Documentation Verification Report

ModEq

πŸ“ Source: Mathlib/Data/Nat/ModEq.lean

Statistics

MetricCount
DefinitionsModEq, instTrans, chineseRemainder, chineseRemainder', instDecidableModEq, Β«term_≑_[MOD_]Β»
6
TheoremsnatCast, modEq_iff_natModEq, natCast_modEq_natCast, modEq_zero_nat, zero_modEq_nat, add, add_iff_left, add_iff_right, add_le_of_lt, add_left, add_left_cancel, add_left_cancel', add_right, add_right_cancel, add_right_cancel', cancel_left_div_gcd, cancel_left_div_gcd', cancel_left_of_coprime, cancel_right_div_gcd, cancel_right_div_gcd', cancel_right_of_coprime, comm, dvd, dvd_iff, eq_of_abs_lt, eq_of_lt_of_lt, gcd_eq, instRefl, le_of_lt_add, modulus_mul_add, mul, mul_left, mul_left', mul_left_cancel', mul_left_cancel_iff', mul_right, mul_right', mul_right_cancel', mul_right_cancel_iff', of_div, of_dvd, of_mul_left, of_mul_right, of_natCast, pow, refl, rfl, self_mul_add, sub, sub', sub_left, sub_left', sub_right, sub_right', trans, add_div_eq_of_add_mod_lt, add_div_eq_of_le_mod_add_mod, add_div_le_add_div, add_div_of_dvd_left, add_div_of_dvd_right, add_modEq_left, add_modEq_left_iff, add_modEq_right, add_modEq_right_iff, add_mod_add_ite, add_mod_add_of_le_add_mod, add_mod_of_add_mod_lt, add_modulus_modEq_iff, add_modulus_mul_modEq_iff, add_mul_modulus_modEq_iff, chineseRemainder'_lt_lcm, chineseRemainder_lt_mul, chineseRemainder_modEq_unique, coprime_of_mul_modEq_one, ext_div_modEq, ext_div_modEq_iff, le_mod_add_mod_of_dvd_add_of_not_dvd, left_modEq_add_iff, modEq_add_modulus_iff, modEq_add_modulus_mul_iff, modEq_add_mul_modulus_iff, modEq_and_modEq_iff_modEq_mul, modEq_iff_dvd, modEq_iff_dvd', modEq_iff_eq_of_div_eq, modEq_iff_exists_eq_add, modEq_modulus_add_iff, modEq_modulus_mul_add_iff, modEq_mul_modulus_add_iff, modEq_of_dvd, modEq_one, modEq_sub, modEq_sub_modulus_iff, modEq_zero_iff, modEq_zero_iff_dvd, mod_eq_of_modEq, mod_lcm, mod_modEq, mod_sub_of_le, modulus_add_modEq_iff, modulus_modEq_zero, modulus_mul_add_modEq_iff, mul_modulus_add_modEq_iff, odd_mod_four_iff, odd_mul_odd, odd_mul_odd_div_two, odd_of_mod_four_eq_one, odd_of_mod_four_eq_three, right_modEq_add_iff, sub_modulus_modEq_iff
110
Total116

AddCommGroup

Theorems

NameKindAssumesProvesValidatesDepends On
modEq_iff_natModEq πŸ“–mathematicalβ€”ModEq
Nat.instAddCommMonoid
Nat.ModEq
β€”modEq_iff_nsmul
Nat.ModEq.eq_1
nsmul_eq_mul
Nat.nsmul_eq_mul
ModEq.trans
nsmul_add_modEq
ModEq.symm
natCast_modEq_natCast πŸ“–mathematicalβ€”ModEq
AddCommMonoidWithOne.toAddCommMonoid
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
Nat.ModEq
β€”map_modEq_iff
AddMonoidHom.instAddMonoidHomClass
Nat.cast_injective

AddCommGroup.ModEq

Theorems

NameKindAssumesProvesValidatesDepends On
natCast πŸ“–mathematicalNat.ModEqAddCommGroup.ModEq
AddCommMonoidWithOne.toAddCommMonoid
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
β€”map
AddMonoidHom.instAddMonoidHomClass
AddCommGroup.modEq_iff_natModEq

Dvd.dvd

Theorems

NameKindAssumesProvesValidatesDepends On
modEq_zero_nat πŸ“–mathematicalβ€”Nat.ModEqβ€”Nat.modEq_zero_iff_dvd
zero_modEq_nat πŸ“–mathematicalβ€”Nat.ModEqβ€”Nat.ModEq.symm
modEq_zero_nat

Nat

Definitions

NameCategoryTheorems
ModEq πŸ“–MathDef
105 mathmath: modEq_one, add_modulus_mul_modEq_iff, modulus_mul_add_modEq_iff, infinite_setOf_prime_modEq_one, ModEq.instRefl, Ioc_filter_modEq_cast, Pell.xn_modEq_x4n_add, ModEq.mul_right_cancel_iff', mod_modEq, ModEq.refl, pow_pow_modEq_one, modEq_digits_sum, frequently_atTop_modEq_one, modEq_mul_modulus_add_iff, exists_lt_modEq_of_infinite, modEq_list_map_prod_iff, Dvd.dvd.modEq_zero_nat, Dioph.modEq_dioph, add_modEq_left, nsmul_eq_nsmul_iff_modEq, add_modulus_modEq_iff, ModEq.pow_card_sub_one_eq_one, ext_div_modEq_iff, Int.natCast_modEq_iff, modEq_iff_dvd, Equiv.Perm.card_fixedPoints_modEq, AddCommGroup.modEq_iff_natModEq, frequently_atTop_prime_and_modEq, mul_modulus_add_modEq_iff, ModEq.modulus_mul_add, Pell.xy_modEq_yn, Ioc_filter_modEq_card, AddCommGroup.natCast_modEq_natCast, pow_eq_pow_iff_modEq, probablePrime_iff_modEq, IsPGroup.card_modEq_card_fixedPoints, IsOfFinAddOrder.nsmul_eq_nsmul_iff_modEq, modEq_iff_dvd', forall_exists_prime_gt_and_modEq, Ico_filter_modEq_card, left_modEq_add_iff, Equiv.Perm.card_compl_support_modEq, Pell.xn_modEq_x2n_sub, ModEq.pow_totient, modEq_sub, count_modEq_card, Ico_filter_modEq_cast, Pell.matiyasevic, add_modEq_left_iff, Choose.lucas_theorem_nat, chineseRemainder_lt_mul, ModEq.mul_left_cancel_iff', ModEq.rfl, add_modEq_right, Pell.xn_modEq_x2n_add, modEq_iff_eq_of_div_eq, ModEq.comm, CharP.natCast_eq_natCast, Equiv.Perm.IsCycleOn.pow_apply_eq_pow_apply, nsmul_eq_zero_iff_modEq, infinite_setOf_prime_and_modEq, modEq_list_prod_iff, modEq_modulus_add_iff, modEq_iff_exists_eq_add, exists_prime_gt_modEq_one, right_modEq_add_iff, card_sylow_modEq_one, add_mul_modulus_modEq_iff, modEq_of_dvd, ModEq.of_natCast, Sylow.card_quotient_normalizer_modEq_card_quotient, Pell.xn_modEq_x4n_sub, modulus_modEq_zero, modEq_zero_iff, IsOfFinOrder.pow_eq_pow_iff_modEq, modEq_add_modulus_mul_iff, schnirelmannDensity_setOf_modeq_one, modEq_mersenne, modEq_add_modulus_iff, Pell.yn_modEq_two, ZMod.natCast_eq_natCast_iff, modulus_add_modEq_iff, add_modEq_right_iff, modEq_three_digits_sum, sq_add_sq_modEq, count_modEq_card_eq_ceil, Choose.choose_modEq_choose_mod_mul_choose_div_nat, modEq_zero_iff_dvd, ofDigits_modEq, modEq_nine_digits_sum, modEq_and_modEq_iff_modEq_mul, Pell.xn_modEq_x2n_sub_lem, Dvd.dvd.zero_modEq_nat, Choose.choose_modEq_prod_range_choose_nat, ModEq.self_mul_add, modEq_modulus_mul_add_iff, Pell.yn_modEq_a_sub_one, ZMod.eisenstein_lemma_aux, modEq_sub_modulus_iff, Pell.eq_pow_of_pell, frequently_modEq, Sylow.card_normalizer_modEq_card, sub_modulus_modEq_iff, pow_eq_one_iff_modEq, modEq_add_mul_modulus_iff
chineseRemainder πŸ“–CompOp
2 mathmath: chineseRemainder_modEq_unique, chineseRemainder_lt_mul
chineseRemainder' πŸ“–CompOp
1 mathmath: chineseRemainder'_lt_lcm
instDecidableModEq πŸ“–CompOp
7 mathmath: Ioc_filter_modEq_cast, Ioc_filter_modEq_card, Ico_filter_modEq_card, count_modEq_card, Ico_filter_modEq_cast, schnirelmannDensity_setOf_modeq_one, count_modEq_card_eq_ceil

Theorems

NameKindAssumesProvesValidatesDepends On
add_div_eq_of_add_mod_lt πŸ“–β€”β€”β€”β€”add_zero
not_le_of_gt
add_div_eq_of_le_mod_add_mod πŸ“–β€”β€”β€”β€”β€”
add_div_le_add_div πŸ“–β€”β€”β€”β€”add_zero
add_div_of_dvd_left πŸ“–β€”β€”β€”β€”add_comm
add_div_of_dvd_right
add_div_of_dvd_right πŸ“–β€”β€”β€”β€”add_zero
add_div_eq_of_add_mod_lt
zero_add
add_modEq_left πŸ“–mathematicalβ€”ModEqβ€”β€”
add_modEq_left_iff πŸ“–mathematicalβ€”ModEqβ€”sub_add_cancel_left
add_modEq_right πŸ“–mathematicalβ€”ModEqβ€”β€”
add_modEq_right_iff πŸ“–mathematicalβ€”ModEqβ€”add_comm
add_modEq_left_iff
add_mod_add_ite πŸ“–β€”β€”β€”β€”ModEq.symm
ModEq.add
mod_modEq
add_zero
instAtLeastTwoHAddOfNat
mul_two
add_right_cancel_iff
AddRightCancelSemigroup.toIsRightCancelAdd
add_comm
add_assoc
le_antisymm
mul_one
lt_of_not_ge
add_mod_add_of_le_add_mod πŸ“–β€”β€”β€”β€”add_mod_add_ite
add_mod_of_add_mod_lt πŸ“–β€”β€”β€”β€”add_mod_add_ite
not_le_of_gt
add_zero
add_modulus_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”β€”
add_modulus_mul_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”β€”
add_mul_modulus_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”β€”
chineseRemainder'_lt_lcm πŸ“–mathematicalModEqchineseRemainder'β€”xgcd_val
chineseRemainder_lt_mul πŸ“–mathematicalβ€”ModEq
chineseRemainder
β€”lt_of_lt_of_le
chineseRemainder'_lt_lcm
le_of_eq
Coprime.lcm_eq_mul
chineseRemainder_modEq_unique πŸ“–mathematicalModEqchineseRemainderβ€”Coprime.lcm_eq_mul
mod_lcm
ModEq.trans
ModEq.symm
Subtype.prop
coprime_of_mul_modEq_one πŸ“–β€”ModEqβ€”β€”modEq_zero_iff_dvd
ModEq.of_mul_right
ModEq.symm
ModEq.mul_right
MulZeroClass.zero_mul
ext_div_modEq πŸ“–β€”ModEqβ€”β€”ext_div_mod
ext_div_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”ext_div_mod_iff
le_mod_add_mod_of_dvd_add_of_not_dvd πŸ“–β€”β€”β€”β€”by_contradiction
add_mod_of_add_mod_lt
lt_of_not_ge
left_modEq_add_iff πŸ“–mathematicalβ€”ModEqβ€”ModEq.comm
add_modEq_left_iff
modEq_add_modulus_iff πŸ“–mathematicalβ€”ModEqβ€”β€”
modEq_add_modulus_mul_iff πŸ“–mathematicalβ€”ModEqβ€”β€”
modEq_add_mul_modulus_iff πŸ“–mathematicalβ€”ModEqβ€”β€”
modEq_and_modEq_iff_modEq_mul πŸ“–mathematicalβ€”ModEqβ€”modEq_iff_dvd
ModEq.of_mul_right
ModEq.of_mul_left
modEq_iff_dvd πŸ“–mathematicalβ€”ModEqβ€”ModEq.eq_1
Int.natCast_mod
modEq_iff_dvd' πŸ“–mathematicalβ€”ModEqβ€”modEq_iff_dvd
modEq_iff_eq_of_div_eq πŸ“–mathematicalβ€”ModEqβ€”β€”
modEq_iff_exists_eq_add πŸ“–mathematicalβ€”ModEqβ€”modEq_iff_dvd'
dvd_def
modEq_modulus_add_iff πŸ“–mathematicalβ€”ModEqβ€”β€”
modEq_modulus_mul_add_iff πŸ“–mathematicalβ€”ModEqβ€”add_comm
modEq_add_modulus_mul_iff
modEq_mul_modulus_add_iff πŸ“–mathematicalβ€”ModEqβ€”add_comm
modEq_add_mul_modulus_iff
modEq_of_dvd πŸ“–mathematicalβ€”ModEqβ€”modEq_iff_dvd
modEq_one πŸ“–mathematicalβ€”ModEqβ€”modEq_of_dvd
one_dvd
modEq_sub πŸ“–mathematicalβ€”ModEqβ€”ModEq.symm
modEq_of_dvd
dvd_refl
modEq_sub_modulus_iff πŸ“–mathematicalβ€”ModEqβ€”modEq_add_modulus_iff
modEq_zero_iff πŸ“–mathematicalβ€”ModEqβ€”ModEq.eq_1
modEq_zero_iff_dvd πŸ“–mathematicalβ€”ModEqβ€”ModEq.eq_1
mod_eq_of_modEq πŸ“–β€”ModEqβ€”β€”β€”
mod_lcm πŸ“–β€”ModEqβ€”β€”modEq_iff_dvd
mod_modEq πŸ“–mathematicalβ€”ModEqβ€”β€”
mod_sub_of_le πŸ“–β€”β€”β€”β€”LE.le.trans_lt
modulus_add_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”add_comm
add_modulus_modEq_iff
modulus_modEq_zero πŸ“–mathematicalβ€”ModEqβ€”β€”
modulus_mul_add_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”add_comm
add_modulus_mul_modEq_iff
mul_modulus_add_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”add_comm
add_mul_modulus_modEq_iff
odd_mod_four_iff πŸ“–β€”β€”β€”β€”odd_of_mod_four_eq_one
odd_of_mod_four_eq_three
odd_mul_odd πŸ“–β€”β€”β€”β€”mul_one
ModEq.mul
odd_mul_odd_div_two πŸ“–β€”β€”β€”β€”mul_right_injectiveβ‚€
IsCancelMulZero.toIsLeftCancelMulZero
instIsCancelMulZero
two_ne_zero
mul_add
Distrib.leftDistribClass
two_mul_odd_div_two
mul_left_comm
odd_mul_odd
mul_one
odd_of_mod_four_eq_one πŸ“–β€”β€”β€”β€”ModEq.of_mul_left
odd_of_mod_four_eq_three πŸ“–β€”β€”β€”β€”ModEq.of_mul_left
right_modEq_add_iff πŸ“–mathematicalβ€”ModEqβ€”ModEq.comm
add_modEq_right_iff
sub_modulus_modEq_iff πŸ“–mathematicalβ€”ModEqβ€”add_modulus_modEq_iff

Nat.ModEq

Definitions

NameCategoryTheorems
instTrans πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
add πŸ“–β€”Nat.ModEqβ€”β€”Nat.modEq_iff_dvd
add_sub_add_comm
dvd
add_iff_left πŸ“–β€”Nat.ModEqβ€”β€”add_left_cancel
add
add_iff_right πŸ“–β€”Nat.ModEqβ€”β€”add_right_cancel
add
add_le_of_lt πŸ“–β€”Nat.ModEqβ€”β€”le_of_lt_add
trans
Nat.add_modEq_right
add_left πŸ“–β€”Nat.ModEqβ€”β€”add
rfl
add_left_cancel πŸ“–β€”Nat.ModEqβ€”β€”add_sub_cancel_left
add_sub_add_comm
add_left_cancel' πŸ“–β€”Nat.ModEqβ€”β€”add_left_cancel
rfl
add_right πŸ“–β€”Nat.ModEqβ€”β€”add
rfl
add_right_cancel πŸ“–β€”Nat.ModEqβ€”β€”add_left_cancel
add_comm
add_right_cancel' πŸ“–β€”Nat.ModEqβ€”β€”add_right_cancel
rfl
cancel_left_div_gcd πŸ“–β€”Nat.ModEqβ€”β€”Nat.modEq_iff_dvd
Int.dvd_of_dvd_mul_right_of_gcd_one
mul_comm
cancel_left_div_gcd' πŸ“–β€”Nat.ModEqβ€”β€”cancel_left_div_gcd
trans
mul_right
symm
cancel_left_of_coprime πŸ“–β€”Nat.ModEqβ€”β€”one_mul
cancel_left_div_gcd
cancel_right_div_gcd πŸ“–β€”Nat.ModEqβ€”β€”cancel_left_div_gcd
mul_comm
cancel_right_div_gcd' πŸ“–β€”Nat.ModEqβ€”β€”cancel_right_div_gcd
trans
mul_left
symm
cancel_right_of_coprime πŸ“–β€”Nat.ModEqβ€”β€”cancel_left_of_coprime
mul_comm
comm πŸ“–mathematicalβ€”Nat.ModEqβ€”symm
dvd πŸ“–β€”Nat.ModEqβ€”β€”Nat.modEq_iff_dvd
dvd_iff πŸ“–β€”Nat.ModEqβ€”β€”of_dvd
trans
symm
eq_of_abs_lt πŸ“–β€”Nat.ModEq
abs
instLatticeInt
Int.instAddGroup
β€”β€”sub_eq_zero
Int.eq_zero_of_abs_lt_dvd
dvd
eq_of_lt_of_lt πŸ“–β€”Nat.ModEqβ€”β€”eq_of_abs_lt
Int.abs_sub_lt_of_lt_lt
gcd_eq πŸ“–β€”Nat.ModEqβ€”β€”dvd_antisymm
Nat.instIsCancelMulZero
Unique.instSubsingleton
dvd_iff
instRefl πŸ“–mathematicalβ€”Nat.ModEqβ€”refl
le_of_lt_add πŸ“–β€”Nat.ModEqβ€”β€”le_total
Nat.modEq_iff_dvd'
symm
modulus_mul_add πŸ“–mathematicalβ€”Nat.ModEqβ€”β€”
mul πŸ“–β€”Nat.ModEqβ€”β€”trans
mul_left
mul_right
mul_left πŸ“–β€”Nat.ModEqβ€”β€”of_dvd
dvd_mul_left
mul_left'
mul_left' πŸ“–β€”Nat.ModEqβ€”β€”β€”
mul_left_cancel' πŸ“–β€”Nat.ModEqβ€”β€”β€”
mul_left_cancel_iff' πŸ“–mathematicalβ€”Nat.ModEqβ€”mul_left_cancel'
mul_left'
mul_right πŸ“–β€”Nat.ModEqβ€”β€”mul_comm
mul_left
mul_right' πŸ“–β€”Nat.ModEqβ€”β€”mul_comm
mul_left'
mul_right_cancel' πŸ“–β€”Nat.ModEqβ€”β€”β€”
mul_right_cancel_iff' πŸ“–mathematicalβ€”Nat.ModEqβ€”mul_right_cancel'
mul_right'
of_div πŸ“–β€”Nat.ModEqβ€”β€”mul_left'
of_dvd πŸ“–β€”Nat.ModEqβ€”β€”Nat.modEq_of_dvd
Dvd.dvd.trans
dvd
of_mul_left πŸ“–β€”Nat.ModEqβ€”β€”Nat.modEq_iff_dvd
Dvd.dvd.trans
dvd_mul_left
of_mul_right πŸ“–β€”Nat.ModEqβ€”β€”of_mul_left
mul_comm
of_natCast πŸ“–mathematicalAddCommGroup.ModEq
AddCommMonoidWithOne.toAddCommMonoid
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
Nat.ModEqβ€”AddCommGroup.natCast_modEq_natCast
pow πŸ“–mathematicalNat.ModEqMonoid.toNatPow
Nat.instMonoid
β€”mul
refl πŸ“–mathematicalβ€”Nat.ModEqβ€”β€”
rfl πŸ“–mathematicalβ€”Nat.ModEqβ€”refl
self_mul_add πŸ“–mathematicalβ€”Nat.ModEqβ€”modulus_mul_add
sub πŸ“–β€”Nat.ModEqβ€”β€”sub'
sub' πŸ“–β€”Nat.ModEqβ€”β€”lt_or_ge
LT.lt.le
lt_iff_lt_of_le_iff_le
refl
Nat.modEq_iff_dvd
sub_sub_sub_comm
dvd
sub_left πŸ“–β€”Nat.ModEqβ€”β€”sub
rfl
sub_left' πŸ“–β€”Nat.ModEqβ€”β€”sub'
rfl
sub_right πŸ“–β€”Nat.ModEqβ€”β€”sub
rfl
sub_right' πŸ“–β€”Nat.ModEqβ€”β€”sub'
rfl
trans πŸ“–β€”Nat.ModEqβ€”β€”β€”

(root)

Definitions

NameCategoryTheorems
Β«term_≑_[MOD_]Β» πŸ“–CompOpβ€”

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