209t_char @ X )
=> ~ ( ~ Y
=> ~ ( accp_P224314924008452035t_char @ empty_2805327505229121209t_char @ X ) ) ) ) ).
% empty_bmask.pelims(1)
thf(fact_67_empty__bmask_Opelims_I1_J,axiom,
! [X: produc1457211279475724562t_char > prat,Y: $o] :
( ( ( empty_7634413185502380953r_prat @ X )
= Y )
=> ( ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X )
=> ~ ( ~ Y
=> ~ ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X ) ) ) ) ).
% empty_bmask.pelims(1)
thf(fact_68_val_Osize__gen_I3_J,axiom,
! [X3: prat] :
( ( size_val @ ( rat @ X3 ) )
= zero_zero_nat ) ).
% val.size_gen(3)
thf(fact_69_val_Osize__gen_I1_J,axiom,
! [X1: $o] :
( ( size_val @ ( bool @ X1 ) )
= zero_zero_nat ) ).
% val.size_gen(1)
thf(fact_70_val_Osize__gen_I2_J,axiom,
! [X2: nat] :
( ( size_val @ ( address @ X2 ) )
= zero_zero_nat ) ).
% val.size_gen(2)
thf(fact_71_upper__bounded__def,axiom,
( upper_bounded
= ( ^ [Pi: produc1457211279475724562t_char > prat,P2: prat] :
! [Hl2: produc1457211279475724562t_char] : ( pgte @ P2 @ ( Pi @ Hl2 ) ) ) ) ).
% upper_bounded_def
thf(fact_72_p__greater__exists,axiom,
( pgte
= ( ^ [A4: prat,B2: prat] :
? [R: prat] :
( A4
= ( padd @ B2 @ R ) ) ) ) ).
% p_greater_exists
thf(fact_73_greater__sum__both,axiom,
! [A: prat,B: prat,C: prat] :
( ( pgte @ A @ ( padd @ B @ C ) )
=> ? [A1: prat,A2: prat] :
( ( A
= ( padd @ A1 @ A2 ) )
& ( pgte @ A1 @ B )
& ( pgte @ A2 @ C ) ) ) ).
% greater_sum_both
thf(fact_74_Mask_Onull__def,axiom,
null_nat = zero_zero_nat ).
% Mask.null_def
thf(fact_75_Mask_Onull__def,axiom,
null_int = zero_zero_int ).
% Mask.null_def
thf(fact_76_val_Osize_I4_J,axiom,
! [X1: $o] :
( ( size_size_val @ ( bool @ X1 ) )
= zero_zero_nat ) ).
% val.size(4)
thf(fact_77_val_Osize_I5_J,axiom,
! [X2: nat] :
( ( size_size_val @ ( address @ X2 ) )
= zero_zero_nat ) ).
% val.size(5)
thf(fact_78_val_Osize_I6_J,axiom,
! [X3: prat] :
( ( size_size_val @ ( rat @ X3 ) )
= zero_zero_nat ) ).
% val.size(6)
thf(fact_79_padd__cancellative,axiom,
! [A: prat,X: prat,B: prat,Y: prat] :
( ( A
= ( padd @ X @ B ) )
=> ( ( A
= ( padd @ Y @ B ) )
=> ( X = Y ) ) ) ).
% padd_cancellative
thf(fact_80_padd__comm,axiom,
( padd
= ( ^ [A4: prat,B2: prat] : ( padd @ B2 @ A4 ) ) ) ).
% padd_comm
thf(fact_81_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_82_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_83_pgte__antisym,axiom,
! [A: prat,B: prat] :
( ( pgte @ A @ B )
=> ( ( pgte @ B @ A )
=> ( A = B ) ) ) ).
% pgte_antisym
thf(fact_84_padd__asso,axiom,
! [A: prat,B: prat,C: prat] :
( ( padd @ ( padd @ A @ B ) @ C )
= ( padd @ A @ ( padd @ B @ C ) ) ) ).
% padd_asso
thf(fact_85_pgte__pgt,axiom,
! [A: prat,B: prat,C: prat,D: prat] :
( ( pgt @ A @ B )
=> ( ( pgte @ C @ D )
=> ( pgt @ ( padd @ A @ C ) @ ( padd @ B @ D ) ) ) ) ).
% pgte_pgt
thf(fact_86_uni__mask_Oelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: prat,Y: ( produc1457211279475724562t_char > prat ) > prat] :
( ( ( uni_ma8757825773229499086r_prat @ X @ Xa )
= Y )
=> ( Y
= ( fun_up9120501182914450918t_prat @ empty_6616431780912394577r_prat @ X @ Xa ) ) ) ).
% uni_mask.elims
thf(fact_87_uni__mask_Oelims,axiom,
! [X: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
( ( ( uni_ma6578582744724537101t_char @ X @ Xa )
= Y )
=> ( Y
= ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X @ Xa ) ) ) ).
% uni_mask.elims
thf(fact_88_uni__mask_Osimps,axiom,
( uni_ma8757825773229499086r_prat
= ( fun_up9120501182914450918t_prat @ empty_6616431780912394577r_prat ) ) ).
% uni_mask.simps
thf(fact_89_uni__mask_Osimps,axiom,
( uni_ma6578582744724537101t_char
= ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char ) ) ).
% uni_mask.simps
thf(fact_90_accp_Ocases,axiom,
! [R2: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,A: produc3933091914578511633r_prat] :
( ( accp_P8956218777217113800r_prat @ R2 @ A )
=> ! [Y4: produc3933091914578511633r_prat] :
( ( R2 @ Y4 @ A )
=> ( accp_P8956218777217113800r_prat @ R2 @ Y4 ) ) ) ).
% accp.cases
thf(fact_91_accp_Ocases,axiom,
! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char] :
( ( accp_P8892882183480857371t_char @ R2 @ A )
=> ! [Y4: produc1457211279475724562t_char] :
( ( R2 @ Y4 @ A )
=> ( accp_P8892882183480857371t_char @ R2 @ Y4 ) ) ) ).
% accp.cases
thf(fact_92_accp_Ocases,axiom,
! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A: produc5803078220529002682t_char] :
( ( accp_P224314924008452035t_char @ R2 @ A )
=> ! [Y4: produc5803078220529002682t_char] :
( ( R2 @ Y4 @ A )
=> ( accp_P224314924008452035t_char @ R2 @ Y4 ) ) ) ).
% accp.cases
thf(fact_93_accp_Ocases,axiom,
! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R2 @ A )
=> ! [Y4: produc1457211279475724562t_char > prat] :
( ( R2 @ Y4 @ A )
=> ( accp_P4765339447491148764r_prat @ R2 @ Y4 ) ) ) ).
% accp.cases
thf(fact_94_accp_Osimps,axiom,
( accp_P8956218777217113800r_prat
= ( ^ [R: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,A4: produc3933091914578511633r_prat] :
? [X4: produc3933091914578511633r_prat] :
( ( A4 = X4 )
& ! [Y5: produc3933091914578511633r_prat] :
( ( R @ Y5 @ X4 )
=> ( accp_P8956218777217113800r_prat @ R @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_95_accp_Osimps,axiom,
( accp_P8892882183480857371t_char
= ( ^ [R: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A4: produc1457211279475724562t_char] :
? [X4: produc1457211279475724562t_char] :
( ( A4 = X4 )
& ! [Y5: produc1457211279475724562t_char] :
( ( R @ Y5 @ X4 )
=> ( accp_P8892882183480857371t_char @ R @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_96_accp_Osimps,axiom,
( accp_P224314924008452035t_char
= ( ^ [R: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A4: produc5803078220529002682t_char] :
? [X4: produc5803078220529002682t_char] :
( ( A4 = X4 )
& ! [Y5: produc5803078220529002682t_char] :
( ( R @ Y5 @ X4 )
=> ( accp_P224314924008452035t_char @ R @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_97_accp_Osimps,axiom,
( accp_P4765339447491148764r_prat
= ( ^ [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A4: produc1457211279475724562t_char > prat] :
? [X4: produc1457211279475724562t_char > prat] :
( ( A4 = X4 )
& ! [Y5: produc1457211279475724562t_char > prat] :
( ( R @ Y5 @ X4 )
=> ( accp_P4765339447491148764r_prat @ R @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_98_accpI,axiom,
! [R2: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,X: produc3933091914578511633r_prat] :
( ! [Y6: produc3933091914578511633r_prat] :
( ( R2 @ Y6 @ X )
=> ( accp_P8956218777217113800r_prat @ R2 @ Y6 ) )
=> ( accp_P8956218777217113800r_prat @ R2 @ X ) ) ).
% accpI
thf(fact_99_accpI,axiom,
! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char] :
( ! [Y6: produc1457211279475724562t_char] :
( ( R2 @ Y6 @ X )
=> ( accp_P8892882183480857371t_char @ R2 @ Y6 ) )
=> ( accp_P8892882183480857371t_char @ R2 @ X ) ) ).
% accpI
thf(fact_100_accpI,axiom,
! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,X: produc5803078220529002682t_char] :
( ! [Y6: produc5803078220529002682t_char] :
( ( R2 @ Y6 @ X )
=> ( accp_P224314924008452035t_char @ R2 @ Y6 ) )
=> ( accp_P224314924008452035t_char @ R2 @ X ) ) ).
% accpI
thf(fact_101_accpI,axiom,
! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
( ! [Y6: produc1457211279475724562t_char > prat] :
( ( R2 @ Y6 @ X )
=> ( accp_P4765339447491148764r_prat @ R2 @ Y6 ) )
=> ( accp_P4765339447491148764r_prat @ R2 @ X ) ) ).
% accpI
thf(fact_102_accp__induct,axiom,
! [R2: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,A: produc3933091914578511633r_prat,P: produc3933091914578511633r_prat > $o] :
( ( accp_P8956218777217113800r_prat @ R2 @ A )
=> ( ! [X5: produc3933091914578511633r_prat] :
( ( accp_P8956218777217113800r_prat @ R2 @ X5 )
=> ( ! [Y4: produc3933091914578511633r_prat] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_103_accp__induct,axiom,
! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char,P: produc1457211279475724562t_char > $o] :
( ( accp_P8892882183480857371t_char @ R2 @ A )
=> ( ! [X5: produc1457211279475724562t_char] :
( ( accp_P8892882183480857371t_char @ R2 @ X5 )
=> ( ! [Y4: produc1457211279475724562t_char] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_104_accp__induct,axiom,
! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A: produc5803078220529002682t_char,P: produc5803078220529002682t_char > $o] :
( ( accp_P224314924008452035t_char @ R2 @ A )
=> ( ! [X5: produc5803078220529002682t_char] :
( ( accp_P224314924008452035t_char @ R2 @ X5 )
=> ( ! [Y4: produc5803078220529002682t_char] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_105_accp__induct,axiom,
! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P: ( produc1457211279475724562t_char > prat ) > $o] :
( ( accp_P4765339447491148764r_prat @ R2 @ A )
=> ( ! [X5: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R2 @ X5 )
=> ( ! [Y4: produc1457211279475724562t_char > prat] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_106_accp__downward,axiom,
! [R2: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,B: produc3933091914578511633r_prat,A: produc3933091914578511633r_prat] :
( ( accp_P8956218777217113800r_prat @ R2 @ B )
=> ( ( R2 @ A @ B )
=> ( accp_P8956218777217113800r_prat @ R2 @ A ) ) ) ).
% accp_downward
thf(fact_107_accp__downward,axiom,
! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,B: produc1457211279475724562t_char,A: produc1457211279475724562t_char] :
( ( accp_P8892882183480857371t_char @ R2 @ B )
=> ( ( R2 @ A @ B )
=> ( accp_P8892882183480857371t_char @ R2 @ A ) ) ) ).
% accp_downward
thf(fact_108_accp__downward,axiom,
! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,B: produc5803078220529002682t_char,A: produc5803078220529002682t_char] :
( ( accp_P224314924008452035t_char @ R2 @ B )
=> ( ( R2 @ A @ B )
=> ( accp_P224314924008452035t_char @ R2 @ A ) ) ) ).
% accp_downward
thf(fact_109_accp__downward,axiom,
! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,B: produc1457211279475724562t_char > prat,A: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R2 @ B )
=> ( ( R2 @ A @ B )
=> ( accp_P4765339447491148764r_prat @ R2 @ A ) ) ) ).
% accp_downward
thf(fact_110_not__accp__down,axiom,
! [R3: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,X: produc3933091914578511633r_prat] :
( ~ ( accp_P8956218777217113800r_prat @ R3 @ X )
=> ~ ! [Z: produc3933091914578511633r_prat] :
( ( R3 @ Z @ X )
=> ( accp_P8956218777217113800r_prat @ R3 @ Z ) ) ) ).
% not_accp_down
thf(fact_111_not__accp__down,axiom,
! [R3: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char] :
( ~ ( accp_P8892882183480857371t_char @ R3 @ X )
=> ~ ! [Z: produc1457211279475724562t_char] :
( ( R3 @ Z @ X )
=> ( accp_P8892882183480857371t_char @ R3 @ Z ) ) ) ).
% not_accp_down
thf(fact_112_not__accp__down,axiom,
! [R3: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,X: produc5803078220529002682t_char] :
( ~ ( accp_P224314924008452035t_char @ R3 @ X )
=> ~ ! [Z: produc5803078220529002682t_char] :
( ( R3 @ Z @ X )
=> ( accp_P224314924008452035t_char @ R3 @ Z ) ) ) ).
% not_accp_down
thf(fact_113_not__accp__down,axiom,
! [R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
( ~ ( accp_P4765339447491148764r_prat @ R3 @ X )
=> ~ ! [Z: produc1457211279475724562t_char > prat] :
( ( R3 @ Z @ X )
=> ( accp_P4765339447491148764r_prat @ R3 @ Z ) ) ) ).
% not_accp_down
thf(fact_114_pgt__implies__pgte,axiom,
! [A: prat,B: prat] :
( ( pgt @ A @ B )
=> ( pgte @ A @ B ) ) ).
% pgt_implies_pgte
thf(fact_115_not__pgte__charact,axiom,
! [A: prat,B: prat] :
( ( ~ ( pgte @ A @ B ) )
= ( pgt @ B @ A ) ) ).
% not_pgte_charact
thf(fact_116_add__acc_Osimps,axiom,
( add_ac6043276672051819257t_char
= ( ^ [Pi: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char,P2: prat] : ( fun_up829850848796697765r_prat @ Pi @ Hl2 @ ( padd @ ( Pi @ Hl2 ) @ P2 ) ) ) ) ).
% add_acc.simps
thf(fact_117_add__acc_Oelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char,Xb: prat,Y: produc1457211279475724562t_char > prat] :
( ( ( add_ac6043276672051819257t_char @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( fun_up829850848796697765r_prat @ X @ Xa @ ( padd @ ( X @ Xa ) @ Xb ) ) ) ) ).
% add_acc.elims
thf(fact_118_accp__induct__rule,axiom,
! [R2: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,A: produc3933091914578511633r_prat,P: produc3933091914578511633r_prat > $o] :
( ( accp_P8956218777217113800r_prat @ R2 @ A )
=> ( ! [X5: produc3933091914578511633r_prat] :
( ( accp_P8956218777217113800r_prat @ R2 @ X5 )
=> ( ! [Y4: produc3933091914578511633r_prat] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_119_accp__induct__rule,axiom,
! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char,P: produc1457211279475724562t_char > $o] :
( ( accp_P8892882183480857371t_char @ R2 @ A )
=> ( ! [X5: produc1457211279475724562t_char] :
( ( accp_P8892882183480857371t_char @ R2 @ X5 )
=> ( ! [Y4: produc1457211279475724562t_char] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_120_accp__induct__rule,axiom,
! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A: produc5803078220529002682t_char,P: produc5803078220529002682t_char > $o] :
( ( accp_P224314924008452035t_char @ R2 @ A )
=> ( ! [X5: produc5803078220529002682t_char] :
( ( accp_P224314924008452035t_char @ R2 @ X5 )
=> ( ! [Y4: produc5803078220529002682t_char] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_121_accp__induct__rule,axiom,
! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P: ( produc1457211279475724562t_char > prat ) > $o] :
( ( accp_P4765339447491148764r_prat @ R2 @ A )
=> ( ! [X5: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R2 @ X5 )
=> ( ! [Y4: produc1457211279475724562t_char > prat] :
( ( R2 @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_122_rm__acc_Ointros,axiom,
! [Pi3: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char,P3: prat,R2: prat] :
( ( ( Pi3 @ Hl )
= ( padd @ P3 @ R2 ) )
=> ( rm_acc1853229909052044931t_char @ Pi3 @ Hl @ P3 @ ( fun_up829850848796697765r_prat @ Pi3 @ Hl @ R2 ) ) ) ).
% rm_acc.intros
thf(fact_123_rm__acc_Osimps,axiom,
( rm_acc1853229909052044931t_char
= ( ^ [A12: produc1457211279475724562t_char > prat,A22: produc1457211279475724562t_char,A32: prat,A42: produc1457211279475724562t_char > prat] :
? [Pi: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char,P2: prat,R: prat] :
( ( A12 = Pi )
& ( A22 = Hl2 )
& ( A32 = P2 )
& ( A42
= ( fun_up829850848796697765r_prat @ Pi @ Hl2 @ R ) )
& ( ( Pi @ Hl2 )
= ( padd @ P2 @ R ) ) ) ) ) ).
% rm_acc.simps
thf(fact_124_rm__acc_Ocases,axiom,
! [A13: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char,A33: prat,A43: produc1457211279475724562t_char > prat] :
( ( rm_acc1853229909052044931t_char @ A13 @ A23 @ A33 @ A43 )
=> ~ ! [R4: prat] :
( ( A43
= ( fun_up829850848796697765r_prat @ A13 @ A23 @ R4 ) )
=> ( ( A13 @ A23 )
!= ( padd @ A33 @ R4 ) ) ) ) ).
% rm_acc.cases
thf(fact_125_uni__mask_Opelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: prat,Y: ( produc1457211279475724562t_char > prat ) > prat] :
( ( ( uni_ma8757825773229499086r_prat @ X @ Xa )
= Y )
=> ( ( accp_P6822921472154139145t_prat @ uni_ma4504258969416521949r_prat @ ( produc5508603645230141260t_prat @ X @ Xa ) )
=> ~ ( ( Y
= ( fun_up9120501182914450918t_prat @ empty_6616431780912394577r_prat @ X @ Xa ) )
=> ~ ( accp_P6822921472154139145t_prat @ uni_ma4504258969416521949r_prat @ ( produc5508603645230141260t_prat @ X @ Xa ) ) ) ) ) ).
% uni_mask.pelims
thf(fact_126_uni__mask_Opelims,axiom,
! [X: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
( ( ( uni_ma6578582744724537101t_char @ X @ Xa )
= Y )
=> ( ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X @ Xa ) )
=> ~ ( ( Y
= ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X @ Xa ) )
=> ~ ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X @ Xa ) ) ) ) ) ).
% uni_mask.pelims
thf(fact_127_size__neq__size__imp__neq,axiom,
! [X: val,Y: val] :
( ( ( size_size_val @ X )
!= ( size_size_val @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_128_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_129_size__neq__size__imp__neq,axiom,
! [X: typerep,Y: typerep] :
( ( ( size_size_typerep @ X )
!= ( size_size_typerep @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_130_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_131_empty__mask_Opelims,axiom,
! [X: produc3933091914578511633r_prat,Y: prat] :
( ( ( empty_7962835221509435283r_prat @ X )
= Y )
=> ( ( accp_P8956218777217113800r_prat @ empty_9147938987322397194r_prat @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P8956218777217113800r_prat @ empty_9147938987322397194r_prat @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_132_empty__mask_Opelims,axiom,
! [X: produc5803078220529002682t_char,Y: prat] :
( ( ( empty_1134373636806348472t_char @ X )
= Y )
=> ( ( accp_P224314924008452035t_char @ empty_5292206184366095105t_char @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P224314924008452035t_char @ empty_5292206184366095105t_char @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_133_empty__mask_Opelims,axiom,
! [X: produc1457211279475724562t_char,Y: prat] :
( ( ( empty_3446695950879338768t_char @ X )
= Y )
=> ( ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_134_empty__mask_Opelims,axiom,
! [X: produc1457211279475724562t_char > prat,Y: prat] :
( ( ( empty_6616431780912394577r_prat @ X )
= Y )
=> ( ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_135_uni__mask_Ocases,axiom,
! [X: produc3933091914578511633r_prat] :
~ ! [Hl3: produc1457211279475724562t_char,P4: prat] :
( X
!= ( produc2920179824973321483r_prat @ Hl3 @ P4 ) ) ).
% uni_mask.cases
thf(fact_136_padd__zero,axiom,
! [A: prat,B: prat] :
( ( pnone
= ( padd @ A @ B ) )
= ( ( A = pnone )
& ( B = pnone ) ) ) ).
% padd_zero
thf(fact_137_empty__mask_Oelims,axiom,
! [X: produc1457211279475724562t_char > prat,Y: prat] :
( ( ( empty_6616431780912394577r_prat @ X )
= Y )
=> ( Y = pnone ) ) ).
% empty_mask.elims
thf(fact_138_empty__mask_Oelims,axiom,
! [X: produc1457211279475724562t_char,Y: prat] :
( ( ( empty_3446695950879338768t_char @ X )
= Y )
=> ( Y = pnone ) ) ).
% empty_mask.elims
thf(fact_139_empty__mask_Osimps,axiom,
( empty_6616431780912394577r_prat
= ( ^ [Hl2: produc1457211279475724562t_char > prat] : pnone ) ) ).
% empty_mask.simps
thf(fact_140_empty__mask_Osimps,axiom,
( empty_3446695950879338768t_char
= ( ^ [Hl2: produc1457211279475724562t_char] : pnone ) ) ).
% empty_mask.simps
thf(fact_141_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_142_add__acc_Opelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char,Xb: prat,Y: produc1457211279475724562t_char > prat] :
( ( ( add_ac6043276672051819257t_char @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P5529878251199943754r_prat @ add_ac7939326059656485296t_char @ ( produc5928798746291773453r_prat @ X @ ( produc2920179824973321483r_prat @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( fun_up829850848796697765r_prat @ X @ Xa @ ( padd @ ( X @ Xa ) @ Xb ) ) )
=> ~ ( accp_P5529878251199943754r_prat @ add_ac7939326059656485296t_char @ ( produc5928798746291773453r_prat @ X @ ( produc2920179824973321483r_prat @ Xa @ Xb ) ) ) ) ) ) ).
% add_acc.pelims
thf(fact_143_divides__aux__eq,axiom,
! [Q: nat,R2: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_144_divides__aux__eq,axiom,
! [Q: int,R2: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_145_typerep_Osize__neq,axiom,
! [X: typerep] :
( ( size_size_typerep @ X )
!= zero_zero_nat ) ).
% typerep.size_neq
thf(fact_146_add__masks_Ocases,axiom,
! [X: produc5803078220529002682t_char] :
~ ! [Pi5: produc1457211279475724562t_char > prat,Pi6: produc1457211279475724562t_char > prat,Hl3: produc1457211279475724562t_char] :
( X
!= ( produc2957999048406202538t_char @ Pi5 @ ( produc2591414526589101846t_char @ Pi6 @ Hl3 ) ) ) ).
% add_masks.cases
thf(fact_147_add__acc_Ocases,axiom,
! [X: produc6235724684372434707r_prat] :
~ ! [Pi6: produc1457211279475724562t_char > prat,Hl3: produc1457211279475724562t_char,P4: prat] :
( X
!= ( produc5928798746291773453r_prat @ Pi6 @ ( produc2920179824973321483r_prat @ Hl3 @ P4 ) ) ) ).
% add_acc.cases
thf(fact_148_add__masks_Opelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) )
=> ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) ) ) ) ) ).
% add_masks.pelims
thf(fact_149_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_150_char_Osize_I2_J,axiom,
! [X1: $o,X2: $o,X3: $o,X42: $o,X52: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X2 @ X3 @ X42 @ X52 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_151_valid__null__def,axiom,
( valid_null
= ( ^ [Pi: produc1457211279475724562t_char > prat] :
! [F: list_char] :
( ( Pi @ ( produc120197956887798346t_char @ null_nat @ F ) )
= pnone ) ) ) ).
% valid_null_def
thf(fact_152_divides__aux__def,axiom,
( unique5332122412489317741ux_nat
= ( ^ [Qr: product_prod_nat_nat] :
( ( product_snd_nat_nat @ Qr )
= zero_zero_nat ) ) ) ).
% divides_aux_def
thf(fact_153_divides__aux__def,axiom,
( unique5329631941980267465ux_int
= ( ^ [Qr: product_prod_int_int] :
( ( product_snd_int_int @ Qr )
= zero_zero_int ) ) ) ).
% divides_aux_def
thf(fact_154_pmin__greater,axiom,
! [A: prat,B: prat] : ( pgte @ A @ ( pmin @ A @ B ) ) ).
% pmin_greater
thf(fact_155_pmin__is,axiom,
! [A: prat,B: prat] :
( ( pgte @ A @ B )
=> ( ( pmin @ A @ B )
= B ) ) ).
% pmin_is
thf(fact_156_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_157_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_158_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_159_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_160_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_161_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_162_pmin__comm,axiom,
( pmin
= ( ^ [A4: prat,B2: prat] : ( pmin @ B2 @ A4 ) ) ) ).
% pmin_comm
thf(fact_163_add__masks__equiv__valid__null,axiom,
! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
( ( valid_null @ ( add_ma6586698345353345811t_char @ A @ B ) )
= ( ( valid_null @ A )
& ( valid_null @ B ) ) ) ).
% add_masks_equiv_valid_null
thf(fact_164_char_Osize__gen,axiom,
! [X1: $o,X2: $o,X3: $o,X42: $o,X52: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X2 @ X3 @ X42 @ X52 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_165_pmin__pmax,axiom,
! [X: prat,A: prat,B: prat] :
( ( pgte @ X @ ( pmin @ A @ B ) )
=> ( X
= ( pmin @ ( pmax @ X @ A ) @ ( pmax @ X @ B ) ) ) ) ).
% pmin_pmax
thf(fact_166_pre__get__h_Oelims,axiom,
! [X: product_prod_int_int,Y: int] :
( ( ( pre_get_h_int_int @ X )
= Y )
=> ( Y
= ( product_snd_int_int @ X ) ) ) ).
% pre_get_h.elims
thf(fact_167_pre__get__h_Oelims,axiom,
! [X: product_prod_nat_nat,Y: nat] :
( ( ( pre_get_h_nat_nat @ X )
= Y )
=> ( Y
= ( product_snd_nat_nat @ X ) ) ) ).
% pre_get_h.elims
thf(fact_168_pre__get__h_Osimps,axiom,
pre_get_h_int_int = product_snd_int_int ).
% pre_get_h.simps
thf(fact_169_pre__get__h_Osimps,axiom,
pre_get_h_nat_nat = product_snd_nat_nat ).
% pre_get_h.simps
thf(fact_170_pre__get__h_Opelims,axiom,
! [X: produc3933091914578511633r_prat,Y: prat] :
( ( ( pre_ge2177556708334386373r_prat @ X )
= Y )
=> ( ( accp_P8956218777217113800r_prat @ pre_ge525017830379303046r_prat @ X )
=> ~ ( ( Y
= ( produc2879476908335557213r_prat @ X ) )
=> ~ ( accp_P8956218777217113800r_prat @ pre_ge525017830379303046r_prat @ X ) ) ) ) ).
% pre_get_h.pelims
thf(fact_171_pre__get__h_Opelims,axiom,
! [X: produc1457211279475724562t_char,Y: list_char] :
( ( ( pre_ge1487422843495280912t_char @ X )
= Y )
=> ( ( accp_P8892882183480857371t_char @ pre_ge7016768201484263823t_char @ X )
=> ~ ( ( Y
= ( produc1900778367047502200t_char @ X ) )
=> ~ ( accp_P8892882183480857371t_char @ pre_ge7016768201484263823t_char @ X ) ) ) ) ).
% pre_get_h.pelims
thf(fact_172_pre__get__h_Opelims,axiom,
! [X: produc5803078220529002682t_char,Y: produc2489117125269924006t_char] :
( ( ( pre_ge3831893988676007280t_char @ X )
= Y )
=> ( ( accp_P224314924008452035t_char @ pre_ge5134124999753271535t_char @ X )
=> ~ ( ( Y
= ( produc5882964721085011416t_char @ X ) )
=> ~ ( accp_P224314924008452035t_char @ pre_ge5134124999753271535t_char @ X ) ) ) ) ).
% pre_get_h.pelims
thf(fact_173_pre__get__h_Opelims,axiom,
! [X: product_prod_int_int,Y: int] :
( ( ( pre_get_h_int_int @ X )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ pre_ge5444797914478895542nt_int @ X )
=> ~ ( ( Y
= ( product_snd_int_int @ X ) )
=> ~ ( accp_P1096762738010456898nt_int @ pre_ge5444797914478895542nt_int @ X ) ) ) ) ).
% pre_get_h.pelims
thf(fact_174_pre__get__h_Opelims,axiom,
! [X: product_prod_nat_nat,Y: nat] :
( ( ( pre_get_h_nat_nat @ X )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ pre_ge4447934673086891262at_nat @ X )
=> ~ ( ( Y
= ( product_snd_nat_nat @ X ) )
=> ~ ( accp_P4275260045618599050at_nat @ pre_ge4447934673086891262at_nat @ X ) ) ) ) ).
% pre_get_h.pelims
thf(fact_175_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_176_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_177_valid__mask_Oelims_I3_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ~ ( valid_mask @ X )
=> ~ ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
& ! [F2: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
= pnone ) ) ) ).
% valid_mask.elims(3)
thf(fact_178_valid__mask_Oelims_I2_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ( valid_mask @ X )
=> ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl4 ) )
& ! [F3: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
= pnone ) ) ) ).
% valid_mask.elims(2)
thf(fact_179_valid__mask_Oelims_I1_J,axiom,
! [X: produc1457211279475724562t_char > prat,Y: $o] :
( ( ( valid_mask @ X )
= Y )
=> ( Y
= ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
& ! [F: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
= pnone ) ) ) ) ).
% valid_mask.elims(1)
thf(fact_180_valid__mask_Osimps,axiom,
( valid_mask
= ( ^ [Pi: produc1457211279475724562t_char > prat] :
( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi @ Hl2 ) )
& ! [F: list_char] :
( ( Pi @ ( produc120197956887798346t_char @ null_nat @ F ) )
= pnone ) ) ) ) ).
% valid_mask.simps
thf(fact_181_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_182_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_183_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_184_accp__subset,axiom,
! [R1: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,R22: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o] :
( ( ord_le5632367588805359236prat_o @ R1 @ R22 )
=> ( ord_le1173062544651188894prat_o @ ( accp_P8956218777217113800r_prat @ R22 ) @ ( accp_P8956218777217113800r_prat @ R1 ) ) ) ).
% accp_subset
thf(fact_185_accp__subset,axiom,
! [R1: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,R22: produc1457211279475724562t_char > produc1457211279475724562t_char > $o] :
( ( ord_le2827606955515295502char_o @ R1 @ R22 )
=> ( ord_le5722155653058504523char_o @ ( accp_P8892882183480857371t_char @ R22 ) @ ( accp_P8892882183480857371t_char @ R1 ) ) ) ).
% accp_subset
thf(fact_186_accp__subset,axiom,
! [R1: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,R22: produc5803078220529002682t_char > produc5803078220529002682t_char > $o] :
( ( ord_le6348261579570775310char_o @ R1 @ R22 )
=> ( ord_le7815076450745241763char_o @ ( accp_P224314924008452035t_char @ R22 ) @ ( accp_P224314924008452035t_char @ R1 ) ) ) ).
% accp_subset
thf(fact_187_accp__subset,axiom,
! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R22: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
( ( ord_le6723339807950303054prat_o @ R1 @ R22 )
=> ( ord_le2830794348966088778prat_o @ ( accp_P4765339447491148764r_prat @ R22 ) @ ( accp_P4765339447491148764r_prat @ R1 ) ) ) ).
% accp_subset
thf(fact_188_pmax__comm,axiom,
( pmax
= ( ^ [A4: prat,B2: prat] : ( pmax @ B2 @ A4 ) ) ) ).
% pmax_comm
thf(fact_189_upper__valid,axiom,
! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( valid_mask @ A )
=> ( ( A
= ( add_ma6586698345353345811t_char @ B @ C ) )
=> ( ( valid_mask @ B )
& ( valid_mask @ C ) ) ) ) ).
% upper_valid
thf(fact_190_upper__valid__aux,axiom,
! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( valid_mask @ A )
=> ( ( A
= ( add_ma6586698345353345811t_char @ B @ C ) )
=> ( valid_mask @ B ) ) ) ).
% upper_valid_aux
thf(fact_191_valid__empty,axiom,
valid_mask @ empty_3446695950879338768t_char ).
% valid_empty
thf(fact_192_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_193_ssubsetI,axiom,
! [A3: set_Pr2722754889679711815r_prat,B3: set_Pr2722754889679711815r_prat] :
( ! [Pi6: produc1457211279475724562t_char,H3: prat] :
( ( member7501365625282528168r_prat @ ( produc2920179824973321483r_prat @ Pi6 @ H3 ) @ A3 )
=> ( member7501365625282528168r_prat @ ( produc2920179824973321483r_prat @ Pi6 @ H3 ) @ B3 ) )
=> ( ord_le9052479809666887335r_prat @ A3 @ B3 ) ) ).
% ssubsetI
thf(fact_194_ssubsetI,axiom,
! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ! [Pi6: nat,H3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi6 @ H3 ) @ A3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi6 @ H3 ) @ B3 ) )
=> ( ord_le3146513528884898305at_nat @ A3 @ B3 ) ) ).
% ssubsetI
thf(fact_195_ssubsetI,axiom,
! [A3: set_Pr1935176096852982554t_char,B3: set_Pr1935176096852982554t_char] :
( ! [Pi6: produc1457211279475724562t_char > prat,H3: produc2489117125269924006t_char] :
( ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ Pi6 @ H3 ) @ A3 )
=> ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ Pi6 @ H3 ) @ B3 ) )
=> ( ord_le5742415681470231226t_char @ A3 @ B3 ) ) ).
% ssubsetI
thf(fact_196_ssubsetI,axiom,
! [A3: set_Pr7632435056502277254t_char,B3: set_Pr7632435056502277254t_char] :
( ! [Pi6: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char] :
( ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ Pi6 @ H3 ) @ A3 )
=> ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ Pi6 @ H3 ) @ B3 ) )
=> ( ord_le3803256517986266150t_char @ A3 @ B3 ) ) ).
% ssubsetI
thf(fact_197_ssubsetI,axiom,
! [A3: set_Pr2507339680178222706t_char,B3: set_Pr2507339680178222706t_char] :
( ! [Pi6: nat,H3: list_char] :
( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi6 @ H3 ) @ A3 )
=> ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi6 @ H3 ) @ B3 ) )
=> ( ord_le6315511645215477266t_char @ A3 @ B3 ) ) ).
% ssubsetI
thf(fact_198_ssubsetI,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ! [Pi6: int,H3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Pi6 @ H3 ) @ A3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Pi6 @ H3 ) @ B3 ) )
=> ( ord_le2843351958646193337nt_int @ A3 @ B3 ) ) ).
% ssubsetI
thf(fact_199_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_200_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_201_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_202_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_203_accp__subset__induct,axiom,
! [D2: produc3933091914578511633r_prat > $o,R3: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o,X: produc3933091914578511633r_prat,P: produc3933091914578511633r_prat > $o] :
( ( ord_le1173062544651188894prat_o @ D2 @ ( accp_P8956218777217113800r_prat @ R3 ) )
=> ( ! [X5: produc3933091914578511633r_prat,Z: produc3933091914578511633r_prat] :
( ( D2 @ X5 )
=> ( ( R3 @ Z @ X5 )
=> ( D2 @ Z ) ) )
=> ( ( D2 @ X )
=> ( ! [X5: produc3933091914578511633r_prat] :
( ( D2 @ X5 )
=> ( ! [Z2: produc3933091914578511633r_prat] :
( ( R3 @ Z2 @ X5 )
=> ( P @ Z2 ) )
=> ( P @ X5 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_204_accp__subset__induct,axiom,
! [D2: produc1457211279475724562t_char > $o,R3: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char,P: produc1457211279475724562t_char > $o] :
( ( ord_le5722155653058504523char_o @ D2 @ ( accp_P8892882183480857371t_char @ R3 ) )
=> ( ! [X5: produc1457211279475724562t_char,Z: produc1457211279475724562t_char] :
( ( D2 @ X5 )
=> ( ( R3 @ Z @ X5 )
=> ( D2 @ Z ) ) )
=> ( ( D2 @ X )
=> ( ! [X5: produc1457211279475724562t_char] :
( ( D2 @ X5 )
=> ( ! [Z2: produc1457211279475724562t_char] :
( ( R3 @ Z2 @ X5 )
=> ( P @ Z2 ) )
=> ( P @ X5 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_205_accp__subset__induct,axiom,
! [D2: produc5803078220529002682t_char > $o,R3: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,X: produc5803078220529002682t_char,P: produc5803078220529002682t_char > $o] :
( ( ord_le7815076450745241763char_o @ D2 @ ( accp_P224314924008452035t_char @ R3 ) )
=> ( ! [X5: produc5803078220529002682t_char,Z: produc5803078220529002682t_char] :
( ( D2 @ X5 )
=> ( ( R3 @ Z @ X5 )
=> ( D2 @ Z ) ) )
=> ( ( D2 @ X )
=> ( ! [X5: produc5803078220529002682t_char] :
( ( D2 @ X5 )
=> ( ! [Z2: produc5803078220529002682t_char] :
( ( R3 @ Z2 @ X5 )
=> ( P @ Z2 ) )
=> ( P @ X5 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_206_accp__subset__induct,axiom,
! [D2: ( produc1457211279475724562t_char > prat ) > $o,R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,P: ( produc1457211279475724562t_char > prat ) > $o] :
( ( ord_le2830794348966088778prat_o @ D2 @ ( accp_P4765339447491148764r_prat @ R3 ) )
=> ( ! [X5: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > prat] :
( ( D2 @ X5 )
=> ( ( R3 @ Z @ X5 )
=> ( D2 @ Z ) ) )
=> ( ( D2 @ X )
=> ( ! [X5: produc1457211279475724562t_char > prat] :
( ( D2 @ X5 )
=> ( ! [Z2: produc1457211279475724562t_char > prat] :
( ( R3 @ Z2 @ X5 )
=> ( P @ Z2 ) )
=> ( P @ X5 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_207_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_208_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_209_pmax__is__smaller,axiom,
! [X: prat,A: prat,B: prat] :
( ( pgte @ X @ A )
=> ( ( pgte @ X @ B )
=> ( pgte @ X @ ( pmax @ A @ B ) ) ) ) ).
% pmax_is_smaller
thf(fact_210_pmax__smaller,axiom,
! [A: prat,B: prat] : ( pgte @ ( pmax @ A @ B ) @ A ) ).
% pmax_smaller
thf(fact_211_pmax__is,axiom,
! [A: prat,B: prat] :
( ( pgte @ A @ B )
=> ( ( pmax @ A @ B )
= A ) ) ).
% pmax_is
thf(fact_212_valid__maskI,axiom,
! [Pi3: produc1457211279475724562t_char > prat] :
( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi3 @ Hl3 ) )
=> ( ! [F2: list_char] :
( ( Pi3 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
= pnone )
=> ( valid_mask @ Pi3 ) ) ) ).
% valid_maskI
thf(fact_213_valid__mask_Opelims_I1_J,axiom,
! [X: produc1457211279475724562t_char > prat,Y: $o] :
( ( ( valid_mask @ X )
= Y )
=> ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ( Y
= ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
& ! [F: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
= pnone ) ) )
=> ~ ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X ) ) ) ) ).
% valid_mask.pelims(1)
thf(fact_214_valid__mask_Opelims_I2_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ( valid_mask @ X )
=> ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl4 ) )
& ! [F3: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
= pnone ) ) ) ) ) ).
% valid_mask.pelims(2)
thf(fact_215_valid__mask_Opelims_I3_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ~ ( valid_mask @ X )
=> ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
& ! [F2: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
= pnone ) ) ) ) ) ).
% valid_mask.pelims(3)
thf(fact_216_padd__comp__one,axiom,
! [X: prat] :
( ( pgte @ pwrite @ X )
=> ( ( padd @ X @ ( comp_one @ X ) )
= pwrite ) ) ).
% padd_comp_one
thf(fact_217_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_218_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_219_half__plus__half,axiom,
( ( padd @ half @ half )
= pwrite ) ).
% half_plus_half
thf(fact_220_padd__one__ineq__sum,axiom,
! [A: prat,B: prat,X: prat,Aa: prat,Bb: prat] :
( ( ( padd @ A @ B )
= pwrite )
=> ( ( pgte @ X @ Aa )
=> ( ( pgte @ X @ Bb )
=> ( pgte @ X @ ( padd @ ( pmult @ A @ Aa ) @ ( pmult @ B @ Bb ) ) ) ) ) ) ).
% padd_one_ineq_sum
thf(fact_221_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_222_double__inclusion,axiom,
! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
=> ( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% double_inclusion
thf(fact_223_double__inclusion,axiom,
! [A3: set_Pr2507339680178222706t_char,B3: set_Pr2507339680178222706t_char] :
( ( ord_le6315511645215477266t_char @ A3 @ B3 )
=> ( ( ord_le6315511645215477266t_char @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% double_inclusion
thf(fact_224_pmult__comm,axiom,
( pmult
= ( ^ [A4: prat,B2: prat] : ( pmult @ B2 @ A4 ) ) ) ).
% pmult_comm
thf(fact_225_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_226_pmult__order,axiom,
! [A: prat,B: prat,P3: prat] :
( ( pgte @ A @ B )
=> ( pgte @ ( pmult @ P3 @ A ) @ ( pmult @ B @ P3 ) ) ) ).
% pmult_order
thf(fact_227_pmult__padd,axiom,
! [A: prat,B: prat,X: prat,C: prat,Y: prat] :
( ( pmult @ A @ ( padd @ ( pmult @ B @ X ) @ ( pmult @ C @ Y ) ) )
= ( padd @ ( pmult @ ( pmult @ A @ B ) @ X ) @ ( pmult @ ( pmult @ A @ C ) @ Y ) ) ) ).
% pmult_padd
thf(fact_228_pmult__distr,axiom,
! [A: prat,B: prat,C: prat] :
( ( pmult @ A @ ( padd @ B @ C ) )
= ( padd @ ( pmult @ A @ B ) @ ( pmult @ A @ C ) ) ) ).
% pmult_distr
thf(fact_229_pmult__special_I1_J,axiom,
! [X: prat] :
( ( pmult @ pwrite @ X )
= X ) ).
% pmult_special(1)
thf(fact_230_pmult__special_I2_J,axiom,
! [X: prat] :
( ( pmult @ pnone @ X )
= pnone ) ).
% pmult_special(2)
thf(fact_231_multiply__smaller__pwrite,axiom,
! [A: prat,B: prat] :
( ( pgte @ pwrite @ A )
=> ( ( pgte @ pwrite @ B )
=> ( pgte @ pwrite @ ( pmult @ A @ B ) ) ) ) ).
% multiply_smaller_pwrite
thf(fact_232_multiply__mask__def,axiom,
( multiply_mask
= ( ^ [P2: prat,Pi: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( pmult @ P2 @ ( Pi @ Hl2 ) ) ) ) ).
% multiply_mask_def
thf(fact_233_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_234_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_235_half__between__0__1,axiom,
( ( ppos @ half )
& ( pgt @ pwrite @ half ) ) ).
% half_between_0_1
thf(fact_236_conj__le__cong,axiom,
! [X: int,X9: int,P: $o,P5: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_237_imp__le__cong,axiom,
! [X: int,X9: int,P: $o,P5: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_238_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_239_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_240_ppos__add,axiom,
! [A: prat,B: prat] :
( ( ppos @ A )
=> ( ppos @ ( padd @ A @ B ) ) ) ).
% ppos_add
thf(fact_241_pmult__ppos,axiom,
! [A: prat,B: prat] :
( ( ppos @ A )
=> ( ( ppos @ B )
=> ( ppos @ ( pmult @ A @ B ) ) ) ) ).
% pmult_ppos
thf(fact_242_ppos__eq__pnone,axiom,
( ppos
= ( ^ [P2: prat] : ( P2 != pnone ) ) ) ).
% ppos_eq_pnone
thf(fact_243_pinv__double__half,axiom,
! [P3: prat] :
( ( ppos @ P3 )
=> ( ( pmult @ half @ ( pinv @ P3 ) )
= ( pinv @ ( padd @ P3 @ P3 ) ) ) ) ).
% pinv_double_half
thf(fact_244_pinv__pmult__ok,axiom,
! [P3: prat] :
( ( ppos @ P3 )
=> ( ( pmult @ P3 @ ( pinv @ P3 ) )
= pwrite ) ) ).
% pinv_pmult_ok
thf(fact_245_sum__coeff,axiom,
! [A: prat,B: prat] :
( ( ppos @ A )
=> ( ( ppos @ B )
=> ( ( padd @ ( pdiv @ A @ ( padd @ A @ B ) ) @ ( pdiv @ B @ ( padd @ A @ B ) ) )
= pwrite ) ) ) ).
% sum_coeff
thf(fact_246_pdiv__smaller,axiom,
! [A: prat,B: prat] :
( ( pgte @ A @ B )
=> ( ( ppos @ A )
=> ( pgte @ pwrite @ ( pdiv @ B @ A ) ) ) ) ).
% pdiv_smaller
thf(fact_247_pinv__inverts,axiom,
! [A: prat,B: prat] :
( ( pgte @ A @ B )
=> ( ( ppos @ B )
=> ( pgte @ ( pinv @ B ) @ ( pinv @ A ) ) ) ) ).
% pinv_inverts
thf(fact_248_pmult__pdiv__cancel,axiom,
! [A: prat,X: prat] :
( ( ppos @ A )
=> ( ( pmult @ A @ ( pdiv @ X @ A ) )
= X ) ) ).
% pmult_pdiv_cancel
thf(fact_249_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_250_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_int @ Z3 )
= zero_zero_int )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_251_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_252_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_253_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_254_pinv__def,axiom,
( pinv
= ( pdiv @ pwrite ) ) ).
% pinv_def
thf(fact_255_pinv__pwrite,axiom,
( ( pinv @ pwrite )
= pwrite ) ).
% pinv_pwrite
thf(fact_256_ppos__inv,axiom,
! [P3: prat] :
( ( ppos @ P3 )
=> ( ppos @ ( pinv @ P3 ) ) ) ).
% ppos_inv
thf(fact_257_of__int__nonneg,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_nonneg
thf(fact_258_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_259_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_260_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_261_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_262_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_263_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_264_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_265_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_266_eq__nat__nat__iff,axiom,
! [Z3: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z4 ) )
= ( Z3 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_267_all__nat,axiom,
( ( ^ [P6: nat > $o] :
! [X10: nat] : ( P6 @ X10 ) )
= ( ^ [P7: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P7 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_268_ex__nat,axiom,
( ( ^ [P6: nat > $o] :
? [X10: nat] : ( P6 @ X10 ) )
= ( ^ [P7: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P7 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_269_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_270_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_271_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_272_nat__le__eq__zle,axiom,
! [W: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_273_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_274_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_275_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_276_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_277_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_278_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_279_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_280_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_281_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_282_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_293_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_294_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_295_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_296_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_297_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_298_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_299_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_300_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_301_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_302_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_303_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_304_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_305_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_306_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_307_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_308_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_309_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_310_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_311_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_312_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_313_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_314_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_315_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_316_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_317_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_318_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_319_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_320_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_321_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_322_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_323_of__int__pos,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_pos
thf(fact_324_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_325_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_326_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_327_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_328_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_329_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_330_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_331_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_332_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_333_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_334_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_335_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_336_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_337_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_338_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_339_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_340_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_341_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_342_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_343_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_344_zless__nat__conj,axiom,
! [W: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_345_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_346_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_347_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_348_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_349_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_350_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_351_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_352_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_353_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_354_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_355_nat__mono__iff,axiom,
! [Z3: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_356_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_357_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_358_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_359_nat__less__eq__zless,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_360_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_361_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_362_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_363_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_364_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_365_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_366_sgn__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% sgn_neg
thf(fact_367_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_368_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_369_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_370_sgn__0,axiom,
( ( sgn_sgn_int @ zero_zero_int )
= zero_zero_int ) ).
% sgn_0
thf(fact_371_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_372_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_373_sgn__greater,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_greater
thf(fact_374_sgn__less,axiom,
! [A: int] :
( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_less
thf(fact_375_sgn__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( sgn_sgn_int @ A )
= one_one_int ) ) ).
% sgn_pos
thf(fact_376_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_377_sgn__0__0,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_0_0
thf(fact_378_sgn__eq__0__iff,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_eq_0_iff
thf(fact_379_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_380_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_381_old_Onat_Odistinct_I2_J,axiom,
! [Nat: nat] :
( ( suc @ Nat )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_382_old_Onat_Odistinct_I1_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
% old.nat.distinct(1)
thf(fact_383_nat_OdiscI,axiom,
! [Nat2: nat,X2: nat] :
( ( Nat2
= ( suc @ X2 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_384_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_385_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_386_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y6: nat] : ( P @ zero_zero_nat @ ( suc @ Y6 ) )
=> ( ! [X5: nat,Y6: nat] :
( ( P @ X5 @ Y6 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y6 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_387_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_388_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_389_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_390_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_391_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_392_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_393_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_394_sgn__not__eq__imp,axiom,
! [B: int,A: int] :
( ( ( sgn_sgn_int @ B )
!= ( sgn_sgn_int @ A ) )
=> ( ( ( sgn_sgn_int @ A )
!= zero_zero_int )
=> ( ( ( sgn_sgn_int @ B )
!= zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_395_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_396_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_397_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_398_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_399_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_400_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_401_sgn__1__pos,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= one_one_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_1_pos
thf(fact_402_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_403_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_404_sgn__1__neg,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_1_neg
thf(fact_405_sgn__if,axiom,
( sgn_sgn_int
= ( ^ [X4: int] : ( if_int @ ( X4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% sgn_if
thf(fact_406_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_407_Suc__nat__eq__nat__zadd1,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( suc @ ( nat2 @ Z3 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_408_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_409_sgn__of__nat,axiom,
! [N: nat] :
( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% sgn_of_nat
thf(fact_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_420_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_421_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_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_432_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_433_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_434_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_435_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_436_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_437_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_438_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_439_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_440_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_441_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_442_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_443_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_444_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_445_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_446_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_447_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_448_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_449_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_450_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_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_460_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_461_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= zero_zero_nat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_462_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= zero_zero_int )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_463_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_464_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_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_488_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_489_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_490_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_491_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_492_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_493_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_494_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_495_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_496_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_497_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_498_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_499_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_500_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_501_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_502_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_503_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_504_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_505_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_506_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_507_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_508_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_509_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_510_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_511_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_512_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B2: int] : ( plus_plus_int @ B2 @ A4 ) ) ) ).
% add.commute
thf(fact_524_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B2: nat] : ( plus_plus_nat @ B2 @ A4 ) ) ) ).
% add.commute
thf(fact_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_551_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B2: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_552_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_553_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_554_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B2: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_555_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_556_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_557_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_558_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_559_eq__iff__diff__eq__0,axiom,
( ( ^ [Y7: int,Z5: int] : ( Y7 = Z5 ) )
= ( ^ [A4: int,B2: int] :
( ( minus_minus_int @ A4 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_560_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_561_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_562_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_563_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_564_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_565_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_566_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_567_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( I = J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_568_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_569_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_570_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_571_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_572_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_573_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_574_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_575_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_576_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_577_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_578_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_579_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_580_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_581_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_582_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_583_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_584_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_585_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_586_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_587_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_588_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_589_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_590_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_591_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_592_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( I = J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_593_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_594_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J2 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_595_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_596_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_597_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_598_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_599_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_600_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_601_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_602_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_603_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_604_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_605_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_606_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_607_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_608_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_609_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_610_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_611_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_612_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_613_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_614_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P2: $o] : ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_615_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P2: $o] : ( if_int @ P2 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_616_split__of__bool,axiom,
! [P: nat > $o,P3: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P3 ) )
= ( ( P3
=> ( P @ one_one_nat ) )
& ( ~ P3
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_617_split__of__bool,axiom,
! [P: int > $o,P3: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P3 ) )
= ( ( P3
=> ( P @ one_one_int ) )
& ( ~ P3
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_618_split__of__bool__asm,axiom,
! [P: nat > $o,P3: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_nat ) )
| ( ~ P3
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_619_split__of__bool__asm,axiom,
! [P: int > $o,P3: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_int ) )
| ( ~ P3
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_620_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_621_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_622_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_623_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_624_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_625_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_626_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_627_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_628_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_629_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_630_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_631_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_632_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_633_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_634_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_635_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_636_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_637_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_638_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_639_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_640_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_641_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_642_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_643_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_644_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_645_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_646_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_647_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_648_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_649_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_650_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_651_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_652_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_653_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_654_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_655_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_656_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_657_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_658_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_659_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_660_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_661_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_662_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_663_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_664_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_665_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_666_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_667_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_668_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_669_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_670_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_671_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_672_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_673_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_674_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_675_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_676_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_677_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_678_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_679_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_680_Parity_Oadjust__div__eq,axiom,
! [Q: int,R2: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q @ R2 ) )
= ( plus_plus_int @ Q @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% Parity.adjust_div_eq
thf(fact_681_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_682_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_683_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_684_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_685_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_686_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_687_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_688_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_689_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_690_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_691_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_692_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_693_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_694_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 ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_695_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 ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_696_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_697_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_698_nat__diff__distrib,axiom,
! [Z4: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ Z4 @ Z3 )
=> ( ( nat2 @ ( minus_minus_int @ Z3 @ Z4 ) )
= ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_699_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_700_nat__add__distrib,axiom,
! [Z3: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( nat2 @ ( plus_plus_int @ Z3 @ Z4 ) )
= ( plus_plus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_701_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_702_nat0__intermed__int__val,axiom,
! [N: nat,F4: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F4 @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F4 @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F4 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F4 @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( F4 @ I4 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_703_nat__ivt__aux,axiom,
! [N: nat,F4: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F4 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F4 @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( F4 @ I4 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_704_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_705_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_706_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_707_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_708_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_709_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_710_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_711_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_712_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_713_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_714_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_715_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_716_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_717_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_718_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_719_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_720_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_721_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_722_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_723_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_724_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_725_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_726_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_727_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_728_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_729_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_730_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_731_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_732_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_733_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_734_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_735_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_736_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_737_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_738_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_739_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_740_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_741_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_742_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_743_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_744_abs__sgn__eq__1,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ).
% abs_sgn_eq_1
thf(fact_745_left__sgn__mult__self__eq,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( times_times_int @ ( sgn_sgn_int @ A ) @ B ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) @ B ) ) ).
% left_sgn_mult_self_eq
thf(fact_746_sgn__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% sgn_mult_self_eq
thf(fact_747_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
= ( Z3 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_748_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_749_sgn__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% sgn_abs
thf(fact_750_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_751_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_752_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_753_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_754_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_755_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_756_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_757_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_758_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_759_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_760_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_761_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_762_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_763_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_764_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_765_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_766_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_767_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_768_nat__mult__distrib,axiom,
! [Z3: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z4 ) )
= ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z4 ) ) ) ) ).
% nat_mult_distrib
thf(fact_769_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_770_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_771_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_772_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_773_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B2: nat] : ( times_times_nat @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_774_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B2: int] : ( times_times_int @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_775_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_776_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_777_abs__mult__pos_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ X @ ( abs_abs_int @ Y ) )
= ( abs_abs_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_778_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
| ( ord_less_eq_int @ A @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_779_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_780_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_781_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_782_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_783_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_784_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_785_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_786_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_787_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_788_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_789_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_790_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_791_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_792_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_793_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_794_decr__lemma,axiom,
! [D: int,X: int,Z3: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D ) ) @ Z3 ) ) ).
% decr_lemma
thf(fact_795_incr__lemma,axiom,
! [D: int,Z3: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z3 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_796_nat__mult__distrib__neg,axiom,
! [Z3: int,Z4: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z4 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z4 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_797_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_798_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_799_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_800_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_801_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_802_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_803_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_804_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_805_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_806_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_807_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_808_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_809_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_810_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_811_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_812_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_813_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_814_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_815_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_816_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_817_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_818_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_819_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_820_mult__mono_H,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 @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_821_mult__mono_H,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 @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_822_mult__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 @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_823_mult__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 @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_824_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_825_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_826_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_827_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_828_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_829_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_830_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_831_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_832_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_833_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_834_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_835_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_836_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_837_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_838_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_839_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_840_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_841_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_842_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_843_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_844_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_845_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_846_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_847_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_848_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_849_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_850_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_851_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_852_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_853_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_854_mult__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_855_mult__less__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_856_zmult__zless__mono2,axiom,
! [I: int,J2: int,K: int] :
( ( ord_less_int @ I @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_857_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int,K3: int] :
( ( P1 @ X5 )
= ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( P @ X5 )
= ( P1 @ X5 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_858_plusinfinity,axiom,
! [D: int,P5: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int,K3: int] :
( ( P5 @ X5 )
= ( P5 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [X_1: int] : ( P5 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_859_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_860_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_861_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_862_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_863_abs__if,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_864_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_865_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_866_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_867_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_868_abs__sgn__eq,axiom,
! [A: int] :
( ( ( A = zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= zero_zero_int ) )
& ( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ) ).
% abs_sgn_eq
thf(fact_869_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_870_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_871_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_872_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_873_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_874_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_875_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_876_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_877_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_878_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_879_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_880_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_881_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_882_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_883_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_884_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_885_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_886_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_887_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_888_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_889_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_890_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_891_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_892_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_893_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_894_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_895_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_896_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_897_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_898_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_899_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_900_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_901_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_902_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_903_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X11: int] :
( ( P @ X11 )
=> ( P @ ( minus_minus_int @ X11 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_904_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_905_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_906_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_907_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_908_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_909_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_910_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_911_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_912_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_913_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_914_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_915_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_916_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_917_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_918_zmult__zless__mono2__lemma,axiom,
! [I: int,J2: int,K: nat] :
( ( ord_less_int @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_919_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X11: int] :
( ( P @ X11 )
=> ( P @ ( plus_plus_int @ X11 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_920_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_921_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_922_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_923_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_924_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_925_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_926_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_927_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_928_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_929_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_930_mult__le__cancel__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_931_mult__le__cancel__iff2,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_932_mult__less__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_933_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_934_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_935_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_936_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_937_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_938_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_939_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_940_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_941_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_942_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_943_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_944_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_945_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_946_verit__less__mono__div__int2,axiom,
! [A3: int,B3: int,N: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_947_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_948_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_949_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_950_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_951_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_952_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_953_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_954_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_955_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_956_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_957_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_958_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_959_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_960_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_961_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_962_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_963_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_964_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_965_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_966_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_967_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_968_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_969_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_970_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_971_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_972_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_973_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_zdiv__mono2__neg,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_984_zdiv__mono1__neg,axiom,
! [A: int,A6: int,B: int] :
( ( ord_less_eq_int @ A @ A6 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_985_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_986_zdiv__mono2,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_987_zdiv__mono1,axiom,
! [A: int,A6: int,B: int] :
( ( ord_less_eq_int @ A @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_988_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_989_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_990_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_991_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_992_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_993_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_994_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_995_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_996_div__if,axiom,
( divide_divide_nat
= ( ^ [M4: nat,N3: nat] :
( if_nat
@ ( ( ord_less_nat @ M4 @ N3 )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M4 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_997_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_998_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J: nat] :
( ( ( ord_less_nat @ J @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_999_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1000_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1001_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1002_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
= ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_1003_sgn__div__eq__sgn__mult,axiom,
! [K: int,L: int] :
( ( sgn_sgn_int @ ( divide_divide_int @ K @ L ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( divide_divide_int @ K @ L )
!= zero_zero_int ) )
@ ( sgn_sgn_int @ ( times_times_int @ K @ L ) ) ) ) ).
% sgn_div_eq_sgn_mult
thf(fact_1004_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1005_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
& ( P @ Q2 ) ) ) ) ).
% split_div'
thf(fact_1006_int__div__pos__eq,axiom,
! [A: int,B: int,Q: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_pos_eq
thf(fact_1007_int__div__neg__eq,axiom,
! [A: int,B: int,Q: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_neg_eq
thf(fact_1008_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J )
& ( ord_less_int @ J @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J: int] :
( ( ( ord_less_int @ K @ J )
& ( ord_less_eq_int @ J @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1009_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1010_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1011_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1012_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1013_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1014_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1015_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K2: int,L2: int] :
( minus_minus_int @ ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( sgn_sgn_int @ L2 ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
@ ( zero_n2684676970156552555ol_int
@ ( ( L2 != zero_zero_int )
& ( ( sgn_sgn_int @ K2 )
!= ( sgn_sgn_int @ L2 ) )
& ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ).
% divide_int_def
thf(fact_1016_divide__int__unfold,axiom,
! [K: int,M: nat,L: int,N: nat] :
( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( minus_minus_int @ ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) )
@ ( zero_n2684676970156552555ol_int
@ ( ( ( K = zero_zero_int )
= ( M = zero_zero_nat ) )
& ( L != zero_zero_int )
& ( N != zero_zero_nat )
& ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
& ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ).
% divide_int_unfold
thf(fact_1017_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_1018_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_1019_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_1020_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_1021_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1022_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1023_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1024_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1025_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1026_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1027_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1028_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1029_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1030_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
= ( ( dvd_dvd_int @ L @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_1031_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
= ( ( dvd_dvd_int @ L @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_1032_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
= ( ( dvd_dvd_int @ L @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_1033_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
= ( ( dvd_dvd_int @ L @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_1034_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_1035_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1036_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1037_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X5: nat,Y6: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( times_times_nat @ A @ X5 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y6 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1038_zdvd__imp__le,axiom,
! [Z3: int,N: int] :
( ( dvd_dvd_int @ Z3 @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z3 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1039_dvd__imp__le__int,axiom,
! [I: int,D: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_1040_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1041_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1042_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1043_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_1044_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1045_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_1046_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_1047_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1048_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_1049_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1050_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_1051_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_1052_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1053_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_1054_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_1055_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1056_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
% Helper facts (5)
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_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
equal_on_bmask_a_b @ pi @ h @ h2 ).
%------------------------------------------------------------------------------