statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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normalize : α →*₀ α | { to_fun := λ x, x * norm_unit x,
map_zero' := by simp,
map_one' := by rw [norm_unit_one, units.coe_one, mul_one],
map_mul' := λ x y,
classical.by_cases (λ hx : x = 0, by rw [hx, zero_mul, zero_mul, zero_mul]) $ λ hx,
classical.by_cases (λ hy : y = 0, by rw [hy, mul_zero, zero_mul, mul_zero]) $ λ hy,
by sim... | def | normalize | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"mul_assoc",
"mul_left_comm",
"mul_one",
"mul_zero",
"norm_unit_one",
"units.coe_mul",
"units.coe_one",
"zero_mul"
] | Chooses an element of each associate class, by multiplying by `norm_unit` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_normalize (x : α) : associated x (normalize x) | ⟨_, rfl⟩ | theorem | associated_normalize | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_associated (x : α) : associated (normalize x) x | (associated_normalize _).symm | theorem | normalize_associated | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_normalize",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated_normalize_iff {x y : α} :
associated x (normalize y) ↔ associated x y | ⟨λ h, h.trans (normalize_associated y), λ h, h.trans (associated_normalize y)⟩ | lemma | associated_normalize_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_normalize",
"normalize",
"normalize_associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_associated_iff {x y : α} :
associated (normalize x) y ↔ associated x y | ⟨λ h, (associated_normalize _).trans h, λ h, (normalize_associated _).trans h⟩ | lemma | normalize_associated_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_normalize",
"normalize",
"normalize_associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates.mk_normalize (x : α) : associates.mk (normalize x) = associates.mk x | associates.mk_eq_mk_iff_associated.2 (normalize_associated _) | lemma | associates.mk_normalize | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates.mk",
"normalize",
"normalize_associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_apply (x : α) : normalize x = x * norm_unit x | rfl | lemma | normalize_apply | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_zero : normalize (0 : α) = 0 | normalize.map_zero | lemma | normalize_zero | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_one : normalize (1 : α) = 1 | normalize.map_one | lemma | normalize_one | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_coe_units (u : αˣ) : normalize (u : α) = 1 | by simp | lemma | normalize_coe_units | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_eq_zero {x : α} : normalize x = 0 ↔ x = 0 | ⟨λ hx, (associated_zero_iff_eq_zero x).1 $ hx ▸ associated_normalize _,
by rintro rfl; exact normalize_zero⟩ | lemma | normalize_eq_zero | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated_normalize",
"associated_zero_iff_eq_zero",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_eq_one {x : α} : normalize x = 1 ↔ is_unit x | ⟨λ hx, is_unit_iff_exists_inv.2 ⟨_, hx⟩, λ ⟨u, hu⟩, hu ▸ normalize_coe_units u⟩ | lemma | normalize_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"is_unit",
"normalize",
"normalize_coe_units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_unit_mul_norm_unit (a : α) : norm_unit (a * norm_unit a) = 1 | begin
nontriviality α using [subsingleton.elim a 0],
obtain rfl|h := eq_or_ne a 0,
{ rw [norm_unit_zero, zero_mul, norm_unit_zero] },
{ rw [norm_unit_mul h (units.ne_zero _), norm_unit_coe_units, mul_inv_eq_one] }
end | theorem | norm_unit_mul_norm_unit | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"eq_or_ne",
"mul_inv_eq_one",
"units.ne_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_idem (x : α) : normalize (normalize x) = normalize x | by simp | theorem | normalize_idem | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_eq_normalize {a b : α}
(hab : a ∣ b) (hba : b ∣ a) : normalize a = normalize b | begin
nontriviality α,
rcases associated_of_dvd_dvd hab hba with ⟨u, rfl⟩,
refine classical.by_cases (by rintro rfl; simp only [zero_mul]) (assume ha : a ≠ 0, _),
suffices : a * ↑(norm_unit a) = a * ↑u * ↑(norm_unit a) * ↑u⁻¹,
by simpa only [normalize_apply, mul_assoc, norm_unit_mul ha u.ne_zero, norm_unit_... | theorem | normalize_eq_normalize | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated_of_dvd_dvd",
"mul_assoc",
"mul_right_comm",
"normalize",
"normalize_apply",
"units.mul_inv_cancel_right",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_eq_normalize_iff {x y : α} : normalize x = normalize y ↔ x ∣ y ∧ y ∣ x | ⟨λ h, ⟨units.dvd_mul_right.1 ⟨_, h.symm⟩, units.dvd_mul_right.1 ⟨_, h⟩⟩,
λ ⟨hxy, hyx⟩, normalize_eq_normalize hxy hyx⟩ | lemma | normalize_eq_normalize_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"normalize_eq_normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_antisymm_of_normalize_eq {a b : α}
(ha : normalize a = a) (hb : normalize b = b) (hab : a ∣ b) (hba : b ∣ a) :
a = b | ha ▸ hb ▸ normalize_eq_normalize hab hba | theorem | dvd_antisymm_of_normalize_eq | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"normalize_eq_normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_normalize_iff {a b : α} : a ∣ normalize b ↔ a ∣ b | units.dvd_mul_right | lemma | dvd_normalize_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"units.dvd_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_dvd_iff {a b : α} : normalize a ∣ b ↔ a ∣ b | units.mul_right_dvd | lemma | normalize_dvd_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"units.mul_right_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out : associates α → α | quotient.lift (normalize : α → α) $ λ a b ⟨u, hu⟩, hu ▸
normalize_eq_normalize ⟨_, rfl⟩ (units.mul_right_dvd.2 $ dvd_refl a) | def | associates.out | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"dvd_refl",
"normalize",
"normalize_eq_normalize"
] | Maps an element of `associates` back to the normalized element of its associate class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
out_mk (a : α) : (associates.mk a).out = normalize a | rfl | lemma | associates.out_mk | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates.mk",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_one : (1 : associates α).out = 1 | normalize_one | lemma | associates.out_one | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"normalize_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_mul (a b : associates α) : (a * b).out = a.out * b.out | quotient.induction_on₂ a b $ assume a b,
by simp only [associates.quotient_mk_eq_mk, out_mk, mk_mul_mk, normalize.map_mul] | lemma | associates.out_mul | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"associates.quotient_mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_out_iff (a : α) (b : associates α) : a ∣ b.out ↔ associates.mk a ≤ b | quotient.induction_on b $
by simp [associates.out_mk, associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff] | lemma | associates.dvd_out_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"associates.mk",
"associates.out_mk",
"associates.quotient_mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_dvd_iff (a : α) (b : associates α) : b.out ∣ a ↔ b ≤ associates.mk a | quotient.induction_on b $
by simp [associates.out_mk, associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff] | lemma | associates.out_dvd_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"associates.mk",
"associates.out_mk",
"associates.quotient_mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_top : (⊤ : associates α).out = 0 | normalize_zero | lemma | associates.out_top | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"normalize_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_out (a : associates α) : normalize a.out = a.out | quotient.induction_on a normalize_idem | lemma | associates.normalize_out | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"normalize",
"normalize_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_out (a : associates α) : associates.mk (a.out) = a | quotient.induction_on a mk_normalize | lemma | associates.mk_out | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"associates.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_injective : function.injective (associates.out : _ → α) | function.left_inverse.injective mk_out | lemma | associates.out_injective | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_monoid (α : Type*) [cancel_comm_monoid_with_zero α] | (gcd : α → α → α)
(lcm : α → α → α)
(gcd_dvd_left : ∀a b, gcd a b ∣ a)
(gcd_dvd_right : ∀a b, gcd a b ∣ b)
(dvd_gcd : ∀{a b c}, a ∣ c → a ∣ b → a ∣ gcd c b)
(gcd_mul_lcm : ∀a b, associated (gcd a b * lcm a b) (a * b))
(lcm_zero_left : ∀a, lcm 0 a = 0)
(lcm_zero_right : ∀a, lcm a 0 = 0) | class | gcd_monoid | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"gcd_mul_lcm"
] | GCD monoid: a `cancel_comm_monoid_with_zero` with `gcd` (greatest common divisor) and
`lcm` (least common multiple) operations, determined up to a unit. The type class focuses on `gcd`
and we derive the corresponding `lcm` facts from `gcd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_gcd_monoid (α : Type*) [cancel_comm_monoid_with_zero α]
extends normalization_monoid α, gcd_monoid α | (normalize_gcd : ∀a b, normalize (gcd a b) = gcd a b)
(normalize_lcm : ∀a b, normalize (lcm a b) = lcm a b) | class | normalized_gcd_monoid | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"cancel_comm_monoid_with_zero",
"gcd_monoid",
"normalization_monoid",
"normalize",
"normalize_gcd",
"normalize_lcm"
] | Normalized GCD monoid: a `cancel_comm_monoid_with_zero` with normalization and `gcd`
(greatest common divisor) and `lcm` (least common multiple) operations. In this setting `gcd` and
`lcm` form a bounded lattice on the associated elements where `gcd` is the infimum, `lcm` is the
supremum, `1` is bottom, and `0` is top.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_gcd [normalized_gcd_monoid α] : ∀a b:α, normalize (gcd a b) = gcd a b | normalized_gcd_monoid.normalize_gcd | theorem | normalize_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_lcm [gcd_monoid α] : ∀a b:α, associated (gcd a b * lcm a b) (a * b) | gcd_monoid.gcd_mul_lcm | theorem | gcd_mul_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_gcd_iff [gcd_monoid α] (a b c : α) : a ∣ gcd b c ↔ (a ∣ b ∧ a ∣ c) | iff.intro
(assume h, ⟨h.trans (gcd_dvd_left _ _), h.trans (gcd_dvd_right _ _)⟩)
(assume ⟨hab, hac⟩, dvd_gcd hab hac) | theorem | dvd_gcd_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_comm [normalized_gcd_monoid α] (a b : α) : gcd a b = gcd b a | dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
(dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
(dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _)) | theorem | gcd_comm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"normalize_gcd",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_comm' [gcd_monoid α] (a b : α) : associated (gcd a b) (gcd b a) | associated_of_dvd_dvd
(dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _))
(dvd_gcd (gcd_dvd_right _ _) (gcd_dvd_left _ _)) | theorem | gcd_comm' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_assoc [normalized_gcd_monoid α] (m n k : α) : gcd (gcd m n) k = gcd m (gcd n k) | dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
(dvd_gcd
((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_left m n))
(dvd_gcd ((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_right m n))
(gcd_dvd_right (gcd m n) k)))
(dvd_gcd
(dvd_gcd (gcd_dvd_left m (gcd n k)) ((gcd_dvd_right m (gcd n k)... | theorem | gcd_assoc | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"normalize_gcd",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_assoc' [gcd_monoid α] (m n k : α) : associated (gcd (gcd m n) k) (gcd m (gcd n k)) | associated_of_dvd_dvd
(dvd_gcd
((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_left m n))
(dvd_gcd ((gcd_dvd_left (gcd m n) k).trans (gcd_dvd_right m n))
(gcd_dvd_right (gcd m n) k)))
(dvd_gcd
(dvd_gcd (gcd_dvd_left m (gcd n k)) ((gcd_dvd_right m (gcd n k)).trans (gcd_dvd_left n k)))
((gcd_dvd_righ... | theorem | gcd_assoc' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_normalize [normalized_gcd_monoid α] {a b c : α}
(habc : gcd a b ∣ c) (hcab : c ∣ gcd a b) :
gcd a b = normalize c | normalize_gcd a b ▸ normalize_eq_normalize habc hcab | theorem | gcd_eq_normalize | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"normalize_eq_normalize",
"normalize_gcd",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_zero_left [normalized_gcd_monoid α] (a : α) : gcd 0 a = normalize a | gcd_eq_normalize (gcd_dvd_right 0 a) (dvd_gcd (dvd_zero _) (dvd_refl a)) | theorem | gcd_zero_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_refl",
"dvd_zero",
"gcd_eq_normalize",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_zero_left' [gcd_monoid α] (a : α) : associated (gcd 0 a) a | associated_of_dvd_dvd (gcd_dvd_right 0 a) (dvd_gcd (dvd_zero _) (dvd_refl a)) | theorem | gcd_zero_left' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"dvd_refl",
"dvd_zero",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_zero_right [normalized_gcd_monoid α] (a : α) : gcd a 0 = normalize a | gcd_eq_normalize (gcd_dvd_left a 0) (dvd_gcd (dvd_refl a) (dvd_zero _)) | theorem | gcd_zero_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_refl",
"dvd_zero",
"gcd_eq_normalize",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_zero_right' [gcd_monoid α] (a : α) : associated (gcd a 0) a | associated_of_dvd_dvd (gcd_dvd_left a 0) (dvd_gcd (dvd_refl a) (dvd_zero _)) | theorem | gcd_zero_right' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"dvd_refl",
"dvd_zero",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_zero_iff [gcd_monoid α] (a b : α) : gcd a b = 0 ↔ a = 0 ∧ b = 0 | iff.intro
(assume h, let ⟨ca, ha⟩ := gcd_dvd_left a b, ⟨cb, hb⟩ := gcd_dvd_right a b in
by rw [h, zero_mul] at ha hb; exact ⟨ha, hb⟩)
(assume ⟨ha, hb⟩, by
{ rw [ha, hb, ←zero_dvd_iff],
apply dvd_gcd; refl }) | theorem | gcd_eq_zero_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_monoid",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_one_left [normalized_gcd_monoid α] (a : α) : gcd 1 a = 1 | dvd_antisymm_of_normalize_eq (normalize_gcd _ _) normalize_one (gcd_dvd_left _ _) (one_dvd _) | theorem | gcd_one_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"normalize_gcd",
"normalize_one",
"normalized_gcd_monoid",
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_one_left' [gcd_monoid α] (a : α) : associated (gcd 1 a) 1 | associated_of_dvd_dvd (gcd_dvd_left _ _) (one_dvd _) | theorem | gcd_one_left' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"gcd_monoid",
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_one_right [normalized_gcd_monoid α] (a : α) : gcd a 1 = 1 | dvd_antisymm_of_normalize_eq (normalize_gcd _ _) normalize_one (gcd_dvd_right _ _) (one_dvd _) | theorem | gcd_one_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"normalize_gcd",
"normalize_one",
"normalized_gcd_monoid",
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_one_right' [gcd_monoid α] (a : α) : associated (gcd a 1) 1 | associated_of_dvd_dvd (gcd_dvd_right _ _) (one_dvd _) | theorem | gcd_one_right' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"gcd_monoid",
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_gcd [gcd_monoid α] {a b c d: α} (hab : a ∣ b) (hcd : c ∣ d) : gcd a c ∣ gcd b d | dvd_gcd ((gcd_dvd_left _ _).trans hab) ((gcd_dvd_right _ _).trans hcd) | theorem | gcd_dvd_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_same [normalized_gcd_monoid α] (a : α) : gcd a a = normalize a | gcd_eq_normalize (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) (dvd_refl a)) | theorem | gcd_same | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_refl",
"gcd_eq_normalize",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_left [normalized_gcd_monoid α] (a b c : α) :
gcd (a * b) (a * c) = normalize a * gcd b c | classical.by_cases (by rintro rfl; simp only [zero_mul, gcd_zero_left, normalize_zero]) $
assume ha : a ≠ 0,
suffices gcd (a * b) (a * c) = normalize (a * gcd b c),
by simpa only [normalize.map_mul, normalize_gcd],
let ⟨d, eq⟩ := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c) in
gcd_eq_normalize
(eq.symm ▸ mul_dvd... | theorem | gcd_mul_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_right",
"gcd_eq_normalize",
"gcd_zero_left",
"mul_dvd_mul_iff_left",
"mul_dvd_mul_left",
"normalize",
"normalize_gcd",
"normalize_zero",
"normalized_gcd_monoid",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_left' [gcd_monoid α] (a b c : α) : associated (gcd (a * b) (a * c)) (a * gcd b c) | begin
obtain rfl|ha := eq_or_ne a 0,
{ simp only [zero_mul, gcd_zero_left'] },
obtain ⟨d, eq⟩ := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c),
apply associated_of_dvd_dvd,
{ rw eq,
apply mul_dvd_mul_left,
exact dvd_gcd
((mul_dvd_mul_iff_left ha).1 $ eq ▸ gcd_dvd_left _ _)
((mul_dvd_mul_... | theorem | gcd_mul_left' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"dvd_mul_right",
"eq_or_ne",
"gcd_monoid",
"gcd_zero_left'",
"mul_dvd_mul_iff_left",
"mul_dvd_mul_left",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_right [normalized_gcd_monoid α] (a b c : α) :
gcd (b * a) (c * a) = gcd b c * normalize a | by simp only [mul_comm, gcd_mul_left] | theorem | gcd_mul_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_mul_left",
"mul_comm",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_right' [gcd_monoid α] (a b c : α) :
associated (gcd (b * a) (c * a)) (gcd b c * a) | by simp only [mul_comm, gcd_mul_left'] | theorem | gcd_mul_right' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"gcd_monoid",
"gcd_mul_left'",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_left_iff [normalized_gcd_monoid α] (a b : α) (h : normalize a = a) :
gcd a b = a ↔ a ∣ b | iff.intro (assume eq, eq ▸ gcd_dvd_right _ _) $
assume hab, dvd_antisymm_of_normalize_eq (normalize_gcd _ _) h (gcd_dvd_left _ _)
(dvd_gcd (dvd_refl a) hab) | theorem | gcd_eq_left_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"dvd_refl",
"normalize",
"normalize_gcd",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_right_iff [normalized_gcd_monoid α] (a b : α) (h : normalize b = b) :
gcd a b = b ↔ b ∣ a | by simpa only [gcd_comm a b] using gcd_eq_left_iff b a h | theorem | gcd_eq_right_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_comm",
"gcd_eq_left_iff",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_gcd_mul_left [gcd_monoid α] (m n k : α) : gcd m n ∣ gcd (k * m) n | gcd_dvd_gcd (dvd_mul_left _ _) dvd_rfl | theorem | gcd_dvd_gcd_mul_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_left",
"dvd_rfl",
"gcd_dvd_gcd",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_gcd_mul_right [gcd_monoid α] (m n k : α) : gcd m n ∣ gcd (m * k) n | gcd_dvd_gcd (dvd_mul_right _ _) dvd_rfl | theorem | gcd_dvd_gcd_mul_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_right",
"dvd_rfl",
"gcd_dvd_gcd",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_gcd_mul_left_right [gcd_monoid α] (m n k : α) : gcd m n ∣ gcd m (k * n) | gcd_dvd_gcd dvd_rfl (dvd_mul_left _ _) | theorem | gcd_dvd_gcd_mul_left_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_left",
"dvd_rfl",
"gcd_dvd_gcd",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_gcd_mul_right_right [gcd_monoid α] (m n k : α) : gcd m n ∣ gcd m (n * k) | gcd_dvd_gcd dvd_rfl (dvd_mul_right _ _) | theorem | gcd_dvd_gcd_mul_right_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_right",
"dvd_rfl",
"gcd_dvd_gcd",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated.gcd_eq_left [normalized_gcd_monoid α] {m n : α} (h : associated m n) (k : α) :
gcd m k = gcd n k | dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
(gcd_dvd_gcd h.dvd dvd_rfl)
(gcd_dvd_gcd h.symm.dvd dvd_rfl) | theorem | associated.gcd_eq_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"dvd_antisymm_of_normalize_eq",
"dvd_rfl",
"gcd_dvd_gcd",
"normalize_gcd",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated.gcd_eq_right [normalized_gcd_monoid α] {m n : α} (h : associated m n) (k : α) :
gcd k m = gcd k n | dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)
(gcd_dvd_gcd dvd_rfl h.dvd)
(gcd_dvd_gcd dvd_rfl h.symm.dvd) | theorem | associated.gcd_eq_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"dvd_antisymm_of_normalize_eq",
"dvd_rfl",
"gcd_dvd_gcd",
"normalize_gcd",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_gcd_mul_of_dvd_mul [gcd_monoid α] {m n k : α} (H : k ∣ m * n) : k ∣ (gcd k m) * n | (dvd_gcd (dvd_mul_right _ n) H).trans (gcd_mul_right' n k m).dvd | lemma | dvd_gcd_mul_of_dvd_mul | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_right",
"gcd_monoid",
"gcd_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_mul_gcd_of_dvd_mul [gcd_monoid α] {m n k : α} (H : k ∣ m * n) : k ∣ m * gcd k n | by { rw mul_comm at H ⊢, exact dvd_gcd_mul_of_dvd_mul H } | lemma | dvd_mul_gcd_of_dvd_mul | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_gcd_mul_of_dvd_mul",
"gcd_monoid",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_dvd_and_dvd_of_dvd_mul [gcd_monoid α] {m n k : α} (H : k ∣ m * n) :
∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂ | begin
by_cases h0 : gcd k m = 0,
{ rw gcd_eq_zero_iff at h0,
rcases h0 with ⟨rfl, rfl⟩,
refine ⟨0, n, dvd_refl 0, dvd_refl n, _⟩,
simp },
{ obtain ⟨a, ha⟩ := gcd_dvd_left k m,
refine ⟨gcd k m, a, gcd_dvd_right _ _, _, ha⟩,
suffices h : gcd k m * a ∣ gcd k m * n,
{ cases h with b hb,
... | lemma | exists_dvd_and_dvd_of_dvd_mul | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_gcd_mul_of_dvd_mul",
"dvd_refl",
"gcd_eq_zero_iff",
"gcd_monoid",
"mul_assoc",
"mul_left_cancel₀"
] | Represent a divisor of `m * n` as a product of a divisor of `m` and a divisor of `n`.
In other words, the nonzero elements of a `gcd_monoid` form a decomposition monoid
(more widely known as a pre-Schreier domain in the context of rings).
Note: In general, this representation is highly non-unique.
See `nat.prod_dvd_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_mul [gcd_monoid α] {k m n : α} :
k ∣ (m * n) ↔ ∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂ | begin
refine ⟨exists_dvd_and_dvd_of_dvd_mul, _⟩,
rintro ⟨d₁, d₂, hy, hz, rfl⟩,
exact mul_dvd_mul hy hz,
end | lemma | dvd_mul | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_monoid",
"mul_dvd_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_dvd_mul_gcd [gcd_monoid α] (k m n : α) : gcd k (m * n) ∣ gcd k m * gcd k n | begin
obtain ⟨m', n', hm', hn', h⟩ := exists_dvd_and_dvd_of_dvd_mul (gcd_dvd_right k (m * n)),
replace h : gcd k (m * n) = m' * n' := h,
rw h,
have hm'n' : m' * n' ∣ k := h ▸ gcd_dvd_left _ _,
apply mul_dvd_mul,
{ have hm'k : m' ∣ k := (dvd_mul_right m' n').trans hm'n',
exact dvd_gcd hm'k hm' },
{ hav... | theorem | gcd_mul_dvd_mul_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_left",
"dvd_mul_right",
"exists_dvd_and_dvd_of_dvd_mul",
"gcd_monoid",
"mul_dvd_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_pow_right_dvd_pow_gcd [gcd_monoid α] {a b : α} {k : ℕ} :
gcd a (b ^ k) ∣ (gcd a b) ^ k | begin
by_cases hg : gcd a b = 0,
{ rw gcd_eq_zero_iff at hg,
rcases hg with ⟨rfl, rfl⟩,
exact (gcd_zero_left' (0 ^ k : α)).dvd.trans
(pow_dvd_pow_of_dvd (gcd_zero_left' (0 : α)).symm.dvd _) },
{ induction k with k hk,
{ simp only [pow_zero],
exact (gcd_one_right' a).dvd, },
rw [pow_suc... | theorem | gcd_pow_right_dvd_pow_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_eq_zero_iff",
"gcd_monoid",
"gcd_mul_dvd_mul_gcd",
"gcd_one_right'",
"gcd_zero_left'",
"mul_dvd_mul_iff_left",
"pow_dvd_pow_of_dvd",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_pow_left_dvd_pow_gcd [gcd_monoid α] {a b : α} {k : ℕ} :
gcd (a ^ k) b ∣ (gcd a b) ^ k | calc gcd (a ^ k) b
∣ gcd b (a ^ k) : (gcd_comm' _ _).dvd
... ∣ (gcd b a) ^ k : gcd_pow_right_dvd_pow_gcd
... ∣ (gcd a b) ^ k : pow_dvd_pow_of_dvd (gcd_comm' _ _).dvd _ | theorem | gcd_pow_left_dvd_pow_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_comm'",
"gcd_monoid",
"gcd_pow_right_dvd_pow_gcd",
"pow_dvd_pow_of_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_dvd_of_mul_eq_pow [gcd_monoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0)
(hab : is_unit (gcd a b)) {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂)
(hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a | begin
have h1 : is_unit (gcd (d₁ ^ k) b),
{ apply is_unit_of_dvd_one,
transitivity (gcd d₁ b) ^ k,
{ exact gcd_pow_left_dvd_pow_gcd },
{ apply is_unit.dvd, apply is_unit.pow, apply is_unit_of_dvd_one,
apply dvd_trans _ hab.dvd,
apply gcd_dvd_gcd hd₁ (dvd_refl b) } },
have h2 : d₁ ^ k ∣ a *... | theorem | pow_dvd_of_mul_eq_pow | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_gcd_mul_of_dvd_mul",
"dvd_refl",
"dvd_trans",
"gcd_dvd_gcd",
"gcd_monoid",
"gcd_pow_left_dvd_pow_gcd",
"is_unit",
"is_unit.dvd",
"is_unit.mul_left_dvd",
"is_unit.pow",
"is_unit_of_dvd_one",
"mul_comm",
"mul_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_associated_pow_of_mul_eq_pow [gcd_monoid α] {a b c : α}
(hab : is_unit (gcd a b)) {k : ℕ}
(h : a * b = c ^ k) : ∃ (d : α), associated (d ^ k) a | begin
casesI subsingleton_or_nontrivial α,
{ use 0, rw [subsingleton.elim a (0 ^ k)] },
by_cases ha : a = 0,
{ use 0, rw ha,
obtain (rfl | hk) := k.eq_zero_or_pos,
{ exfalso, revert h, rw [ha, zero_mul, pow_zero], apply zero_ne_one },
{ rw zero_pow hk } },
by_cases hb : b = 0,
{ use 1, rw [one_p... | theorem | exists_associated_pow_of_mul_eq_pow | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"dvd_pow_self",
"exists_dvd_and_dvd_of_dvd_mul",
"gcd_comm'",
"gcd_monoid",
"gcd_zero_right'",
"is_unit",
"mul_assoc",
"mul_comm",
"mul_one",
"mul_pow",
"mul_right_inj'",
"one_mul",
"one_pow",
"pow_dvd_of_mul_eq_pow",
"pow_zero",
"subsingleton_or_nontrivial",
"units.c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_pow_of_mul_eq_pow [gcd_monoid α] [unique αˣ] {a b c : α}
(hab : is_unit (gcd a b)) {k : ℕ}
(h : a * b = c ^ k) : ∃ (d : α), a = d ^ k | let ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow hab h in ⟨d, (associated_iff_eq.mp hd).symm⟩ | theorem | exists_eq_pow_of_mul_eq_pow | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"exists_associated_pow_of_mul_eq_pow",
"gcd_monoid",
"is_unit",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_greatest {α : Type*} [cancel_comm_monoid_with_zero α] [normalized_gcd_monoid α]
{a b d : α} (hda : d ∣ a) (hdb : d ∣ b)
(hd : ∀ e : α, e ∣ a → e ∣ b → e ∣ d) : gcd_monoid.gcd a b = normalize d | begin
have h := hd _ (gcd_monoid.gcd_dvd_left a b) (gcd_monoid.gcd_dvd_right a b),
exact gcd_eq_normalize h (gcd_monoid.dvd_gcd hda hdb),
end | lemma | gcd_greatest | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"cancel_comm_monoid_with_zero",
"gcd_eq_normalize",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_greatest_associated {α : Type*} [cancel_comm_monoid_with_zero α] [gcd_monoid α]
{a b d : α} (hda : d ∣ a) (hdb : d ∣ b)
(hd : ∀ e : α, e ∣ a → e ∣ b → e ∣ d) : associated d (gcd_monoid.gcd a b) | begin
have h := hd _ (gcd_monoid.gcd_dvd_left a b) (gcd_monoid.gcd_dvd_right a b),
exact associated_of_dvd_dvd (gcd_monoid.dvd_gcd hda hdb) h,
end | lemma | gcd_greatest_associated | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"cancel_comm_monoid_with_zero",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_gcd_of_eq_mul_gcd {α : Type*} [cancel_comm_monoid_with_zero α] [gcd_monoid α]
{x y x' y' : α} (ex : x = gcd x y * x') (ey : y = gcd x y * y') (h : gcd x y ≠ 0) :
is_unit (gcd x' y') | begin
rw ← associated_one_iff_is_unit,
refine associated.of_mul_left _ (associated.refl $ gcd x y) h,
convert (gcd_mul_left' _ _ _).symm using 1,
rw [← ex, ← ey, mul_one],
end | lemma | is_unit_gcd_of_eq_mul_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated.of_mul_left",
"associated.refl",
"associated_one_iff_is_unit",
"cancel_comm_monoid_with_zero",
"gcd_monoid",
"gcd_mul_left'",
"is_unit",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extract_gcd {α : Type*} [cancel_comm_monoid_with_zero α] [gcd_monoid α] (x y : α) :
∃ x' y', x = gcd x y * x' ∧ y = gcd x y * y' ∧ is_unit (gcd x' y') | begin
by_cases h : gcd x y = 0,
{ obtain ⟨rfl, rfl⟩ := (gcd_eq_zero_iff x y).1 h,
simp_rw ← associated_one_iff_is_unit,
exact ⟨1, 1, by rw [h, zero_mul], by rw [h, zero_mul], gcd_one_left' 1⟩ },
obtain ⟨x', ex⟩ := gcd_dvd_left x y,
obtain ⟨y', ey⟩ := gcd_dvd_right x y,
exact ⟨x', y', ex, ey, is_unit_g... | lemma | extract_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated_one_iff_is_unit",
"cancel_comm_monoid_with_zero",
"gcd_eq_zero_iff",
"gcd_monoid",
"gcd_one_left'",
"is_unit",
"is_unit_gcd_of_eq_mul_gcd",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd_iff [gcd_monoid α] {a b c : α} : lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c | begin
by_cases this : a = 0 ∨ b = 0,
{ rcases this with rfl | rfl;
simp only [iff_def, lcm_zero_left, lcm_zero_right, zero_dvd_iff, dvd_zero,
eq_self_iff_true, and_true, imp_true_iff] {contextual:=tt} },
{ obtain ⟨h1, h2⟩ := not_or_distrib.1 this,
have h : gcd a b ≠ 0, from λ H, h1 ((gcd_eq_zero_iff... | lemma | lcm_dvd_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_gcd_iff",
"dvd_zero",
"gcd_eq_zero_iff",
"gcd_monoid",
"gcd_mul_lcm",
"gcd_mul_right'",
"iff_def",
"imp_true_iff",
"mul_comm",
"mul_dvd_mul_iff_left",
"mul_dvd_mul_iff_right",
"zero_dvd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_lcm_left [gcd_monoid α] (a b : α) : a ∣ lcm a b | (lcm_dvd_iff.1 (dvd_refl (lcm a b))).1 | lemma | dvd_lcm_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_refl",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_lcm_right [gcd_monoid α] (a b : α) : b ∣ lcm a b | (lcm_dvd_iff.1 (dvd_refl (lcm a b))).2 | lemma | dvd_lcm_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_refl",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd [gcd_monoid α] {a b c : α} (hab : a ∣ b) (hcb : c ∣ b) : lcm a c ∣ b | lcm_dvd_iff.2 ⟨hab, hcb⟩ | lemma | lcm_dvd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_zero_iff [gcd_monoid α] (a b : α) : lcm a b = 0 ↔ a = 0 ∨ b = 0 | iff.intro
(assume h : lcm a b = 0,
have associated (a * b) 0 := (gcd_mul_lcm a b).symm.trans $
by rw [h, mul_zero],
by simpa only [associated_zero_iff_eq_zero, mul_eq_zero])
(by rintro (rfl | rfl); [apply lcm_zero_left, apply lcm_zero_right]) | theorem | lcm_eq_zero_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_zero_iff_eq_zero",
"gcd_monoid",
"gcd_mul_lcm",
"mul_eq_zero",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_lcm [normalized_gcd_monoid α] (a b : α) : normalize (lcm a b) = lcm a b | normalized_gcd_monoid.normalize_lcm a b | lemma | normalize_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_comm [normalized_gcd_monoid α] (a b : α) : lcm a b = lcm b a | dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
(lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
(lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _)) | theorem | lcm_comm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"dvd_lcm_left",
"dvd_lcm_right",
"lcm_dvd",
"normalize_lcm",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_comm' [gcd_monoid α] (a b : α) : associated (lcm a b) (lcm b a) | associated_of_dvd_dvd
(lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _))
(lcm_dvd (dvd_lcm_right _ _) (dvd_lcm_left _ _)) | theorem | lcm_comm' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"dvd_lcm_left",
"dvd_lcm_right",
"gcd_monoid",
"lcm_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_assoc [normalized_gcd_monoid α] (m n k : α) : lcm (lcm m n) k = lcm m (lcm n k) | dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
(lcm_dvd
(lcm_dvd (dvd_lcm_left _ _) ((dvd_lcm_left _ _).trans (dvd_lcm_right _ _)))
((dvd_lcm_right _ _).trans (dvd_lcm_right _ _)))
(lcm_dvd
((dvd_lcm_left _ _).trans (dvd_lcm_left _ _))
(lcm_dvd ((dvd_lcm_right _ _).trans (dvd_l... | theorem | lcm_assoc | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"dvd_lcm_left",
"dvd_lcm_right",
"lcm_dvd",
"normalize_lcm",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_assoc' [gcd_monoid α] (m n k : α) : associated (lcm (lcm m n) k) (lcm m (lcm n k)) | associated_of_dvd_dvd
(lcm_dvd
(lcm_dvd (dvd_lcm_left _ _) ((dvd_lcm_left _ _).trans (dvd_lcm_right _ _)))
((dvd_lcm_right _ _).trans (dvd_lcm_right _ _)))
(lcm_dvd
((dvd_lcm_left _ _).trans (dvd_lcm_left _ _))
(lcm_dvd ((dvd_lcm_right _ _).trans (dvd_lcm_left _ _)) (dvd_lcm_right _ _))) | theorem | lcm_assoc' | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated",
"associated_of_dvd_dvd",
"dvd_lcm_left",
"dvd_lcm_right",
"gcd_monoid",
"lcm_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_normalize [normalized_gcd_monoid α] {a b c : α}
(habc : lcm a b ∣ c) (hcab : c ∣ lcm a b) :
lcm a b = normalize c | normalize_lcm a b ▸ normalize_eq_normalize habc hcab | lemma | lcm_eq_normalize | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"normalize_eq_normalize",
"normalize_lcm",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd_lcm [gcd_monoid α] {a b c d : α} (hab : a ∣ b) (hcd : c ∣ d) :
lcm a c ∣ lcm b d | lcm_dvd (hab.trans (dvd_lcm_left _ _)) (hcd.trans (dvd_lcm_right _ _)) | theorem | lcm_dvd_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_lcm_left",
"dvd_lcm_right",
"gcd_monoid",
"lcm_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_units_coe_left [normalized_gcd_monoid α] (u : αˣ) (a : α) :
lcm ↑u a = normalize a | lcm_eq_normalize (lcm_dvd units.coe_dvd dvd_rfl) (dvd_lcm_right _ _) | theorem | lcm_units_coe_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_lcm_right",
"dvd_rfl",
"lcm_dvd",
"lcm_eq_normalize",
"normalize",
"normalized_gcd_monoid",
"units.coe_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_units_coe_right [normalized_gcd_monoid α] (a : α) (u : αˣ) :
lcm a ↑u = normalize a | (lcm_comm a u).trans $ lcm_units_coe_left _ _ | theorem | lcm_units_coe_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"lcm_comm",
"lcm_units_coe_left",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_one_left [normalized_gcd_monoid α] (a : α) : lcm 1 a = normalize a | lcm_units_coe_left 1 a | theorem | lcm_one_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"lcm_units_coe_left",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_one_right [normalized_gcd_monoid α] (a : α) : lcm a 1 = normalize a | lcm_units_coe_right a 1 | theorem | lcm_one_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"lcm_units_coe_right",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_same [normalized_gcd_monoid α] (a : α) : lcm a a = normalize a | lcm_eq_normalize (lcm_dvd dvd_rfl dvd_rfl) (dvd_lcm_left _ _) | theorem | lcm_same | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_lcm_left",
"dvd_rfl",
"lcm_dvd",
"lcm_eq_normalize",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_one_iff [normalized_gcd_monoid α] (a b : α) : lcm a b = 1 ↔ a ∣ 1 ∧ b ∣ 1 | iff.intro
(assume eq, eq ▸ ⟨dvd_lcm_left _ _, dvd_lcm_right _ _⟩)
(assume ⟨⟨c, hc⟩, ⟨d, hd⟩⟩,
show lcm (units.mk_of_mul_eq_one a c hc.symm : α) (units.mk_of_mul_eq_one b d hd.symm) = 1,
by rw [lcm_units_coe_left, normalize_coe_units]) | theorem | lcm_eq_one_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_lcm_right",
"lcm_units_coe_left",
"normalize_coe_units",
"normalized_gcd_monoid",
"units.mk_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_mul_left [normalized_gcd_monoid α] (a b c : α) :
lcm (a * b) (a * c) = normalize a * lcm b c | classical.by_cases (by rintro rfl; simp only [zero_mul, lcm_zero_left, normalize_zero]) $
assume ha : a ≠ 0,
suffices lcm (a * b) (a * c) = normalize (a * lcm b c),
by simpa only [normalize.map_mul, normalize_lcm],
have a ∣ lcm (a * b) (a * c), from (dvd_mul_right _ _).trans (dvd_lcm_left _ _),
let ⟨d, eq⟩ := this in... | theorem | lcm_mul_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_lcm_left",
"dvd_lcm_right",
"dvd_mul_right",
"lcm_dvd",
"lcm_eq_normalize",
"mul_dvd_mul_iff_left",
"mul_dvd_mul_left",
"normalize",
"normalize_lcm",
"normalize_zero",
"normalized_gcd_monoid",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_mul_right [normalized_gcd_monoid α] (a b c : α) :
lcm (b * a) (c * a) = lcm b c * normalize a | by simp only [mul_comm, lcm_mul_left] | theorem | lcm_mul_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"lcm_mul_left",
"mul_comm",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_left_iff [normalized_gcd_monoid α] (a b : α) (h : normalize a = a) :
lcm a b = a ↔ b ∣ a | iff.intro (assume eq, eq ▸ dvd_lcm_right _ _) $
assume hab, dvd_antisymm_of_normalize_eq (normalize_lcm _ _) h (lcm_dvd (dvd_refl a) hab)
(dvd_lcm_left _ _) | theorem | lcm_eq_left_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"dvd_lcm_left",
"dvd_lcm_right",
"dvd_refl",
"lcm_dvd",
"normalize",
"normalize_lcm",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_right_iff [normalized_gcd_monoid α] (a b : α) (h : normalize b = b) :
lcm a b = b ↔ a ∣ b | by simpa only [lcm_comm b a] using lcm_eq_left_iff b a h | theorem | lcm_eq_right_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"lcm_comm",
"lcm_eq_left_iff",
"normalize",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd_lcm_mul_left [gcd_monoid α] (m n k : α) : lcm m n ∣ lcm (k * m) n | lcm_dvd_lcm (dvd_mul_left _ _) dvd_rfl | theorem | lcm_dvd_lcm_mul_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_left",
"dvd_rfl",
"gcd_monoid",
"lcm_dvd_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd_lcm_mul_right [gcd_monoid α] (m n k : α) : lcm m n ∣ lcm (m * k) n | lcm_dvd_lcm (dvd_mul_right _ _) dvd_rfl | theorem | lcm_dvd_lcm_mul_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_mul_right",
"dvd_rfl",
"gcd_monoid",
"lcm_dvd_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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