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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
lcm_dvd_lcm_mul_left_right [gcd_monoid α] (m n k : α) : lcm m n ∣ lcm m (k * n) | lcm_dvd_lcm dvd_rfl (dvd_mul_left _ _) | theorem | lcm_dvd_lcm_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_monoid",
"lcm_dvd_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd_lcm_mul_right_right [gcd_monoid α] (m n k : α) : lcm m n ∣ lcm m (n * k) | lcm_dvd_lcm dvd_rfl (dvd_mul_right _ _) | theorem | lcm_dvd_lcm_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_monoid",
"lcm_dvd_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_of_associated_left [normalized_gcd_monoid α] {m n : α}
(h : associated m n) (k : α) : lcm m k = lcm n k | dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
(lcm_dvd_lcm h.dvd dvd_rfl)
(lcm_dvd_lcm h.symm.dvd dvd_rfl) | theorem | lcm_eq_of_associated_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",
"lcm_dvd_lcm",
"normalize_lcm",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_of_associated_right [normalized_gcd_monoid α] {m n : α}
(h : associated m n) (k : α) : lcm k m = lcm k n | dvd_antisymm_of_normalize_eq (normalize_lcm _ _) (normalize_lcm _ _)
(lcm_dvd_lcm dvd_rfl h.dvd)
(lcm_dvd_lcm dvd_rfl h.symm.dvd) | theorem | lcm_eq_of_associated_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",
"lcm_dvd_lcm",
"normalize_lcm",
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_of_irreducible [gcd_monoid α] {x : α} (hi: irreducible x) : prime x | ⟨hi.ne_zero, ⟨hi.1, λ a b h,
begin
cases gcd_dvd_left x a with y hy,
cases hi.is_unit_or_is_unit hy with hu hu,
{ right, transitivity (gcd (x * b) (a * b)), apply dvd_gcd (dvd_mul_right x b) h,
rw (gcd_mul_right' b x a).dvd_iff_dvd_left,
exact (associated_unit_mul_left _ _ hu).dvd },
{ left,
rw hy,
... | theorem | gcd_monoid.prime_of_irreducible | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated_mul_unit_left",
"associated_unit_mul_left",
"dvd_mul_right",
"dvd_trans",
"gcd_monoid",
"gcd_mul_right'",
"irreducible",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_iff_prime [gcd_monoid α] {p : α} : irreducible p ↔ prime p | ⟨prime_of_irreducible, prime.irreducible⟩ | theorem | gcd_monoid.irreducible_iff_prime | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_monoid",
"irreducible",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalization_monoid_of_unique_units : normalization_monoid α | { norm_unit := λ x, 1,
norm_unit_zero := rfl,
norm_unit_mul := λ x y hx hy, (mul_one 1).symm,
norm_unit_coe_units := λ u, subsingleton.elim _ _ } | instance | normalization_monoid_of_unique_units | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"mul_one",
"normalization_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_normalization_monoid_of_unique_units : unique (normalization_monoid α) | { default := normalization_monoid_of_unique_units,
uniq := λ ⟨u, _, _, _⟩, by simpa only [(subsingleton.elim _ _ : u = λ _, 1)] } | instance | unique_normalization_monoid_of_unique_units | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalization_monoid",
"normalization_monoid_of_unique_units",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_gcd_monoid_of_unique_units : subsingleton (gcd_monoid α) | ⟨λ g₁ g₂, begin
have hgcd : g₁.gcd = g₂.gcd,
{ ext a b,
refine associated_iff_eq.mp (associated_of_dvd_dvd _ _);
apply dvd_gcd (gcd_dvd_left _ _) (gcd_dvd_right _ _) },
have hlcm : g₁.lcm = g₂.lcm,
{ ext a b,
refine associated_iff_eq.mp (associated_of_dvd_dvd _ _);
apply lcm_dvd_iff.2 ⟨dvd_lcm_l... | instance | subsingleton_gcd_monoid_of_unique_units | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated_of_dvd_dvd",
"dvd_lcm_right",
"gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_normalized_gcd_monoid_of_unique_units :
subsingleton (normalized_gcd_monoid α) | ⟨begin
intros a b,
cases a with a_norm a_gcd,
cases b with b_norm b_gcd,
have := subsingleton.elim a_gcd b_gcd,
subst this,
have := subsingleton.elim a_norm b_norm,
subst this
end⟩ | instance | subsingleton_normalized_gcd_monoid_of_unique_units | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_unit_eq_one (x : α) : norm_unit x = 1 | rfl | lemma | norm_unit_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_eq (x : α) : normalize x = x | mul_one x | lemma | normalize_eq | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"mul_one",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates_equiv_of_unique_units : associates α ≃* α | { to_fun := associates.out,
inv_fun := associates.mk,
left_inv := associates.mk_out,
right_inv := λ t, (associates.out_mk _).trans $ normalize_eq _,
map_mul' := associates.out_mul } | def | associates_equiv_of_unique_units | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"associates.mk",
"associates.mk_out",
"associates.out",
"associates.out_mk",
"associates.out_mul",
"inv_fun",
"normalize_eq"
] | If a monoid's only unit is `1`, then it is isomorphic to its associates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_eq_of_dvd_sub_right {a b c : α} (h : a ∣ b - c) : gcd a b = gcd a c | begin
apply dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _);
rw dvd_gcd_iff; refine ⟨gcd_dvd_left _ _, _⟩,
{ rcases h with ⟨d, hd⟩,
rcases gcd_dvd_right a b with ⟨e, he⟩,
rcases gcd_dvd_left a b with ⟨f, hf⟩,
use e - f * d,
rw [mul_sub, ← he, ← mul_assoc, ← hf, ← hd, sub_sub_c... | lemma | gcd_eq_of_dvd_sub_right | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_antisymm_of_normalize_eq",
"dvd_gcd_iff",
"mul_assoc",
"normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_of_dvd_sub_left {a b c : α} (h : a ∣ b - c) : gcd b a = gcd c a | by rw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h] | lemma | gcd_eq_of_dvd_sub_left | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"gcd_comm",
"gcd_eq_of_dvd_sub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mk_unit_aux [decidable_eq α] {f : associates α →* α}
(hinv : function.right_inverse f associates.mk) (a : α) :
a * ↑(classical.some (associated_map_mk hinv a)) = f (associates.mk a) | classical.some_spec (associated_map_mk hinv a) | lemma | map_mk_unit_aux | 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 | |
normalization_monoid_of_monoid_hom_right_inverse [decidable_eq α] (f : associates α →* α)
(hinv : function.right_inverse f associates.mk) :
normalization_monoid α | { norm_unit := λ a, if a = 0 then 1 else
classical.some (associates.mk_eq_mk_iff_associated.1 (hinv (associates.mk a)).symm),
norm_unit_zero := if_pos rfl,
norm_unit_mul := λ a b ha hb, by
{ rw [if_neg (mul_ne_zero ha hb), if_neg ha, if_neg hb, units.ext_iff, units.coe_mul],
suffices : (a * b) * ↑(classic... | def | normalization_monoid_of_monoid_hom_right_inverse | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associates",
"associates.mk",
"associates.mk_mul_mk",
"associates.mk_one",
"map_mk_unit_aux",
"monoid_hom.map_mul",
"monoid_hom.map_one",
"mul_assoc",
"mul_comm",
"mul_left_cancel₀",
"mul_left_comm",
"mul_ne_zero",
"normalization_monoid",
"units.coe_mul",
"units.ext_iff",
"units.mul_i... | Define `normalization_monoid` on a structure from a `monoid_hom` inverse to `associates.mk`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_monoid_of_gcd [decidable_eq α] (gcd : 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_monoid α | { gcd := gcd,
gcd_dvd_left := gcd_dvd_left,
gcd_dvd_right := gcd_dvd_right,
dvd_gcd := λ a b c, dvd_gcd,
lcm := λ a b, if a = 0 then 0 else classical.some ((gcd_dvd_left a b).trans (dvd.intro b rfl)),
gcd_mul_lcm := λ a b, by
{ split_ifs with a0,
{ rw [mul_zero, a0, zero_mul] },
{ rw ←classical.some... | def | gcd_monoid_of_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"associated_of_dvd_dvd",
"dvd.intro",
"dvd_refl",
"dvd_zero",
"gcd_monoid",
"gcd_mul_lcm",
"mul_zero",
"zero_mul"
] | Define `gcd_monoid` on a structure just from the `gcd` and its properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_gcd_monoid_of_gcd [normalization_monoid α] [decidable_eq α]
(gcd : 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)
(normalize_gcd : ∀a b, normalize (gcd a b) = gcd a b) :
normalized_gcd_monoid α | { gcd := gcd,
gcd_dvd_left := gcd_dvd_left,
gcd_dvd_right := gcd_dvd_right,
dvd_gcd := λ a b c, dvd_gcd,
normalize_gcd := normalize_gcd,
lcm := λ a b, if a = 0 then 0 else classical.some (dvd_normalize_iff.2
((gcd_dvd_left a b).trans (dvd.intro b rfl))),
normalize_lcm := λ a b, by
{ dsimp [norma... | def | normalized_gcd_monoid_of_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd.intro",
"dvd_refl",
"dvd_zero",
"eq_or_ne",
"gcd_mul_lcm",
"mul_eq_zero",
"mul_ne_zero",
"mul_right_inj'",
"mul_zero",
"normalization_monoid",
"normalize",
"normalize_associated",
"normalize_eq_normalize",
"normalize_eq_zero",
"normalize_gcd",
"normalize_idem",
"normalize_lcm",
... | Define `normalized_gcd_monoid` on a structure just from the `gcd` and its properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_monoid_of_lcm [decidable_eq α] (lcm : α → α → α)
(dvd_lcm_left : ∀a b, a ∣ lcm a b)
(dvd_lcm_right : ∀a b, b ∣ lcm a b)
(lcm_dvd : ∀{a b c}, c ∣ a → b ∣ a → lcm c b ∣ a):
gcd_monoid α | let exists_gcd := λ a b, lcm_dvd (dvd.intro b rfl) (dvd.intro_left a rfl) in
{ lcm := lcm,
gcd := λ a b, if a = 0 then b else (if b = 0 then a else
classical.some (exists_gcd a b)),
gcd_mul_lcm := λ a b, by
{ split_ifs,
{ rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul] },
{ rw [h_1,... | def | gcd_monoid_of_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"con",
"dvd.intro",
"dvd.intro_left",
"dvd_lcm_left",
"dvd_lcm_right",
"dvd_rfl",
"dvd_zero",
"eq_zero_of_zero_dvd",
"gcd_monoid",
"gcd_mul_lcm",
"lcm_dvd",
"mul_assoc",
"mul_comm",
"mul_dvd_mul_iff_left",
"mul_dvd_mul_iff_right",
"mul_eq_zero",
"mul_zero",
"zero_dvd_iff",
"zero_... | Define `gcd_monoid` on a structure just from the `lcm` and its properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_gcd_monoid_of_lcm [normalization_monoid α] [decidable_eq α]
(lcm : α → α → α)
(dvd_lcm_left : ∀a b, a ∣ lcm a b)
(dvd_lcm_right : ∀a b, b ∣ lcm a b)
(lcm_dvd : ∀{a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
(normalize_lcm : ∀a b, normalize (lcm a b) = lcm a b) :
normalized_gcd_monoid α | let exists_gcd := λ a b, dvd_normalize_iff.2 (lcm_dvd (dvd.intro b rfl) (dvd.intro_left a rfl)) in
{ lcm := lcm,
gcd := λ a b, if a = 0 then normalize b else (if b = 0 then normalize a else
classical.some (exists_gcd a b)),
gcd_mul_lcm := λ a b, by
{ split_ifs with h h_1,
{ rw [h, eq_zero_of_zero_dvd (dvd... | def | normalized_gcd_monoid_of_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"con",
"dvd.intro",
"dvd.intro_left",
"dvd_lcm_left",
"dvd_lcm_right",
"dvd_normalize_iff",
"dvd_zero",
"eq_zero_of_zero_dvd",
"gcd_mul_lcm",
"lcm_dvd",
"mul_assoc",
"mul_comm",
"mul_dvd_mul_iff_left",
"mul_dvd_mul_iff_right",
"mul_eq_zero",
"mul_left_cancel₀",
"mul_zero",
"normali... | Define `normalized_gcd_monoid` on a structure just from the `lcm` and its properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_monoid_of_exists_gcd [decidable_eq α]
(h : ∀ a b : α, ∃ c : α, ∀ d : α, d ∣ a ∧ d ∣ b ↔ d ∣ c) :
gcd_monoid α | gcd_monoid_of_gcd
(λ a b, (classical.some (h a b)))
(λ a b,
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).1)
(λ a b,
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).2)
(λ a b c ac ab, ((classical.some_spec (h c b) a).1 ⟨ac, ab⟩)) | def | gcd_monoid_of_exists_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_rfl",
"gcd_monoid",
"gcd_monoid_of_gcd"
] | Define a `gcd_monoid` structure on a monoid just from the existence of a `gcd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_gcd_monoid_of_exists_gcd [normalization_monoid α] [decidable_eq α]
(h : ∀ a b : α, ∃ c : α, ∀ d : α, d ∣ a ∧ d ∣ b ↔ d ∣ c) :
normalized_gcd_monoid α | normalized_gcd_monoid_of_gcd
(λ a b, normalize (classical.some (h a b)))
(λ a b, normalize_dvd_iff.2
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).1)
(λ a b, normalize_dvd_iff.2
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).2)
(λ a b c ac ab, dvd_normalize_i... | def | normalized_gcd_monoid_of_exists_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_rfl",
"normalization_monoid",
"normalize",
"normalize_idem",
"normalized_gcd_monoid",
"normalized_gcd_monoid_of_gcd"
] | Define a `normalized_gcd_monoid` structure on a monoid just from the existence of a `gcd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_monoid_of_exists_lcm [decidable_eq α]
(h : ∀ a b : α, ∃ c : α, ∀ d : α, a ∣ d ∧ b ∣ d ↔ c ∣ d) :
gcd_monoid α | gcd_monoid_of_lcm
(λ a b, (classical.some (h a b)))
(λ a b,
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).1)
(λ a b,
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).2)
(λ a b c ac ab, ((classical.some_spec (h c b) a).1 ⟨ac, ab⟩)) | def | gcd_monoid_of_exists_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_rfl",
"gcd_monoid",
"gcd_monoid_of_lcm"
] | Define a `gcd_monoid` structure on a monoid just from the existence of an `lcm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_gcd_monoid_of_exists_lcm [normalization_monoid α] [decidable_eq α]
(h : ∀ a b : α, ∃ c : α, ∀ d : α, a ∣ d ∧ b ∣ d ↔ c ∣ d) :
normalized_gcd_monoid α | normalized_gcd_monoid_of_lcm
(λ a b, normalize (classical.some (h a b)))
(λ a b, dvd_normalize_iff.2
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).1)
(λ a b, dvd_normalize_iff.2
(((classical.some_spec (h a b) (classical.some (h a b))).2 dvd_rfl)).2)
(λ a b c ac ab, normalize_dvd_i... | def | normalized_gcd_monoid_of_exists_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"dvd_rfl",
"normalization_monoid",
"normalize",
"normalize_idem",
"normalized_gcd_monoid",
"normalized_gcd_monoid_of_lcm"
] | Define a `normalized_gcd_monoid` structure on a monoid just from the existence of an `lcm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_norm_unit {a : G₀} (h0 : a ≠ 0) : (↑(norm_unit a) : G₀) = a⁻¹ | by simp [norm_unit, h0] | lemma | comm_group_with_zero.coe_norm_unit | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_eq_one {a : G₀} (h0 : a ≠ 0) : normalize a = 1 | by simp [normalize_apply, h0] | lemma | comm_group_with_zero.normalize_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"normalize",
"normalize_apply",
"normalize_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_div_eq_one {β : Type*} {f : β → ℕ} (s : finset β) {x : β} (hx : x ∈ s)
(hfz : f x ≠ 0) : s.gcd (λ b, f b / s.gcd f) = 1 | begin
obtain ⟨g, he, hg⟩ := finset.extract_gcd f ⟨x, hx⟩,
refine (finset.gcd_congr rfl $ λ a ha, _).trans hg,
rw [he a ha, nat.mul_div_cancel_left],
exact nat.pos_of_ne_zero (mt finset.gcd_eq_zero_iff.1 (λ h, hfz $ h x hx)),
end | theorem | finset.nat.gcd_div_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/div.lean | [
"algebra.gcd_monoid.finset",
"algebra.gcd_monoid.basic",
"ring_theory.int.basic",
"ring_theory.polynomial.content"
] | [
"finset",
"finset.extract_gcd",
"finset.gcd_congr"
] | Given a nonempty finset `s` and a function `f` from `s` to `ℕ`, if `d = s.gcd`,
then the `gcd` of `(f i) / d` is equal to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_div_id_eq_one {s : finset ℕ} {x : ℕ} (hx : x ∈ s) (hnz : x ≠ 0) :
s.gcd (λ b, b / s.gcd id) = 1 | gcd_div_eq_one s hx hnz | theorem | finset.nat.gcd_div_id_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/div.lean | [
"algebra.gcd_monoid.finset",
"algebra.gcd_monoid.basic",
"ring_theory.int.basic",
"ring_theory.polynomial.content"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_div_eq_one {β : Type*} {f : β → ℤ} (s : finset β) {x : β} (hx : x ∈ s)
(hfz : f x ≠ 0) : s.gcd (λ b, f b / s.gcd f) = 1 | begin
obtain ⟨g, he, hg⟩ := finset.extract_gcd f ⟨x, hx⟩,
refine (finset.gcd_congr rfl $ λ a ha, _).trans hg,
rw [he a ha, int.mul_div_cancel_left],
exact mt finset.gcd_eq_zero_iff.1 (λ h, hfz $ h x hx),
end | theorem | finset.int.gcd_div_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/div.lean | [
"algebra.gcd_monoid.finset",
"algebra.gcd_monoid.basic",
"ring_theory.int.basic",
"ring_theory.polynomial.content"
] | [
"finset",
"finset.extract_gcd",
"finset.gcd_congr",
"int.mul_div_cancel_left"
] | Given a nonempty finset `s` and a function `f` from `s` to `ℤ`, if `d = s.gcd`,
then the `gcd` of `(f i) / d` is equal to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_div_id_eq_one {s : finset ℤ} {x : ℤ} (hx : x ∈ s) (hnz : x ≠ 0) :
s.gcd (λ b, b / s.gcd id) = 1 | gcd_div_eq_one s hx hnz | theorem | finset.int.gcd_div_id_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/div.lean | [
"algebra.gcd_monoid.finset",
"algebra.gcd_monoid.basic",
"ring_theory.int.basic",
"ring_theory.polynomial.content"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_div_eq_one {β : Type*} {f : β → K[X]} (s : finset β) {x : β} (hx : x ∈ s)
(hfz : f x ≠ 0) : s.gcd (λ b, f b / s.gcd f) = 1 | begin
obtain ⟨g, he, hg⟩ := finset.extract_gcd f ⟨x, hx⟩,
refine (finset.gcd_congr rfl $ λ a ha, _).trans hg,
rw [he a ha, euclidean_domain.mul_div_cancel_left],
exact mt finset.gcd_eq_zero_iff.1 (λ h, hfz $ h x hx),
end | theorem | finset.polynomial.gcd_div_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/div.lean | [
"algebra.gcd_monoid.finset",
"algebra.gcd_monoid.basic",
"ring_theory.int.basic",
"ring_theory.polynomial.content"
] | [
"euclidean_domain.mul_div_cancel_left",
"finset",
"finset.extract_gcd",
"finset.gcd_congr"
] | Given a nonempty finset `s` and a function `f` from `s` to `K[X]`, if `d = s.gcd f`,
then the `gcd` of `(f i) / d` is equal to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_div_id_eq_one {s : finset K[X]} {x : K[X]} (hx : x ∈ s) (hnz : x ≠ 0) :
s.gcd (λ b, b / s.gcd id) = 1 | gcd_div_eq_one s hx hnz | theorem | finset.polynomial.gcd_div_id_eq_one | algebra.gcd_monoid | src/algebra/gcd_monoid/div.lean | [
"algebra.gcd_monoid.finset",
"algebra.gcd_monoid.basic",
"ring_theory.int.basic",
"ring_theory.polynomial.content"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm (s : finset β) (f : β → α) : α | s.fold gcd_monoid.lcm 1 f | def | finset.lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset"
] | Least common multiple of a finite set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lcm_def : s.lcm f = (s.1.map f).lcm | rfl | lemma | finset.lcm_def | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_empty : (∅ : finset β).lcm f = 1 | fold_empty | lemma | finset.lcm_empty | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd_iff {a : α} : s.lcm f ∣ a ↔ (∀b ∈ s, f b ∣ a) | begin
apply iff.trans multiset.lcm_dvd,
simp only [multiset.mem_map, and_imp, exists_imp_distrib],
exact ⟨λ k b hb, k _ _ hb rfl, λ k a' b hb h, h ▸ k _ hb⟩,
end | lemma | finset.lcm_dvd_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"and_imp",
"exists_imp_distrib",
"lcm_dvd_iff",
"multiset.lcm_dvd",
"multiset.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd {a : α} : (∀b ∈ s, f b ∣ a) → s.lcm f ∣ a | lcm_dvd_iff.2 | lemma | finset.lcm_dvd | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"lcm_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_lcm {b : β} (hb : b ∈ s) : f b ∣ s.lcm f | lcm_dvd_iff.1 dvd_rfl _ hb | lemma | finset.dvd_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"dvd_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_insert [decidable_eq β] {b : β} :
(insert b s : finset β).lcm f = gcd_monoid.lcm (f b) (s.lcm f) | begin
by_cases h : b ∈ s,
{ rw [insert_eq_of_mem h,
(lcm_eq_right_iff (f b) (s.lcm f) (multiset.normalize_lcm (s.1.map f))).2 (dvd_lcm h)] },
apply fold_insert h,
end | lemma | finset.lcm_insert | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset",
"lcm_eq_right_iff",
"multiset.normalize_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_singleton {b : β} : ({b} : finset β).lcm f = normalize (f b) | multiset.lcm_singleton | lemma | finset.lcm_singleton | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset",
"multiset.lcm_singleton",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_lcm : normalize (s.lcm f) = s.lcm f | by simp [lcm_def] | lemma | finset.normalize_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"normalize",
"normalize_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_union [decidable_eq β] : (s₁ ∪ s₂).lcm f = gcd_monoid.lcm (s₁.lcm f) (s₂.lcm f) | finset.induction_on s₁ (by rw [empty_union, lcm_empty, lcm_one_left, normalize_lcm]) $ λ a s has ih,
by rw [insert_union, lcm_insert, lcm_insert, ih, lcm_assoc] | lemma | finset.lcm_union | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset.induction_on",
"ih",
"lcm_assoc",
"lcm_one_left",
"normalize_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀a ∈ s₂, f a = g a) :
s₁.lcm f = s₂.lcm g | by { subst hs, exact finset.fold_congr hfg } | theorem | finset.lcm_congr | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset.fold_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_mono_fun {g : β → α} (h : ∀ b ∈ s, f b ∣ g b) : s.lcm f ∣ s.lcm g | lcm_dvd (λ b hb, (h b hb).trans (dvd_lcm hb)) | lemma | finset.lcm_mono_fun | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"lcm_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_mono (h : s₁ ⊆ s₂) : s₁.lcm f ∣ s₂.lcm f | lcm_dvd $ assume b hb, dvd_lcm (h hb) | lemma | finset.lcm_mono | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"lcm_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_image [decidable_eq β] {g : γ → β} (s : finset γ) : (s.image g).lcm f = s.lcm (f ∘ g) | by { classical, induction s using finset.induction with c s hc ih; simp [*] } | lemma | finset.lcm_image | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset",
"finset.induction",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_lcm_image [decidable_eq α] : s.lcm f = (s.image f).lcm id | eq.symm $ lcm_image _ | lemma | finset.lcm_eq_lcm_image | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_zero_iff [nontrivial α] : s.lcm f = 0 ↔ 0 ∈ f '' s | by simp only [multiset.mem_map, lcm_def, multiset.lcm_eq_zero_iff, set.mem_image, mem_coe,
← finset.mem_def] | theorem | finset.lcm_eq_zero_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset.mem_def",
"lcm_eq_zero_iff",
"multiset.lcm_eq_zero_iff",
"multiset.mem_map",
"nontrivial",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd (s : finset β) (f : β → α) : α | s.fold gcd_monoid.gcd 0 f | def | finset.gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset"
] | Greatest common divisor of a finite set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_def : s.gcd f = (s.1.map f).gcd | rfl | lemma | finset.gcd_def | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_empty : (∅ : finset β).gcd f = 0 | fold_empty | lemma | finset.gcd_empty | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_gcd_iff {a : α} : a ∣ s.gcd f ↔ ∀b ∈ s, a ∣ f b | begin
apply iff.trans multiset.dvd_gcd,
simp only [multiset.mem_map, and_imp, exists_imp_distrib],
exact ⟨λ k b hb, k _ _ hb rfl, λ k a' b hb h, h ▸ k _ hb⟩,
end | lemma | finset.dvd_gcd_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"and_imp",
"dvd_gcd_iff",
"exists_imp_distrib",
"multiset.dvd_gcd",
"multiset.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd {b : β} (hb : b ∈ s) : s.gcd f ∣ f b | dvd_gcd_iff.1 dvd_rfl _ hb | lemma | finset.gcd_dvd | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"dvd_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_gcd {a : α} : (∀b ∈ s, a ∣ f b) → a ∣ s.gcd f | dvd_gcd_iff.2 | lemma | finset.dvd_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_insert [decidable_eq β] {b : β} :
(insert b s : finset β).gcd f = gcd_monoid.gcd (f b) (s.gcd f) | begin
by_cases h : b ∈ s,
{ rw [insert_eq_of_mem h,
(gcd_eq_right_iff (f b) (s.gcd f) (multiset.normalize_gcd (s.1.map f))).2 (gcd_dvd h)] ,},
apply fold_insert h,
end | lemma | finset.gcd_insert | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset",
"gcd_eq_right_iff",
"multiset.normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_singleton {b : β} : ({b} : finset β).gcd f = normalize (f b) | multiset.gcd_singleton | lemma | finset.gcd_singleton | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset",
"multiset.gcd_singleton",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_gcd : normalize (s.gcd f) = s.gcd f | by simp [gcd_def] | lemma | finset.normalize_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"normalize",
"normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_union [decidable_eq β] : (s₁ ∪ s₂).gcd f = gcd_monoid.gcd (s₁.gcd f) (s₂.gcd f) | finset.induction_on s₁ (by rw [empty_union, gcd_empty, gcd_zero_left, normalize_gcd]) $
λ a s has ih, by rw [insert_union, gcd_insert, gcd_insert, ih, gcd_assoc] | lemma | finset.gcd_union | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset.induction_on",
"gcd_assoc",
"gcd_zero_left",
"ih",
"normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀a ∈ s₂, f a = g a) :
s₁.gcd f = s₂.gcd g | by { subst hs, exact finset.fold_congr hfg } | theorem | finset.gcd_congr | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset.fold_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mono_fun {g : β → α} (h : ∀ b ∈ s, f b ∣ g b) : s.gcd f ∣ s.gcd g | dvd_gcd (λ b hb, (gcd_dvd hb).trans (h b hb)) | lemma | finset.gcd_mono_fun | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mono (h : s₁ ⊆ s₂) : s₂.gcd f ∣ s₁.gcd f | dvd_gcd $ assume b hb, gcd_dvd (h hb) | lemma | finset.gcd_mono | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_image [decidable_eq β] {g : γ → β} (s : finset γ) : (s.image g).gcd f = s.gcd (f ∘ g) | by { classical, induction s using finset.induction with c s hc ih; simp [*] } | lemma | finset.gcd_image | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset",
"finset.induction",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_gcd_image [decidable_eq α] : s.gcd f = (s.image f).gcd id | eq.symm $ gcd_image _ | lemma | finset.gcd_eq_gcd_image | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_zero_iff : s.gcd f = 0 ↔ ∀ (x : β), x ∈ s → f x = 0 | begin
rw [gcd_def, multiset.gcd_eq_zero_iff],
split; intro h,
{ intros b bs,
apply h (f b),
simp only [multiset.mem_map, mem_def.1 bs],
use b,
simp [mem_def.1 bs] },
{ intros a as,
rw multiset.mem_map at as,
rcases as with ⟨b, ⟨bs, rfl⟩⟩,
apply h b (mem_def.1 bs) }
end | theorem | finset.gcd_eq_zero_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"gcd_eq_zero_iff",
"multiset.gcd_eq_zero_iff",
"multiset.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_gcd_filter_ne_zero [decidable_pred (λ (x : β), f x = 0)] :
s.gcd f = (s.filter (λ x, f x ≠ 0)).gcd f | begin
classical,
transitivity ((s.filter (λ x, f x = 0)) ∪ (s.filter (λ x, f x ≠ 0))).gcd f,
{ rw filter_union_filter_neg_eq },
rw gcd_union,
transitivity gcd_monoid.gcd (0 : α) _,
{ refine congr (congr rfl _) rfl,
apply s.induction_on, { simp },
intros a s has h,
rw filter_insert,
split_ifs... | lemma | finset.gcd_eq_gcd_filter_ne_zero | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"gcd_zero_left",
"normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_left {a : α} : s.gcd (λ x, a * f x) = normalize a * s.gcd f | begin
classical,
apply s.induction_on,
{ simp },
intros b t hbt h,
rw [gcd_insert, gcd_insert, h, ← gcd_mul_left],
apply ((normalize_associated a).mul_right _).gcd_eq_right
end | lemma | finset.gcd_mul_left | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"gcd_mul_left",
"normalize",
"normalize_associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_right {a : α} : s.gcd (λ x, f x * a) = s.gcd f * normalize a | begin
classical,
apply s.induction_on,
{ simp },
intros b t hbt h,
rw [gcd_insert, gcd_insert, h, ← gcd_mul_right],
apply ((normalize_associated a).mul_left _).gcd_eq_right
end | lemma | finset.gcd_mul_right | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"gcd_mul_right",
"normalize",
"normalize_associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extract_gcd' (f g : β → α) (hs : ∃ x, x ∈ s ∧ f x ≠ 0)
(hg : ∀ b ∈ s, f b = s.gcd f * g b) : s.gcd g = 1 | ((@mul_right_eq_self₀ _ _ (s.gcd f) _).1 $
by conv_lhs { rw [← normalize_gcd, ← gcd_mul_left, ← gcd_congr rfl hg] }).resolve_right $
by {contrapose! hs, exact gcd_eq_zero_iff.1 hs} | lemma | finset.extract_gcd' | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"gcd_mul_left",
"mul_right_eq_self₀",
"normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extract_gcd (f : β → α) (hs : s.nonempty) :
∃ g : β → α, (∀ b ∈ s, f b = s.gcd f * g b) ∧ s.gcd g = 1 | begin
classical,
by_cases h : ∀ x ∈ s, f x = (0 : α),
{ refine ⟨λ b, 1, λ b hb, by rw [h b hb, gcd_eq_zero_iff.2 h, mul_one], _⟩,
rw [gcd_eq_gcd_image, image_const hs, gcd_singleton, id, normalize_one] },
{ choose g' hg using @gcd_dvd _ _ _ _ s f,
have := λ b hb, _, push_neg at h,
refine ⟨λ b, if hb... | lemma | finset.extract_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"extract_gcd",
"mul_one",
"normalize_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_of_dvd_sub {s : finset β} {f g : β → α} {a : α}
(h : ∀ x : β, x ∈ s → a ∣ f x - g x) :
gcd_monoid.gcd a (s.gcd f) = gcd_monoid.gcd a (s.gcd g) | begin
classical,
revert h,
apply s.induction_on,
{ simp },
intros b s bs hi h,
rw [gcd_insert, gcd_insert, gcd_comm (f b), ← gcd_assoc, hi (λ x hx, h _ (mem_insert_of_mem hx)),
gcd_comm a, gcd_assoc, gcd_comm a (gcd_monoid.gcd _ _),
gcd_comm (g b), gcd_assoc _ _ a, gcd_comm _ a],
exact congr_a... | lemma | finset.gcd_eq_of_dvd_sub | algebra.gcd_monoid | src/algebra/gcd_monoid/finset.lean | [
"data.finset.fold",
"algebra.gcd_monoid.multiset"
] | [
"finset",
"gcd_assoc",
"gcd_comm",
"gcd_eq_of_dvd_sub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.surj_of_gcd_domain (M : submonoid R) [is_localization M A] (z : A) :
∃ a b : R, is_unit (gcd a b) ∧ z * algebra_map R A b = algebra_map R A a | begin
obtain ⟨x, ⟨y, hy⟩, rfl⟩ := is_localization.mk'_surjective M z,
obtain ⟨x', y', hx', hy', hu⟩ := extract_gcd x y,
use [x', y', hu],
rw [mul_comm, is_localization.mul_mk'_eq_mk'_of_mul],
convert is_localization.mk'_mul_cancel_left _ _ using 2,
{ rw [subtype.coe_mk, hy', ← mul_comm y', mul_assoc], conv_... | lemma | is_localization.surj_of_gcd_domain | algebra.gcd_monoid | src/algebra/gcd_monoid/integrally_closed.lean | [
"algebra.gcd_monoid.basic",
"ring_theory.integrally_closed",
"ring_theory.polynomial.eisenstein.basic"
] | [
"algebra_map",
"extract_gcd",
"is_localization",
"is_localization.mk'_mul_cancel_left",
"is_localization.mk'_surjective",
"is_localization.mul_mk'_eq_mk'_of_mul",
"is_unit",
"mul_assoc",
"mul_comm",
"submonoid",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_monoid.to_is_integrally_closed : is_integrally_closed R | ⟨λ X ⟨p, hp₁, hp₂⟩, begin
obtain ⟨x, y, hg, he⟩ := is_localization.surj_of_gcd_domain (non_zero_divisors R) X,
have := polynomial.dvd_pow_nat_degree_of_eval₂_eq_zero
(is_fraction_ring.injective R $ fraction_ring R) hp₁ y x _ hp₂ (by rw [mul_comm, he]),
have : is_unit y,
{ rw [is_unit_iff_dvd_one, ← one_pow]... | instance | gcd_monoid.to_is_integrally_closed | algebra.gcd_monoid | src/algebra/gcd_monoid/integrally_closed.lean | [
"algebra.gcd_monoid.basic",
"ring_theory.integrally_closed",
"ring_theory.polynomial.eisenstein.basic"
] | [
"dvd_refl",
"fraction_ring",
"is_fraction_ring.injective",
"is_integrally_closed",
"is_localization.surj_of_gcd_domain",
"is_unit",
"is_unit_iff_dvd_one",
"map_mul",
"mul_comm",
"non_zero_divisors",
"one_pow",
"polynomial.dvd_pow_nat_degree_of_eval₂_eq_zero",
"pow_dvd_pow_of_dvd",
"units.c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm (s : multiset α) : α | s.fold gcd_monoid.lcm 1 | def | multiset.lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | Least common multiple of a multiset | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lcm_zero : (0 : multiset α).lcm = 1 | fold_zero _ _ | lemma | multiset.lcm_zero | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_cons (a : α) (s : multiset α) :
(a ::ₘ s).lcm = gcd_monoid.lcm a s.lcm | fold_cons_left _ _ _ _ | lemma | multiset.lcm_cons | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_singleton {a : α} : ({a} : multiset α).lcm = normalize a | (fold_singleton _ _ _).trans $ lcm_one_right _ | lemma | multiset.lcm_singleton | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"lcm_one_right",
"multiset",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_add (s₁ s₂ : multiset α) : (s₁ + s₂).lcm = gcd_monoid.lcm s₁.lcm s₂.lcm | eq.trans (by simp [lcm]) (fold_add _ _ _ _ _) | lemma | multiset.lcm_add | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dvd {s : multiset α} {a : α} : s.lcm ∣ a ↔ (∀ b ∈ s, b ∣ a) | multiset.induction_on s (by simp)
(by simp [or_imp_distrib, forall_and_distrib, lcm_dvd_iff] {contextual := tt}) | lemma | multiset.lcm_dvd | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"forall_and_distrib",
"lcm_dvd",
"lcm_dvd_iff",
"multiset",
"multiset.induction_on",
"or_imp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_lcm {s : multiset α} {a : α} (h : a ∈ s) : a ∣ s.lcm | lcm_dvd.1 dvd_rfl _ h | lemma | multiset.dvd_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"dvd_rfl",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_mono {s₁ s₂ : multiset α} (h : s₁ ⊆ s₂) : s₁.lcm ∣ s₂.lcm | lcm_dvd.2 $ assume b hb, dvd_lcm (h hb) | lemma | multiset.lcm_mono | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_lcm (s : multiset α) : normalize (s.lcm) = s.lcm | multiset.induction_on s (by simp) $ λ a s IH, by simp | lemma | multiset.normalize_lcm | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset",
"multiset.induction_on",
"normalize",
"normalize_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_zero_iff [nontrivial α] (s : multiset α) : s.lcm = 0 ↔ (0 : α) ∈ s | begin
induction s using multiset.induction_on with a s ihs,
{ simp only [lcm_zero, one_ne_zero, not_mem_zero] },
{ simp only [mem_cons, lcm_cons, lcm_eq_zero_iff, ihs, @eq_comm _ a] },
end | theorem | multiset.lcm_eq_zero_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"lcm_eq_zero_iff",
"mem_cons",
"multiset",
"multiset.induction_on",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_dedup (s : multiset α) : (dedup s).lcm = s.lcm | multiset.induction_on s (by simp) $ λ a s IH, begin
by_cases a ∈ s; simp [IH, h],
unfold lcm,
rw [← cons_erase h, fold_cons_left, ← lcm_assoc, lcm_same],
apply lcm_eq_of_associated_left (associated_normalize _),
end | lemma | multiset.lcm_dedup | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"associated_normalize",
"lcm_assoc",
"lcm_eq_of_associated_left",
"lcm_same",
"multiset",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_ndunion (s₁ s₂ : multiset α) :
(ndunion s₁ s₂).lcm = gcd_monoid.lcm s₁.lcm s₂.lcm | by { rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_add], simp } | lemma | multiset.lcm_ndunion | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_union (s₁ s₂ : multiset α) :
(s₁ ∪ s₂).lcm = gcd_monoid.lcm s₁.lcm s₂.lcm | by { rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_add], simp } | lemma | multiset.lcm_union | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_ndinsert (a : α) (s : multiset α) :
(ndinsert a s).lcm = gcd_monoid.lcm a s.lcm | by { rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_cons], simp } | lemma | multiset.lcm_ndinsert | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd (s : multiset α) : α | s.fold gcd_monoid.gcd 0 | def | multiset.gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | Greatest common divisor of a multiset | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_zero : (0 : multiset α).gcd = 0 | fold_zero _ _ | lemma | multiset.gcd_zero | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_cons (a : α) (s : multiset α) :
(a ::ₘ s).gcd = gcd_monoid.gcd a s.gcd | fold_cons_left _ _ _ _ | lemma | multiset.gcd_cons | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_singleton {a : α} : ({a} : multiset α).gcd = normalize a | (fold_singleton _ _ _).trans $ gcd_zero_right _ | lemma | multiset.gcd_singleton | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"gcd_zero_right",
"multiset",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_add (s₁ s₂ : multiset α) : (s₁ + s₂).gcd = gcd_monoid.gcd s₁.gcd s₂.gcd | eq.trans (by simp [gcd]) (fold_add _ _ _ _ _) | lemma | multiset.gcd_add | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_gcd {s : multiset α} {a : α} : a ∣ s.gcd ↔ (∀ b ∈ s, a ∣ b) | multiset.induction_on s (by simp)
(by simp [or_imp_distrib, forall_and_distrib, dvd_gcd_iff] {contextual := tt}) | lemma | multiset.dvd_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"dvd_gcd_iff",
"forall_and_distrib",
"multiset",
"multiset.induction_on",
"or_imp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd {s : multiset α} {a : α} (h : a ∈ s) : s.gcd ∣ a | dvd_gcd.1 dvd_rfl _ h | lemma | multiset.gcd_dvd | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"dvd_rfl",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mono {s₁ s₂ : multiset α} (h : s₁ ⊆ s₂) : s₂.gcd ∣ s₁.gcd | dvd_gcd.2 $ assume b hb, gcd_dvd (h hb) | lemma | multiset.gcd_mono | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_gcd (s : multiset α) : normalize (s.gcd) = s.gcd | multiset.induction_on s (by simp) $ λ a s IH, by simp | lemma | multiset.normalize_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset",
"multiset.induction_on",
"normalize",
"normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_zero_iff (s : multiset α) : s.gcd = 0 ↔ ∀ (x : α), x ∈ s → x = 0 | begin
split,
{ intros h x hx,
apply eq_zero_of_zero_dvd,
rw ← h,
apply gcd_dvd hx },
{ apply s.induction_on,
{ simp },
intros a s sgcd h,
simp [h a (mem_cons_self a s), sgcd (λ x hx, h x (mem_cons_of_mem hx))] }
end | theorem | multiset.gcd_eq_zero_iff | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"eq_zero_of_zero_dvd",
"gcd_eq_zero_iff",
"mem_cons_of_mem",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_map_mul (a : α) (s : multiset α) :
(s.map ((*) a)).gcd = normalize a * s.gcd | begin
refine s.induction_on _ (λ b s ih, _),
{ simp_rw [map_zero, gcd_zero, mul_zero] },
{ simp_rw [map_cons, gcd_cons, ← gcd_mul_left], rw ih,
apply ((normalize_associated a).mul_right _).gcd_eq_right },
end | lemma | multiset.gcd_map_mul | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"gcd_mul_left",
"ih",
"map_cons",
"mul_zero",
"multiset",
"normalize",
"normalize_associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dedup (s : multiset α) : (dedup s).gcd = s.gcd | multiset.induction_on s (by simp) $ λ a s IH, begin
by_cases a ∈ s; simp [IH, h],
unfold gcd,
rw [← cons_erase h, fold_cons_left, ← gcd_assoc, gcd_same],
apply (associated_normalize _).gcd_eq_left,
end | lemma | multiset.gcd_dedup | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"associated_normalize",
"gcd_assoc",
"gcd_same",
"multiset",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_ndunion (s₁ s₂ : multiset α) :
(ndunion s₁ s₂).gcd = gcd_monoid.gcd s₁.gcd s₂.gcd | by { rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_add], simp } | lemma | multiset.gcd_ndunion | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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