fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
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value | commit stringclasses 1
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irreducible.dvd_symm [monoid α] {p q : α}
(hp : irreducible p) (hq : irreducible q) : p ∣ q → q ∣ p :=
begin
unfreezingI { rintros ⟨q', rfl⟩ },
rw is_unit.mul_right_dvd (or.resolve_left (of_irreducible_mul hq) hp.not_unit),
end | irreducible.dvd_symm [monoid α] {p q : α}
(hp : irreducible p) (hq : irreducible q) : p ∣ q → q ∣ p | begin
unfreezingI { rintros ⟨q', rfl⟩ },
rw is_unit.mul_right_dvd (or.resolve_left (of_irreducible_mul hq) hp.not_unit),
end | lemma | irreducible.dvd_symm | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"is_unit.mul_right_dvd",
"monoid",
"of_irreducible_mul"
] | If `p` and `q` are irreducible, then `p ∣ q` implies `q ∣ p`. | 212 | 217 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible.dvd_comm [monoid α] {p q : α}
(hp : irreducible p) (hq : irreducible q) : p ∣ q ↔ q ∣ p :=
⟨hp.dvd_symm hq, hq.dvd_symm hp⟩ | irreducible.dvd_comm [monoid α] {p q : α}
(hp : irreducible p) (hq : irreducible q) : p ∣ q ↔ q ∣ p | ⟨hp.dvd_symm hq, hq.dvd_symm hp⟩ | lemma | irreducible.dvd_comm | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"monoid"
] | null | 219 | 221 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_units_mul (a : αˣ) (b : α) : irreducible (↑a * b) ↔ irreducible b :=
begin
simp only [irreducible_iff, units.is_unit_units_mul, and.congr_right_iff],
refine λ hu, ⟨λ h A B HAB, _, λ h A B HAB, _⟩,
{ rw [←a.is_unit_units_mul],
apply h,
rw [mul_assoc, ←HAB] },
{ rw [←(a⁻¹).is_unit_units_mul],
... | irreducible_units_mul (a : αˣ) (b : α) : irreducible (↑a * b) ↔ irreducible b | begin
simp only [irreducible_iff, units.is_unit_units_mul, and.congr_right_iff],
refine λ hu, ⟨λ h A B HAB, _, λ h A B HAB, _⟩,
{ rw [←a.is_unit_units_mul],
apply h,
rw [mul_assoc, ←HAB] },
{ rw [←(a⁻¹).is_unit_units_mul],
apply h,
rw [mul_assoc, ←HAB, units.inv_mul_cancel_left] },
end | lemma | irreducible_units_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"and.congr_right_iff",
"irreducible",
"irreducible_iff",
"mul_assoc",
"units.inv_mul_cancel_left",
"units.is_unit_units_mul"
] | null | 226 | 236 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_is_unit_mul {a b : α} (h : is_unit a) : irreducible (a * b) ↔ irreducible b :=
let ⟨a, ha⟩ := h in ha ▸ irreducible_units_mul a b | irreducible_is_unit_mul {a b : α} (h : is_unit a) : irreducible (a * b) ↔ irreducible b | let ⟨a, ha⟩ := h in ha ▸ irreducible_units_mul a b | lemma | irreducible_is_unit_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"irreducible_units_mul",
"is_unit"
] | null | 238 | 239 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_mul_units (a : αˣ) (b : α) : irreducible (b * ↑a) ↔ irreducible b :=
begin
simp only [irreducible_iff, units.is_unit_mul_units, and.congr_right_iff],
refine λ hu, ⟨λ h A B HAB, _, λ h A B HAB, _⟩,
{ rw [←units.is_unit_mul_units B a],
apply h,
rw [←mul_assoc, ←HAB] },
{ rw [←units.is_unit_mul... | irreducible_mul_units (a : αˣ) (b : α) : irreducible (b * ↑a) ↔ irreducible b | begin
simp only [irreducible_iff, units.is_unit_mul_units, and.congr_right_iff],
refine λ hu, ⟨λ h A B HAB, _, λ h A B HAB, _⟩,
{ rw [←units.is_unit_mul_units B a],
apply h,
rw [←mul_assoc, ←HAB] },
{ rw [←units.is_unit_mul_units B a⁻¹],
apply h,
rw [←mul_assoc, ←HAB, units.mul_inv_cancel_right]... | lemma | irreducible_mul_units | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"and.congr_right_iff",
"irreducible",
"irreducible_iff",
"units.is_unit_mul_units",
"units.mul_inv_cancel_right"
] | null | 241 | 251 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_mul_is_unit {a b : α} (h : is_unit a) : irreducible (b * a) ↔ irreducible b :=
let ⟨a, ha⟩ := h in ha ▸ irreducible_mul_units a b | irreducible_mul_is_unit {a b : α} (h : is_unit a) : irreducible (b * a) ↔ irreducible b | let ⟨a, ha⟩ := h in ha ▸ irreducible_mul_units a b | lemma | irreducible_mul_is_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"irreducible_mul_units",
"is_unit"
] | null | 253 | 254 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_mul_iff {a b : α} :
irreducible (a * b) ↔ (irreducible a ∧ is_unit b) ∨ (irreducible b ∧ is_unit a) :=
begin
split,
{ refine λ h, or.imp (λ h', ⟨_, h'⟩) (λ h', ⟨_, h'⟩) (h.is_unit_or_is_unit rfl).symm,
{ rwa [irreducible_mul_is_unit h'] at h },
{ rwa [irreducible_is_unit_mul h'] at h } },
{ ... | irreducible_mul_iff {a b : α} :
irreducible (a * b) ↔ (irreducible a ∧ is_unit b) ∨ (irreducible b ∧ is_unit a) | begin
split,
{ refine λ h, or.imp (λ h', ⟨_, h'⟩) (λ h', ⟨_, h'⟩) (h.is_unit_or_is_unit rfl).symm,
{ rwa [irreducible_mul_is_unit h'] at h },
{ rwa [irreducible_is_unit_mul h'] at h } },
{ rintros (⟨ha, hb⟩|⟨hb, ha⟩),
{ rwa [irreducible_mul_is_unit hb] },
{ rwa [irreducible_is_unit_mul ha] } },
en... | lemma | irreducible_mul_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"irreducible_is_unit_mul",
"irreducible_mul_is_unit",
"is_unit"
] | null | 256 | 266 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible.not_square (ha : irreducible a) : ¬ is_square a :=
by { rintro ⟨b, rfl⟩, simp only [irreducible_mul_iff, or_self] at ha, exact ha.1.not_unit ha.2 } | irreducible.not_square (ha : irreducible a) : ¬ is_square a | by { rintro ⟨b, rfl⟩, simp only [irreducible_mul_iff, or_self] at ha, exact ha.1.not_unit ha.2 } | lemma | irreducible.not_square | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"irreducible_mul_iff",
"is_square"
] | null | 273 | 274 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_square.not_irreducible (ha : is_square a) : ¬ irreducible a := λ h, h.not_square ha | is_square.not_irreducible (ha : is_square a) : ¬ irreducible a | λ h, h.not_square ha | lemma | is_square.not_irreducible | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"is_square"
] | null | 276 | 276 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.irreducible (hp : prime p) : irreducible p :=
⟨hp.not_unit, λ a b hab,
(show a * b ∣ a ∨ a * b ∣ b, from hab ▸ hp.dvd_or_dvd (hab ▸ dvd_rfl)).elim
(λ ⟨x, hx⟩, or.inr (is_unit_iff_dvd_one.2
⟨x, mul_right_cancel₀ (show a ≠ 0, from λ h, by simp [*, prime] at *)
$ by conv {to_lhs, rw hx}; simp [mu... | prime.irreducible (hp : prime p) : irreducible p | ⟨hp.not_unit, λ a b hab,
(show a * b ∣ a ∨ a * b ∣ b, from hab ▸ hp.dvd_or_dvd (hab ▸ dvd_rfl)).elim
(λ ⟨x, hx⟩, or.inr (is_unit_iff_dvd_one.2
⟨x, mul_right_cancel₀ (show a ≠ 0, from λ h, by simp [*, prime] at *)
$ by conv {to_lhs, rw hx}; simp [mul_comm, mul_assoc, mul_left_comm]⟩))
(λ ⟨x, hx⟩,... | lemma | prime.irreducible | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"dvd_rfl",
"irreducible",
"mul_assoc",
"mul_comm",
"mul_left_comm",
"mul_right_cancel₀",
"prime"
] | null | 283 | 291 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : prime p) {a b : α} {k l : ℕ} :
p ^ k ∣ a → p ^ l ∣ b → p ^ ((k + l) + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b :=
λ ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩,
have h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z),
by simpa [mul_comm, pow_add, hx, hy, mul_assoc, mul_left_comm] usi... | succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : prime p) {a b : α} {k l : ℕ} :
p ^ k ∣ a → p ^ l ∣ b → p ^ ((k + l) + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b | λ ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩,
have h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z),
by simpa [mul_comm, pow_add, hx, hy, mul_assoc, mul_left_comm] using hz,
have hp0: p ^ (k + l) ≠ 0, from pow_ne_zero _ hp.ne_zero,
have hpd : p ∣ x * y, from ⟨z, by rwa [mul_right_inj' hp0] at h⟩,
(hp.dvd_or_dvd hpd).elim
(λ ⟨d, hd⟩,... | lemma | succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"mul_assoc",
"mul_comm",
"mul_left_comm",
"mul_right_inj'",
"pow_add",
"pow_ne_zero",
"pow_succ",
"prime"
] | null | 293 | 302 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.not_square (hp : prime p) : ¬ is_square p := hp.irreducible.not_square | prime.not_square (hp : prime p) : ¬ is_square p | hp.irreducible.not_square | lemma | prime.not_square | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_square",
"prime"
] | null | 304 | 304 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_square.not_prime (ha : is_square a) : ¬ prime a := λ h, h.not_square ha | is_square.not_prime (ha : is_square a) : ¬ prime a | λ h, h.not_square ha | lemma | is_square.not_prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_square",
"prime"
] | null | 305 | 305 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_not_prime {n : ℕ} (hn : n ≠ 1) : ¬ prime (a ^ n) :=
λ hp, hp.not_unit $ is_unit.pow _ $ of_irreducible_pow hn $ hp.irreducible | pow_not_prime {n : ℕ} (hn : n ≠ 1) : ¬ prime (a ^ n) | λ hp, hp.not_unit $ is_unit.pow _ $ of_irreducible_pow hn $ hp.irreducible | lemma | pow_not_prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit.pow",
"of_irreducible_pow",
"prime"
] | null | 307 | 308 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated [monoid α] (x y : α) : Prop := ∃u:αˣ, x * u = y | associated [monoid α] (x y : α) : Prop | ∃u:αˣ, x * u = y | def | associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid"
] | Two elements of a `monoid` are `associated` if one of them is another one
multiplied by a unit on the right. | 314 | 314 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl [monoid α] (x : α) : x ~ᵤ x := ⟨1, by simp⟩ | refl [monoid α] (x : α) : x ~ᵤ x | ⟨1, by simp⟩ | theorem | associated.refl | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid"
] | null | 320 | 320 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] : is_refl α associated := ⟨associated.refl⟩ | [monoid α] : is_refl α associated | ⟨associated.refl⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"monoid"
] | null | 321 | 321 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm [monoid α] : ∀{x y : α}, x ~ᵤ y → y ~ᵤ x
| x _ ⟨u, rfl⟩ := ⟨u⁻¹, by rw [mul_assoc, units.mul_inv, mul_one]⟩ | symm [monoid α] : ∀{x y : α}, x ~ᵤ y → y ~ᵤ x
| x _ ⟨u, rfl⟩ | ⟨u⁻¹, by rw [mul_assoc, units.mul_inv, mul_one]⟩ | theorem | associated.symm | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid",
"mul_assoc",
"mul_one",
"units.mul_inv"
] | null | 323 | 324 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] : is_symm α associated := ⟨λ a b, associated.symm⟩ | [monoid α] : is_symm α associated | ⟨λ a b, associated.symm⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"monoid"
] | null | 325 | 325 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm [monoid α] {x y : α} : x ~ᵤ y ↔ y ~ᵤ x :=
⟨associated.symm, associated.symm⟩ | comm [monoid α] {x y : α} : x ~ᵤ y ↔ y ~ᵤ x | ⟨associated.symm, associated.symm⟩ | theorem | associated.comm | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm",
"monoid"
] | null | 327 | 328 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans [monoid α] : ∀{x y z : α}, x ~ᵤ y → y ~ᵤ z → x ~ᵤ z
| x _ _ ⟨u, rfl⟩ ⟨v, rfl⟩ := ⟨u * v, by rw [units.coe_mul, mul_assoc]⟩ | trans [monoid α] : ∀{x y z : α}, x ~ᵤ y → y ~ᵤ z → x ~ᵤ z
| x _ _ ⟨u, rfl⟩ ⟨v, rfl⟩ | ⟨u * v, by rw [units.coe_mul, mul_assoc]⟩ | theorem | associated.trans | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid",
"mul_assoc",
"units.coe_mul"
] | null | 330 | 331 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] : is_trans α associated := ⟨λ a b c, associated.trans⟩ | [monoid α] : is_trans α associated | ⟨λ a b c, associated.trans⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"monoid"
] | null | 332 | 332 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
setoid (α : Type*) [monoid α] : setoid α :=
{ r := associated, iseqv := ⟨associated.refl, λa b, associated.symm, λa b c, associated.trans⟩ } | setoid (α : Type*) [monoid α] : setoid α | { r := associated, iseqv := ⟨associated.refl, λa b, associated.symm, λa b c, associated.trans⟩ } | def | associated.setoid | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated.symm",
"monoid"
] | The setoid of the relation `x ~ᵤ y` iff there is a unit `u` such that `x * u = y` | 335 | 336 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_associated_one [monoid α] {u : αˣ} : (u : α) ~ᵤ 1 := ⟨u⁻¹, units.mul_inv u⟩ | unit_associated_one [monoid α] {u : αˣ} : (u : α) ~ᵤ 1 | ⟨u⁻¹, units.mul_inv u⟩ | theorem | unit_associated_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid",
"units.mul_inv"
] | null | 342 | 342 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_one_iff_is_unit [monoid α] {a : α} : (a : α) ~ᵤ 1 ↔ is_unit a :=
iff.intro
(assume h, let ⟨c, h⟩ := h.symm in h ▸ ⟨c, (one_mul _).symm⟩)
(assume ⟨c, h⟩, associated.symm ⟨c, by simp [h]⟩) | associated_one_iff_is_unit [monoid α] {a : α} : (a : α) ~ᵤ 1 ↔ is_unit a | iff.intro
(assume h, let ⟨c, h⟩ := h.symm in h ▸ ⟨c, (one_mul _).symm⟩)
(assume ⟨c, h⟩, associated.symm ⟨c, by simp [h]⟩) | theorem | associated_one_iff_is_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.symm",
"is_unit",
"monoid",
"one_mul"
] | null | 344 | 347 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_zero_iff_eq_zero [monoid_with_zero α] (a : α) : a ~ᵤ 0 ↔ a = 0 :=
iff.intro
(assume h, let ⟨u, h⟩ := h.symm in by simpa using h.symm)
(assume h, h ▸ associated.refl a) | associated_zero_iff_eq_zero [monoid_with_zero α] (a : α) : a ~ᵤ 0 ↔ a = 0 | iff.intro
(assume h, let ⟨u, h⟩ := h.symm in by simpa using h.symm)
(assume h, h ▸ associated.refl a) | theorem | associated_zero_iff_eq_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.refl",
"monoid_with_zero"
] | null | 349 | 352 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_one_of_mul_eq_one [comm_monoid α] {a : α} (b : α) (hab : a * b = 1) : a ~ᵤ 1 :=
show (units.mk_of_mul_eq_one a b hab : α) ~ᵤ 1, from unit_associated_one | associated_one_of_mul_eq_one [comm_monoid α] {a : α} (b : α) (hab : a * b = 1) : a ~ᵤ 1 | show (units.mk_of_mul_eq_one a b hab : α) ~ᵤ 1, from unit_associated_one | theorem | associated_one_of_mul_eq_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid",
"unit_associated_one",
"units.mk_of_mul_eq_one"
] | null | 354 | 355 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_one_of_associated_mul_one [comm_monoid α] {a b : α} :
a * b ~ᵤ 1 → a ~ᵤ 1
| ⟨u, h⟩ := associated_one_of_mul_eq_one (b * u) $ by simpa [mul_assoc] using h | associated_one_of_associated_mul_one [comm_monoid α] {a b : α} :
a * b ~ᵤ 1 → a ~ᵤ 1
| ⟨u, h⟩ | associated_one_of_mul_eq_one (b * u) $ by simpa [mul_assoc] using h | theorem | associated_one_of_associated_mul_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_one_of_mul_eq_one",
"comm_monoid",
"mul_assoc"
] | null | 357 | 359 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_mul_unit_left {β : Type*} [monoid β] (a u : β) (hu : is_unit u) :
associated (a * u) a :=
let ⟨u', hu⟩ := hu in ⟨u'⁻¹, hu ▸ units.mul_inv_cancel_right _ _⟩ | associated_mul_unit_left {β : Type*} [monoid β] (a u : β) (hu : is_unit u) :
associated (a * u) a | let ⟨u', hu⟩ := hu in ⟨u'⁻¹, hu ▸ units.mul_inv_cancel_right _ _⟩ | lemma | associated_mul_unit_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"is_unit",
"monoid",
"units.mul_inv_cancel_right"
] | null | 361 | 363 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_unit_mul_left {β : Type*} [comm_monoid β] (a u : β) (hu : is_unit u) :
associated (u * a) a :=
begin
rw mul_comm,
exact associated_mul_unit_left _ _ hu
end | associated_unit_mul_left {β : Type*} [comm_monoid β] (a u : β) (hu : is_unit u) :
associated (u * a) a | begin
rw mul_comm,
exact associated_mul_unit_left _ _ hu
end | lemma | associated_unit_mul_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_mul_unit_left",
"comm_monoid",
"is_unit",
"mul_comm"
] | null | 365 | 370 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_mul_unit_right {β : Type*} [monoid β] (a u : β) (hu : is_unit u) :
associated a (a * u) :=
(associated_mul_unit_left a u hu).symm | associated_mul_unit_right {β : Type*} [monoid β] (a u : β) (hu : is_unit u) :
associated a (a * u) | (associated_mul_unit_left a u hu).symm | lemma | associated_mul_unit_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_mul_unit_left",
"is_unit",
"monoid"
] | null | 372 | 374 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_unit_mul_right {β : Type*} [comm_monoid β] (a u : β) (hu : is_unit u) :
associated a (u * a) :=
(associated_unit_mul_left a u hu).symm | associated_unit_mul_right {β : Type*} [comm_monoid β] (a u : β) (hu : is_unit u) :
associated a (u * a) | (associated_unit_mul_left a u hu).symm | lemma | associated_unit_mul_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_unit_mul_left",
"comm_monoid",
"is_unit"
] | null | 376 | 378 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_mul_is_unit_left_iff {β : Type*} [monoid β] {a u b : β} (hu : is_unit u) :
associated (a * u) b ↔ associated a b :=
⟨trans (associated_mul_unit_right _ _ hu), trans (associated_mul_unit_left _ _ hu)⟩ | associated_mul_is_unit_left_iff {β : Type*} [monoid β] {a u b : β} (hu : is_unit u) :
associated (a * u) b ↔ associated a b | ⟨trans (associated_mul_unit_right _ _ hu), trans (associated_mul_unit_left _ _ hu)⟩ | lemma | associated_mul_is_unit_left_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_mul_unit_left",
"associated_mul_unit_right",
"is_unit",
"monoid"
] | null | 380 | 382 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_is_unit_mul_left_iff {β : Type*} [comm_monoid β] {u a b : β} (hu : is_unit u) :
associated (u * a) b ↔ associated a b :=
begin
rw mul_comm,
exact associated_mul_is_unit_left_iff hu
end | associated_is_unit_mul_left_iff {β : Type*} [comm_monoid β] {u a b : β} (hu : is_unit u) :
associated (u * a) b ↔ associated a b | begin
rw mul_comm,
exact associated_mul_is_unit_left_iff hu
end | lemma | associated_is_unit_mul_left_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_mul_is_unit_left_iff",
"comm_monoid",
"is_unit",
"mul_comm"
] | null | 384 | 389 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_mul_is_unit_right_iff {β : Type*} [monoid β] {a b u : β} (hu : is_unit u) :
associated a (b * u) ↔ associated a b :=
associated.comm.trans $ (associated_mul_is_unit_left_iff hu).trans associated.comm | associated_mul_is_unit_right_iff {β : Type*} [monoid β] {a b u : β} (hu : is_unit u) :
associated a (b * u) ↔ associated a b | associated.comm.trans $ (associated_mul_is_unit_left_iff hu).trans associated.comm | lemma | associated_mul_is_unit_right_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated.comm",
"associated_mul_is_unit_left_iff",
"is_unit",
"monoid"
] | null | 391 | 393 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_is_unit_mul_right_iff {β : Type*} [comm_monoid β] {a u b : β} (hu : is_unit u) :
associated a (u * b) ↔ associated a b :=
associated.comm.trans $ (associated_is_unit_mul_left_iff hu).trans associated.comm | associated_is_unit_mul_right_iff {β : Type*} [comm_monoid β] {a u b : β} (hu : is_unit u) :
associated a (u * b) ↔ associated a b | associated.comm.trans $ (associated_is_unit_mul_left_iff hu).trans associated.comm | lemma | associated_is_unit_mul_right_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated.comm",
"associated_is_unit_mul_left_iff",
"comm_monoid",
"is_unit"
] | null | 395 | 397 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_mul_unit_left_iff {β : Type*} [monoid β] {a b : β} {u : units β} :
associated (a * u) b ↔ associated a b :=
associated_mul_is_unit_left_iff u.is_unit | associated_mul_unit_left_iff {β : Type*} [monoid β] {a b : β} {u : units β} :
associated (a * u) b ↔ associated a b | associated_mul_is_unit_left_iff u.is_unit | lemma | associated_mul_unit_left_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_mul_is_unit_left_iff",
"monoid",
"units"
] | null | 399 | 402 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_unit_mul_left_iff {β : Type*} [comm_monoid β] {a b : β} {u : units β} :
associated (↑u * a) b ↔ associated a b :=
associated_is_unit_mul_left_iff u.is_unit | associated_unit_mul_left_iff {β : Type*} [comm_monoid β] {a b : β} {u : units β} :
associated (↑u * a) b ↔ associated a b | associated_is_unit_mul_left_iff u.is_unit | lemma | associated_unit_mul_left_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_is_unit_mul_left_iff",
"comm_monoid",
"units"
] | null | 404 | 407 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_mul_unit_right_iff {β : Type*} [monoid β] {a b : β} {u : units β} :
associated a (b * u) ↔ associated a b :=
associated_mul_is_unit_right_iff u.is_unit | associated_mul_unit_right_iff {β : Type*} [monoid β] {a b : β} {u : units β} :
associated a (b * u) ↔ associated a b | associated_mul_is_unit_right_iff u.is_unit | lemma | associated_mul_unit_right_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_mul_is_unit_right_iff",
"monoid",
"units"
] | null | 409 | 412 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_unit_mul_right_iff {β : Type*} [comm_monoid β] {a b : β} {u : units β} :
associated a (↑u * b) ↔ associated a b :=
associated_is_unit_mul_right_iff u.is_unit | associated_unit_mul_right_iff {β : Type*} [comm_monoid β] {a b : β} {u : units β} :
associated a (↑u * b) ↔ associated a b | associated_is_unit_mul_right_iff u.is_unit | lemma | associated_unit_mul_right_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_is_unit_mul_right_iff",
"comm_monoid",
"units"
] | null | 414 | 417 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.mul_mul [comm_monoid α] {a₁ a₂ b₁ b₂ : α} :
a₁ ~ᵤ b₁ → a₂ ~ᵤ b₂ → (a₁ * a₂) ~ᵤ (b₁ * b₂)
| ⟨c₁, h₁⟩ ⟨c₂, h₂⟩ := ⟨c₁ * c₂, by simp [h₁.symm, h₂.symm, mul_assoc, mul_comm, mul_left_comm]⟩ | associated.mul_mul [comm_monoid α] {a₁ a₂ b₁ b₂ : α} :
a₁ ~ᵤ b₁ → a₂ ~ᵤ b₂ → (a₁ * a₂) ~ᵤ (b₁ * b₂)
| ⟨c₁, h₁⟩ ⟨c₂, h₂⟩ | ⟨c₁ * c₂, by simp [h₁.symm, h₂.symm, mul_assoc, mul_comm, mul_left_comm]⟩ | lemma | associated.mul_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid",
"mul_assoc",
"mul_comm",
"mul_left_comm"
] | null | 419 | 421 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.mul_left [comm_monoid α] (a : α) {b c : α} (h : b ~ᵤ c) :
(a * b) ~ᵤ (a * c) :=
(associated.refl a).mul_mul h | associated.mul_left [comm_monoid α] (a : α) {b c : α} (h : b ~ᵤ c) :
(a * b) ~ᵤ (a * c) | (associated.refl a).mul_mul h | lemma | associated.mul_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.refl",
"comm_monoid"
] | null | 423 | 425 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.mul_right [comm_monoid α] {a b : α} (h : a ~ᵤ b) (c : α) :
(a * c) ~ᵤ (b * c) :=
h.mul_mul (associated.refl c) | associated.mul_right [comm_monoid α] {a b : α} (h : a ~ᵤ b) (c : α) :
(a * c) ~ᵤ (b * c) | h.mul_mul (associated.refl c) | lemma | associated.mul_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.refl",
"comm_monoid"
] | null | 427 | 429 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.pow_pow [comm_monoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) :
a ^ n ~ᵤ b ^ n :=
begin
induction n with n ih, { simp [h] },
convert h.mul_mul ih;
rw pow_succ
end | associated.pow_pow [comm_monoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) :
a ^ n ~ᵤ b ^ n | begin
induction n with n ih, { simp [h] },
convert h.mul_mul ih;
rw pow_succ
end | lemma | associated.pow_pow | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid",
"ih",
"pow_succ"
] | null | 431 | 437 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.dvd [monoid α] {a b : α} : a ~ᵤ b → a ∣ b := λ ⟨u, hu⟩, ⟨u, hu.symm⟩ | associated.dvd [monoid α] {a b : α} : a ~ᵤ b → a ∣ b | λ ⟨u, hu⟩, ⟨u, hu.symm⟩ | lemma | associated.dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid"
] | null | 439 | 439 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.dvd_dvd [monoid α] {a b : α} (h : a ~ᵤ b) : a ∣ b ∧ b ∣ a :=
⟨h.dvd, h.symm.dvd⟩ | associated.dvd_dvd [monoid α] {a b : α} (h : a ~ᵤ b) : a ∣ b ∧ b ∣ a | ⟨h.dvd, h.symm.dvd⟩ | lemma | associated.dvd_dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid"
] | null | 441 | 442 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_of_dvd_dvd [cancel_monoid_with_zero α]
{a b : α} (hab : a ∣ b) (hba : b ∣ a) : a ~ᵤ b :=
begin
rcases hab with ⟨c, rfl⟩,
rcases hba with ⟨d, a_eq⟩,
by_cases ha0 : a = 0,
{ simp [*] at * },
have hac0 : a * c ≠ 0,
{ intro con, rw [con, zero_mul] at a_eq, apply ha0 a_eq, },
have : a * (c * d) = ... | associated_of_dvd_dvd [cancel_monoid_with_zero α]
{a b : α} (hab : a ∣ b) (hba : b ∣ a) : a ~ᵤ b | begin
rcases hab with ⟨c, rfl⟩,
rcases hba with ⟨d, a_eq⟩,
by_cases ha0 : a = 0,
{ simp [*] at * },
have hac0 : a * c ≠ 0,
{ intro con, rw [con, zero_mul] at a_eq, apply ha0 a_eq, },
have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one],
have hcd : (c * d) = 1, from mul_left_cancel₀ ha0 th... | theorem | associated_of_dvd_dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_monoid_with_zero",
"con",
"mul_assoc",
"mul_left_cancel₀",
"mul_one",
"zero_mul"
] | null | 444 | 458 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_dvd_iff_associated [cancel_monoid_with_zero α] {a b : α} : a ∣ b ∧ b ∣ a ↔ a ~ᵤ b :=
⟨λ ⟨h1, h2⟩, associated_of_dvd_dvd h1 h2, associated.dvd_dvd⟩ | dvd_dvd_iff_associated [cancel_monoid_with_zero α] {a b : α} : a ∣ b ∧ b ∣ a ↔ a ~ᵤ b | ⟨λ ⟨h1, h2⟩, associated_of_dvd_dvd h1 h2, associated.dvd_dvd⟩ | theorem | dvd_dvd_iff_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_of_dvd_dvd",
"cancel_monoid_with_zero"
] | null | 460 | 461 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[cancel_monoid_with_zero α] [decidable_rel ((∣) : α → α → Prop)] :
decidable_rel ((~ᵤ) : α → α → Prop) :=
λ a b, decidable_of_iff _ dvd_dvd_iff_associated | [cancel_monoid_with_zero α] [decidable_rel ((∣) : α → α → Prop)] :
decidable_rel ((~ᵤ) : α → α → Prop) | λ a b, decidable_of_iff _ dvd_dvd_iff_associated | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_monoid_with_zero",
"decidable_of_iff",
"dvd_dvd_iff_associated"
] | null | 463 | 465 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated.dvd_iff_dvd_left [monoid α] {a b c : α} (h : a ~ᵤ b) : a ∣ c ↔ b ∣ c :=
let ⟨u, hu⟩ := h in hu ▸ units.mul_right_dvd.symm | associated.dvd_iff_dvd_left [monoid α] {a b c : α} (h : a ~ᵤ b) : a ∣ c ↔ b ∣ c | let ⟨u, hu⟩ := h in hu ▸ units.mul_right_dvd.symm | lemma | associated.dvd_iff_dvd_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid"
] | null | 467 | 468 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.dvd_iff_dvd_right [monoid α] {a b c : α} (h : b ~ᵤ c) : a ∣ b ↔ a ∣ c :=
let ⟨u, hu⟩ := h in hu ▸ units.dvd_mul_right.symm | associated.dvd_iff_dvd_right [monoid α] {a b c : α} (h : b ~ᵤ c) : a ∣ b ↔ a ∣ c | let ⟨u, hu⟩ := h in hu ▸ units.dvd_mul_right.symm | lemma | associated.dvd_iff_dvd_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid"
] | null | 470 | 471 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.eq_zero_iff [monoid_with_zero α] {a b : α} (h : a ~ᵤ b) : a = 0 ↔ b = 0 :=
⟨λ ha, let ⟨u, hu⟩ := h in by simp [hu.symm, ha],
λ hb, let ⟨u, hu⟩ := h.symm in by simp [hu.symm, hb]⟩ | associated.eq_zero_iff [monoid_with_zero α] {a b : α} (h : a ~ᵤ b) : a = 0 ↔ b = 0 | ⟨λ ha, let ⟨u, hu⟩ := h in by simp [hu.symm, ha],
λ hb, let ⟨u, hu⟩ := h.symm in by simp [hu.symm, hb]⟩ | lemma | associated.eq_zero_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid_with_zero"
] | null | 473 | 475 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.ne_zero_iff [monoid_with_zero α] {a b : α} (h : a ~ᵤ b) : a ≠ 0 ↔ b ≠ 0 :=
not_congr h.eq_zero_iff | associated.ne_zero_iff [monoid_with_zero α] {a b : α} (h : a ~ᵤ b) : a ≠ 0 ↔ b ≠ 0 | not_congr h.eq_zero_iff | lemma | associated.ne_zero_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"monoid_with_zero"
] | null | 477 | 478 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.prime [comm_monoid_with_zero α] {p q : α} (h : p ~ᵤ q) (hp : prime p) :
prime q :=
⟨h.ne_zero_iff.1 hp.ne_zero,
let ⟨u, hu⟩ := h in
⟨λ ⟨v, hv⟩, hp.not_unit ⟨v * u⁻¹, by simp [hv, hu.symm]⟩,
hu ▸ by { simp [units.mul_right_dvd], intros a b, exact hp.dvd_or_dvd }⟩⟩ | associated.prime [comm_monoid_with_zero α] {p q : α} (h : p ~ᵤ q) (hp : prime p) :
prime q | ⟨h.ne_zero_iff.1 hp.ne_zero,
let ⟨u, hu⟩ := h in
⟨λ ⟨v, hv⟩, hp.not_unit ⟨v * u⁻¹, by simp [hv, hu.symm]⟩,
hu ▸ by { simp [units.mul_right_dvd], intros a b, exact hp.dvd_or_dvd }⟩⟩ | lemma | associated.prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"prime",
"units.mul_right_dvd"
] | null | 480 | 485 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible.associated_of_dvd [cancel_monoid_with_zero α] {p q : α}
(p_irr : irreducible p) (q_irr : irreducible q) (dvd : p ∣ q) : associated p q :=
associated_of_dvd_dvd dvd (p_irr.dvd_symm q_irr dvd) | irreducible.associated_of_dvd [cancel_monoid_with_zero α] {p q : α}
(p_irr : irreducible p) (q_irr : irreducible q) (dvd : p ∣ q) : associated p q | associated_of_dvd_dvd dvd (p_irr.dvd_symm q_irr dvd) | lemma | irreducible.associated_of_dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_of_dvd_dvd",
"cancel_monoid_with_zero",
"irreducible"
] | null | 487 | 489 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible.dvd_irreducible_iff_associated [cancel_monoid_with_zero α]
{p q : α} (pp : irreducible p) (qp : irreducible q) :
p ∣ q ↔ associated p q :=
⟨irreducible.associated_of_dvd pp qp, associated.dvd⟩ | irreducible.dvd_irreducible_iff_associated [cancel_monoid_with_zero α]
{p q : α} (pp : irreducible p) (qp : irreducible q) :
p ∣ q ↔ associated p q | ⟨irreducible.associated_of_dvd pp qp, associated.dvd⟩ | lemma | irreducible.dvd_irreducible_iff_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"cancel_monoid_with_zero",
"irreducible"
] | null | 491 | 494 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.associated_of_dvd [cancel_comm_monoid_with_zero α] {p q : α}
(p_prime : prime p) (q_prime : prime q) (dvd : p ∣ q) : associated p q :=
p_prime.irreducible.associated_of_dvd q_prime.irreducible dvd | prime.associated_of_dvd [cancel_comm_monoid_with_zero α] {p q : α}
(p_prime : prime p) (q_prime : prime q) (dvd : p ∣ q) : associated p q | p_prime.irreducible.associated_of_dvd q_prime.irreducible dvd | lemma | prime.associated_of_dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"prime"
] | null | 496 | 498 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.dvd_prime_iff_associated [cancel_comm_monoid_with_zero α]
{p q : α} (pp : prime p) (qp : prime q) :
p ∣ q ↔ associated p q :=
pp.irreducible.dvd_irreducible_iff_associated qp.irreducible | prime.dvd_prime_iff_associated [cancel_comm_monoid_with_zero α]
{p q : α} (pp : prime p) (qp : prime q) :
p ∣ q ↔ associated p q | pp.irreducible.dvd_irreducible_iff_associated qp.irreducible | theorem | prime.dvd_prime_iff_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"prime"
] | null | 500 | 503 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.prime_iff [comm_monoid_with_zero α] {p q : α}
(h : p ~ᵤ q) : prime p ↔ prime q :=
⟨h.prime, h.symm.prime⟩ | associated.prime_iff [comm_monoid_with_zero α] {p q : α}
(h : p ~ᵤ q) : prime p ↔ prime q | ⟨h.prime, h.symm.prime⟩ | lemma | associated.prime_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"prime"
] | null | 504 | 506 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.is_unit [monoid α] {a b : α} (h : a ~ᵤ b) : is_unit a → is_unit b :=
let ⟨u, hu⟩ := h in λ ⟨v, hv⟩, ⟨v * u, by simp [hv, hu.symm]⟩ | associated.is_unit [monoid α] {a b : α} (h : a ~ᵤ b) : is_unit a → is_unit b | let ⟨u, hu⟩ := h in λ ⟨v, hv⟩, ⟨v * u, by simp [hv, hu.symm]⟩ | lemma | associated.is_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit",
"monoid"
] | null | 508 | 509 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.is_unit_iff [monoid α] {a b : α} (h : a ~ᵤ b) : is_unit a ↔ is_unit b :=
⟨h.is_unit, h.symm.is_unit⟩ | associated.is_unit_iff [monoid α] {a b : α} (h : a ~ᵤ b) : is_unit a ↔ is_unit b | ⟨h.is_unit, h.symm.is_unit⟩ | lemma | associated.is_unit_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit",
"monoid"
] | null | 511 | 512 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.irreducible [monoid α] {p q : α} (h : p ~ᵤ q)
(hp : irreducible p) : irreducible q :=
⟨mt h.symm.is_unit hp.1,
let ⟨u, hu⟩ := h in λ a b hab,
have hpab : p = a * (b * (u⁻¹ : αˣ)),
from calc p = (p * u) * (u ⁻¹ : αˣ) : by simp
... = _ : by rw hu; simp [hab, mul_assoc],
(hp.is_unit_or_is_unit... | associated.irreducible [monoid α] {p q : α} (h : p ~ᵤ q)
(hp : irreducible p) : irreducible q | ⟨mt h.symm.is_unit hp.1,
let ⟨u, hu⟩ := h in λ a b hab,
have hpab : p = a * (b * (u⁻¹ : αˣ)),
from calc p = (p * u) * (u ⁻¹ : αˣ) : by simp
... = _ : by rw hu; simp [hab, mul_assoc],
(hp.is_unit_or_is_unit hpab).elim or.inl (λ ⟨v, hv⟩, or.inr ⟨v * u, by simp [hv]⟩)⟩ | lemma | associated.irreducible | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"monoid",
"mul_assoc"
] | null | 514 | 521 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.irreducible_iff [monoid α] {p q : α} (h : p ~ᵤ q) :
irreducible p ↔ irreducible q :=
⟨h.irreducible, h.symm.irreducible⟩ | associated.irreducible_iff [monoid α] {p q : α} (h : p ~ᵤ q) :
irreducible p ↔ irreducible q | ⟨h.irreducible, h.symm.irreducible⟩ | lemma | associated.irreducible_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"monoid"
] | null | 523 | 525 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.of_mul_left [cancel_comm_monoid_with_zero α] {a b c d : α}
(h : a * b ~ᵤ c * d) (h₁ : a ~ᵤ c) (ha : a ≠ 0) : b ~ᵤ d :=
let ⟨u, hu⟩ := h in let ⟨v, hv⟩ := associated.symm h₁ in
⟨u * (v : αˣ), mul_left_cancel₀ ha
begin
rw [← hv, mul_assoc c (v : α) d, mul_left_comm c, ← hu],
simp [hv.symm, mul_asso... | associated.of_mul_left [cancel_comm_monoid_with_zero α] {a b c d : α}
(h : a * b ~ᵤ c * d) (h₁ : a ~ᵤ c) (ha : a ≠ 0) : b ~ᵤ d | let ⟨u, hu⟩ := h in let ⟨v, hv⟩ := associated.symm h₁ in
⟨u * (v : αˣ), mul_left_cancel₀ ha
begin
rw [← hv, mul_assoc c (v : α) d, mul_left_comm c, ← hu],
simp [hv.symm, mul_assoc, mul_comm, mul_left_comm]
end⟩ | lemma | associated.of_mul_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.symm",
"cancel_comm_monoid_with_zero",
"mul_assoc",
"mul_comm",
"mul_left_cancel₀",
"mul_left_comm"
] | null | 527 | 534 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.of_mul_right [cancel_comm_monoid_with_zero α] {a b c d : α} :
a * b ~ᵤ c * d → b ~ᵤ d → b ≠ 0 → a ~ᵤ c :=
by rw [mul_comm a, mul_comm c]; exact associated.of_mul_left | associated.of_mul_right [cancel_comm_monoid_with_zero α] {a b c d : α} :
a * b ~ᵤ c * d → b ~ᵤ d → b ≠ 0 → a ~ᵤ c | by rw [mul_comm a, mul_comm c]; exact associated.of_mul_left | lemma | associated.of_mul_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.of_mul_left",
"cancel_comm_monoid_with_zero",
"mul_comm"
] | null | 536 | 538 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.of_pow_associated_of_prime [cancel_comm_monoid_with_zero α] {p₁ p₂ : α}
{k₁ k₂ : ℕ} (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₁ : 0 < k₁) (h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂) :
p₁ ~ᵤ p₂ :=
begin
have : p₁ ∣ p₂ ^ k₂,
{ rw ←h.dvd_iff_dvd_right,
apply dvd_pow_self _ hk₁.ne' },
rw ←hp₁.dvd_prime_iff_associated hp₂,... | associated.of_pow_associated_of_prime [cancel_comm_monoid_with_zero α] {p₁ p₂ : α}
{k₁ k₂ : ℕ} (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₁ : 0 < k₁) (h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂) :
p₁ ~ᵤ p₂ | begin
have : p₁ ∣ p₂ ^ k₂,
{ rw ←h.dvd_iff_dvd_right,
apply dvd_pow_self _ hk₁.ne' },
rw ←hp₁.dvd_prime_iff_associated hp₂,
exact hp₁.dvd_of_dvd_pow this,
end | lemma | associated.of_pow_associated_of_prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"dvd_pow_self",
"prime"
] | null | 540 | 549 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated.of_pow_associated_of_prime' [cancel_comm_monoid_with_zero α] {p₁ p₂ : α}
{k₁ k₂ : ℕ} (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₂ : 0 < k₂) (h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂) :
p₁ ~ᵤ p₂ :=
(h.symm.of_pow_associated_of_prime hp₂ hp₁ hk₂).symm | associated.of_pow_associated_of_prime' [cancel_comm_monoid_with_zero α] {p₁ p₂ : α}
{k₁ k₂ : ℕ} (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₂ : 0 < k₂) (h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂) :
p₁ ~ᵤ p₂ | (h.symm.of_pow_associated_of_prime hp₂ hp₁ hk₂).symm | lemma | associated.of_pow_associated_of_prime' | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"prime"
] | null | 551 | 554 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_eq_one (u : αˣ) : u = 1 := subsingleton.elim u 1 | units_eq_one (u : αˣ) : u = 1 | subsingleton.elim u 1 | lemma | units_eq_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [] | null | 559 | 559 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_iff_eq {x y : α} : x ~ᵤ y ↔ x = y :=
begin
split,
{ rintro ⟨c, rfl⟩, rw [units_eq_one c, units.coe_one, mul_one] },
{ rintro rfl, refl },
end | associated_iff_eq {x y : α} : x ~ᵤ y ↔ x = y | begin
split,
{ rintro ⟨c, rfl⟩, rw [units_eq_one c, units.coe_one, mul_one] },
{ rintro rfl, refl },
end | theorem | associated_iff_eq | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"mul_one",
"units.coe_one",
"units_eq_one"
] | null | 561 | 566 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_eq_eq : (associated : α → α → Prop) = eq :=
by { ext, rw associated_iff_eq } | associated_eq_eq : (associated : α → α → Prop) = eq | by { ext, rw associated_iff_eq } | theorem | associated_eq_eq | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_iff_eq"
] | null | 568 | 569 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_dvd_prime_iff_eq
{M : Type*} [cancel_comm_monoid_with_zero M] [unique Mˣ] {p q : M} (pp : prime p) (qp : prime q) :
p ∣ q ↔ p = q :=
by rw [pp.dvd_prime_iff_associated qp, ←associated_eq_eq] | prime_dvd_prime_iff_eq
{M : Type*} [cancel_comm_monoid_with_zero M] [unique Mˣ] {p q : M} (pp : prime p) (qp : prime q) :
p ∣ q ↔ p = q | by rw [pp.dvd_prime_iff_associated qp, ←associated_eq_eq] | lemma | prime_dvd_prime_iff_eq | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"prime",
"unique"
] | null | 571 | 574 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_prime_pow_eq (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₁ : 0 < k₁) (h : p₁ ^ k₁ = p₂ ^ k₂) :
p₁ = p₂ :=
by { rw [←associated_iff_eq] at h ⊢, apply h.of_pow_associated_of_prime hp₁ hp₂ hk₁ } | eq_of_prime_pow_eq (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₁ : 0 < k₁) (h : p₁ ^ k₁ = p₂ ^ k₂) :
p₁ = p₂ | by { rw [←associated_iff_eq] at h ⊢, apply h.of_pow_associated_of_prime hp₁ hp₂ hk₁ } | lemma | eq_of_prime_pow_eq | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"prime"
] | null | 582 | 584 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_prime_pow_eq' (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₁ : 0 < k₂) (h : p₁ ^ k₁ = p₂ ^ k₂) :
p₁ = p₂ :=
by { rw [←associated_iff_eq] at h ⊢, apply h.of_pow_associated_of_prime' hp₁ hp₂ hk₁ } | eq_of_prime_pow_eq' (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₁ : 0 < k₂) (h : p₁ ^ k₁ = p₂ ^ k₂) :
p₁ = p₂ | by { rw [←associated_iff_eq] at h ⊢, apply h.of_pow_associated_of_prime' hp₁ hp₂ hk₁ } | lemma | eq_of_prime_pow_eq' | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"prime"
] | null | 586 | 588 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associates (α : Type*) [monoid α] : Type* :=
quotient (associated.setoid α) | associates (α : Type*) [monoid α] : Type* | quotient (associated.setoid α) | def | associates | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.setoid",
"monoid"
] | The quotient of a monoid by the `associated` relation. Two elements `x` and `y`
are associated iff there is a unit `u` such that `x * u = y`. There is a natural
monoid structure on `associates α`. | 595 | 596 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {α : Type*} [monoid α] (a : α) : associates α :=
⟦ a ⟧ | mk {α : Type*} [monoid α] (a : α) : associates α | ⟦ a ⟧ | def | associates.mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"monoid"
] | The canonical quotient map from a monoid `α` into the `associates` of `α` | 602 | 603 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] : inhabited (associates α) := ⟨⟦1⟧⟩ | [monoid α] : inhabited (associates α) | ⟨⟦1⟧⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"monoid"
] | null | 605 | 605 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_mk_iff_associated [monoid α] {a b : α} :
associates.mk a = associates.mk b ↔ a ~ᵤ b :=
iff.intro quotient.exact quot.sound | mk_eq_mk_iff_associated [monoid α] {a b : α} :
associates.mk a = associates.mk b ↔ a ~ᵤ b | iff.intro quotient.exact quot.sound | theorem | associates.mk_eq_mk_iff_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"monoid"
] | null | 607 | 609 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_mk_eq_mk [monoid α] (a : α) : ⟦ a ⟧ = associates.mk a := rfl | quotient_mk_eq_mk [monoid α] (a : α) : ⟦ a ⟧ = associates.mk a | rfl | theorem | associates.quotient_mk_eq_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"monoid"
] | null | 611 | 611 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mk_eq_mk [monoid α] (a : α) : quot.mk setoid.r a = associates.mk a := rfl | quot_mk_eq_mk [monoid α] (a : α) : quot.mk setoid.r a = associates.mk a | rfl | theorem | associates.quot_mk_eq_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"monoid"
] | null | 613 | 613 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forall_associated [monoid α] {p : associates α → Prop} :
(∀a, p a) ↔ (∀a, p (associates.mk a)) :=
iff.intro
(assume h a, h _)
(assume h a, quotient.induction_on a h) | forall_associated [monoid α] {p : associates α → Prop} :
(∀a, p a) ↔ (∀a, p (associates.mk a)) | iff.intro
(assume h a, h _)
(assume h a, quotient.induction_on a h) | theorem | associates.forall_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.mk",
"monoid"
] | null | 615 | 619 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_surjective [monoid α] : function.surjective (@associates.mk α _) :=
forall_associated.2 (λ a, ⟨a, rfl⟩) | mk_surjective [monoid α] : function.surjective (@associates.mk α _) | forall_associated.2 (λ a, ⟨a, rfl⟩) | theorem | associates.mk_surjective | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"monoid"
] | null | 621 | 622 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] : has_one (associates α) := ⟨⟦ 1 ⟧⟩ | [monoid α] : has_one (associates α) | ⟨⟦ 1 ⟧⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"monoid"
] | null | 624 | 624 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_one [monoid α] : associates.mk (1 : α) = 1 := rfl | mk_one [monoid α] : associates.mk (1 : α) = 1 | rfl | lemma | associates.mk_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"monoid"
] | null | 626 | 626 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_eq_mk_one [monoid α] : (1 : associates α) = associates.mk 1 := rfl | one_eq_mk_one [monoid α] : (1 : associates α) = associates.mk 1 | rfl | theorem | associates.one_eq_mk_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.mk",
"monoid"
] | null | 628 | 628 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] : has_bot (associates α) := ⟨1⟩ | [monoid α] : has_bot (associates α) | ⟨1⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"has_bot",
"monoid"
] | null | 630 | 630 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_eq_one [monoid α] : (⊥ : associates α) = 1 := rfl | bot_eq_one [monoid α] : (⊥ : associates α) = 1 | rfl | lemma | associates.bot_eq_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"bot_eq_one",
"monoid"
] | null | 632 | 632 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_rep [monoid α] (a : associates α) : ∃ a0 : α, associates.mk a0 = a :=
quot.exists_rep a | exists_rep [monoid α] (a : associates α) : ∃ a0 : α, associates.mk a0 = a | quot.exists_rep a | lemma | associates.exists_rep | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.mk",
"monoid"
] | null | 634 | 635 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] [subsingleton α] : unique (associates α) :=
{ default := 1,
uniq := λ a, by { apply quotient.rec_on_subsingleton₂, intros a b, congr } } | [monoid α] [subsingleton α] : unique (associates α) | { default := 1,
uniq := λ a, by { apply quotient.rec_on_subsingleton₂, intros a b, congr } } | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"monoid",
"unique"
] | null | 637 | 639 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_injective [monoid α] [unique (units α)] : function.injective (@associates.mk α _) :=
λ a b h, associated_iff_eq.mp (associates.mk_eq_mk_iff_associated.mp h) | mk_injective [monoid α] [unique (units α)] : function.injective (@associates.mk α _) | λ a b h, associated_iff_eq.mp (associates.mk_eq_mk_iff_associated.mp h) | lemma | associates.mk_injective | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"monoid",
"unique",
"units"
] | null | 641 | 642 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: has_mul (associates α) :=
⟨λa' b', quotient.lift_on₂ a' b' (λa b, ⟦ a * b ⟧) $
assume a₁ a₂ b₁ b₂ ⟨c₁, h₁⟩ ⟨c₂, h₂⟩,
quotient.sound $ ⟨c₁ * c₂, by simp [h₁.symm, h₂.symm, mul_assoc, mul_comm, mul_left_comm]⟩⟩ | : has_mul (associates α) | ⟨λa' b', quotient.lift_on₂ a' b' (λa b, ⟦ a * b ⟧) $
assume a₁ a₂ b₁ b₂ ⟨c₁, h₁⟩ ⟨c₂, h₂⟩,
quotient.sound $ ⟨c₁ * c₂, by simp [h₁.symm, h₂.symm, mul_assoc, mul_comm, mul_left_comm]⟩⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"mul_assoc",
"mul_comm",
"mul_left_comm"
] | null | 647 | 650 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_mk {x y : α} : associates.mk x * associates.mk y = associates.mk (x * y) :=
rfl | mk_mul_mk {x y : α} : associates.mk x * associates.mk y = associates.mk (x * y) | rfl | theorem | associates.mk_mul_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk"
] | null | 652 | 653 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: comm_monoid (associates α) :=
{ one := 1,
mul := (*),
mul_one := assume a', quotient.induction_on a' $
assume a, show ⟦a * 1⟧ = ⟦ a ⟧, by simp,
one_mul := assume a', quotient.induction_on a' $
assume a, show ⟦1 * a⟧ = ⟦ a ⟧, by simp,
mul_assoc := assume a' b' c', quotient.induction_on₃... | : comm_monoid (associates α) | { one := 1,
mul := (*),
mul_one := assume a', quotient.induction_on a' $
assume a, show ⟦a * 1⟧ = ⟦ a ⟧, by simp,
one_mul := assume a', quotient.induction_on a' $
assume a, show ⟦1 * a⟧ = ⟦ a ⟧, by simp,
mul_assoc := assume a' b' c', quotient.induction_on₃ a' b' c' $
assume a b c, sh... | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"comm_monoid",
"mul_assoc",
"mul_comm",
"mul_one",
"one_mul"
] | null | 655 | 665 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: preorder (associates α) :=
{ le := has_dvd.dvd,
le_refl := dvd_refl,
le_trans := λ a b c, dvd_trans} | : preorder (associates α) | { le := has_dvd.dvd,
le_refl := dvd_refl,
le_trans := λ a b c, dvd_trans} | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"dvd_refl",
"dvd_trans"
] | null | 667 | 670 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_monoid_hom : α →* (associates α) := ⟨associates.mk, mk_one, λ x y, mk_mul_mk⟩ | mk_monoid_hom : α →* (associates α) | ⟨associates.mk, mk_one, λ x y, mk_mul_mk⟩ | def | associates.mk_monoid_hom | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates"
] | `associates.mk` as a `monoid_hom`. | 673 | 673 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_monoid_hom_apply (a : α) : associates.mk_monoid_hom a = associates.mk a := rfl | mk_monoid_hom_apply (a : α) : associates.mk_monoid_hom a = associates.mk a | rfl | lemma | associates.mk_monoid_hom_apply | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"associates.mk_monoid_hom"
] | null | 675 | 675 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_map_mk {f : associates α →* α}
(hinv : function.right_inverse f associates.mk) (a : α) :
a ~ᵤ f (associates.mk a) :=
associates.mk_eq_mk_iff_associated.1 (hinv (associates.mk a)).symm | associated_map_mk {f : associates α →* α}
(hinv : function.right_inverse f associates.mk) (a : α) :
a ~ᵤ f (associates.mk a) | associates.mk_eq_mk_iff_associated.1 (hinv (associates.mk a)).symm | lemma | associates.associated_map_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.mk"
] | null | 677 | 680 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pow (a : α) (n : ℕ) : associates.mk (a ^ n) = (associates.mk a) ^ n :=
by induction n; simp [*, pow_succ, associates.mk_mul_mk.symm] | mk_pow (a : α) (n : ℕ) : associates.mk (a ^ n) = (associates.mk a) ^ n | by induction n; simp [*, pow_succ, associates.mk_mul_mk.symm] | lemma | associates.mk_pow | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"pow_succ"
] | null | 682 | 683 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_eq_le : ((∣) : associates α → associates α → Prop) = (≤) := rfl | dvd_eq_le : ((∣) : associates α → associates α → Prop) = (≤) | rfl | lemma | associates.dvd_eq_le | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates"
] | null | 685 | 685 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq_one_iff {x y : associates α} : x * y = 1 ↔ (x = 1 ∧ y = 1) :=
iff.intro
(quotient.induction_on₂ x y $ assume a b h,
have a * b ~ᵤ 1, from quotient.exact h,
⟨quotient.sound $ associated_one_of_associated_mul_one this,
quotient.sound $ associated_one_of_associated_mul_one $ by rwa [mul_comm] at thi... | mul_eq_one_iff {x y : associates α} : x * y = 1 ↔ (x = 1 ∧ y = 1) | iff.intro
(quotient.induction_on₂ x y $ assume a b h,
have a * b ~ᵤ 1, from quotient.exact h,
⟨quotient.sound $ associated_one_of_associated_mul_one this,
quotient.sound $ associated_one_of_associated_mul_one $ by rwa [mul_comm] at this⟩)
(by simp {contextual := tt}) | theorem | associates.mul_eq_one_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_one_of_associated_mul_one",
"associates",
"mul_comm",
"mul_eq_one_iff"
] | null | 687 | 693 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_eq_one (u : (associates α)ˣ) : u = 1 :=
units.ext (mul_eq_one_iff.1 u.val_inv).1 | units_eq_one (u : (associates α)ˣ) : u = 1 | units.ext (mul_eq_one_iff.1 u.val_inv).1 | theorem | associates.units_eq_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"units.ext",
"units_eq_one"
] | null | 695 | 696 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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