statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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dvd_of_multiplicity_pos {a b : α} (h : (0 : part_enat) < multiplicity a b) : a ∣ b | begin
rw ← pow_one a,
apply pow_dvd_of_le_multiplicity,
simpa only [nat.cast_one, part_enat.pos_iff_one_le] using h
end | lemma | multiplicity.dvd_of_multiplicity_pos | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"nat.cast_one",
"part_enat",
"part_enat.pos_iff_one_le",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_iff_multiplicity_pos {a b : α} : (0 : part_enat) < multiplicity a b ↔ a ∣ b | ⟨dvd_of_multiplicity_pos,
λ hdvd, lt_of_le_of_ne (zero_le _) (λ heq, is_greatest
(show multiplicity a b < ↑1,
by simpa only [heq, nat.cast_zero] using part_enat.coe_lt_coe.mpr zero_lt_one)
(by rwa pow_one a))⟩ | lemma | multiplicity.dvd_iff_multiplicity_pos | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_greatest",
"multiplicity",
"nat.cast_zero",
"part_enat",
"pow_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_nat_iff {a b : ℕ} : finite a b ↔ (a ≠ 1 ∧ 0 < b) | begin
rw [← not_iff_not, not_finite_iff_forall, not_and_distrib, ne.def,
not_not, not_lt, le_zero_iff],
exact ⟨λ h, or_iff_not_imp_right.2 (λ hb,
have ha : a ≠ 0, from λ ha, by simpa [ha] using h 1,
by_contradiction (λ ha1 : a ≠ 1,
have ha_gt_one : 1 < a, from
lt_of_not_ge (λ ha', by { cle... | lemma | multiplicity.finite_nat_iff | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"by_contradiction",
"finite",
"le_zero_iff",
"not_and_distrib",
"not_iff_not",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_of_finite_mul_left {a b c : α} : finite a (b * c) → finite a c | by rw mul_comm; exact finite_of_finite_mul_right | lemma | multiplicity.finite_of_finite_mul_left | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_right {a b : α} (ha : ¬is_unit a) (hb : is_unit b) :
multiplicity a b = 0 | eq_coe_iff.2 ⟨show a ^ 0 ∣ b, by simp only [pow_zero, one_dvd],
by { rw pow_one, exact λ h, mt (is_unit_of_dvd_unit h) ha hb }⟩ | lemma | multiplicity.is_unit_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_unit",
"is_unit_of_dvd_unit",
"multiplicity",
"one_dvd",
"pow_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_right {a : α} (ha : ¬is_unit a) : multiplicity a 1 = 0 | is_unit_right ha is_unit_one | lemma | multiplicity.one_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_unit",
"is_unit_one",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_right {a : α} (ha : ¬is_unit a) (u : αˣ) : multiplicity a u = 0 | is_unit_right ha u.is_unit | lemma | multiplicity.unit_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_unit",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c | multiplicity_le_multiplicity_iff.2 $ λ n h, (pow_dvd_pow_of_dvd hdvd n).trans h | lemma | multiplicity.multiplicity_le_multiplicity_of_dvd_left | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"pow_dvd_pow_of_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_associated_left {a b c : α} (h : associated a b) :
multiplicity b c = multiplicity a c | le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) | lemma | multiplicity.eq_of_associated_left | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"associated",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_finite {a b : α} (h : finite a b) : b ≠ 0 | let ⟨n, hn⟩ := h in λ hb, by simpa [hb] using hn | lemma | multiplicity.ne_zero_of_finite | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero (a : α) : multiplicity a 0 = ⊤ | part.eq_none_iff.2 (λ n ⟨⟨k, hk⟩, _⟩, hk (dvd_zero _)) | lemma | multiplicity.zero | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"dvd_zero",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 | multiplicity.multiplicity_eq_zero.2 $ mt zero_dvd_iff.1 ha | lemma | multiplicity.multiplicity_zero_eq_zero_of_ne_zero | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_mk_eq_multiplicity [decidable_rel ((∣) : associates α → associates α → Prop)]
{a b : α} : multiplicity (associates.mk a) (associates.mk b) = multiplicity a b | begin
by_cases h : finite a b,
{ rw ← part_enat.coe_get (finite_iff_dom.mp h),
refine (multiplicity.unique
(show (associates.mk a)^(((multiplicity a b).get h)) ∣ associates.mk b, from _) _).symm ;
rw [← associates.mk_pow, associates.mk_dvd_mk],
{ exact pow_multiplicity_dvd h },
{ exact is_gr... | lemma | multiplicity.multiplicity_mk_eq_multiplicity | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"associates",
"associates.mk",
"associates.mk_dvd_mk",
"associates.mk_pow",
"finite",
"is_greatest",
"multiplicity",
"multiplicity.unique",
"part_enat.coe_get",
"part_enat.lt_coe_iff",
"part_enat.not_dom_iff_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) | (le_total (multiplicity p a) (multiplicity p b)).elim
(λ h, by rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact λ n hn, dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
(λ h, by rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact λ n hn, dvd_add (multiplicity_le_multiplicity_iff.1... | lemma | multiplicity.min_le_multiplicity_add | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"dvd_add",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg (a b : α) : multiplicity a (-b) = multiplicity a b | part.ext' (by simp only [multiplicity, part_enat.find, dvd_neg])
(λ h₁ h₂, part_enat.coe_inj.1 (by rw [part_enat.coe_get]; exact
eq.symm (unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))))) | lemma | multiplicity.neg | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"dvd_neg",
"multiplicity",
"part.ext'",
"part_enat.coe_get",
"part_enat.find",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.nat_abs (a : ℕ) (b : ℤ) : multiplicity a b.nat_abs = multiplicity (a : ℤ) b | begin
cases int.nat_abs_eq b with h h; conv_rhs { rw h },
{ rw [int.coe_nat_multiplicity], },
{ rw [multiplicity.neg, int.coe_nat_multiplicity], },
end | theorem | multiplicity.int.nat_abs | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"multiplicity.neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | begin
apply le_antisymm,
{ apply part_enat.le_of_lt_add_one,
cases part_enat.ne_top_iff.mp (part_enat.ne_top_of_lt h) with k hk,
rw [hk], rw_mod_cast [multiplicity_lt_iff_neg_dvd, dvd_add_right],
intro h_dvd,
apply multiplicity.is_greatest _ h_dvd, rw [hk], apply_mod_cast nat.lt_succ_self,
rw [p... | lemma | multiplicity.multiplicity_add_of_gt | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"dvd_add_right",
"multiplicity",
"multiplicity.is_greatest",
"nat.cast_add",
"nat.cast_one",
"part_enat.add_one_le_of_lt",
"part_enat.le_of_lt_add_one",
"part_enat.ne_top_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b | by { rw [sub_eq_add_neg, multiplicity_add_of_gt]; rwa [multiplicity.neg] } | lemma | multiplicity.multiplicity_sub_of_gt | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"multiplicity.neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | begin
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with hab|hab|hab,
{ rw [add_comm, multiplicity_add_of_gt hab, min_eq_left], exact le_of_lt hab },
{ contradiction },
{ rw [multiplicity_add_of_gt hab, min_eq_right], exact le_of_lt hab},
end | lemma | multiplicity.multiplicity_add_eq_min | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_mul_aux {p : α} (hp : prime p) : ∀ {n m : ℕ} {a b : α},
¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | | n m := λ a b ha hb ⟨s, hs⟩,
have p ∣ a * b, from ⟨p ^ (n + m) * s,
by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩,
(hp.2.2 a b this).elim
(λ ⟨x, hx⟩, have hn0 : 0 < n,
from nat.pos_of_ne_zero (λ hn0, by clear _fun_match _fun_match; simpa [hx, hn0] using ha),
have wf : (n - 1) < n... | lemma | multiplicity.finite_mul_aux | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"mul_assoc",
"mul_comm",
"mul_left_comm",
"mul_right_cancel₀",
"pow_add",
"prime",
"tsub_add_cancel_of_le",
"tsub_add_eq_add_tsub",
"tsub_lt_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_mul {p a b : α} (hp : prime p) : finite p a → finite p b → finite p (a * b) | λ ⟨n, hn⟩ ⟨m, hm⟩, ⟨n + m, finite_mul_aux hp hn hm⟩ | lemma | multiplicity.finite_mul | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_mul_iff {p a b : α} (hp : prime p) : finite p (a * b) ↔ finite p a ∧ finite p b | ⟨λ h, ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩,
λ h, finite_mul hp h.1 h.2⟩ | lemma | multiplicity.finite_mul_iff | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_pow {p a : α} (hp : prime p) : Π {k : ℕ} (ha : finite p a), finite p (a ^ k) | | 0 ha := ⟨0, by simp [mt is_unit_iff_dvd_one.2 hp.2.1]⟩
| (k+1) ha := by rw [pow_succ]; exact finite_mul hp ha (finite_pow ha) | lemma | multiplicity.finite_pow | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"pow_succ",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_self {a : α} (ha : ¬is_unit a) (ha0 : a ≠ 0) :
multiplicity a a = 1 | by { rw ← nat.cast_one, exact
eq_coe_iff.2 ⟨by simp, λ ⟨b, hb⟩, ha (is_unit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 $ by { clear _fun_match,
simpa [pow_succ, mul_assoc] using hb }⟩)⟩ } | lemma | multiplicity.multiplicity_self | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_unit",
"mul_assoc",
"mul_left_cancel₀",
"multiplicity",
"nat.cast_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
get_multiplicity_self {a : α} (ha : finite a a) :
get (multiplicity a a) ha = 1 | part_enat.get_eq_iff_eq_coe.2 (eq_coe_iff.2
⟨by simp, λ ⟨b, hb⟩,
by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact mt is_unit_iff_dvd_one.2 (not_unit_of_finite ha)
⟨b, by clear _fun_match; simp * at *⟩⟩) | lemma | multiplicity.get_multiplicity_self | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"mul_assoc",
"mul_one",
"mul_right_inj'",
"multiplicity",
"pow_add",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul' {p a b : α} (hp : prime p)
(h : (multiplicity p (a * b)).dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a,
from pow_multiplicity_dvd _,
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b,
from pow_multiplicity_dvd _,
have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_... | lemma | multiplicity.mul' | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"mul_dvd_mul",
"multiplicity",
"part_enat.coe_get",
"part_enat.coe_inj",
"pow_add",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul {p a b : α} (hp : prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b | if h : finite p a ∧ finite p b then
by rw [← part_enat.coe_get (finite_iff_dom.1 h.1), ← part_enat.coe_get (finite_iff_dom.1 h.2),
← part_enat.coe_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)),
← nat.cast_add, part_enat.coe_inj, multiplicity.mul' hp]; refl
else begin
rw [eq_top_iff_not_finite.2 (mt (finite_mul... | lemma | multiplicity.mul | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"multiplicity.mul'",
"nat.cast_add",
"part_enat.coe_get",
"part_enat.coe_inj",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.prod {β : Type*} {p : α} (hp : prime p) (s : finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | begin
classical,
induction s using finset.induction with a s has ih h,
{ simp only [finset.sum_empty, finset.prod_empty],
convert one_right hp.not_unit },
{ simp [has, ← ih],
convert multiplicity.mul hp }
end | lemma | multiplicity.finset.prod | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finset",
"finset.induction",
"finset.prod",
"finset.prod_empty",
"ih",
"multiplicity",
"multiplicity.mul",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow' {p a : α} (hp : prime p) (ha : finite p a) : ∀ {k : ℕ},
get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha | | 0 := by simp [one_right hp.not_unit]
| (k+1) := have multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k), by rw pow_succ,
by rw [get_eq_get_of_eq _ _ this, multiplicity.mul' hp, pow', add_mul, one_mul, add_comm] | lemma | multiplicity.pow' | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"multiplicity.mul'",
"one_mul",
"pow_succ",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow {p a : α} (hp : prime p) : ∀ {k : ℕ},
multiplicity p (a ^ k) = k • (multiplicity p a) | | 0 := by simp [one_right hp.not_unit]
| (succ k) := by simp [pow_succ, succ_nsmul, pow, multiplicity.mul hp] | lemma | multiplicity.pow | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"multiplicity.mul",
"pow_succ",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬ is_unit p) (n : ℕ) :
multiplicity p (p ^ n) = n | by { rw [eq_coe_iff], use dvd_rfl, rw [pow_dvd_pow_iff h0 hu], apply nat.not_succ_le_self } | lemma | multiplicity.multiplicity_pow_self | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"dvd_rfl",
"is_unit",
"multiplicity",
"pow_dvd_pow_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_pow_self_of_prime {p : α} (hp : prime p) (n : ℕ) :
multiplicity p (p ^ n) = n | multiplicity_pow_self hp.ne_zero hp.not_unit n | lemma | multiplicity.multiplicity_pow_self_of_prime | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_valuation (hp : prime p) : add_valuation R part_enat | add_valuation.of (multiplicity p) (multiplicity.zero _) (one_right hp.not_unit)
(λ _ _, min_le_multiplicity_add) (λ a b, multiplicity.mul hp) | def | multiplicity.add_valuation | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"add_valuation",
"add_valuation.of",
"multiplicity",
"multiplicity.mul",
"multiplicity.zero",
"part_enat",
"prime"
] | `multiplicity` of a prime inan integral domain as an additive valuation to `part_enat`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_valuation_apply {hp : prime p} {r : R} : add_valuation hp r = multiplicity p r | rfl | lemma | multiplicity.add_valuation_apply | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"add_valuation",
"multiplicity",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b)
(hab : nat.coprime a b) : multiplicity p a = 0 | begin
rw [multiplicity_le_multiplicity_iff] at hle,
rw [← nonpos_iff_eq_zero, ← not_lt, part_enat.pos_iff_one_le, ← nat.cast_one,
← pow_dvd_iff_le_multiplicity],
assume h,
have := nat.dvd_gcd h (hle _ h),
rw [coprime.gcd_eq_one hab, nat.dvd_one, pow_one] at this,
exact hp this
end | lemma | multiplicity_eq_zero_of_coprime | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"nat.cast_one",
"nat.dvd_gcd",
"nat.dvd_one",
"part_enat.pos_iff_one_le",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_smul_of_le_smul_of_le_jacobson {I J : ideal R} {N : submodule R M}
(hN : N.fg) (hIN : N ≤ I • N) (hIjac : I ≤ jacobson J) : N = J • N | begin
refine le_antisymm _ (submodule.smul_le.2 (λ _ _ _, submodule.smul_mem _ _)),
intros n hn,
cases submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hN hIN with r hr,
cases exists_mul_sub_mem_of_sub_one_mem_jacobson r (hIjac hr.1) with s hs,
have : n = (-(s * r - 1) • n),
{ rw [neg_sub,... | lemma | submodule.eq_smul_of_le_smul_of_le_jacobson | ring_theory | src/ring_theory/nakayama.lean | [
"ring_theory.jacobson_ideal"
] | [
"ideal",
"one_smul",
"smul_zero",
"sub_smul",
"submodule",
"submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul",
"submodule.neg_mem",
"submodule.smul_mem",
"submodule.smul_mem_smul"
] | *Nakayama's Lemma** - A slightly more general version of (2) in
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV).
See also `eq_bot_of_le_smul_of_le_jacobson_bot` for the special case when `J = ⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_bot_of_le_smul_of_le_jacobson_bot (I : ideal R) (N : submodule R M)
(hN : N.fg) (hIN : N ≤ I • N) (hIjac : I ≤ jacobson ⊥) : N = ⊥ | by rw [eq_smul_of_le_smul_of_le_jacobson hN hIN hIjac, submodule.bot_smul] | lemma | submodule.eq_bot_of_le_smul_of_le_jacobson_bot | ring_theory | src/ring_theory/nakayama.lean | [
"ring_theory.jacobson_ideal"
] | [
"ideal",
"submodule",
"submodule.bot_smul"
] | *Nakayama's Lemma** - Statement (2) in
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV).
See also `eq_smul_of_le_smul_of_le_jacobson` for a generalisation
to the `jacobson` of any ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson {I J : ideal R}
{N N' : submodule R M} (hN' : N'.fg) (hIJ : I ≤ jacobson J)
(hNN : N ⊔ N' ≤ N ⊔ I • N') : N ⊔ I • N' = N ⊔ J • N' | begin
have hNN' : N ⊔ N' = N ⊔ I • N',
from le_antisymm hNN
(sup_le_sup_left (submodule.smul_le.2 (λ _ _ _, submodule.smul_mem _ _)) _),
have h_comap := submodule.comap_injective_of_surjective (linear_map.range_eq_top.1 (N.range_mkq)),
have : (I • N').map N.mkq = N'.map N.mkq,
{ rw ←h_comap.eq_iff,
... | lemma | submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson | ring_theory | src/ring_theory/nakayama.lean | [
"ring_theory.jacobson_ideal"
] | [
"ideal",
"le_rfl",
"submodule",
"submodule.comap_injective_of_surjective",
"submodule.eq_smul_of_le_smul_of_le_jacobson",
"submodule.ker_mkq",
"submodule.smul_mem",
"sup_comm",
"sup_le_sup_left"
] | *Nakayama's Lemma** - A slightly more general version of (4) in
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV).
See also `smul_sup_eq_of_le_smul_of_le_jacobson_bot` for the special case when `J = ⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_sup_le_of_le_smul_of_le_jacobson_bot {I : ideal R}
{N N' : submodule R M} (hN' : N'.fg) (hIJ : I ≤ jacobson ⊥)
(hNN : N ⊔ N' ≤ N ⊔ I • N') : I • N' ≤ N | by rw [← sup_eq_left, smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson hN' hIJ hNN,
bot_smul, sup_bot_eq] | lemma | submodule.smul_sup_le_of_le_smul_of_le_jacobson_bot | ring_theory | src/ring_theory/nakayama.lean | [
"ring_theory.jacobson_ideal"
] | [
"ideal",
"submodule",
"sup_bot_eq",
"sup_eq_left"
] | *Nakayama's Lemma** - Statement (4) in
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV).
See also `smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson` for a generalisation
to the `jacobson` of any ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_nilpotent [has_zero R] [has_pow R ℕ] (x : R) : Prop | ∃ (n : ℕ), x^n = 0 | def | is_nilpotent | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [] | An element is said to be nilpotent if some natural-number-power of it equals zero.
Note that we require only the bare minimum assumptions for the definition to make sense. Even
`monoid_with_zero` is too strong since nilpotency is important in the study of rings that are only
power-associative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_nilpotent.mk [has_zero R] [has_pow R ℕ] (x : R) (n : ℕ)
(e : x ^ n = 0) : is_nilpotent x | ⟨n, e⟩ | lemma | is_nilpotent.mk | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent.zero [monoid_with_zero R] : is_nilpotent (0 : R) | ⟨1, pow_one 0⟩ | lemma | is_nilpotent.zero | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"monoid_with_zero",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent.neg [ring R] (h : is_nilpotent x) : is_nilpotent (-x) | begin
obtain ⟨n, hn⟩ := h,
use n,
rw [neg_pow, hn, mul_zero],
end | lemma | is_nilpotent.neg | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"mul_zero",
"neg_pow",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_neg_iff [ring R] : is_nilpotent (-x) ↔ is_nilpotent x | ⟨λ h, neg_neg x ▸ h.neg, λ h, h.neg⟩ | lemma | is_nilpotent_neg_iff | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent.map [monoid_with_zero R] [monoid_with_zero S] {r : R}
{F : Type*} [monoid_with_zero_hom_class F R S] (hr : is_nilpotent r) (f : F) :
is_nilpotent (f r) | by { use hr.some, rw [← map_pow, hr.some_spec, map_zero] } | lemma | is_nilpotent.map | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"map_pow",
"monoid_with_zero",
"monoid_with_zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_reduced (R : Type*) [has_zero R] [has_pow R ℕ] : Prop | (eq_zero : ∀ (x : R), is_nilpotent x → x = 0) | class | is_reduced | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent"
] | A structure that has zero and pow is reduced if it has no nonzero nilpotent elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_reduced_of_no_zero_divisors [monoid_with_zero R] [no_zero_divisors R] : is_reduced R | ⟨λ _ ⟨_, hn⟩, pow_eq_zero hn⟩ | instance | is_reduced_of_no_zero_divisors | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_reduced",
"monoid_with_zero",
"no_zero_divisors",
"pow_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_reduced_of_subsingleton [has_zero R] [has_pow R ℕ] [subsingleton R] :
is_reduced R | ⟨λ _ _, subsingleton.elim _ _⟩ | instance | is_reduced_of_subsingleton | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_reduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent.eq_zero [has_zero R] [has_pow R ℕ] [is_reduced R]
(h : is_nilpotent x) : x = 0 | is_reduced.eq_zero x h | lemma | is_nilpotent.eq_zero | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"is_reduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_iff_eq_zero [monoid_with_zero R] [is_reduced R] :
is_nilpotent x ↔ x = 0 | ⟨λ h, h.eq_zero, λ h, h.symm ▸ is_nilpotent.zero⟩ | lemma | is_nilpotent_iff_eq_zero | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"is_reduced",
"monoid_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_reduced_of_injective [monoid_with_zero R] [monoid_with_zero S]
{F : Type*} [monoid_with_zero_hom_class F R S] (f : F)
(hf : function.injective f) [_root_.is_reduced S] : _root_.is_reduced R | begin
constructor,
intros x hx,
apply hf,
rw map_zero,
exact (hx.map f).eq_zero,
end | lemma | is_reduced_of_injective | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"monoid_with_zero",
"monoid_with_zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.ker_is_radical_iff_reduced_of_surjective {S F} [comm_semiring R] [comm_ring S]
[ring_hom_class F R S] {f : F} (hf : function.surjective f) :
(ring_hom.ker f).is_radical ↔ is_reduced S | by simp_rw [is_reduced_iff, hf.forall, is_nilpotent, ← map_pow, ← ring_hom.mem_ker]; refl | lemma | ring_hom.ker_is_radical_iff_reduced_of_surjective | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"comm_ring",
"comm_semiring",
"is_nilpotent",
"is_radical",
"is_reduced",
"map_pow",
"ring_hom.ker",
"ring_hom.mem_ker",
"ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical [has_dvd R] [has_pow R ℕ] (y : R) : Prop | ∀ (n : ℕ) x, y ∣ x ^ n → y ∣ x | def | is_radical | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [] | An element `y` in a monoid is radical if for any element `x`, `y` divides `x` whenever it
divides a power of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_is_radical_iff [monoid_with_zero R] : is_radical (0 : R) ↔ is_reduced R | by { simp_rw [is_reduced_iff, is_nilpotent, exists_imp_distrib, ← zero_dvd_iff], exact forall_swap } | lemma | zero_is_radical_iff | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"exists_imp_distrib",
"forall_swap",
"is_nilpotent",
"is_radical",
"is_reduced",
"monoid_with_zero",
"zero_dvd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical_iff_span_singleton [comm_semiring R] :
is_radical y ↔ (ideal.span ({y} : set R)).is_radical | begin
simp_rw [is_radical, ← ideal.mem_span_singleton],
exact forall_swap.trans (forall_congr $ λ r, exists_imp_distrib.symm),
end | lemma | is_radical_iff_span_singleton | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"comm_semiring",
"ideal.mem_span_singleton",
"ideal.span",
"is_radical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical_iff_pow_one_lt [monoid_with_zero R] (k : ℕ) (hk : 1 < k) :
is_radical y ↔ ∀ x, y ∣ x ^ k → y ∣ x | ⟨λ h x, h k x, λ h, k.cauchy_induction_mul
(λ n h x hd, h x $ (pow_succ' x n).symm ▸ hd.mul_right x) 0 hk
(λ x hd, pow_one x ▸ hd) (λ n _ hn x hd, h x $ hn _ $ (pow_mul x k n).subst hd)⟩ | lemma | is_radical_iff_pow_one_lt | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_radical",
"monoid_with_zero",
"pow_mul",
"pow_one",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_reduced_iff_pow_one_lt [monoid_with_zero R] (k : ℕ) (hk : 1 < k) :
is_reduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 | by simp_rw [← zero_is_radical_iff, is_radical_iff_pow_one_lt k hk, zero_dvd_iff] | lemma | is_reduced_iff_pow_one_lt | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_radical_iff_pow_one_lt",
"is_reduced",
"monoid_with_zero",
"zero_dvd_iff",
"zero_is_radical_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_add (hx : is_nilpotent x) (hy : is_nilpotent y) : is_nilpotent (x + y) | begin
obtain ⟨n, hn⟩ := hx,
obtain ⟨m, hm⟩ := hy,
use n + m - 1,
rw h_comm.add_pow',
apply finset.sum_eq_zero,
rintros ⟨i, j⟩ hij,
suffices : x^i * y^j = 0, { simp only [this, nsmul_eq_mul, mul_zero], },
cases nat.le_or_le_of_add_eq_add_pred (finset.nat.mem_antidiagonal.mp hij) with hi hj,
{ rw [pow_e... | lemma | commute.is_nilpotent_add | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"mul_zero",
"nat.le_or_le_of_add_eq_add_pred",
"nsmul_eq_mul",
"pow_eq_zero_of_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_mul_left (h : is_nilpotent x) : is_nilpotent (x * y) | begin
obtain ⟨n, hn⟩ := h,
use n,
rw [h_comm.mul_pow, hn, zero_mul],
end | lemma | commute.is_nilpotent_mul_left | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_mul_right (h : is_nilpotent y) : is_nilpotent (x * y) | by { rw h_comm.eq, exact h_comm.symm.is_nilpotent_mul_left h, } | lemma | commute.is_nilpotent_mul_right | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_sub (hx : is_nilpotent x) (hy : is_nilpotent y) : is_nilpotent (x - y) | begin
rw ← neg_right_iff at h_comm,
rw ← is_nilpotent_neg_iff at hy,
rw sub_eq_add_neg,
exact h_comm.is_nilpotent_add hx hy,
end | lemma | commute.is_nilpotent_sub | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"is_nilpotent_neg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nilradical (R : Type*) [comm_semiring R] : ideal R | (0 : ideal R).radical | def | nilradical | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"comm_semiring",
"ideal"
] | The nilradical of a commutative semiring is the ideal of nilpotent elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nilradical : x ∈ nilradical R ↔ is_nilpotent x | iff.rfl | lemma | mem_nilradical | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent",
"nilradical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nilradical_eq_Inf (R : Type*) [comm_semiring R] :
nilradical R = Inf { J : ideal R | J.is_prime } | (ideal.radical_eq_Inf ⊥).trans $ by simp_rw and_iff_right bot_le | lemma | nilradical_eq_Inf | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"bot_le",
"comm_semiring",
"ideal",
"ideal.radical_eq_Inf",
"nilradical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nilpotent_iff_mem_prime : is_nilpotent x ↔ ∀ (J : ideal R), J.is_prime → x ∈ J | by { rw [← mem_nilradical, nilradical_eq_Inf, submodule.mem_Inf], refl } | lemma | nilpotent_iff_mem_prime | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"ideal",
"is_nilpotent",
"mem_nilradical",
"nilradical_eq_Inf",
"submodule.mem_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nilradical_le_prime (J : ideal R) [H : J.is_prime] : nilradical R ≤ J | (nilradical_eq_Inf R).symm ▸ Inf_le H | lemma | nilradical_le_prime | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"Inf_le",
"ideal",
"nilradical",
"nilradical_eq_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nilradical_eq_zero (R : Type*) [comm_semiring R] [is_reduced R] : nilradical R = 0 | ideal.ext $ λ _, is_nilpotent_iff_eq_zero | lemma | nilradical_eq_zero | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"comm_semiring",
"ideal.ext",
"is_nilpotent_iff_eq_zero",
"is_reduced",
"nilradical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_mul_left_iff (a : A) :
is_nilpotent (mul_left R a) ↔ is_nilpotent a | begin
split; rintros ⟨n, hn⟩; use n;
simp only [mul_left_eq_zero_iff, pow_mul_left] at ⊢ hn;
exact hn,
end | lemma | linear_map.is_nilpotent_mul_left_iff | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_mul_right_iff (a : A) :
is_nilpotent (mul_right R a) ↔ is_nilpotent a | begin
split; rintros ⟨n, hn⟩; use n;
simp only [mul_right_eq_zero_iff, pow_mul_right] at ⊢ hn;
exact hn,
end | lemma | linear_map.is_nilpotent_mul_right_iff | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent.mapq (hnp : is_nilpotent f) : is_nilpotent (p.mapq p f hp) | begin
obtain ⟨k, hk⟩ := hnp,
use k,
simp [← p.mapq_pow, hk],
end | lemma | module.End.is_nilpotent.mapq | ring_theory | src/ring_theory/nilpotent.lean | [
"data.nat.choose.sum",
"algebra.algebra.bilinear",
"ring_theory.ideal.operations"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian (R M) [semiring R] [add_comm_monoid M] [module R M] : Prop | (noetherian : ∀ (s : submodule R M), s.fg) | class | is_noetherian | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_monoid",
"module",
"semiring",
"submodule"
] | `is_noetherian R M` is the proposition that `M` is a Noetherian `R`-module,
implemented as the predicate that all `R`-submodules of `M` are finitely generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_def : is_noetherian R M ↔ ∀ (s : submodule R M), s.fg | ⟨λ h, h.noetherian, is_noetherian.mk⟩ | lemma | is_noetherian_def | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"submodule"
] | An R-module is Noetherian iff all its submodules are finitely-generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_submodule {N : submodule R M} :
is_noetherian R N ↔ ∀ s : submodule R M, s ≤ N → s.fg | begin
refine ⟨λ ⟨hn⟩, λ s hs, have s ≤ N.subtype.range, from (N.range_subtype).symm ▸ hs,
submodule.map_comap_eq_self this ▸ (hn _).map _, λ h, ⟨λ s, _⟩⟩,
have f := (submodule.equiv_map_of_injective N.subtype subtype.val_injective s).symm,
have h₁ := h (s.map N.subtype) (submodule.map_subtype_le N s),
have ... | theorem | is_noetherian_submodule | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"submodule",
"submodule.equiv_map_of_injective",
"submodule.fg_top",
"submodule.map_comap_eq_self",
"submodule.map_subtype_le",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_submodule_left {N : submodule R M} :
is_noetherian R N ↔ ∀ s : submodule R M, (N ⊓ s).fg | is_noetherian_submodule.trans
⟨λ H s, H _ inf_le_left, λ H s hs, (inf_of_le_right hs) ▸ H _⟩ | theorem | is_noetherian_submodule_left | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"inf_le_left",
"is_noetherian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_submodule_right {N : submodule R M} :
is_noetherian R N ↔ ∀ s : submodule R M, (s ⊓ N).fg | is_noetherian_submodule.trans
⟨λ H s, H _ inf_le_right, λ H s hs, (inf_of_le_left hs) ▸ H _⟩ | theorem | is_noetherian_submodule_right | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"inf_le_right",
"is_noetherian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_submodule' [is_noetherian R M] (N : submodule R M) : is_noetherian R N | is_noetherian_submodule.2 $ λ _ _, is_noetherian.noetherian _ | instance | is_noetherian_submodule' | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_le {s t : submodule R M} [ht : is_noetherian R t]
(h : s ≤ t) : is_noetherian R s | is_noetherian_submodule.mpr (λ s' hs', is_noetherian_submodule.mp ht _ (le_trans hs' h)) | lemma | is_noetherian_of_le | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_surjective (f : M →ₗ[R] P) (hf : f.range = ⊤)
[is_noetherian R M] : is_noetherian R P | ⟨λ s, have (s.comap f).map f = s, from submodule.map_comap_eq_self $ hf.symm ▸ le_top,
this ▸ (noetherian _).map _⟩ | theorem | is_noetherian_of_surjective | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"le_top",
"submodule.map_comap_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_linear_equiv (f : M ≃ₗ[R] P)
[is_noetherian R M] : is_noetherian R P | is_noetherian_of_surjective _ f.to_linear_map f.range | theorem | is_noetherian_of_linear_equiv | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_top_iff :
is_noetherian R (⊤ : submodule R M) ↔ is_noetherian R M | begin
unfreezingI { split; assume h },
{ exact is_noetherian_of_linear_equiv (linear_equiv.of_top (⊤ : submodule R M) rfl) },
{ exact is_noetherian_of_linear_equiv (linear_equiv.of_top (⊤ : submodule R M) rfl).symm },
end | lemma | is_noetherian_top_iff | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_of_linear_equiv",
"linear_equiv.of_top",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_injective [is_noetherian R P] (f : M →ₗ[R] P) (hf : function.injective f) :
is_noetherian R M | is_noetherian_of_linear_equiv (linear_equiv.of_injective f hf).symm | lemma | is_noetherian_of_injective | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_of_linear_equiv",
"linear_equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_injective [is_noetherian R P] {N : submodule R M} (f : M →ₗ[R] P)
(hf : function.injective f) : N.fg | @@is_noetherian.noetherian _ _ _ (is_noetherian_of_injective f hf) N | lemma | fg_of_injective | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_of_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian.finite [is_noetherian R M] : finite R M | ⟨is_noetherian.noetherian ⊤⟩ | instance | module.is_noetherian.finite | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"finite",
"is_noetherian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.of_injective [is_noetherian R N] (f : M →ₗ[R] N)
(hf : function.injective f) : finite R M | ⟨fg_of_injective f hf⟩ | lemma | module.finite.of_injective | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"finite",
"finite.of_injective",
"is_noetherian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_ker_bot [is_noetherian R P] (f : M →ₗ[R] P) (hf : f.ker = ⊥) :
is_noetherian R M | is_noetherian_of_linear_equiv (linear_equiv.of_injective f $ linear_map.ker_eq_bot.mp hf).symm | lemma | is_noetherian_of_ker_bot | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_of_linear_equiv",
"linear_equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_ker_bot [is_noetherian R P] {N : submodule R M} (f : M →ₗ[R] P) (hf : f.ker = ⊥) :
N.fg | @@is_noetherian.noetherian _ _ _ (is_noetherian_of_ker_bot f hf) N | lemma | fg_of_ker_bot | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_of_ker_bot",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_prod [is_noetherian R M]
[is_noetherian R P] : is_noetherian R (M × P) | ⟨λ s, submodule.fg_of_fg_map_of_fg_inf_ker (linear_map.snd R M P) (noetherian _) $
have s ⊓ linear_map.ker (linear_map.snd R M P) ≤ linear_map.range (linear_map.inl R M P),
from λ x ⟨hx1, hx2⟩, ⟨x.1, prod.ext rfl $ eq.symm $ linear_map.mem_ker.1 hx2⟩,
submodule.map_comap_eq_self this ▸ (noetherian _).map _⟩ | instance | is_noetherian_prod | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"linear_map.inl",
"linear_map.ker",
"linear_map.range",
"linear_map.snd",
"prod.ext",
"submodule.fg_of_fg_map_of_fg_inf_ker",
"submodule.map_comap_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_pi {R ι : Type*} {M : ι → Type*} [ring R]
[Π i, add_comm_group (M i)] [Π i, module R (M i)] [finite ι]
[∀ i, is_noetherian R (M i)] : is_noetherian R (Π i, M i) | begin
casesI nonempty_fintype ι,
haveI := classical.dec_eq ι,
suffices on_finset : ∀ s : finset ι, is_noetherian R (Π i : s, M i),
{ let coe_e := equiv.subtype_univ_equiv finset.mem_univ,
letI : is_noetherian R (Π i : finset.univ, M (coe_e i)) := on_finset finset.univ,
exact is_noetherian_of_linear_equi... | instance | is_noetherian_pi | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_group",
"classical.dec_eq",
"equiv.subtype_univ_equiv",
"finite",
"finset",
"finset.induction",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.mem_univ",
"finset.univ",
"ih",
"is_noetherian",
"is_noetherian_of_linear_equiv",
"is_noetherian_prod",
"linear_equiv.Pi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_pi' {R ι M : Type*} [ring R] [add_comm_group M] [module R M] [finite ι]
[is_noetherian R M] : is_noetherian R (ι → M) | is_noetherian_pi | instance | is_noetherian_pi' | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_group",
"finite",
"is_noetherian",
"is_noetherian_pi",
"module",
"ring"
] | A version of `is_noetherian_pi` for non-dependent functions. We need this instance because
sometimes Lean fails to apply the dependent version in non-dependent settings (e.g., it fails to
prove that `ι → ℝ` is finite dimensional over `ℝ`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_iff_well_founded :
is_noetherian R M ↔ well_founded ((>) : submodule R M → submodule R M → Prop) | begin
rw (complete_lattice.well_founded_characterisations $ submodule R M).out 0 3,
exact ⟨λ ⟨h⟩, λ k, (fg_iff_compact k).mp (h k), λ h, ⟨λ k, (fg_iff_compact k).mpr (h k)⟩⟩,
end | theorem | is_noetherian_iff_well_founded | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"complete_lattice.well_founded_characterisations",
"is_noetherian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_iff_fg_well_founded :
is_noetherian R M ↔ well_founded
((>) : { N : submodule R M // N.fg } → { N : submodule R M // N.fg } → Prop) | begin
let α := { N : submodule R M // N.fg },
split,
{ introI H,
let f : α ↪o submodule R M := order_embedding.subtype _,
exact order_embedding.well_founded f.dual (is_noetherian_iff_well_founded.mp H) },
{ intro H,
constructor,
intro N,
obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ := well_founded.has_... | lemma | is_noetherian_iff_fg_well_founded | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"by_contra",
"eq_of_le_of_not_lt",
"finset.coe_singleton",
"is_noetherian",
"le_sup_left",
"le_sup_right",
"order_embedding.subtype",
"order_embedding.well_founded",
"submodule",
"submodule.fg.sup",
"submodule.mem_span_singleton_self",
"submodule.span_singleton_le_iff_mem",
"sup_le",
"well... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_founded_submodule_gt (R M) [semiring R] [add_comm_monoid M] [module R M] :
∀ [is_noetherian R M], well_founded ((>) : submodule R M → submodule R M → Prop) | is_noetherian_iff_well_founded.mp | lemma | well_founded_submodule_gt | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_monoid",
"is_noetherian",
"module",
"semiring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_has_maximal_iff_noetherian :
(∀ a : set $ submodule R M, a.nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬ M' < I) ↔ is_noetherian R M | by rw [is_noetherian_iff_well_founded, well_founded.well_founded_iff_has_min] | theorem | set_has_maximal_iff_noetherian | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_iff_well_founded",
"submodule",
"well_founded.well_founded_iff_has_min"
] | A module is Noetherian iff every nonempty set of submodules has a maximal submodule among them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_stabilizes_iff_noetherian :
(∀ (f : ℕ →o submodule R M), ∃ n, ∀ m, n ≤ m → f n = f m) ↔ is_noetherian R M | by rw [is_noetherian_iff_well_founded, well_founded.monotone_chain_condition] | theorem | monotone_stabilizes_iff_noetherian | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_iff_well_founded",
"submodule",
"well_founded.monotone_chain_condition"
] | A module is Noetherian iff every increasing chain of submodules stabilizes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian.induction [is_noetherian R M] {P : submodule R M → Prop}
(hgt : ∀ I, (∀ J > I, P J) → P I) (I : submodule R M) : P I | well_founded.recursion (well_founded_submodule_gt R M) I hgt | lemma | is_noetherian.induction | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"submodule",
"well_founded_submodule_gt"
] | If `∀ I > J, P I` implies `P J`, then `P` holds for all submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_of_linear_independent [nontrivial R] [is_noetherian R M]
{s : set M} (hs : linear_independent R (coe : s → M)) : s.finite | begin
refine classical.by_contradiction (λ hf, (rel_embedding.well_founded_iff_no_descending_seq.1
(well_founded_submodule_gt R M)).elim' _),
have f : ℕ ↪ s, from set.infinite.nat_embedding s hf,
have : ∀ n, (coe ∘ f) '' {m | m ≤ n} ⊆ s,
{ rintros n x ⟨y, hy₁, rfl⟩, exact (f y).2 },
have : ∀ a b : ℕ, a ≤ ... | lemma | finite_of_linear_independent | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"linear_independent",
"nontrivial",
"set.image_subset_image_iff",
"set.infinite.nat_embedding",
"set.subset_def",
"span_le_span_iff",
"well_founded_submodule_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_range_eq_ker
[is_noetherian R M] [is_noetherian R P]
(f : M →ₗ[R] N) (g : N →ₗ[R] P)
(hf : function.injective f)
(hg : function.surjective g)
(h : f.range = g.ker) :
is_noetherian R N | is_noetherian_iff_well_founded.2 $
well_founded_gt_exact_sequence
(well_founded_submodule_gt R M)
(well_founded_submodule_gt R P)
f.range
(submodule.map f)
(submodule.comap f)
(submodule.comap g)
(submodule.map g)
(submodule.gci_map_comap hf)
(submodule.gi_map_comap hg)
(by simp [submodule.map_comap... | theorem | is_noetherian_of_range_eq_ker | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"inf_comm",
"is_noetherian",
"submodule.comap",
"submodule.comap_map_eq",
"submodule.gci_map_comap",
"submodule.gi_map_comap",
"submodule.map",
"submodule.map_comap_eq",
"well_founded_gt_exact_sequence",
"well_founded_submodule_gt"
] | If the first and final modules in a short exact sequence are noetherian,
then the middle module is also noetherian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian.exists_endomorphism_iterate_ker_inf_range_eq_bot
[I : is_noetherian R M] (f : M →ₗ[R] M) : ∃ n : ℕ, n ≠ 0 ∧ (f ^ n).ker ⊓ (f ^ n).range = ⊥ | begin
obtain ⟨n, w⟩ := monotone_stabilizes_iff_noetherian.mpr I
(f.iterate_ker.comp ⟨λ n, n+1, λ n m w, by linarith⟩),
specialize w (2 * n + 1) (by linarith only),
dsimp at w,
refine ⟨n+1, nat.succ_ne_zero _, _⟩,
rw eq_bot_iff,
rintros - ⟨h, ⟨y, rfl⟩⟩,
rw [mem_bot, ←linear_map.mem_ker, w],
erw linea... | theorem | is_noetherian.exists_endomorphism_iterate_ker_inf_range_eq_bot | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"eq_bot_iff",
"is_noetherian",
"linear_map.mem_ker",
"ring"
] | For any endomorphism of a Noetherian module, there is some nontrivial iterate
with disjoint kernel and range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian.injective_of_surjective_endomorphism [is_noetherian R M]
(f : M →ₗ[R] M) (s : surjective f) : injective f | begin
obtain ⟨n, ne, w⟩ := is_noetherian.exists_endomorphism_iterate_ker_inf_range_eq_bot f,
rw [linear_map.range_eq_top.mpr (linear_map.iterate_surjective s n), inf_top_eq,
linear_map.ker_eq_bot] at w,
exact linear_map.injective_of_iterate_injective ne w,
end | theorem | is_noetherian.injective_of_surjective_endomorphism | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"inf_top_eq",
"is_noetherian",
"is_noetherian.exists_endomorphism_iterate_ker_inf_range_eq_bot",
"linear_map.injective_of_iterate_injective",
"linear_map.iterate_surjective",
"linear_map.ker_eq_bot"
] | Any surjective endomorphism of a Noetherian module is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian.bijective_of_surjective_endomorphism [is_noetherian R M]
(f : M →ₗ[R] M) (s : surjective f) : bijective f | ⟨is_noetherian.injective_of_surjective_endomorphism f s, s⟩ | theorem | is_noetherian.bijective_of_surjective_endomorphism | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian"
] | Any surjective endomorphism of a Noetherian module is bijective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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