TPTP Problem File: SLH0407^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Multiset_Ordering_NPC/0002_Multiset_Ordering_in_NP/prob_00451_019500__13666936_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1327 ( 652 unt; 76 typ; 0 def)
% Number of atoms : 3158 (1375 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 8872 ( 308 ~; 84 |; 160 &;7215 @)
% ( 0 <=>;1105 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 229 ( 229 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 66 usr; 9 con; 0-8 aty)
% Number of variables : 2946 ( 89 ^;2771 !; 86 ?;2946 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:29:32.709
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__b_J_J,type,
multiset_multiset_b: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__b_J_J,type,
set_multiset_b: $tType ).
thf(ty_n_t__Multiset____Ordering____in____NP__OPropVar,type,
multis3193088007478089820ropVar: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__b_J,type,
multiset_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Typerep__Otyperep,type,
typerep: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (66)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Cancellation_Oiterate__add_001t__Int__Oint,type,
iterate_add_int: nat > int > int ).
thf(sy_c_Cancellation_Oiterate__add_001t__Multiset__Omultiset_Itf__b_J,type,
iterat743893162068676942iset_b: nat > multiset_b > multiset_b ).
thf(sy_c_Cancellation_Oiterate__add_001t__Nat__Onat,type,
iterate_add_nat: nat > nat > nat ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
euclid4774559944035922753ze_int: int > nat ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
euclid4777050414544973029ze_nat: nat > nat ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
euclid3395696857347342551nt_int: int > int ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
euclid3398187327856392827nt_nat: nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__b_J,type,
minus_3765977311343717292iset_b: multiset_b > multiset_b > multiset_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__b_J,type,
plus_plus_multiset_b: multiset_b > multiset_b > multiset_b ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__b_J,type,
zero_zero_multiset_b: multiset_b ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_Itf__b_J,type,
comm_m1977238987320879926iset_b: ( multiset_b > multiset_b > multiset_b ) > multiset_b > multiset_multiset_b > multiset_b ).
thf(sy_c_Multiset_Ois__empty_001tf__b,type,
is_empty_b: multiset_b > $o ).
thf(sy_c_Multiset_Oreplicate__mset_001tf__b,type,
replicate_mset_b: nat > b > multiset_b ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__b_J,type,
set_mset_multiset_b: multiset_multiset_b > set_multiset_b ).
thf(sy_c_Multiset_Oset__mset_001tf__b,type,
set_mset_b: multiset_b > set_b ).
thf(sy_c_Multiset_Osize__multiset_001tf__b,type,
size_multiset_b: ( b > nat ) > multiset_b > nat ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxOneIJ,type,
multis6646701651571498855xOneIJ: nat > nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxOneJI,type,
multis6646701651571564453xOneJI: nat > nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxZeroIJ,type,
multis2983220944385456105ZeroIJ: nat > nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxZeroJI,type,
multis2983220944385521703ZeroJI: nat > nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OEpsilon,type,
multis2544335231667181926psilon: nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OGamma,type,
multis387687052011358179_Gamma: nat > nat > multis3193088007478089820ropVar ).
thf(sy_c_Multiset__Ordering__in__NP_OPropVar_Osize__PropVar,type,
multis2955979900537361535ropVar: multis3193088007478089820ropVar > nat ).
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__Multiset__Omultiset_Itf__b_J,type,
size_size_multiset_b: multiset_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset____Ordering____in____NP__OPropVar,type,
size_s6253272723116879048ropVar: multis3193088007478089820ropVar > 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__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_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_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
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_member_001t__Multiset__Omultiset_Itf__b_J,type,
member_multiset_b: multiset_b > set_multiset_b > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_P,type,
p: b > $o ).
thf(sy_v_Q,type,
q: b > $o ).
thf(sy_v_X,type,
x: set_b ).
thf(sy_v_Y,type,
y: set_b ).
% Relevant facts (1243)
thf(fact_0_PropVar_Oinject_I6_J,axiom,
! [X61: nat,X62: nat,Y61: nat,Y62: nat] :
( ( ( multis6646701651571498855xOneIJ @ X61 @ X62 )
= ( multis6646701651571498855xOneIJ @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% PropVar.inject(6)
thf(fact_1_PropVar_Oinject_I5_J,axiom,
! [X51: nat,X52: nat,Y51: nat,Y52: nat] :
( ( ( multis2983220944385456105ZeroIJ @ X51 @ X52 )
= ( multis2983220944385456105ZeroIJ @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% PropVar.inject(5)
thf(fact_2_PropVar_Oinject_I4_J,axiom,
! [X41: nat,X42: nat,Y41: nat,Y42: nat] :
( ( ( multis6646701651571564453xOneJI @ X41 @ X42 )
= ( multis6646701651571564453xOneJI @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% PropVar.inject(4)
thf(fact_3_PropVar_Oinject_I3_J,axiom,
! [X31: nat,X32: nat,Y31: nat,Y32: nat] :
( ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
= ( multis2983220944385521703ZeroJI @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% PropVar.inject(3)
thf(fact_4_PropVar_Oinject_I1_J,axiom,
! [X11: nat,X12: nat,Y11: nat,Y12: nat] :
( ( ( multis387687052011358179_Gamma @ X11 @ X12 )
= ( multis387687052011358179_Gamma @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% PropVar.inject(1)
thf(fact_5_PropVar_Odistinct_I29_J,axiom,
! [X51: nat,X52: nat,X61: nat,X62: nat] :
( ( multis2983220944385456105ZeroIJ @ X51 @ X52 )
!= ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).
% PropVar.distinct(29)
thf(fact_6_PropVar_Odistinct_I27_J,axiom,
! [X41: nat,X42: nat,X61: nat,X62: nat] :
( ( multis6646701651571564453xOneJI @ X41 @ X42 )
!= ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).
% PropVar.distinct(27)
thf(fact_7_PropVar_Odistinct_I25_J,axiom,
! [X41: nat,X42: nat,X51: nat,X52: nat] :
( ( multis6646701651571564453xOneJI @ X41 @ X42 )
!= ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).
% PropVar.distinct(25)
thf(fact_8_PropVar_Odistinct_I23_J,axiom,
! [X31: nat,X32: nat,X61: nat,X62: nat] :
( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
!= ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).
% PropVar.distinct(23)
thf(fact_9_PropVar_Odistinct_I21_J,axiom,
! [X31: nat,X32: nat,X51: nat,X52: nat] :
( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
!= ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).
% PropVar.distinct(21)
thf(fact_10_PropVar_Odistinct_I19_J,axiom,
! [X31: nat,X32: nat,X41: nat,X42: nat] :
( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
!= ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).
% PropVar.distinct(19)
thf(fact_11_PropVar_Odistinct_I9_J,axiom,
! [X11: nat,X12: nat,X61: nat,X62: nat] :
( ( multis387687052011358179_Gamma @ X11 @ X12 )
!= ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).
% PropVar.distinct(9)
thf(fact_12_PropVar_Odistinct_I7_J,axiom,
! [X11: nat,X12: nat,X51: nat,X52: nat] :
( ( multis387687052011358179_Gamma @ X11 @ X12 )
!= ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).
% PropVar.distinct(7)
thf(fact_13_PropVar_Odistinct_I5_J,axiom,
! [X11: nat,X12: nat,X41: nat,X42: nat] :
( ( multis387687052011358179_Gamma @ X11 @ X12 )
!= ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).
% PropVar.distinct(5)
thf(fact_14_PropVar_Odistinct_I3_J,axiom,
! [X11: nat,X12: nat,X31: nat,X32: nat] :
( ( multis387687052011358179_Gamma @ X11 @ X12 )
!= ( multis2983220944385521703ZeroJI @ X31 @ X32 ) ) ).
% PropVar.distinct(3)
thf(fact_15_PropVar_Oexhaust,axiom,
! [Y: multis3193088007478089820ropVar] :
( ! [X112: nat,X122: nat] :
( Y
!= ( multis387687052011358179_Gamma @ X112 @ X122 ) )
=> ( ! [X2: nat] :
( Y
!= ( multis2544335231667181926psilon @ X2 ) )
=> ( ! [X312: nat,X322: nat] :
( Y
!= ( multis2983220944385521703ZeroJI @ X312 @ X322 ) )
=> ( ! [X412: nat,X422: nat] :
( Y
!= ( multis6646701651571564453xOneJI @ X412 @ X422 ) )
=> ( ! [X512: nat,X522: nat] :
( Y
!= ( multis2983220944385456105ZeroIJ @ X512 @ X522 ) )
=> ~ ! [X612: nat,X622: nat] :
( Y
!= ( multis6646701651571498855xOneIJ @ X612 @ X622 ) ) ) ) ) ) ) ).
% PropVar.exhaust
thf(fact_16_PropVar_Oinject_I2_J,axiom,
! [X22: nat,Y2: nat] :
( ( ( multis2544335231667181926psilon @ X22 )
= ( multis2544335231667181926psilon @ Y2 ) )
= ( X22 = Y2 ) ) ).
% PropVar.inject(2)
thf(fact_17_PropVar_Odistinct_I1_J,axiom,
! [X11: nat,X12: nat,X22: nat] :
( ( multis387687052011358179_Gamma @ X11 @ X12 )
!= ( multis2544335231667181926psilon @ X22 ) ) ).
% PropVar.distinct(1)
thf(fact_18_PropVar_Odistinct_I11_J,axiom,
! [X22: nat,X31: nat,X32: nat] :
( ( multis2544335231667181926psilon @ X22 )
!= ( multis2983220944385521703ZeroJI @ X31 @ X32 ) ) ).
% PropVar.distinct(11)
thf(fact_19_PropVar_Odistinct_I13_J,axiom,
! [X22: nat,X41: nat,X42: nat] :
( ( multis2544335231667181926psilon @ X22 )
!= ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).
% PropVar.distinct(13)
thf(fact_20_PropVar_Odistinct_I15_J,axiom,
! [X22: nat,X51: nat,X52: nat] :
( ( multis2544335231667181926psilon @ X22 )
!= ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).
% PropVar.distinct(15)
thf(fact_21_PropVar_Odistinct_I17_J,axiom,
! [X22: nat,X61: nat,X62: nat] :
( ( multis2544335231667181926psilon @ X22 )
!= ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).
% PropVar.distinct(17)
thf(fact_22_PropVar_Osize__gen_I3_J,axiom,
! [X31: nat,X32: nat] :
( ( multis2955979900537361535ropVar @ ( multis2983220944385521703ZeroJI @ X31 @ X32 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(3)
thf(fact_23_PropVar_Osize__gen_I4_J,axiom,
! [X41: nat,X42: nat] :
( ( multis2955979900537361535ropVar @ ( multis6646701651571564453xOneJI @ X41 @ X42 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(4)
thf(fact_24_PropVar_Osize__gen_I5_J,axiom,
! [X51: nat,X52: nat] :
( ( multis2955979900537361535ropVar @ ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(5)
thf(fact_25_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_26_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_27_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_28_PropVar_Osize__gen_I6_J,axiom,
! [X61: nat,X62: nat] :
( ( multis2955979900537361535ropVar @ ( multis6646701651571498855xOneIJ @ X61 @ X62 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(6)
thf(fact_29_PropVar_Osize__gen_I1_J,axiom,
! [X11: nat,X12: nat] :
( ( multis2955979900537361535ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(1)
thf(fact_30_PropVar_Osize__gen_I2_J,axiom,
! [X22: nat] :
( ( multis2955979900537361535ropVar @ ( multis2544335231667181926psilon @ X22 ) )
= zero_zero_nat ) ).
% PropVar.size_gen(2)
thf(fact_31_PropVar_Osize_I9_J,axiom,
! [X31: nat,X32: nat] :
( ( size_s6253272723116879048ropVar @ ( multis2983220944385521703ZeroJI @ X31 @ X32 ) )
= zero_zero_nat ) ).
% PropVar.size(9)
thf(fact_32_PropVar_Osize_I10_J,axiom,
! [X41: nat,X42: nat] :
( ( size_s6253272723116879048ropVar @ ( multis6646701651571564453xOneJI @ X41 @ X42 ) )
= zero_zero_nat ) ).
% PropVar.size(10)
thf(fact_33_PropVar_Osize_I11_J,axiom,
! [X51: nat,X52: nat] :
( ( size_s6253272723116879048ropVar @ ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) )
= zero_zero_nat ) ).
% PropVar.size(11)
thf(fact_34_PropVar_Osize_I12_J,axiom,
! [X61: nat,X62: nat] :
( ( size_s6253272723116879048ropVar @ ( multis6646701651571498855xOneIJ @ X61 @ X62 ) )
= zero_zero_nat ) ).
% PropVar.size(12)
thf(fact_35_PropVar_Osize_I7_J,axiom,
! [X11: nat,X12: nat] :
( ( size_s6253272723116879048ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
= zero_zero_nat ) ).
% PropVar.size(7)
thf(fact_36_PropVar_Osize_I8_J,axiom,
! [X22: nat] :
( ( size_s6253272723116879048ropVar @ ( multis2544335231667181926psilon @ X22 ) )
= zero_zero_nat ) ).
% PropVar.size(8)
thf(fact_37_size__neq__size__imp__neq,axiom,
! [X4: multiset_b,Y: multiset_b] :
( ( ( size_size_multiset_b @ X4 )
!= ( size_size_multiset_b @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_38_size__neq__size__imp__neq,axiom,
! [X4: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar] :
( ( ( size_s6253272723116879048ropVar @ X4 )
!= ( size_s6253272723116879048ropVar @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_39_size__neq__size__imp__neq,axiom,
! [X4: char,Y: char] :
( ( ( size_size_char @ X4 )
!= ( size_size_char @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_40_size__neq__size__imp__neq,axiom,
! [X4: typerep,Y: typerep] :
( ( ( size_size_typerep @ X4 )
!= ( size_size_typerep @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_41_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_42_zero__reorient,axiom,
! [X4: multiset_b] :
( ( zero_zero_multiset_b = X4 )
= ( X4 = zero_zero_multiset_b ) ) ).
% zero_reorient
thf(fact_43_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_44_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_45_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_46_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_47_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_48_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_49_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_50_iterate__add__simps_I1_J,axiom,
! [A: multiset_b] :
( ( iterat743893162068676942iset_b @ zero_zero_nat @ A )
= zero_zero_multiset_b ) ).
% iterate_add_simps(1)
thf(fact_51_iterate__add__simps_I1_J,axiom,
! [A: nat] :
( ( iterate_add_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% iterate_add_simps(1)
thf(fact_52_iterate__add__simps_I1_J,axiom,
! [A: int] :
( ( iterate_add_int @ zero_zero_nat @ A )
= zero_zero_int ) ).
% iterate_add_simps(1)
thf(fact_53_size__0,axiom,
( ( euclid4777050414544973029ze_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% size_0
thf(fact_54_size__0,axiom,
( ( euclid4774559944035922753ze_int @ zero_zero_int )
= zero_zero_nat ) ).
% size_0
thf(fact_55_euclidean__size__eq__0__iff,axiom,
! [B: nat] :
( ( ( euclid4777050414544973029ze_nat @ B )
= zero_zero_nat )
= ( B = zero_zero_nat ) ) ).
% euclidean_size_eq_0_iff
thf(fact_56_euclidean__size__eq__0__iff,axiom,
! [B: int] :
( ( ( euclid4774559944035922753ze_int @ B )
= zero_zero_nat )
= ( B = zero_zero_int ) ) ).
% euclidean_size_eq_0_iff
thf(fact_57_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_58_iterate__add__empty,axiom,
! [N: nat] :
( ( iterat743893162068676942iset_b @ N @ zero_zero_multiset_b )
= zero_zero_multiset_b ) ).
% iterate_add_empty
thf(fact_59_iterate__add__empty,axiom,
! [N: nat] :
( ( iterate_add_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% iterate_add_empty
thf(fact_60_iterate__add__empty,axiom,
! [N: nat] :
( ( iterate_add_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% iterate_add_empty
thf(fact_61_size__empty,axiom,
( ( size_size_multiset_b @ zero_zero_multiset_b )
= zero_zero_nat ) ).
% size_empty
thf(fact_62_size__eq__0__iff__empty,axiom,
! [M: multiset_b] :
( ( ( size_size_multiset_b @ M )
= zero_zero_nat )
= ( M = zero_zero_multiset_b ) ) ).
% size_eq_0_iff_empty
thf(fact_63_euclidean__size__greater__0__iff,axiom,
! [B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4777050414544973029ze_nat @ B ) )
= ( B != zero_zero_nat ) ) ).
% euclidean_size_greater_0_iff
thf(fact_64_euclidean__size__greater__0__iff,axiom,
! [B: int] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4774559944035922753ze_int @ B ) )
= ( B != zero_zero_int ) ) ).
% euclidean_size_greater_0_iff
thf(fact_65_char_Osize__gen,axiom,
! [X1: $o,X22: $o,X33: $o,X43: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X22 @ X33 @ X43 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_66_typerep_Osize__neq,axiom,
! [X4: typerep] :
( ( size_size_typerep @ X4 )
!= zero_zero_nat ) ).
% typerep.size_neq
thf(fact_67_char_Oinject,axiom,
! [X1: $o,X22: $o,X33: $o,X43: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y2: $o,Y3: $o,Y4: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
( ( ( char2 @ X1 @ X22 @ X33 @ X43 @ X5 @ X6 @ X7 @ X8 )
= ( char2 @ Y1 @ Y2 @ Y3 @ Y4 @ Y5 @ Y6 @ Y7 @ Y8 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 )
& ( X33 = Y3 )
& ( X43 = Y4 )
& ( X5 = Y5 )
& ( X6 = Y6 )
& ( X7 = Y7 )
& ( X8 = Y8 ) ) ) ).
% char.inject
thf(fact_68_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_69_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_70_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_71_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_72_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_73_linorder__neqE__nat,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_74_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_75_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_76_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_77_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_78_less__not__refl2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( M3 != N ) ) ).
% less_not_refl2
thf(fact_79_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_80_nat__neq__iff,axiom,
! [M3: nat,N: nat] :
( ( M3 != N )
= ( ( ord_less_nat @ M3 @ N )
| ( ord_less_nat @ N @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_81_char_Oexhaust,axiom,
! [Y: char] :
~ ! [X13: $o,X2: $o,X34: $o,X44: $o,X53: $o,X63: $o,X72: $o,X82: $o] :
( Y
!= ( char2 @ X13 @ X2 @ X34 @ X44 @ X53 @ X63 @ X72 @ X82 ) ) ).
% char.exhaust
thf(fact_82_nonempty__has__size,axiom,
! [S2: multiset_b] :
( ( S2 != zero_zero_multiset_b )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_b @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_83_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_84_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_85_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_86_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_87_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_88_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_89_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_90_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_91_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_92_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_93_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_94_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X33: $o,X43: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X33 @ X43 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_95_Multiset_Ois__empty__def,axiom,
( is_empty_b
= ( ^ [A3: multiset_b] : ( A3 = zero_zero_multiset_b ) ) ) ).
% Multiset.is_empty_def
thf(fact_96_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_97_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_98_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_99_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_100_size__multiset__eq__0__iff__empty,axiom,
! [F: b > nat,M: multiset_b] :
( ( ( size_multiset_b @ F @ M )
= zero_zero_nat )
= ( M = zero_zero_multiset_b ) ) ).
% size_multiset_eq_0_iff_empty
thf(fact_101_size__multiset__empty,axiom,
! [F: b > nat] :
( ( size_multiset_b @ F @ zero_zero_multiset_b )
= zero_zero_nat ) ).
% size_multiset_empty
thf(fact_102_replicate__mset__eq__empty__iff,axiom,
! [N: nat,A: b] :
( ( ( replicate_mset_b @ N @ A )
= zero_zero_multiset_b )
= ( N = zero_zero_nat ) ) ).
% replicate_mset_eq_empty_iff
thf(fact_103_replicate__mset__0,axiom,
! [X4: b] :
( ( replicate_mset_b @ zero_zero_nat @ X4 )
= zero_zero_multiset_b ) ).
% replicate_mset_0
thf(fact_104_of__nat__eq__iff,axiom,
! [M3: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M3 )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M3 = N ) ) ).
% of_nat_eq_iff
thf(fact_105_of__nat__eq__iff,axiom,
! [M3: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M3 = N ) ) ).
% of_nat_eq_iff
thf(fact_106_size__replicate__mset,axiom,
! [N: nat,M: b] :
( ( size_size_multiset_b @ ( replicate_mset_b @ N @ M ) )
= N ) ).
% size_replicate_mset
thf(fact_107_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1316708129612266289at_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_108_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_109_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_110_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_111_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_112_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_113_of__nat__less__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_iff
thf(fact_114_of__nat__less__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_iff
thf(fact_115_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% of_nat_mask_eq
thf(fact_116_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_nat_mask_eq
thf(fact_117_replicate__mset__eq__iff,axiom,
! [M3: nat,A: b,N: nat,B: b] :
( ( ( replicate_mset_b @ M3 @ A )
= ( replicate_mset_b @ N @ B ) )
= ( ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) )
| ( ( M3 = N )
& ( A = B ) ) ) ) ).
% replicate_mset_eq_iff
thf(fact_118_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_119_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_120_less__imp__of__nat__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_121_less__imp__of__nat__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_122_of__nat__less__imp__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_123_of__nat__less__imp__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_124_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ( ( P @ X9 )
& ( Q @ X9 ) )
= ( ( P2 @ X9 )
& ( Q2 @ X9 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_125_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ( ( P @ X9 )
& ( Q @ X9 ) )
= ( ( P2 @ X9 )
& ( Q2 @ X9 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_126_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ( ( P @ X9 )
| ( Q @ X9 ) )
= ( ( P2 @ X9 )
| ( Q2 @ X9 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_127_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ( ( P @ X9 )
| ( Q @ X9 ) )
= ( ( P2 @ X9 )
| ( Q2 @ X9 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_128_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( X9 != T ) ) ).
% pinf(3)
thf(fact_129_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( X9 != T ) ) ).
% pinf(3)
thf(fact_130_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( X9 != T ) ) ).
% pinf(4)
thf(fact_131_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( X9 != T ) ) ).
% pinf(4)
thf(fact_132_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ~ ( ord_less_nat @ X9 @ T ) ) ).
% pinf(5)
thf(fact_133_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ~ ( ord_less_int @ X9 @ T ) ) ).
% pinf(5)
thf(fact_134_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ord_less_nat @ T @ X9 ) ) ).
% pinf(7)
thf(fact_135_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ord_less_int @ T @ X9 ) ) ).
% pinf(7)
thf(fact_136_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ( ( P @ X9 )
& ( Q @ X9 ) )
= ( ( P2 @ X9 )
& ( Q2 @ X9 ) ) ) ) ) ) ).
% minf(1)
thf(fact_137_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ( ( P @ X9 )
& ( Q @ X9 ) )
= ( ( P2 @ X9 )
& ( Q2 @ X9 ) ) ) ) ) ) ).
% minf(1)
thf(fact_138_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ( ( P @ X9 )
| ( Q @ X9 ) )
= ( ( P2 @ X9 )
| ( Q2 @ X9 ) ) ) ) ) ) ).
% minf(2)
thf(fact_139_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ( ( P @ X9 )
| ( Q @ X9 ) )
= ( ( P2 @ X9 )
| ( Q2 @ X9 ) ) ) ) ) ) ).
% minf(2)
thf(fact_140_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( X9 != T ) ) ).
% minf(3)
thf(fact_141_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( X9 != T ) ) ).
% minf(3)
thf(fact_142_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( X9 != T ) ) ).
% minf(4)
thf(fact_143_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( X9 != T ) ) ).
% minf(4)
thf(fact_144_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ord_less_nat @ X9 @ T ) ) ).
% minf(5)
thf(fact_145_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ord_less_int @ X9 @ T ) ) ).
% minf(5)
thf(fact_146_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ~ ( ord_less_nat @ T @ X9 ) ) ).
% minf(7)
thf(fact_147_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ~ ( ord_less_int @ T @ X9 ) ) ).
% minf(7)
thf(fact_148_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid4777050414544973029ze_nat @ ( semiri1316708129612266289at_nat @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_149_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid4774559944035922753ze_int @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_150_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_151_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_152_of__nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_153_of__nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X4 ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_154_in__replicate__mset,axiom,
! [X4: b,N: nat,Y: b] :
( ( member_b @ X4 @ ( set_mset_b @ ( replicate_mset_b @ N @ Y ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( X4 = Y ) ) ) ).
% in_replicate_mset
thf(fact_155_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_156_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_157_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_158_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_159_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_160_Suc__le__mono,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ).
% Suc_le_mono
thf(fact_161_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_162_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_163_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_164_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_165_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_166_nat__power__eq__Suc__0__iff,axiom,
! [X4: nat,M3: nat] :
( ( ( power_power_nat @ X4 @ M3 )
= ( suc @ zero_zero_nat ) )
= ( ( M3 = zero_zero_nat )
| ( X4
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_167_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_168_nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_169_Suc__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_eq
thf(fact_170_Suc__mono,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_171_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_172_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_173_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_174_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_175_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_176_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_177_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_178_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_179_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_180_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X4: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_181_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X4: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_182_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X4: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_183_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X4: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_184_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_185_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_186_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_187_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_188_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_189_imp__le__cong,axiom,
! [X4: int,X10: int,P: $o,P2: $o] :
( ( X4 = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_190_conj__le__cong,axiom,
! [X4: int,X10: int,P: $o,P2: $o] :
( ( X4 = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_191_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_192_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_193_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_194_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_195_Suc__leI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_leI
thf(fact_196_Suc__le__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_eq
thf(fact_197_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_198_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_199_Suc__le__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_lessD
thf(fact_200_le__less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_201_less__Suc__eq__le,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_Suc_eq_le
thf(fact_202_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_203_le__imp__less__Suc,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_204_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_205_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_206_size__eq__Suc__imp__elem,axiom,
! [M: multiset_b,N: nat] :
( ( ( size_size_multiset_b @ M )
= ( suc @ N ) )
=> ? [A4: b] : ( member_b @ A4 @ ( set_mset_b @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_207_transitive__stepwise__le,axiom,
! [M3: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y9: nat,Z2: nat] :
( ( R @ X3 @ Y9 )
=> ( ( R @ Y9 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M3 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_208_nat__induct__at__least,axiom,
! [M3: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P @ M3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_209_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_210_not__less__eq__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_211_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_212_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_213_Suc__inject,axiom,
! [X4: nat,Y: nat] :
( ( ( suc @ X4 )
= ( suc @ Y ) )
=> ( X4 = Y ) ) ).
% Suc_inject
thf(fact_214_le__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M3 @ N )
| ( M3
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_215_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_216_le__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_217_le__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N )
=> ( M3
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_218_Suc__leD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_leD
thf(fact_219_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_220_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_221_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_222_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_223_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_224_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_225_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_226_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_227_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_228_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_229_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_230_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_231_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_232_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_233_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_234_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_235_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_236_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_237_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_238_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_239_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_240_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_241_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_242_multiset__nonemptyE,axiom,
! [A2: multiset_b] :
( ( A2 != zero_zero_multiset_b )
=> ~ ! [X3: b] :
~ ( member_b @ X3 @ ( set_mset_b @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_243_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_244_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_245_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_246_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X9 ) ) ).
% minf(8)
thf(fact_247_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X9 ) ) ).
% minf(8)
thf(fact_248_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ X9 @ Z2 )
=> ( ord_less_eq_nat @ X9 @ T ) ) ).
% minf(6)
thf(fact_249_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ X9 @ Z2 )
=> ( ord_less_eq_int @ X9 @ T ) ) ).
% minf(6)
thf(fact_250_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ( ord_less_eq_nat @ T @ X9 ) ) ).
% pinf(8)
thf(fact_251_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ( ord_less_eq_int @ T @ X9 ) ) ).
% pinf(8)
thf(fact_252_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X9: nat] :
( ( ord_less_nat @ Z2 @ X9 )
=> ~ ( ord_less_eq_nat @ X9 @ T ) ) ).
% pinf(6)
thf(fact_253_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X9: int] :
( ( ord_less_int @ Z2 @ X9 )
=> ~ ( ord_less_eq_int @ X9 @ T ) ) ).
% pinf(6)
thf(fact_254_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_255_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_256_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_257_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_258_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_259_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_260_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y9: nat] : ( P @ zero_zero_nat @ ( suc @ Y9 ) )
=> ( ! [X3: nat,Y9: nat] :
( ( P @ X3 @ Y9 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y9 ) ) )
=> ( P @ M3 @ N ) ) ) ) ).
% diff_induct
thf(fact_261_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_262_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_263_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_264_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_265_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_266_not__less__less__Suc__eq,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_267_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_268_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_269_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_270_Suc__less__SucD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_SucD
thf(fact_271_less__antisym,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
=> ( M3 = N ) ) ) ).
% less_antisym
thf(fact_272_Suc__less__eq2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M3 )
= ( ? [M6: nat] :
( ( M3
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_273_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_274_not__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_nat @ M3 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_275_less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ).
% less_Suc_eq
thf(fact_276_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_277_less__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_278_less__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M3 @ N )
=> ( M3 = N ) ) ) ).
% less_SucE
thf(fact_279_Suc__lessI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ( suc @ M3 )
!= N )
=> ( ord_less_nat @ ( suc @ M3 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_280_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_281_Suc__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_lessD
thf(fact_282_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_283_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_284_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_285_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_286_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_287_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_288_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_289_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_290_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M7: nat,N3: nat] :
( ( ord_less_nat @ M7 @ N3 )
| ( M7 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_291_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_292_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M7: nat,N3: nat] :
( ( ord_less_eq_nat @ M7 @ N3 )
& ( M7 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_293_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_294_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_295_nat__power__less__imp__less,axiom,
! [I: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M3 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_296_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_297_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_298_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_299_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_300_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_301_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less_nat @ N @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_302_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less_nat @ N @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_303_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_304_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_305_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_306_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_307_less__Suc__eq__0__disj,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( M3 = zero_zero_nat )
| ? [J3: nat] :
( ( M3
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_308_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_309_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_310_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_311_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_312_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_313_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_314_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_315_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_316_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_317_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y9: nat] :
( ( P @ Y9 )
=> ( ord_less_eq_nat @ Y9 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_318_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_319_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_320_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_321_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_322_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_323_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_324_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_325_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A6 ) )
= ( ord_less_nat @ A6 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_326_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A6 ) )
= ( ord_less_int @ A6 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_327_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_328_leD,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y ) ) ).
% leD
thf(fact_329_leD,axiom,
! [Y: int,X4: int] :
( ( ord_less_eq_int @ Y @ X4 )
=> ~ ( ord_less_int @ X4 @ Y ) ) ).
% leD
thf(fact_330_leI,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% leI
thf(fact_331_leI,axiom,
! [X4: int,Y: int] :
( ~ ( ord_less_int @ X4 @ Y )
=> ( ord_less_eq_int @ Y @ X4 ) ) ).
% leI
thf(fact_332_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_333_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_334_lt__ex,axiom,
! [X4: int] :
? [Y9: int] : ( ord_less_int @ Y9 @ X4 ) ).
% lt_ex
thf(fact_335_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_336_gt__ex,axiom,
! [X4: int] :
? [X_1: int] : ( ord_less_int @ X4 @ X_1 ) ).
% gt_ex
thf(fact_337_less__imp__neq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% less_imp_neq
thf(fact_338_less__imp__neq,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( X4 != Y ) ) ).
% less_imp_neq
thf(fact_339_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_340_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_341_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_342_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_343_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_344_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_345_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y10: nat] :
( ( ord_less_nat @ Y10 @ X3 )
=> ( P @ Y10 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_346_antisym__conv3,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_nat @ Y @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv3
thf(fact_347_antisym__conv3,axiom,
! [Y: int,X4: int] :
( ~ ( ord_less_int @ Y @ X4 )
=> ( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv3
thf(fact_348_linorder__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( X4 != Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_cases
thf(fact_349_linorder__cases,axiom,
! [X4: int,Y: int] :
( ~ ( ord_less_int @ X4 @ Y )
=> ( ( X4 != Y )
=> ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_cases
thf(fact_350_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_351_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_352_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_353_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_354_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X14: nat] : ( P3 @ X14 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M7: nat] :
( ( ord_less_nat @ M7 @ N3 )
=> ~ ( P4 @ M7 ) ) ) ) ) ).
% exists_least_iff
thf(fact_355_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_356_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_357_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_358_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_359_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ( ord_less_nat @ Y @ X4 )
| ( X4 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_360_not__less__iff__gr__or__eq,axiom,
! [X4: int,Y: int] :
( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( ( ord_less_int @ Y @ X4 )
| ( X4 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_361_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_362_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_363_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_364_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_365_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_366_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_367_linorder__neqE,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE
thf(fact_368_linorder__neqE,axiom,
! [X4: int,Y: int] :
( ( X4 != Y )
=> ( ~ ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_neqE
thf(fact_369_order__less__asym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_asym
thf(fact_370_order__less__asym,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ~ ( ord_less_int @ Y @ X4 ) ) ).
% order_less_asym
thf(fact_371_linorder__neq__iff,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
= ( ( ord_less_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_372_linorder__neq__iff,axiom,
! [X4: int,Y: int] :
( ( X4 != Y )
= ( ( ord_less_int @ X4 @ Y )
| ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_373_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_374_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_375_order__less__trans,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_376_order__less__trans,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X4 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_377_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_378_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_379_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_380_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_381_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_382_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_383_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_384_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_385_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_386_order__less__irrefl,axiom,
! [X4: int] :
~ ( ord_less_int @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_387_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_388_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_389_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_390_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_391_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_392_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_393_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_394_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_395_order__less__not__sym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_not_sym
thf(fact_396_order__less__not__sym,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ~ ( ord_less_int @ Y @ X4 ) ) ).
% order_less_not_sym
thf(fact_397_order__less__imp__triv,axiom,
! [X4: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_398_order__less__imp__triv,axiom,
! [X4: int,Y: int,P: $o] :
( ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_int @ Y @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_399_linorder__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_less_linear
thf(fact_400_linorder__less__linear,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
| ( X4 = Y )
| ( ord_less_int @ Y @ X4 ) ) ).
% linorder_less_linear
thf(fact_401_order__less__imp__not__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_402_order__less__imp__not__eq,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( X4 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_403_order__less__imp__not__eq2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( Y != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_404_order__less__imp__not__eq2,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( Y != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_405_order__less__imp__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_406_order__less__imp__not__less,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ~ ( ord_less_int @ Y @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_407_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_408_order__le__imp__less__or__eq,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ord_less_int @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_409_linorder__le__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_410_linorder__le__less__linear,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
| ( ord_less_int @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_411_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_412_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_413_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_414_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_415_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_eq_nat @ X3 @ Y9 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_416_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_eq_nat @ X3 @ Y9 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_417_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_eq_int @ X3 @ Y9 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_418_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_eq_int @ X3 @ Y9 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_419_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_eq_nat @ X3 @ Y9 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_420_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_eq_nat @ X3 @ Y9 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_421_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_eq_int @ X3 @ Y9 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_422_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_eq_int @ X3 @ Y9 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_423_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_424_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_425_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y9: nat] :
( ( ord_less_nat @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_426_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y9: int] :
( ( ord_less_int @ X3 @ Y9 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y9 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_427_order__less__le__trans,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_428_order__less__le__trans,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X4 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_429_order__le__less__trans,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_430_order__le__less__trans,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X4 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_431_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_432_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_433_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_434_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_435_order__less__imp__le,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_436_order__less__imp__le,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_eq_int @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_437_linorder__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_438_linorder__not__less,axiom,
! [X4: int,Y: int] :
( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( ord_less_eq_int @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_439_linorder__not__le,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y ) )
= ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_440_linorder__not__le,axiom,
! [X4: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X4 @ Y ) )
= ( ord_less_int @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_441_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y13: nat] :
( ( ord_less_eq_nat @ X @ Y13 )
& ( X != Y13 ) ) ) ) ).
% order_less_le
thf(fact_442_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y13: int] :
( ( ord_less_eq_int @ X @ Y13 )
& ( X != Y13 ) ) ) ) ).
% order_less_le
thf(fact_443_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y13: nat] :
( ( ord_less_nat @ X @ Y13 )
| ( X = Y13 ) ) ) ) ).
% order_le_less
thf(fact_444_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y13: int] :
( ( ord_less_int @ X @ Y13 )
| ( X = Y13 ) ) ) ) ).
% order_le_less
thf(fact_445_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_446_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_447_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_448_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_449_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A5: nat] :
( ( ord_less_eq_nat @ B2 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_450_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A5: int] :
( ( ord_less_eq_int @ B2 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_451_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_452_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_453_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_454_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_455_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A5: nat] :
( ( ord_less_eq_nat @ B2 @ A5 )
& ( A5 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_456_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A5: int] :
( ( ord_less_eq_int @ B2 @ A5 )
& ( A5 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_457_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( ( ord_less_nat @ B2 @ A5 )
| ( A5 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_458_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A5: int] :
( ( ord_less_int @ B2 @ A5 )
| ( A5 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_459_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ A5 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_460_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B2: int] :
( ( ord_less_eq_int @ A5 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_461_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_462_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_463_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_464_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_465_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ A5 @ B2 )
& ( A5 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_466_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B2: int] :
( ( ord_less_eq_int @ A5 @ B2 )
& ( A5 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_467_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_nat @ A5 @ B2 )
| ( A5 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_468_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] :
( ( ord_less_int @ A5 @ B2 )
| ( A5 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_469_not__le__imp__less,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y @ X4 )
=> ( ord_less_nat @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_470_not__le__imp__less,axiom,
! [Y: int,X4: int] :
( ~ ( ord_less_eq_int @ Y @ X4 )
=> ( ord_less_int @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_471_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y13: nat] :
( ( ord_less_eq_nat @ X @ Y13 )
& ~ ( ord_less_eq_nat @ Y13 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_472_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y13: int] :
( ( ord_less_eq_int @ X @ Y13 )
& ~ ( ord_less_eq_int @ Y13 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_473_antisym__conv2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_474_antisym__conv2,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_475_antisym__conv1,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_476_antisym__conv1,axiom,
! [X4: int,Y: int] :
( ~ ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_eq_int @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_477_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X9: nat] :
( ( ( ord_less_eq_nat @ A @ X9 )
& ( ord_less_nat @ X9 @ C3 ) )
=> ( P @ X9 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_478_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X9: int] :
( ( ( ord_less_eq_int @ A @ X9 )
& ( ord_less_int @ X9 @ C3 ) )
=> ( P @ X9 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_479_power__decreasing__iff,axiom,
! [B: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M3 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ) ) ).
% power_decreasing_iff
thf(fact_480_power__decreasing__iff,axiom,
! [B: int,M3: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M3 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ) ) ).
% power_decreasing_iff
thf(fact_481_power__increasing__iff,axiom,
! [B: nat,X4: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X4 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_482_power__increasing__iff,axiom,
! [B: int,X4: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X4 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_483_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_484_power__strict__decreasing__iff,axiom,
! [B: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M3 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M3 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_485_power__strict__decreasing__iff,axiom,
! [B: int,M3: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M3 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M3 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_486_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_487_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_488_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_489_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_490_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_491_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_492_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_493_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_494_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_495_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_496_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_497_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_498_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_499_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_500_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_501_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_502_euclidean__size__1,axiom,
( ( euclid4777050414544973029ze_nat @ one_one_nat )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_503_euclidean__size__1,axiom,
( ( euclid4774559944035922753ze_int @ one_one_int )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_power__inject__exp,axiom,
! [A: nat,M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M3 )
= ( power_power_nat @ A @ N ) )
= ( M3 = N ) ) ) ).
% power_inject_exp
thf(fact_513_power__inject__exp,axiom,
! [A: int,M3: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M3 )
= ( power_power_int @ A @ N ) )
= ( M3 = N ) ) ) ).
% power_inject_exp
thf(fact_514_negative__eq__positive,axiom,
! [N: nat,M3: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M3 ) )
= ( ( N = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_515_power__strict__increasing__iff,axiom,
! [B: nat,X4: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X4 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_516_power__strict__increasing__iff,axiom,
! [B: int,X4: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X4 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_517_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_518_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_519_negative__zless,axiom,
! [N: nat,M3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zless
thf(fact_520_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_521_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_522_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_523_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_524_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_525_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_526_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_535_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_536_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_537_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_538_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_539_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_540_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_541_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_542_not__int__zless__negative,axiom,
! [N: nat,M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% not_int_zless_negative
thf(fact_543_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_544_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_545_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_546_iterate__add__1,axiom,
! [N: nat] :
( ( iterate_add_nat @ N @ one_one_nat )
= ( semiri1316708129612266289at_nat @ N ) ) ).
% iterate_add_1
thf(fact_547_iterate__add__1,axiom,
! [N: nat] :
( ( iterate_add_int @ N @ one_one_int )
= ( semiri1314217659103216013at_int @ N ) ) ).
% iterate_add_1
thf(fact_548_int__cases4,axiom,
! [M3: int] :
( ! [N2: nat] :
( M3
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_549_int__zle__neg,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) )
= ( ( N = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_550_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_551_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_552_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_553_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_554_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_555_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_556_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_557_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_558_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_559_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_560_power__less__imp__less__exp,axiom,
! [A: nat,M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M3 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_561_power__less__imp__less__exp,axiom,
! [A: int,M3: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M3 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_562_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_563_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_564_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_565_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_566_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_567_negD,axiom,
! [X4: int] :
( ( ord_less_int @ X4 @ zero_zero_int )
=> ? [N2: nat] :
( X4
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_568_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_569_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_570_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_571_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_572_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_573_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_574_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_575_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_576_power__le__imp__le__exp,axiom,
! [A: nat,M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M3 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_577_power__le__imp__le__exp,axiom,
! [A: int,M3: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M3 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_578_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_579_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_580_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_581_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_582_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_583_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_584_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_585_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_586_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_587_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_588_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_589_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_590_linorder__neqE__linordered__idom,axiom,
! [X4: int,Y: int] :
( ( X4 != Y )
=> ( ~ ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_591_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_592_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_593_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_594_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_595_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_596_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_597_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_598_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_599_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_600_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_601_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_602_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_603_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_604_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_605_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_606_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_607_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_608_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X14: nat] : ( P3 @ X14 ) )
= ( ^ [P4: nat > $o] :
? [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
& ( P4 @ ( nat2 @ X ) ) ) ) ) ).
% ex_nat
thf(fact_609_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X14: nat] : ( P3 @ X14 ) )
= ( ^ [P4: nat > $o] :
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P4 @ ( nat2 @ X ) ) ) ) ) ).
% all_nat
thf(fact_610_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_611_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% nat_mask_eq
thf(fact_612_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_613_zless__nat__eq__int__zless,axiom,
! [M3: nat,Z3: int] :
( ( ord_less_nat @ M3 @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_614_int__eq__iff,axiom,
! [M3: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M3 )
= Z3 )
= ( ( M3
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_615_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_616_nat__power__eq,axiom,
! [Z3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( power_power_int @ Z3 @ N ) )
= ( power_power_nat @ ( nat2 @ Z3 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_617_nat__eq__iff2,axiom,
! [M3: nat,W: int] :
( ( M3
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_618_nat__eq__iff,axiom,
! [W: int,M3: nat] :
( ( ( nat2 @ W )
= M3 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_619_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_620_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_621_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_622_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_623_nat__less__iff,axiom,
! [W: int,M3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M3 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M3 ) ) ) ) ).
% nat_less_iff
thf(fact_624_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_625_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_626_division__segment__int__def,axiom,
( euclid3395696857347342551nt_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% division_segment_int_def
thf(fact_627_add__right__cancel,axiom,
! [B: multiset_b,A: multiset_b,C2: multiset_b] :
( ( ( plus_plus_multiset_b @ B @ A )
= ( plus_plus_multiset_b @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_628_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_629_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_630_add__left__cancel,axiom,
! [A: multiset_b,B: multiset_b,C2: multiset_b] :
( ( ( plus_plus_multiset_b @ A @ B )
= ( plus_plus_multiset_b @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_631_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_632_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_633_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_634_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_635_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_636_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_637_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_638_add_Oright__neutral,axiom,
! [A: multiset_b] :
( ( plus_plus_multiset_b @ A @ zero_zero_multiset_b )
= A ) ).
% add.right_neutral
thf(fact_639_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_640_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_641_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_642_add__cancel__left__left,axiom,
! [B: multiset_b,A: multiset_b] :
( ( ( plus_plus_multiset_b @ B @ A )
= A )
= ( B = zero_zero_multiset_b ) ) ).
% add_cancel_left_left
thf(fact_643_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_644_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_645_add__cancel__left__right,axiom,
! [A: multiset_b,B: multiset_b] :
( ( ( plus_plus_multiset_b @ A @ B )
= A )
= ( B = zero_zero_multiset_b ) ) ).
% add_cancel_left_right
thf(fact_646_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_647_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_648_add__cancel__right__left,axiom,
! [A: multiset_b,B: multiset_b] :
( ( A
= ( plus_plus_multiset_b @ B @ A ) )
= ( B = zero_zero_multiset_b ) ) ).
% add_cancel_right_left
thf(fact_649_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_650_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_651_add__cancel__right__right,axiom,
! [A: multiset_b,B: multiset_b] :
( ( A
= ( plus_plus_multiset_b @ A @ B ) )
= ( B = zero_zero_multiset_b ) ) ).
% add_cancel_right_right
thf(fact_652_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_653_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_654_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_655_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y ) )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_656_add__0,axiom,
! [A: multiset_b] :
( ( plus_plus_multiset_b @ zero_zero_multiset_b @ A )
= A ) ).
% add_0
thf(fact_657_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_658_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_659_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_660_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_661_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_662_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_663_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_664_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_665_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_666_of__nat__add,axiom,
! [M3: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_667_of__nat__add,axiom,
! [M3: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M3 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_668_iterate__add__distrib,axiom,
! [M3: nat,N: nat,A: multiset_b] :
( ( iterat743893162068676942iset_b @ ( plus_plus_nat @ M3 @ N ) @ A )
= ( plus_plus_multiset_b @ ( iterat743893162068676942iset_b @ M3 @ A ) @ ( iterat743893162068676942iset_b @ N @ A ) ) ) ).
% iterate_add_distrib
thf(fact_669_iterate__add__distrib,axiom,
! [M3: nat,N: nat,A: int] :
( ( iterate_add_int @ ( plus_plus_nat @ M3 @ N ) @ A )
= ( plus_plus_int @ ( iterate_add_int @ M3 @ A ) @ ( iterate_add_int @ N @ A ) ) ) ).
% iterate_add_distrib
thf(fact_670_iterate__add__distrib,axiom,
! [M3: nat,N: nat,A: nat] :
( ( iterate_add_nat @ ( plus_plus_nat @ M3 @ N ) @ A )
= ( plus_plus_nat @ ( iterate_add_nat @ M3 @ A ) @ ( iterate_add_nat @ N @ A ) ) ) ).
% iterate_add_distrib
thf(fact_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_692_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_693_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_694_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_695_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_696_of__int__less__iff,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% of_int_less_iff
thf(fact_697_iterate__add__simps_I2_J,axiom,
! [N: nat,A: multiset_b] :
( ( iterat743893162068676942iset_b @ ( suc @ N ) @ A )
= ( plus_plus_multiset_b @ A @ ( iterat743893162068676942iset_b @ N @ A ) ) ) ).
% iterate_add_simps(2)
thf(fact_698_iterate__add__simps_I2_J,axiom,
! [N: nat,A: int] :
( ( iterate_add_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ A @ ( iterate_add_int @ N @ A ) ) ) ).
% iterate_add_simps(2)
thf(fact_699_iterate__add__simps_I2_J,axiom,
! [N: nat,A: nat] :
( ( iterate_add_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ A @ ( iterate_add_nat @ N @ A ) ) ) ).
% iterate_add_simps(2)
thf(fact_700_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_701_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_702_of__nat__Suc,axiom,
! [M3: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M3 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M3 ) ) ) ).
% of_nat_Suc
thf(fact_703_of__nat__Suc,axiom,
! [M3: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M3 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% of_nat_Suc
thf(fact_704_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_705_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_706_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_707_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_708_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_709_of__int__less__1__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
= ( ord_less_int @ Z3 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_710_of__int__1__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% of_int_1_less_iff
thf(fact_711_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_712_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X4: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X4 ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_713_of__int__power__less__of__int__cancel__iff,axiom,
! [X4: int,B: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_int @ X4 @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_714_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_715_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_716_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_717_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_718_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_719_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_720_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_721_add__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_722_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_723_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_724_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_725_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_726_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_727_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
? [C: nat] :
( B2
= ( plus_plus_nat @ A5 @ C ) ) ) ) ).
% le_iff_add
thf(fact_728_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_729_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_730_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_731_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_732_verit__sum__simplify,axiom,
! [A: multiset_b] :
( ( plus_plus_multiset_b @ A @ zero_zero_multiset_b )
= A ) ).
% verit_sum_simplify
thf(fact_733_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_734_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_735_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_736_add_Ocomm__neutral,axiom,
! [A: multiset_b] :
( ( plus_plus_multiset_b @ A @ zero_zero_multiset_b )
= A ) ).
% add.comm_neutral
thf(fact_737_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_738_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_739_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_b] :
( ( plus_plus_multiset_b @ zero_zero_multiset_b @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_740_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_741_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_742_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_743_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_744_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_745_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_746_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_747_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_748_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_749_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_750_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_751_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_752_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_753_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_754_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_755_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_756_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_757_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_758_add__right__imp__eq,axiom,
! [B: multiset_b,A: multiset_b,C2: multiset_b] :
( ( ( plus_plus_multiset_b @ B @ A )
= ( plus_plus_multiset_b @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_759_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_760_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_761_add__left__imp__eq,axiom,
! [A: multiset_b,B: multiset_b,C2: multiset_b] :
( ( ( plus_plus_multiset_b @ A @ B )
= ( plus_plus_multiset_b @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_762_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_763_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_764_add_Oleft__commute,axiom,
! [B: multiset_b,A: multiset_b,C2: multiset_b] :
( ( plus_plus_multiset_b @ B @ ( plus_plus_multiset_b @ A @ C2 ) )
= ( plus_plus_multiset_b @ A @ ( plus_plus_multiset_b @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_765_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_766_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_767_add_Ocommute,axiom,
( plus_plus_multiset_b
= ( ^ [A5: multiset_b,B2: multiset_b] : ( plus_plus_multiset_b @ B2 @ A5 ) ) ) ).
% add.commute
thf(fact_768_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B2: int] : ( plus_plus_int @ B2 @ A5 ) ) ) ).
% add.commute
thf(fact_769_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B2: nat] : ( plus_plus_nat @ B2 @ A5 ) ) ) ).
% add.commute
thf(fact_770_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_771_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_772_add_Oassoc,axiom,
! [A: multiset_b,B: multiset_b,C2: multiset_b] :
( ( plus_plus_multiset_b @ ( plus_plus_multiset_b @ A @ B ) @ C2 )
= ( plus_plus_multiset_b @ A @ ( plus_plus_multiset_b @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_773_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_774_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_775_group__cancel_Oadd2,axiom,
! [B5: multiset_b,K: multiset_b,B: multiset_b,A: multiset_b] :
( ( B5
= ( plus_plus_multiset_b @ K @ B ) )
=> ( ( plus_plus_multiset_b @ A @ B5 )
= ( plus_plus_multiset_b @ K @ ( plus_plus_multiset_b @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_776_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_777_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_778_group__cancel_Oadd1,axiom,
! [A2: multiset_b,K: multiset_b,A: multiset_b,B: multiset_b] :
( ( A2
= ( plus_plus_multiset_b @ K @ A ) )
=> ( ( plus_plus_multiset_b @ A2 @ B )
= ( plus_plus_multiset_b @ K @ ( plus_plus_multiset_b @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_779_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_780_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_781_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_782_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_783_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_b,B: multiset_b,C2: multiset_b] :
( ( plus_plus_multiset_b @ ( plus_plus_multiset_b @ A @ B ) @ C2 )
= ( plus_plus_multiset_b @ A @ ( plus_plus_multiset_b @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_784_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_785_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_786_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_787_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_788_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_789_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_790_division__segment__not__0,axiom,
! [A: int] :
( ( euclid3395696857347342551nt_int @ A )
!= zero_zero_int ) ).
% division_segment_not_0
thf(fact_791_division__segment__not__0,axiom,
! [A: nat] :
( ( euclid3398187327856392827nt_nat @ A )
!= zero_zero_nat ) ).
% division_segment_not_0
thf(fact_792_division__segment__eq__iff,axiom,
! [A: int,B: int] :
( ( ( euclid3395696857347342551nt_int @ A )
= ( euclid3395696857347342551nt_int @ B ) )
=> ( ( ( euclid4774559944035922753ze_int @ A )
= ( euclid4774559944035922753ze_int @ B ) )
=> ( A = B ) ) ) ).
% division_segment_eq_iff
thf(fact_793_division__segment__eq__iff,axiom,
! [A: nat,B: nat] :
( ( ( euclid3398187327856392827nt_nat @ A )
= ( euclid3398187327856392827nt_nat @ B ) )
=> ( ( ( euclid4777050414544973029ze_nat @ A )
= ( euclid4777050414544973029ze_nat @ B ) )
=> ( A = B ) ) ) ).
% division_segment_eq_iff
thf(fact_794_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_795_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_796_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_797_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_798_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_799_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_800_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_801_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_802_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_803_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_804_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_805_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_806_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_807_add__nonneg__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_808_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_809_add__nonpos__eq__0__iff,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X4 @ Y )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_810_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_811_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_812_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_813_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_814_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_815_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_816_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_817_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_818_add__less__zeroD,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X4 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X4 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_819_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_820_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_821_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_822_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_823_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_824_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_825_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_826_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_827_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_828_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_829_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_830_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_831_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_832_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_833_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_834_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_835_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_836_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_837_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_838_of__int__mask__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_int_mask_eq
thf(fact_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_848_add__strict__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_849_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_850_add__strict__increasing2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_851_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_852_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_853_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_854_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_855_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_856_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 )
= ( ord_less_int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_857_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_858_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_859_of__nat__less__of__int__iff,axiom,
! [N: nat,X4: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X4 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).
% of_nat_less_of_int_iff
thf(fact_860_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_861_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_862_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_863_union__eq__empty,axiom,
! [M: multiset_b,N5: multiset_b] :
( ( ( plus_plus_multiset_b @ M @ N5 )
= zero_zero_multiset_b )
= ( ( M = zero_zero_multiset_b )
& ( N5 = zero_zero_multiset_b ) ) ) ).
% union_eq_empty
thf(fact_864_empty__eq__union,axiom,
! [M: multiset_b,N5: multiset_b] :
( ( zero_zero_multiset_b
= ( plus_plus_multiset_b @ M @ N5 ) )
= ( ( M = zero_zero_multiset_b )
& ( N5 = zero_zero_multiset_b ) ) ) ).
% empty_eq_union
thf(fact_865_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X4: multiset_b,Y: multiset_b] :
( ( zero_zero_multiset_b
= ( plus_plus_multiset_b @ X4 @ Y ) )
= ( ( X4 = zero_zero_multiset_b )
& ( Y = zero_zero_multiset_b ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_866_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X4: multiset_b,Y: multiset_b] :
( ( ( plus_plus_multiset_b @ X4 @ Y )
= zero_zero_multiset_b )
= ( ( X4 = zero_zero_multiset_b )
& ( Y = zero_zero_multiset_b ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_867_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_868_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_869_add__Suc__right,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc_right
thf(fact_870_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_871_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_872_size__union,axiom,
! [M: multiset_b,N5: multiset_b] :
( ( size_size_multiset_b @ ( plus_plus_multiset_b @ M @ N5 ) )
= ( plus_plus_nat @ ( size_size_multiset_b @ M ) @ ( size_size_multiset_b @ N5 ) ) ) ).
% size_union
thf(fact_873_size__multiset__union,axiom,
! [F: b > nat,M: multiset_b,N5: multiset_b] :
( ( size_multiset_b @ F @ ( plus_plus_multiset_b @ M @ N5 ) )
= ( plus_plus_nat @ ( size_multiset_b @ F @ M ) @ ( size_multiset_b @ F @ N5 ) ) ) ).
% size_multiset_union
thf(fact_874_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_875_union__iff,axiom,
! [A: b,A2: multiset_b,B5: multiset_b] :
( ( member_b @ A @ ( set_mset_b @ ( plus_plus_multiset_b @ A2 @ B5 ) ) )
= ( ( member_b @ A @ ( set_mset_b @ A2 ) )
| ( member_b @ A @ ( set_mset_b @ B5 ) ) ) ) ).
% union_iff
thf(fact_876_empty__neutral_I1_J,axiom,
! [X4: multiset_b] :
( ( plus_plus_multiset_b @ zero_zero_multiset_b @ X4 )
= X4 ) ).
% empty_neutral(1)
thf(fact_877_empty__neutral_I2_J,axiom,
! [X4: multiset_b] :
( ( plus_plus_multiset_b @ X4 @ zero_zero_multiset_b )
= X4 ) ).
% empty_neutral(2)
thf(fact_878_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M7: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M7 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_879_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_880_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_881_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_882_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_883_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_884_add__leD2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_885_add__leD1,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_886_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_887_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_888_add__leE,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_889_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_890_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_891_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_892_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_893_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_894_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_895_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_896_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% less_add_eq_less
thf(fact_897_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_898_add__Suc,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc
thf(fact_899_add__Suc__shift,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( plus_plus_nat @ M3 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_900_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_901_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_902_division__segment__nat__def,axiom,
( euclid3398187327856392827nt_nat
= ( ^ [N3: nat] : one_one_nat ) ) ).
% division_segment_nat_def
thf(fact_903_one__is__add,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_904_add__is__1,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_905_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_906_less__imp__Suc__add,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_907_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M7: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M7 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_908_less__add__Suc2,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M3 @ I ) ) ) ).
% less_add_Suc2
thf(fact_909_less__add__Suc1,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_910_less__natE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M3 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_911_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_912_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_913_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_914_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_915_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_916_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_917_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_918_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_919_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_920_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_921_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_922_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_923_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_924_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_925_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_926_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_927_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_928_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_929_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_930_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_931_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_932_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_933_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_934_multi__union__self__other__eq,axiom,
! [A2: multiset_b,X15: multiset_b,Y14: multiset_b] :
( ( ( plus_plus_multiset_b @ A2 @ X15 )
= ( plus_plus_multiset_b @ A2 @ Y14 ) )
=> ( X15 = Y14 ) ) ).
% multi_union_self_other_eq
thf(fact_935_union__right__cancel,axiom,
! [M: multiset_b,K4: multiset_b,N5: multiset_b] :
( ( ( plus_plus_multiset_b @ M @ K4 )
= ( plus_plus_multiset_b @ N5 @ K4 ) )
= ( M = N5 ) ) ).
% union_right_cancel
thf(fact_936_union__left__cancel,axiom,
! [K4: multiset_b,M: multiset_b,N5: multiset_b] :
( ( ( plus_plus_multiset_b @ K4 @ M )
= ( plus_plus_multiset_b @ K4 @ N5 ) )
= ( M = N5 ) ) ).
% union_left_cancel
thf(fact_937_union__commute,axiom,
( plus_plus_multiset_b
= ( ^ [M8: multiset_b,N6: multiset_b] : ( plus_plus_multiset_b @ N6 @ M8 ) ) ) ).
% union_commute
thf(fact_938_union__lcomm,axiom,
! [M: multiset_b,N5: multiset_b,K4: multiset_b] :
( ( plus_plus_multiset_b @ M @ ( plus_plus_multiset_b @ N5 @ K4 ) )
= ( plus_plus_multiset_b @ N5 @ ( plus_plus_multiset_b @ M @ K4 ) ) ) ).
% union_lcomm
thf(fact_939_union__assoc,axiom,
! [M: multiset_b,N5: multiset_b,K4: multiset_b] :
( ( plus_plus_multiset_b @ ( plus_plus_multiset_b @ M @ N5 ) @ K4 )
= ( plus_plus_multiset_b @ M @ ( plus_plus_multiset_b @ N5 @ K4 ) ) ) ).
% union_assoc
thf(fact_940_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_zdiv__mono2,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_950_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_951_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_952_zdiv__mono2__neg,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_int__div__less__self,axiom,
! [X4: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X4 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).
% int_div_less_self
thf(fact_961_of__nat__euclidean__size,axiom,
! [A: int] :
( ( semiri1314217659103216013at_int @ ( euclid4774559944035922753ze_int @ A ) )
= ( divide_divide_int @ A @ ( euclid3395696857347342551nt_int @ A ) ) ) ).
% of_nat_euclidean_size
thf(fact_962_of__nat__euclidean__size,axiom,
! [A: nat] :
( ( semiri1316708129612266289at_nat @ ( euclid4777050414544973029ze_nat @ A ) )
= ( divide_divide_nat @ A @ ( euclid3398187327856392827nt_nat @ A ) ) ) ).
% of_nat_euclidean_size
thf(fact_963_verit__less__mono__div__int2,axiom,
! [A2: int,B5: int,N: int] :
( ( ord_less_eq_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_964_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_965_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( euclid3395696857347342551nt_int @ A )
= ( euclid3395696857347342551nt_int @ B ) )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ B ) )
| ( B = zero_zero_int ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_966_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( euclid3398187327856392827nt_nat @ A )
= ( euclid3398187327856392827nt_nat @ B ) )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) )
| ( B = zero_zero_nat ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_967_subset__mset_Osum__mset__0__iff,axiom,
! [M: multiset_multiset_b] :
( ( ( comm_m1977238987320879926iset_b @ plus_plus_multiset_b @ zero_zero_multiset_b @ M )
= zero_zero_multiset_b )
= ( ! [X: multiset_b] :
( ( member_multiset_b @ X @ ( set_mset_multiset_b @ M ) )
=> ( X = zero_zero_multiset_b ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_968_int__power__div__base,axiom,
! [M3: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M3 ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_969_div__bounded,axiom,
! [B: int,R2: int,Q4: int] :
( ( B != zero_zero_int )
=> ( ( ( euclid3395696857347342551nt_int @ R2 )
= ( euclid3395696857347342551nt_int @ B ) )
=> ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R2 ) @ ( euclid4774559944035922753ze_int @ B ) )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ Q4 @ B ) @ R2 ) @ B )
= Q4 ) ) ) ) ).
% div_bounded
thf(fact_970_div__bounded,axiom,
! [B: nat,R2: nat,Q4: nat] :
( ( B != zero_zero_nat )
=> ( ( ( euclid3398187327856392827nt_nat @ R2 )
= ( euclid3398187327856392827nt_nat @ B ) )
=> ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R2 ) @ ( euclid4777050414544973029ze_nat @ B ) )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ Q4 @ B ) @ R2 ) @ B )
= Q4 ) ) ) ) ).
% div_bounded
thf(fact_971_mult__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_972_mult__0__right,axiom,
! [M3: nat] :
( ( times_times_nat @ M3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_973_mult__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N ) )
= ( ( M3 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_974_mult__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M3 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M3 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_975_nat__1__eq__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N ) )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_976_nat__mult__eq__1__iff,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_977_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_978_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_979_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_980_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_981_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_982_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_983_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_984_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_985_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_986_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_987_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_988_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_989_zero__diff,axiom,
! [A: multiset_b] :
( ( minus_3765977311343717292iset_b @ zero_zero_multiset_b @ A )
= zero_zero_multiset_b ) ).
% zero_diff
thf(fact_990_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_991_diff__zero,axiom,
! [A: multiset_b] :
( ( minus_3765977311343717292iset_b @ A @ zero_zero_multiset_b )
= A ) ).
% diff_zero
thf(fact_992_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_993_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_994_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_b] :
( ( minus_3765977311343717292iset_b @ A @ A )
= zero_zero_multiset_b ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_995_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_996_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_997_add__diff__cancel__right_H,axiom,
! [A: multiset_b,B: multiset_b] :
( ( minus_3765977311343717292iset_b @ ( plus_plus_multiset_b @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_998_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_999_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_1000_add__diff__cancel__right,axiom,
! [A: multiset_b,C2: multiset_b,B: multiset_b] :
( ( minus_3765977311343717292iset_b @ ( plus_plus_multiset_b @ A @ C2 ) @ ( plus_plus_multiset_b @ B @ C2 ) )
= ( minus_3765977311343717292iset_b @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1001_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1002_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1003_add__diff__cancel__left_H,axiom,
! [A: multiset_b,B: multiset_b] :
( ( minus_3765977311343717292iset_b @ ( plus_plus_multiset_b @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1004_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_1005_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_1006_add__diff__cancel__left,axiom,
! [C2: multiset_b,A: multiset_b,B: multiset_b] :
( ( minus_3765977311343717292iset_b @ ( plus_plus_multiset_b @ C2 @ A ) @ ( plus_plus_multiset_b @ C2 @ B ) )
= ( minus_3765977311343717292iset_b @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1007_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1008_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1009_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1010_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1011_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1012_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1013_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1014_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1015_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_1016_div__by__Suc__0,axiom,
! [M3: nat] :
( ( divide_divide_nat @ M3 @ ( suc @ zero_zero_nat ) )
= M3 ) ).
% div_by_Suc_0
thf(fact_1017_one__eq__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M3 @ N ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1018_mult__eq__1__iff,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1019_div__mult__self1__is__m,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M3 ) @ N )
= M3 ) ) ).
% div_mult_self1_is_m
thf(fact_1020_div__mult__self__is__m,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M3 @ N ) @ N )
= M3 ) ) ).
% div_mult_self_is_m
thf(fact_1021_div__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( divide_divide_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1022_mult__less__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1023_nat__0__less__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1024_of__nat__mult,axiom,
! [M3: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M3 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1025_of__nat__mult,axiom,
! [M3: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M3 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1026_mult__Suc__right,axiom,
! [M3: nat,N: nat] :
( ( times_times_nat @ M3 @ ( suc @ N ) )
= ( plus_plus_nat @ M3 @ ( times_times_nat @ M3 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1027_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1028_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1029_Suc__diff__diff,axiom,
! [M3: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1030_diff__Suc__Suc,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1031_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1032_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1033_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_1034_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_1035_diff__add__zero,axiom,
! [A: multiset_b,B: multiset_b] :
( ( minus_3765977311343717292iset_b @ A @ ( plus_plus_multiset_b @ A @ B ) )
= zero_zero_multiset_b ) ).
% diff_add_zero
thf(fact_1036_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_1037_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1038_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1039_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1040_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1041_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1042_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1043_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_1044_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_1045_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_1046_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_1047_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_1048_div__mult__mult1,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1049_div__mult__mult1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1050_div__mult__mult2,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1051_div__mult__mult2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1052_div__mult__mult1__if,axiom,
! [C2: int,A: int,B: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1053_div__mult__mult1__if,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1054_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_1055_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_1056_one__le__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M3 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1057_mult__le__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1058_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_1059_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1060_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_1061_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1062_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1063_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1064_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1065_div__mult__self4,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1066_div__mult__self4,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1067_div__mult__self3,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1068_div__mult__self3,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1069_div__mult__self2,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1070_div__mult__self2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1071_div__mult__self1,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1072_div__mult__self1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1073_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_1074_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_1075_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1076_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1077_division__segment__euclidean__size,axiom,
! [A: int] :
( ( times_times_int @ ( euclid3395696857347342551nt_int @ A ) @ ( semiri1314217659103216013at_int @ ( euclid4774559944035922753ze_int @ A ) ) )
= A ) ).
% division_segment_euclidean_size
thf(fact_1078_division__segment__euclidean__size,axiom,
! [A: nat] :
( ( times_times_nat @ ( euclid3398187327856392827nt_nat @ A ) @ ( semiri1316708129612266289at_nat @ ( euclid4777050414544973029ze_nat @ A ) ) )
= A ) ).
% division_segment_euclidean_size
thf(fact_1079_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_1080_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_1081_add__mult__distrib2,axiom,
! [K: nat,M3: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M3 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1082_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_1083_add__mult__distrib,axiom,
! [M3: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M3 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1084_diff__diff__eq,axiom,
! [A: multiset_b,B: multiset_b,C2: multiset_b] :
( ( minus_3765977311343717292iset_b @ ( minus_3765977311343717292iset_b @ A @ B ) @ C2 )
= ( minus_3765977311343717292iset_b @ A @ ( plus_plus_multiset_b @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1085_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1086_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1087_diff__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_1088_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1089_add__implies__diff,axiom,
! [C2: multiset_b,B: multiset_b,A: multiset_b] :
( ( ( plus_plus_multiset_b @ C2 @ B )
= A )
=> ( C2
= ( minus_3765977311343717292iset_b @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1090_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1091_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1092_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1093_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_1094_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1095_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_1096_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_1097_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_1098_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1099_dividend__less__times__div,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M3 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M3 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1100_dividend__less__div__times,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M3 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1101_split__div,axiom,
! [P: nat > $o,M3: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M3 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M3
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_div
thf(fact_1102_less__add__iff1,axiom,
! [A: int,E: int,C2: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_1103_less__add__iff2,axiom,
! [A: int,E: int,C2: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_1104_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M7: nat,N3: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M7 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1105_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_eq_nat @ M3 @ ( divide_divide_nat @ N @ Q4 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M3 @ Q4 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1106_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M3 )
=> ( ( ord_less_nat @ M3 @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
=> ( ( divide_divide_nat @ M3 @ N )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_1107_iterate__add__add__eq1,axiom,
! [J: nat,I: nat,U: multiset_b,M3: multiset_b,N: multiset_b] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_3765977311343717292iset_b @ ( plus_plus_multiset_b @ ( iterat743893162068676942iset_b @ I @ U ) @ M3 ) @ ( plus_plus_multiset_b @ ( iterat743893162068676942iset_b @ J @ U ) @ N ) )
= ( minus_3765977311343717292iset_b @ ( plus_plus_multiset_b @ ( iterat743893162068676942iset_b @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N ) ) ) ).
% iterate_add_add_eq1
thf(fact_1108_iterate__add__add__eq1,axiom,
! [J: nat,I: nat,U: int,M3: int,N: int] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_int @ ( plus_plus_int @ ( iterate_add_int @ I @ U ) @ M3 ) @ ( plus_plus_int @ ( iterate_add_int @ J @ U ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( iterate_add_int @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N ) ) ) ).
% iterate_add_add_eq1
thf(fact_1109_iterate__add__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( iterate_add_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( iterate_add_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( iterate_add_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N ) ) ) ).
% iterate_add_add_eq1
thf(fact_1110_iterate__add__diff__add__eq2,axiom,
! [I: nat,J: nat,U: multiset_b,M3: multiset_b,N: multiset_b] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_3765977311343717292iset_b @ ( plus_plus_multiset_b @ ( iterat743893162068676942iset_b @ I @ U ) @ M3 ) @ ( plus_plus_multiset_b @ ( iterat743893162068676942iset_b @ J @ U ) @ N ) )
= ( minus_3765977311343717292iset_b @ M3 @ ( plus_plus_multiset_b @ ( iterat743893162068676942iset_b @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% iterate_add_diff_add_eq2
thf(fact_1111_iterate__add__diff__add__eq2,axiom,
! [I: nat,J: nat,U: int,M3: int,N: int] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_int @ ( plus_plus_int @ ( iterate_add_int @ I @ U ) @ M3 ) @ ( plus_plus_int @ ( iterate_add_int @ J @ U ) @ N ) )
= ( minus_minus_int @ M3 @ ( plus_plus_int @ ( iterate_add_int @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% iterate_add_diff_add_eq2
thf(fact_1112_iterate__add__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( iterate_add_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( iterate_add_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M3 @ ( plus_plus_nat @ ( iterate_add_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% iterate_add_diff_add_eq2
thf(fact_1113_div__less__iff__less__mult,axiom,
! [Q4: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M3 @ Q4 ) @ N )
= ( ord_less_nat @ M3 @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1114_times__div__less__eq__dividend,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M3 @ N ) ) @ M3 ) ).
% times_div_less_eq_dividend
thf(fact_1115_div__times__less__eq__dividend,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N ) @ N ) @ M3 ) ).
% div_times_less_eq_dividend
thf(fact_1116_less__mult__imp__div__less,axiom,
! [M3: nat,I: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M3 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1117_div__le__dividend,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M3 @ N ) @ M3 ) ).
% div_le_dividend
thf(fact_1118_div__le__mono,axiom,
! [M3: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M3 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1119_div__if,axiom,
( divide_divide_nat
= ( ^ [M7: nat,N3: nat] :
( if_nat
@ ( ( ord_less_nat @ M7 @ N3 )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M7 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_1120_le__div__geq,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( ( divide_divide_nat @ M3 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1121_split__div_H,axiom,
! [P: nat > $o,M3: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M3 @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q5: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M3 )
& ( ord_less_nat @ M3 @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1122_power__eq__if,axiom,
( power_power_nat
= ( ^ [P5: nat,M7: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1123_power__eq__if,axiom,
( power_power_int
= ( ^ [P5: int,M7: nat] : ( if_int @ ( M7 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M7 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1124_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1125_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1126_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1127_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1128_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1129_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_1130_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1131_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1132_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1133_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1134_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1135_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1136_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1137_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1138_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_1139_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_1140_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_1141_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_1142_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1143_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1144_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B2: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1145_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B2: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1146_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1147_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_1148_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1149_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_mult__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_1163_mult__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_1164_mult__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1165_mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1166_mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1167_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_1168_mult__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1169_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_1170_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1171_mult__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1172_mult__mono_H,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1173_mult__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1174_mult__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1175_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_1176_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1177_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_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_mult__less__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1192_mult__less__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1193_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1194_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1195_mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1196_mult__less__cancel__left__disj,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1197_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1198_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1199_mult__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1200_mult__less__cancel__right__disj,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1201_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1202_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1203_sum__squares__eq__zero__iff,axiom,
! [X4: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1204_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C2: nat,D2: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C2 != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1205_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C2: int,D2: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C2 != D2 ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C2 ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1206_less__1__mult,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M3 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M3 @ N ) ) ) ) ).
% less_1_mult
thf(fact_1207_less__1__mult,axiom,
! [M3: int,N: int] :
( ( ord_less_int @ one_one_int @ M3 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M3 @ N ) ) ) ) ).
% less_1_mult
thf(fact_1208_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M3: nat,N: nat] :
( ( ( divide_divide_nat @ M3 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M3 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1209_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_1210_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1211_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1212_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_1213_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_1214_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_1215_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_1216_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_1217_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_1218_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1219_mult__left__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1220_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1221_mult__right__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1222_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1223_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1224_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1225_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1226_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1227_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_1228_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_1229_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1230_diff__less__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1231_eq__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1232_le__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1233_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1234_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1235_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_1236_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1237_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1238_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1239_Suc__mult__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M3 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M3 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1240_mult__of__nat__commute,axiom,
! [X4: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X4 ) ) ) ).
% mult_of_nat_commute
thf(fact_1241_mult__of__nat__commute,axiom,
! [X4: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X4 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X4 ) ) ) ).
% mult_of_nat_commute
thf(fact_1242_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $true @ X4 @ Y )
= X4 ) ).
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,
! [X4: nat,Y: nat] :
( ( if_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
x = y ).
thf(conj_1,hypothesis,
! [X9: b] :
( ( member_b @ X9 @ y )
=> ( ( p @ X9 )
= ( q @ X9 ) ) ) ).
thf(conj_2,conjecture,
( ( ? [X: b] :
( ( member_b @ X @ x )
& ( p @ X )
& ! [Y13: b] :
( ( ( member_b @ Y13 @ x )
& ( p @ Y13 ) )
=> ( Y13 = X ) ) ) )
!= ( ~ ? [X: b] :
( ( member_b @ X @ y )
& ( q @ X )
& ! [Y13: b] :
( ( ( member_b @ Y13 @ y )
& ( q @ Y13 ) )
=> ( Y13 = X ) ) ) ) ) ).
%------------------------------------------------------------------------------