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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hsum_of_finsupp {f : α →₀ (hahn_series Γ R)} :
(of_finsupp f).hsum = f.sum (λ a, id) | begin
ext g,
simp only [hsum_coeff, coe_of_finsupp, finsupp.sum, ne.def],
simp_rw [← coeff.add_monoid_hom_apply, id.def],
rw [add_monoid_hom.map_sum, finsum_eq_sum_of_support_subset],
intros x h,
simp only [coeff.add_monoid_hom_apply, mem_coe, finsupp.mem_support_iff, ne.def],
contrapose! h,
simp [h]
en... | lemma | hahn_series.summable_family.hsum_of_finsupp | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"finsupp.mem_support_iff",
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain (s : summable_family Γ R α) (f : α ↪ β) : summable_family Γ R β | { to_fun := λ b, if h : b ∈ set.range f then s (classical.some h) else 0,
is_pwo_Union_support' := begin
refine s.is_pwo_Union_support.mono (set.Union_subset (λ b g h, _)),
by_cases hb : b ∈ set.range f,
{ rw dif_pos hb at h,
exact set.mem_Union.2 ⟨classical.some hb, h⟩ },
{ contrapose! h,
... | def | hahn_series.summable_family.emb_domain | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"not_not",
"set.Union_subset",
"set.range"
] | A summable family can be reindexed by an embedding without changing its sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emb_domain_apply :
s.emb_domain f b = if h : b ∈ set.range f then s (classical.some h) else 0 | rfl | lemma | hahn_series.summable_family.emb_domain_apply | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_image : s.emb_domain f (f a) = s a | begin
rw [emb_domain_apply, dif_pos (set.mem_range_self a)],
exact congr rfl (f.injective (classical.some_spec (set.mem_range_self a)))
end | lemma | hahn_series.summable_family.emb_domain_image | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"set.mem_range_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_notin_range (h : b ∉ set.range f) : s.emb_domain f b = 0 | by rw [emb_domain_apply, dif_neg h] | lemma | hahn_series.summable_family.emb_domain_notin_range | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hsum_emb_domain :
(s.emb_domain f).hsum = s.hsum | begin
ext g,
simp only [hsum_coeff, emb_domain_apply, apply_dite hahn_series.coeff, dite_apply, zero_coeff],
exact finsum_emb_domain f (λ a, (s a).coeff g)
end | lemma | hahn_series.summable_family.hsum_emb_domain | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"apply_dite",
"dite_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powers (x : hahn_series Γ R) (hx : 0 < add_val Γ R x) :
summable_family Γ R ℕ | { to_fun := λ n, x ^ n,
is_pwo_Union_support' := is_pwo_Union_support_powers hx,
finite_co_support' := λ g, begin
have hpwo := (is_pwo_Union_support_powers hx),
by_cases hg : g ∈ ⋃ n : ℕ, {g | (x ^ n).coeff g ≠ 0 },
swap, { exact set.finite_empty.subset (λ n hn, hg (set.mem_Union.2 ⟨n, hn⟩)) },
appl... | def | hahn_series.summable_family.powers | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series",
"powers",
"set.finite_singleton",
"set.mem_Union",
"set.mem_singleton",
"set.mem_union_left",
"set.mem_union_right"
] | The powers of an element of positive valuation form a summable family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_powers : ⇑(powers x hx) = pow x | rfl | lemma | hahn_series.summable_family.coe_powers | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_succ_smul_powers :
(x • powers x hx).emb_domain ⟨nat.succ, nat.succ_injective⟩ =
powers x hx - of_finsupp (finsupp.single 0 1) | begin
apply summable_family.ext (λ n, _),
cases n,
{ rw [emb_domain_notin_range, sub_apply, coe_powers, pow_zero, coe_of_finsupp,
finsupp.single_eq_same, sub_self],
rw [set.mem_range, not_exists],
exact nat.succ_ne_zero },
{ refine eq.trans (emb_domain_image _ ⟨nat.succ, nat.succ_injective⟩) _,
... | lemma | hahn_series.summable_family.emb_domain_succ_smul_powers | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"finsupp.single",
"finsupp.single_eq_of_ne",
"finsupp.single_eq_same",
"not_exists",
"pow_succ",
"pow_zero",
"powers",
"set.mem_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_self_mul_hsum_powers :
(1 - x) * (powers x hx).hsum = 1 | begin
rw [← hsum_smul, sub_smul, one_smul, hsum_sub,
← hsum_emb_domain (x • powers x hx) ⟨nat.succ, nat.succ_injective⟩,
emb_domain_succ_smul_powers],
simp,
end | lemma | hahn_series.summable_family.one_sub_self_mul_hsum_powers | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"one_smul",
"powers",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_aux (x : hahn_series Γ R) {r : R} (hr : r * x.coeff x.order = 1) :
0 < add_val Γ R (1 - C r * (single (- x.order) 1) * x) | begin
have h10 : (1 : R) ≠ 0 := one_ne_zero,
have x0 : x ≠ 0 := ne_zero_of_coeff_ne_zero (right_ne_zero_of_mul_eq_one hr),
refine lt_of_le_of_ne ((add_val Γ R).map_le_sub (ge_of_eq (add_val Γ R).map_one) _) _,
{ simp only [add_valuation.map_mul],
rw [add_val_apply_of_ne x0, add_val_apply_of_ne (single_ne_ze... | lemma | hahn_series.unit_aux | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"add_valuation.map_mul",
"con",
"ge_of_eq",
"hahn_series",
"left_ne_zero_of_mul_eq_one",
"map_one",
"mul_assoc",
"one_mul",
"one_ne_zero",
"right_ne_zero_of_mul_eq_one",
"smul_eq_mul",
"with_top.coe_add",
"with_top.coe_eq_coe",
"with_top.coe_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff {x : hahn_series Γ R} :
is_unit x ↔ is_unit (x.coeff x.order) | begin
split,
{ rintro ⟨⟨u, i, ui, iu⟩, rfl⟩,
refine is_unit_of_mul_eq_one (u.coeff u.order) (i.coeff i.order)
((mul_coeff_order_add_order u i).symm.trans _),
rw [ui, one_coeff, if_pos],
rw [← order_mul (left_ne_zero_of_mul_eq_one ui)
(right_ne_zero_of_mul_eq_one ui), ui, order_one] },
{ ri... | lemma | hahn_series.is_unit_iff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series",
"is_unit",
"is_unit_of_mul_eq_one",
"is_unit_of_mul_is_unit_right",
"left_ne_zero_of_mul_eq_one",
"right_ne_zero_of_mul_eq_one",
"units.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_of_le_jacobson_bot {R : Type*} [comm_ring R]
(I : ideal R) (h : I ≤ ideal.jacobson ⊥) :
is_local_ring_hom (ideal.quotient.mk I) | begin
constructor,
intros a h,
have : is_unit (ideal.quotient.mk (ideal.jacobson ⊥) a),
{ rw [is_unit_iff_exists_inv] at *,
obtain ⟨b, hb⟩ := h,
obtain ⟨b, rfl⟩ := ideal.quotient.mk_surjective b,
use ideal.quotient.mk _ b,
rw [←(ideal.quotient.mk _).map_one, ←(ideal.quotient.mk _).map_mul, ideal... | lemma | is_local_ring_hom_of_le_jacobson_bot | ring_theory | src/ring_theory/henselian.lean | [
"data.polynomial.taylor",
"ring_theory.ideal.local_ring",
"linear_algebra.adic_completion"
] | [
"comm_ring",
"ideal",
"ideal.jacobson",
"ideal.mem_jacobson_bot",
"ideal.quotient.eq",
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"is_local_ring_hom",
"is_unit",
"is_unit_iff_exists_inv",
"map_mul",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
henselian_ring (R : Type*) [comm_ring R] (I : ideal R) : Prop | (jac : I ≤ ideal.jacobson ⊥)
(is_henselian : ∀ (f : R[X]) (hf : f.monic) (a₀ : R) (h₁ : f.eval a₀ ∈ I)
(h₂ : is_unit (ideal.quotient.mk I (f.derivative.eval a₀))),
∃ a : R, f.is_root a ∧ (a - a₀ ∈ I)) | class | henselian_ring | ring_theory | src/ring_theory/henselian.lean | [
"data.polynomial.taylor",
"ring_theory.ideal.local_ring",
"linear_algebra.adic_completion"
] | [
"comm_ring",
"ideal",
"ideal.jacobson",
"ideal.quotient.mk",
"is_unit"
] | A ring `R` is *Henselian* at an ideal `I` if the following condition holds:
for every polynomial `f` over `R`, with a *simple* root `a₀` over the quotient ring `R/I`,
there exists a lift `a : R` of `a₀` that is a root of `f`.
(Here, saying that a root `b` of a polynomial `g` is *simple* means that `g.derivative.eval b... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
henselian_local_ring (R : Type*) [comm_ring R] extends local_ring R : Prop | (is_henselian : ∀ (f : R[X]) (hf : f.monic) (a₀ : R) (h₁ : f.eval a₀ ∈ maximal_ideal R)
(h₂ : is_unit (f.derivative.eval a₀)),
∃ a : R, f.is_root a ∧ (a - a₀ ∈ maximal_ideal R)) | class | henselian_local_ring | ring_theory | src/ring_theory/henselian.lean | [
"data.polynomial.taylor",
"ring_theory.ideal.local_ring",
"linear_algebra.adic_completion"
] | [
"comm_ring",
"is_unit",
"local_ring"
] | A local ring `R` is *Henselian* if the following condition holds:
for every polynomial `f` over `R`, with a *simple* root `a₀` over the residue field,
there exists a lift `a : R` of `a₀` that is a root of `f`.
(Recall that a root `b` of a polynomial `g` is *simple* if it is not a double root, so if
`g.derivative.eval b... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
field.henselian (K : Type*) [field K] : henselian_local_ring K | { is_henselian := λ f hf a₀ h₁ h₂,
begin
refine ⟨a₀, _, _⟩;
rwa [(maximal_ideal K).eq_bot_of_prime, ideal.mem_bot] at *,
rw sub_self,
end } | instance | field.henselian | ring_theory | src/ring_theory/henselian.lean | [
"data.polynomial.taylor",
"ring_theory.ideal.local_ring",
"linear_algebra.adic_completion"
] | [
"field",
"henselian_local_ring",
"ideal.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
henselian_local_ring.tfae (R : Type u) [comm_ring R] [local_ring R] :
tfae [
henselian_local_ring R,
∀ (f : R[X]) (hf : f.monic) (a₀ : residue_field R) (h₁ : aeval a₀ f = 0)
(h₂ : aeval a₀ f.derivative ≠ 0),
∃ a : R, f.is_root a ∧ (residue R a = a₀),
∀ {K : Type u} [field K], by exactI ∀ (φ : R →+* K) (... | begin
tfae_have _3_2 : 3 → 2, { intro H, exact H (residue R) ideal.quotient.mk_surjective, },
tfae_have _2_1 : 2 → 1,
{ intros H, constructor, intros f hf a₀ h₁ h₂,
specialize H f hf (residue R a₀),
have aux := flip mem_nonunits_iff.mp h₂,
simp only [aeval_def, residue_field.algebra_map_eq, eval₂_at_a... | lemma | henselian_local_ring.tfae | ring_theory | src/ring_theory/henselian.lean | [
"data.polynomial.taylor",
"ring_theory.ideal.local_ring",
"linear_algebra.adic_completion"
] | [
"aux",
"comm_ring",
"field",
"henselian_local_ring",
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.mk_surjective",
"local_ring",
"local_ring.ker_eq_maximal_ideal",
"local_ring.mem_maximal_ideal",
"mem_nonunits_iff",
"ring_hom.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_adic_complete.henselian_ring
(R : Type*) [comm_ring R] (I : ideal R) [is_adic_complete I R] :
henselian_ring R I | { jac := is_adic_complete.le_jacobson_bot _,
is_henselian :=
begin
intros f hf a₀ h₁ h₂,
classical,
let f' := f.derivative,
-- we define a sequence `c n` by starting at `a₀` and then continually
-- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
-- Note that `f'.eval... | instance | is_adic_complete.henselian_ring | ring_theory | src/ring_theory/henselian.lean | [
"data.polynomial.taylor",
"ring_theory.ideal.local_ring",
"linear_algebra.adic_completion"
] | [
"aux",
"comm_ring",
"finset.mem_Ico",
"finset.range_one",
"henselian_ring",
"ideal",
"ideal.add_mem",
"ideal.mul_mem_left",
"ideal.mul_mem_right",
"ideal.neg_mem_iff",
"ideal.one_eq_top",
"ideal.pow_le_pow",
"ideal.pow_mem_pow",
"ideal.quotient.mk",
"ideal.smul_eq_mul",
"ideal.zero_mem... | A ring `R` that is `I`-adically complete is Henselian at `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integrally_closed (R : Type*) [comm_ring R] [is_domain R] : Prop | (algebra_map_eq_of_integral :
∀ {x : fraction_ring R}, is_integral R x → ∃ y, algebra_map R (fraction_ring R) y = x) | class | is_integrally_closed | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"algebra_map",
"comm_ring",
"fraction_ring",
"is_domain",
"is_integral"
] | `R` is integrally closed if all integral elements of `Frac(R)` are also elements of `R`.
This definition uses `fraction_ring R` to denote `Frac(R)`. See `is_integrally_closed_iff`
if you want to choose another field of fractions for `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integrally_closed_iff :
is_integrally_closed R ↔ ∀ {x : K}, is_integral R x → ∃ y, algebra_map R K y = x | begin
let e : K ≃ₐ[R] fraction_ring R := is_localization.alg_equiv R⁰_ _,
split,
{ rintros ⟨cl⟩,
refine λ x hx, _,
obtain ⟨y, hy⟩ := cl ((is_integral_alg_equiv e).mpr hx),
exact ⟨y, e.algebra_map_eq_apply.mp hy⟩ },
{ rintros cl,
refine ⟨λ x hx, _⟩,
obtain ⟨y, hy⟩ := cl ((is_integral_alg_equi... | theorem | is_integrally_closed_iff | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"algebra_map",
"fraction_ring",
"is_integral",
"is_integral_alg_equiv",
"is_integrally_closed",
"is_localization.alg_equiv"
] | `R` is integrally closed iff all integral elements of its fraction field `K`
are also elements of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integrally_closed_iff_is_integral_closure :
is_integrally_closed R ↔ is_integral_closure R R K | (is_integrally_closed_iff K).trans $
begin
let e : K ≃ₐ[R] fraction_ring R := is_localization.alg_equiv R⁰_ _,
split,
{ intros cl,
refine ⟨is_fraction_ring.injective _ _, λ x, ⟨cl, _⟩⟩,
rintros ⟨y, y_eq⟩,
rw ← y_eq,
exact is_integral_algebra_map },
{ rintros ⟨-, cl⟩ x hx,
exact cl.mp hx }
en... | theorem | is_integrally_closed_iff_is_integral_closure | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"fraction_ring",
"is_integral_algebra_map",
"is_integral_closure",
"is_integrally_closed",
"is_integrally_closed_iff",
"is_localization.alg_equiv"
] | `R` is integrally closed iff it is the integral closure of itself in its field of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_iff {x : K} : is_integral R x ↔ ∃ y : R, algebra_map R K y = x | is_integral_closure.is_integral_iff | lemma | is_integrally_closed.is_integral_iff | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_algebra_map_eq_of_is_integral_pow {x : K} {n : ℕ} (hn : 0 < n)
(hx : is_integral R $ x ^ n) : ∃ y : R, algebra_map R K y = x | is_integral_iff.mp $ is_integral_of_pow hn hx | lemma | is_integrally_closed.exists_algebra_map_eq_of_is_integral_pow | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"algebra_map",
"is_integral",
"is_integral_of_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_algebra_map_eq_of_pow_mem_subalgebra {K : Type*} [field K] [algebra R K]
{S : subalgebra R K} [is_integrally_closed S] [is_fraction_ring S K] {x : K} {n : ℕ} (hn : 0 < n)
(hx : x ^ n ∈ S) : ∃ y : S, algebra_map S K y = x | exists_algebra_map_eq_of_is_integral_pow hn $ is_integral_iff.mpr ⟨⟨x ^ n, hx⟩, rfl⟩ | lemma | is_integrally_closed.exists_algebra_map_eq_of_pow_mem_subalgebra | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"algebra",
"algebra_map",
"field",
"is_fraction_ring",
"is_integrally_closed",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure_eq_bot_iff :
integral_closure R K = ⊥ ↔ is_integrally_closed R | begin
refine eq_bot_iff.trans _,
split,
{ rw is_integrally_closed_iff K,
intros h x hx,
exact set.mem_range.mp (algebra.mem_bot.mp (h hx)),
assumption },
{ intros h x hx,
rw [algebra.mem_bot, set.mem_range],
exactI is_integral_iff.mp hx },
end | lemma | is_integrally_closed.integral_closure_eq_bot_iff | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"algebra.mem_bot",
"integral_closure",
"is_integrally_closed",
"is_integrally_closed_iff",
"set.mem_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure_eq_bot : integral_closure R K = ⊥ | (integral_closure_eq_bot_iff K).mpr ‹_› | lemma | is_integrally_closed.integral_closure_eq_bot | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"integral_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integrally_closed_of_finite_extension [finite_dimensional K L] :
is_integrally_closed (integral_closure R L) | begin
letI : is_fraction_ring (integral_closure R L) L := is_fraction_ring_of_finite_extension K L,
exact (integral_closure_eq_bot_iff L).mp integral_closure_idem
end | lemma | integral_closure.is_integrally_closed_of_finite_extension | ring_theory | src/ring_theory/integrally_closed.lean | [
"ring_theory.integral_closure",
"ring_theory.localization.integral"
] | [
"finite_dimensional",
"integral_closure",
"integral_closure_idem",
"is_fraction_ring",
"is_integrally_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_elem (f : R →+* A) (x : A) | ∃ p : R[X], monic p ∧ eval₂ f x p = 0 | def | ring_hom.is_integral_elem | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | An element `x` of `A` is said to be integral over `R` with respect to `f`
if it is a root of a monic polynomial `p : R[X]` evaluated under `f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_integral (f : R →+* A) | ∀ x : A, f.is_integral_elem x | def | ring_hom.is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | A ring homomorphism `f : R →+* A` is said to be integral
if every element `A` is integral with respect to the map `f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral (x : A) : Prop | (algebra_map R A).is_integral_elem x | def | is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map"
] | An element `x` of an algebra `A` over a commutative ring `R` is said to be *integral*,
if it is a root of some monic polynomial `p : R[X]`.
Equivalently, the element is integral over `R` with respect to the induced `algebra_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.is_integral : Prop | (algebra_map R A).is_integral | def | algebra.is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | An algebra is integral if every element of the extension is integral over the base ring | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_integral_map {x : R} : f.is_integral_elem (f x) | ⟨X - C x, monic_X_sub_C _, by simp⟩ | lemma | ring_hom.is_integral_map | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_algebra_map {x : R} : is_integral R (algebra_map R A x) | (algebra_map R A).is_integral_map | theorem | is_integral_algebra_map | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_noetherian (H : is_noetherian R A) (x : A) :
is_integral R x | begin
let leval : (R[X] →ₗ[R] A) := (aeval x).to_linear_map,
let D : ℕ → submodule R A := λ n, (degree_le R n).map leval,
let M := well_founded.min (is_noetherian_iff_well_founded.1 H)
(set.range D) ⟨_, ⟨0, rfl⟩⟩,
have HM : M ∈ set.range D := well_founded.min_mem _ _ _,
cases HM with N HN,
have HM : ¬M ... | theorem | is_integral_of_noetherian | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"is_noetherian",
"linear_map.map_sub",
"set.range",
"submodule",
"well_founded.min",
"well_founded.min_mem",
"well_founded.not_lt_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_submodule_noetherian (S : subalgebra R A)
(H : is_noetherian R S.to_submodule) (x : A) (hx : x ∈ S) :
is_integral R x | begin
suffices : is_integral R (show S, from ⟨x, hx⟩),
{ rcases this with ⟨p, hpm, hpx⟩,
replace hpx := congr_arg S.val hpx,
refine ⟨p, hpm, eq.trans _ hpx⟩,
simp only [aeval_def, eval₂, sum_def],
rw S.val.map_sum,
refine finset.sum_congr rfl (λ n hn, _),
rw [S.val.map_mul, S.val.map_pow, S.... | theorem | is_integral_of_submodule_noetherian | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"is_integral_of_noetherian",
"is_noetherian",
"subalgebra",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_is_integral {B C F : Type*} [ring B] [ring C] [algebra R B] [algebra A B]
[algebra R C] [is_scalar_tower R A B] [algebra A C] [is_scalar_tower R A C] {b : B}
[alg_hom_class F A B C] (f : F) (hb : is_integral R b) : is_integral R (f b) | begin
obtain ⟨P, hP⟩ := hb,
refine ⟨P, hP.1, _⟩,
rw [← aeval_def, show (aeval (f b)) P = (aeval (f b)) (P.map (algebra_map R A)), by simp,
aeval_alg_hom_apply, aeval_map_algebra_map, aeval_def, hP.2, _root_.map_zero]
end | lemma | map_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"alg_hom_class",
"algebra",
"algebra_map",
"is_integral",
"is_scalar_tower",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_map_of_comp_eq_of_is_integral {R S T U : Type*} [comm_ring R] [comm_ring S]
[comm_ring T] [comm_ring U] [algebra R S] [algebra T U] (φ : R →+* T) (ψ : S →+* U)
(h : (algebra_map T U).comp φ = ψ.comp (algebra_map R S)) {a : S} (ha : is_integral R a) :
is_integral T (ψ a) | begin
rw [is_integral, ring_hom.is_integral_elem] at ⊢ ha,
obtain ⟨p, hp⟩ := ha,
refine ⟨p.map φ, hp.left.map _, _⟩,
rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply,
eval_map, hp.right, ring_hom.map_zero],
end | lemma | is_integral_map_of_comp_eq_of_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_integral",
"ring_hom.is_integral_elem",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_alg_hom_iff {A B : Type*} [ring A] [ring B] [algebra R A] [algebra R B]
(f : A →ₐ[R] B) (hf : function.injective f) {x : A} : is_integral R (f x) ↔ is_integral R x | begin
refine ⟨_, map_is_integral f⟩,
rintros ⟨p, hp, hx⟩,
use [p, hp],
rwa [← f.comp_algebra_map, ← alg_hom.coe_to_ring_hom, ← polynomial.hom_eval₂,
alg_hom.coe_to_ring_hom, map_eq_zero_iff f hf] at hx
end | theorem | is_integral_alg_hom_iff | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"alg_hom.coe_to_ring_hom",
"algebra",
"is_integral",
"map_is_integral",
"polynomial.hom_eval₂",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_alg_equiv {A B : Type*} [ring A] [ring B] [algebra R A] [algebra R B]
(f : A ≃ₐ[R] B) {x : A} : is_integral R (f x) ↔ is_integral R x | ⟨λ h, by simpa using map_is_integral f.symm.to_alg_hom h, map_is_integral f.to_alg_hom⟩ | theorem | is_integral_alg_equiv | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"is_integral",
"map_is_integral",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_is_scalar_tower [algebra A B] [is_scalar_tower R A B]
{x : B} (hx : is_integral R x) : is_integral A x | let ⟨p, hp, hpx⟩ := hx in
⟨p.map $ algebra_map R A, hp.map _,
by rw [← aeval_def, aeval_map_algebra_map, aeval_def, hpx]⟩ | theorem | is_integral_of_is_scalar_tower | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"is_integral",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_is_integral_int {B C F : Type*} [ring B] [ring C] {b : B}
[ring_hom_class F B C] (f : F) (hb : is_integral ℤ b) :
is_integral ℤ (f b) | map_is_integral (f : B →+* C).to_int_alg_hom hb | lemma | map_is_integral_int | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"map_is_integral",
"ring",
"ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_subring {x : A} (T : subring R)
(hx : is_integral T x) : is_integral R x | is_integral_of_is_scalar_tower hx | theorem | is_integral_of_subring | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"is_integral_of_is_scalar_tower",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.algebra_map [algebra A B] [is_scalar_tower R A B]
{x : A} (h : is_integral R x) :
is_integral R (algebra_map A B x) | begin
rcases h with ⟨f, hf, hx⟩,
use [f, hf],
rw [is_scalar_tower.algebra_map_eq R A B, ← hom_eval₂, hx, ring_hom.map_zero]
end | lemma | is_integral.algebra_map | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"is_integral",
"is_scalar_tower",
"is_scalar_tower.algebra_map_eq",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_algebra_map_iff [algebra A B] [is_scalar_tower R A B]
{x : A} (hAB : function.injective (algebra_map A B)) :
is_integral R (algebra_map A B x) ↔ is_integral R x | is_integral_alg_hom_iff (is_scalar_tower.to_alg_hom R A B) hAB | lemma | is_integral_algebra_map_iff | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"is_integral",
"is_integral_alg_hom_iff",
"is_scalar_tower",
"is_scalar_tower.to_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_iff_is_integral_closure_finite {r : A} :
is_integral R r ↔ ∃ s : set R, s.finite ∧ is_integral (subring.closure s) r | begin
split; intro hr,
{ rcases hr with ⟨p, hmp, hpr⟩,
refine ⟨_, finset.finite_to_set _, p.restriction, monic_restriction.2 hmp, _⟩,
rw [← aeval_def, ← aeval_map_algebra_map R r p.restriction,
map_restriction, aeval_def, hpr], },
rcases hr with ⟨s, hs, hsr⟩,
exact is_integral_of_subring _ hsr
end | theorem | is_integral_iff_is_integral_closure_finite | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"finset.finite_to_set",
"is_integral",
"is_integral_of_subring",
"subring.closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_adjoin_singleton_of_integral (x : A) (hx : is_integral R x) :
(algebra.adjoin R ({x} : set A)).to_submodule.fg | begin
rcases hx with ⟨f, hfm, hfx⟩,
existsi finset.image ((^) x) (finset.range (nat_degree f + 1)),
apply le_antisymm,
{ rw span_le, intros s hs, rw finset.mem_coe at hs,
rcases finset.mem_image.1 hs with ⟨k, hk, rfl⟩, clear hk,
exact (algebra.adjoin R {x}).pow_mem (algebra.subset_adjoin (set.mem_single... | theorem | fg_adjoin_singleton_of_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"algebra.adjoin",
"algebra.adjoin_singleton_eq_range_aeval",
"algebra.smul_def",
"algebra.subset_adjoin",
"finset.image",
"finset.mem_coe",
"finset.mem_range",
"finset.range",
"is_integral",
"nat.lt_succ_iff",
"pow_mem",
"set.mem_singleton",
"zero_mu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_adjoin_of_finite {s : set A} (hfs : s.finite)
(his : ∀ x ∈ s, is_integral R x) : (algebra.adjoin R s).to_submodule.fg | set.finite.induction_on hfs (λ _, ⟨{1}, submodule.ext $ λ x,
by { erw [algebra.adjoin_empty, finset.coe_singleton, ← one_eq_span, one_eq_range,
linear_map.mem_range, algebra.mem_bot], refl }⟩)
(λ a s has hs ih his, by rw [← set.union_singleton, algebra.adjoin_union_coe_submodule]; exact
fg.mul (ih $ λ i... | theorem | fg_adjoin_of_finite | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.adjoin",
"algebra.adjoin_empty",
"algebra.adjoin_union_coe_submodule",
"algebra.mem_bot",
"fg_adjoin_singleton_of_integral",
"finset.coe_singleton",
"ih",
"is_integral",
"linear_map.mem_range",
"set.finite.induction_on",
"set.mem_insert",
"set.mem_insert_of_mem",
"set.union_singleto... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_adjoin_finset [is_noetherian_ring R] (s : finset A)
(hs : ∀ x ∈ s, is_integral R x) :
is_noetherian R (algebra.adjoin R (↑s : set A)) | is_noetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_to_set hs) | lemma | is_noetherian_adjoin_finset | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.adjoin",
"fg_adjoin_of_finite",
"finset",
"is_integral",
"is_noetherian",
"is_noetherian_of_fg_of_noetherian",
"is_noetherian_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_mem_of_fg (S : subalgebra R A)
(HS : S.to_submodule.fg) (x : A) (hx : x ∈ S) : is_integral R x | begin
-- say `x ∈ S`. We want to prove that `x` is integral over `R`.
-- Say `S` is generated as an `R`-module by the set `y`.
cases HS with y hy,
-- We can write `x` as `∑ rᵢ yᵢ` for `yᵢ ∈ Y`.
obtain ⟨lx, hlx1, hlx2⟩ :
∃ (l : A →₀ R) (H : l ∈ finsupp.supported R R ↑y), (finsupp.total A A R id) l = x,
{... | theorem | is_integral_of_mem_of_fg | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.adjoin",
"algebra.algebra_map_eq_smul_one",
"algebra.mem_adjoin_iff",
"algebra.of_subring",
"algebra.subset_adjoin",
"algebra_map",
"comm_ring",
"finset.bUnion",
"finset.coe_insert",
"finset.finite_to_set",
"finset.mem_image_of_mem",
"finset.mem_union_left",
"finset.mem... | If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
then all elements of `S` are integral over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.End.is_integral {M : Type*} [add_comm_group M] [module R M] [module.finite R M] :
algebra.is_integral R (module.End R M) | linear_map.exists_monic_and_aeval_eq_zero R | lemma | module.End.is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"add_comm_group",
"algebra.is_integral",
"linear_map.exists_monic_and_aeval_eq_zero",
"module",
"module.End",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_smul_mem_submodule {M : Type*} [add_comm_group M] [module R M]
[module A M] [is_scalar_tower R A M] [no_zero_smul_divisors A M]
(N : submodule R M) (hN : N ≠ ⊥) (hN' : N.fg) (x : A)
(hx : ∀ n ∈ N, x • n ∈ N) : is_integral R x | begin
let A' : subalgebra R A :=
{ carrier := { x | ∀ n ∈ N, x • n ∈ N },
mul_mem' := λ a b ha hb n hn, smul_smul a b n ▸ ha _ (hb _ hn),
one_mem' := λ n hn, (one_smul A n).symm ▸ hn,
add_mem' := λ a b ha hb n hn, (add_smul a b n).symm ▸ N.add_mem (ha _ hn) (hb _ hn),
zero_mem' := λ n hn, (zero_smul... | lemma | is_integral_of_smul_mem_submodule | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"add_comm_group",
"add_smul",
"alg_hom.of_linear_map",
"algebra_map_smul",
"by_contra",
"distrib_mul_action.to_linear_map",
"eq_bot_iff",
"is_integral",
"is_integral_alg_hom_iff",
"is_scalar_tower",
"linear_map.congr_fun",
"linear_map.ext",
"linear_map.ker_eq_bot",
"module",
"module.End"... | Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.finite.to_is_integral (h : f.finite) : f.is_integral | by { letI := f.to_algebra, exact λ x, is_integral_of_mem_of_fg ⊤ h.1 _ trivial } | lemma | ring_hom.finite.to_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral_of_mem_of_fg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral.to_finite (h : f.is_integral) (h' : f.finite_type) : f.finite | begin
letI := f.to_algebra,
unfreezingI { obtain ⟨s, hs⟩ := h' },
constructor,
change (⊤ : subalgebra R S).to_submodule.fg,
rw ← hs,
exact fg_adjoin_of_finite (set.to_finite _) (λ x _, h x)
end | lemma | ring_hom.is_integral.to_finite | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"fg_adjoin_of_finite",
"set.to_finite",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.finite_iff_is_integral_and_finite_type :
f.finite ↔ f.is_integral ∧ f.finite_type | ⟨λ h, ⟨h.to_is_integral, h.to_finite_type⟩, λ ⟨h, h'⟩, h.to_finite h'⟩ | lemma | ring_hom.finite_iff_is_integral_and_finite_type | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | finite = integral + finite type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.is_integral.finite (h : algebra.is_integral R A) [h' : algebra.finite_type R A] :
module.finite R A | begin
have := h.to_finite
(by { delta ring_hom.finite_type, convert h', ext, exact (algebra.smul_def _ _).symm }),
delta ring_hom.finite at this, convert this, ext, exact algebra.smul_def _ _,
end | lemma | algebra.is_integral.finite | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.finite_type",
"algebra.is_integral",
"algebra.smul_def",
"module.finite",
"ring_hom.finite",
"ring_hom.finite_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.is_integral.of_finite [h : module.finite R A] : algebra.is_integral R A | begin
apply ring_hom.finite.to_is_integral,
delta ring_hom.finite, convert h, ext, exact (algebra.smul_def _ _).symm,
end | lemma | algebra.is_integral.of_finite | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral",
"algebra.smul_def",
"module.finite",
"ring_hom.finite",
"ring_hom.finite.to_is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.finite_iff_is_integral_and_finite_type :
module.finite R A ↔ algebra.is_integral R A ∧ algebra.finite_type R A | ⟨λ h, by exactI ⟨algebra.is_integral.of_finite, infer_instance⟩, λ ⟨h, h'⟩, by exactI h.finite⟩ | lemma | algebra.finite_iff_is_integral_and_finite_type | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.finite_type",
"algebra.is_integral",
"module.finite"
] | finite = integral + finite type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_integral_of_mem_closure {x y z : S}
(hx : f.is_integral_elem x) (hy : f.is_integral_elem y)
(hz : z ∈ subring.closure ({x, y} : set S)) :
f.is_integral_elem z | begin
letI : algebra R S := f.to_algebra,
have := (fg_adjoin_singleton_of_integral x hx).mul (fg_adjoin_singleton_of_integral y hy),
rw [← algebra.adjoin_union_coe_submodule, set.singleton_union] at this,
exact is_integral_of_mem_of_fg (algebra.adjoin R {x, y}) this z
(algebra.mem_adjoin_iff.2 $ subring.clo... | lemma | ring_hom.is_integral_of_mem_closure | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.adjoin",
"algebra.adjoin_union_coe_submodule",
"fg_adjoin_singleton_of_integral",
"is_integral_of_mem_of_fg",
"set.singleton_union",
"set.subset_union_right",
"subring.closure",
"subring.closure_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_mem_closure {x y z : A}
(hx : is_integral R x) (hy : is_integral R y)
(hz : z ∈ subring.closure ({x, y} : set A)) :
is_integral R z | (algebra_map R A).is_integral_of_mem_closure hx hy hz | theorem | is_integral_of_mem_closure | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral",
"subring.closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_zero : f.is_integral_elem 0 | f.map_zero ▸ f.is_integral_map | lemma | ring_hom.is_integral_zero | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_zero : is_integral R (0:A) | (algebra_map R A).is_integral_zero | theorem | is_integral_zero | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_one : f.is_integral_elem 1 | f.map_one ▸ f.is_integral_map | lemma | ring_hom.is_integral_one | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_one : is_integral R (1:A) | (algebra_map R A).is_integral_one | theorem | is_integral_one | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_add {x y : S}
(hx : f.is_integral_elem x) (hy : f.is_integral_elem y) :
f.is_integral_elem (x + y) | f.is_integral_of_mem_closure hx hy $ subring.add_mem _
(subring.subset_closure (or.inl rfl)) (subring.subset_closure (or.inr rfl)) | lemma | ring_hom.is_integral_add | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"subring.add_mem",
"subring.subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_add {x y : A}
(hx : is_integral R x) (hy : is_integral R y) :
is_integral R (x + y) | (algebra_map R A).is_integral_add hx hy | theorem | is_integral_add | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_neg {x : S}
(hx : f.is_integral_elem x) : f.is_integral_elem (-x) | f.is_integral_of_mem_closure hx hx (subring.neg_mem _ (subring.subset_closure (or.inl rfl))) | lemma | ring_hom.is_integral_neg | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"subring.neg_mem",
"subring.subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_neg {x : A}
(hx : is_integral R x) : is_integral R (-x) | (algebra_map R A).is_integral_neg hx | theorem | is_integral_neg | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_sub {x y : S}
(hx : f.is_integral_elem x) (hy : f.is_integral_elem y) : f.is_integral_elem (x - y) | by simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy) | lemma | ring_hom.is_integral_sub | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_sub {x y : A}
(hx : is_integral R x) (hy : is_integral R y) : is_integral R (x - y) | (algebra_map R A).is_integral_sub hx hy | theorem | is_integral_sub | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_mul {x y : S}
(hx : f.is_integral_elem x) (hy : f.is_integral_elem y) : f.is_integral_elem (x * y) | f.is_integral_of_mem_closure hx hy (subring.mul_mem _
(subring.subset_closure (or.inl rfl)) (subring.subset_closure (or.inr rfl))) | lemma | ring_hom.is_integral_mul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"subring.mul_mem",
"subring.subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_mul {x y : A}
(hx : is_integral R x) (hy : is_integral R y) : is_integral R (x * y) | (algebra_map R A).is_integral_mul hx hy | theorem | is_integral_mul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_smul [algebra S A] [algebra R S] [is_scalar_tower R S A] {x : A} (r : R)
(hx : is_integral S x) : is_integral S (r • x) | begin
rw [algebra.smul_def, is_scalar_tower.algebra_map_apply R S A],
exact is_integral_mul is_integral_algebra_map hx,
end | lemma | is_integral_smul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.smul_def",
"is_integral",
"is_integral_algebra_map",
"is_integral_mul",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : is_integral R $ x ^ n) :
is_integral R x | begin
rcases hx with ⟨p, ⟨hmonic, heval⟩⟩,
exact ⟨expand R n p, monic.expand hn hmonic,
by rwa [eval₂_eq_eval_map, map_expand, expand_eval, ← eval₂_eq_eval_map]⟩
end | lemma | is_integral_of_pow | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure : subalgebra R A | { carrier := { r | is_integral R r },
zero_mem' := is_integral_zero,
one_mem' := is_integral_one,
add_mem' := λ _ _, is_integral_add,
mul_mem' := λ _ _, is_integral_mul,
algebra_map_mem' := λ x, is_integral_algebra_map } | def | integral_closure | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"is_integral_add",
"is_integral_algebra_map",
"is_integral_mul",
"is_integral_one",
"is_integral_zero",
"subalgebra"
] | The integral closure of R in an R-algebra A. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_integral_closure_iff_mem_fg {r : A} :
r ∈ integral_closure R A ↔ ∃ M : subalgebra R A, M.to_submodule.fg ∧ r ∈ M | ⟨λ hr, ⟨algebra.adjoin R {r}, fg_adjoin_singleton_of_integral _ hr, algebra.subset_adjoin rfl⟩,
λ ⟨M, Hf, hrM⟩, is_integral_of_mem_of_fg M Hf _ hrM⟩ | theorem | mem_integral_closure_iff_mem_fg | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.subset_adjoin",
"fg_adjoin_singleton_of_integral",
"integral_closure",
"is_integral_of_mem_of_fg",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_le_integral_closure {x : A} (hx : is_integral R x) :
algebra.adjoin R {x} ≤ integral_closure R A | begin
rw [algebra.adjoin_le_iff],
simp only [set_like.mem_coe, set.singleton_subset_iff],
exact hx
end | lemma | adjoin_le_integral_closure | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.adjoin",
"algebra.adjoin_le_iff",
"integral_closure",
"is_integral",
"set.singleton_subset_iff",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_integral_closure_iff_is_integral {S : subalgebra R A} :
S ≤ integral_closure R A ↔ algebra.is_integral R S | set_like.forall.symm.trans (forall_congr (λ x, show is_integral R (algebra_map S A x)
↔ is_integral R x, from is_integral_algebra_map_iff subtype.coe_injective)) | lemma | le_integral_closure_iff_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral",
"algebra_map",
"integral_closure",
"is_integral",
"is_integral_algebra_map_iff",
"subalgebra",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_sup {S T : subalgebra R A} :
algebra.is_integral R ↥(S ⊔ T) ↔ algebra.is_integral R S ∧ algebra.is_integral R T | by simp only [←le_integral_closure_iff_is_integral, sup_le_iff] | lemma | is_integral_sup | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral",
"subalgebra",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure_map_alg_equiv (f : A ≃ₐ[R] B) :
(integral_closure R A).map (f : A →ₐ[R] B) = integral_closure R B | begin
ext y,
rw subalgebra.mem_map,
split,
{ rintros ⟨x, hx, rfl⟩,
exact map_is_integral f hx },
{ intro hy,
use [f.symm y, map_is_integral (f.symm : B →ₐ[R] A) hy],
simp }
end | lemma | integral_closure_map_alg_equiv | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"integral_closure",
"map_is_integral",
"subalgebra.mem_map"
] | Mapping an integral closure along an `alg_equiv` gives the integral closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_closure.is_integral (x : integral_closure R A) : is_integral R x | let ⟨p, hpm, hpx⟩ := x.2 in ⟨p, hpm, subtype.eq $
by rwa [← aeval_def, subtype.val_eq_coe, ← subalgebra.val_apply, aeval_alg_hom_apply] at hpx⟩ | lemma | integral_closure.is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"integral_closure",
"is_integral",
"subalgebra.val_apply",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
(hx : f.is_integral_elem (x * y)) : f.is_integral_elem x | begin
obtain ⟨p, ⟨p_monic, hp⟩⟩ := hx,
refine ⟨scale_roots p r, ⟨(monic_scale_roots_iff r).2 p_monic, _⟩⟩,
convert scale_roots_eval₂_eq_zero f hp,
rw [mul_comm x y, ← mul_assoc, hr, one_mul],
end | lemma | ring_hom.is_integral_of_is_integral_mul_unit | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"mul_assoc",
"mul_comm",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_is_integral_mul_unit {x y : A} {r : R} (hr : algebra_map R A r * y = 1)
(hx : is_integral R (x * y)) : is_integral R x | (algebra_map R A).is_integral_of_is_integral_mul_unit x y r hr hx | theorem | is_integral_of_is_integral_mul_unit | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_mem_closure' (G : set A) (hG : ∀ x ∈ G, is_integral R x) :
∀ x ∈ (subring.closure G), is_integral R x | λ x hx, subring.closure_induction hx hG is_integral_zero is_integral_one
(λ _ _, is_integral_add) (λ _, is_integral_neg) (λ _ _, is_integral_mul) | lemma | is_integral_of_mem_closure' | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"is_integral_add",
"is_integral_mul",
"is_integral_neg",
"is_integral_one",
"is_integral_zero",
"subring.closure",
"subring.closure_induction"
] | Generalization of `is_integral_of_mem_closure` bootstrapped up from that lemma | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_of_mem_closure'' {S : Type*} [comm_ring S] {f : R →+* S} (G : set S)
(hG : ∀ x ∈ G, f.is_integral_elem x) : ∀ x ∈ (subring.closure G), f.is_integral_elem x | λ x hx, @is_integral_of_mem_closure' R S _ _ f.to_algebra G hG x hx | lemma | is_integral_of_mem_closure'' | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"comm_ring",
"is_integral_of_mem_closure'",
"subring.closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.pow {x : A} (h : is_integral R x) (n : ℕ) : is_integral R (x ^ n) | (integral_closure R A).pow_mem h n | lemma | is_integral.pow | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"integral_closure",
"is_integral",
"pow_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.nsmul {x : A} (h : is_integral R x) (n : ℕ) : is_integral R (n • x) | (integral_closure R A).nsmul_mem h n | lemma | is_integral.nsmul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"integral_closure",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.zsmul {x : A} (h : is_integral R x) (n : ℤ) : is_integral R (n • x) | (integral_closure R A).zsmul_mem h n | lemma | is_integral.zsmul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"integral_closure",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.multiset_prod {s : multiset A} (h : ∀ x ∈ s, is_integral R x) :
is_integral R s.prod | (integral_closure R A).multiset_prod_mem h | lemma | is_integral.multiset_prod | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"integral_closure",
"is_integral",
"multiset",
"multiset_prod_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.multiset_sum {s : multiset A} (h : ∀ x ∈ s, is_integral R x) :
is_integral R s.sum | (integral_closure R A).multiset_sum_mem h | lemma | is_integral.multiset_sum | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"integral_closure",
"is_integral",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.prod {α : Type*} {s : finset α} (f : α → A) (h : ∀ x ∈ s, is_integral R (f x)) :
is_integral R (∏ x in s, f x) | (integral_closure R A).prod_mem h | lemma | is_integral.prod | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"finset",
"integral_closure",
"is_integral",
"prod_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.sum {α : Type*} {s : finset α} (f : α → A) (h : ∀ x ∈ s, is_integral R (f x)) :
is_integral R (∑ x in s, f x) | (integral_closure R A).sum_mem h | lemma | is_integral.sum | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"finset",
"integral_closure",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.det {n : Type*} [fintype n] [decidable_eq n] {M : matrix n n A}
(h : ∀ i j, is_integral R (M i j)) :
is_integral R M.det | begin
rw [matrix.det_apply],
exact is_integral.sum _ (λ σ hσ, is_integral.zsmul (is_integral.prod _ (λ i hi, h _ _)) _)
end | lemma | is_integral.det | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"fintype",
"is_integral",
"is_integral.prod",
"is_integral.sum",
"is_integral.zsmul",
"matrix",
"matrix.det_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) :
is_integral R (x ^ n) ↔ is_integral R x | ⟨is_integral_of_pow hn, λ hx, is_integral.pow hx n⟩ | lemma | is_integral.pow_iff | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"is_integral.pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.tmul (x : A) {y : B} (h : is_integral R y) : is_integral A (x ⊗ₜ[R] y) | begin
obtain ⟨p, hp, hp'⟩ := h,
refine ⟨(p.map (algebra_map R A)).scale_roots x, _, _⟩,
{ rw polynomial.monic_scale_roots_iff, exact hp.map _ },
convert @polynomial.scale_roots_eval₂_mul (A ⊗[R] B) A _ _ _
algebra.tensor_product.include_left.to_ring_hom (1 ⊗ₜ y) x using 2,
{ simp only [alg_hom.to_ring_hom... | lemma | is_integral.tmul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"alg_hom.coe_to_ring_hom",
"alg_hom.to_ring_hom_eq_coe",
"algebra.tensor_product.include_left_apply",
"algebra.tensor_product.include_left_comp_algebra_map",
"algebra.tensor_product.tmul_mul_tmul",
"algebra_map",
"is_integral",
"mul_one",
"mul_zero",
"one_mul",
"polynomial.eval_map",
"polynomi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_scale_roots (p : R[X]) : R[X] | ∑ i in p.support, monomial i
(if i = p.nat_degree then 1 else p.coeff i * p.leading_coeff ^ (p.nat_degree - 1 - i)) | def | normalize_scale_roots | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | The monic polynomial whose roots are `p.leading_coeff * x` for roots `x` of `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_scale_roots_coeff_mul_leading_coeff_pow (i : ℕ) (hp : 1 ≤ nat_degree p) :
(normalize_scale_roots p).coeff i * p.leading_coeff ^ i =
p.coeff i * p.leading_coeff ^ (p.nat_degree - 1) | begin
simp only [normalize_scale_roots, finset_sum_coeff, coeff_monomial, finset.sum_ite_eq', one_mul,
zero_mul, mem_support_iff, ite_mul, ne.def, ite_not],
split_ifs with h₁ h₂,
{ simp [h₁], },
{ rw [h₂, leading_coeff, ← pow_succ, tsub_add_cancel_of_le hp], },
{ rw [mul_assoc, ← pow_add, tsub_add_cancel_... | lemma | normalize_scale_roots_coeff_mul_leading_coeff_pow | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"ite_mul",
"ite_not",
"lt_iff_le_and_ne",
"mul_assoc",
"nat.le_pred_of_lt",
"normalize_scale_roots",
"one_mul",
"pow_add",
"pow_succ",
"tsub_add_cancel_of_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_smul_normalize_scale_roots (p : R[X]) :
p.leading_coeff • normalize_scale_roots p = scale_roots p p.leading_coeff | begin
ext,
simp only [coeff_scale_roots, normalize_scale_roots, coeff_monomial, coeff_smul, finset.smul_sum,
ne.def, finset.sum_ite_eq', finset_sum_coeff, smul_ite, smul_zero, mem_support_iff],
split_ifs with h₁ h₂,
{ simp [*] },
{ simp [*] },
{ rw [algebra.id.smul_eq_mul, mul_comm, mul_assoc, ← pow_suc... | lemma | leading_coeff_smul_normalize_scale_roots | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.id.smul_eq_mul",
"finset.smul_sum",
"mul_assoc",
"mul_comm",
"nat.succ_le_iff",
"normalize_scale_roots",
"pow_succ'",
"smul_ite",
"smul_zero",
"tsub_add_cancel_of_le",
"tsub_pos_of_lt",
"tsub_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_scale_roots_support :
(normalize_scale_roots p).support ≤ p.support | begin
intro x,
contrapose,
simp only [not_mem_support_iff, normalize_scale_roots, finset_sum_coeff, coeff_monomial,
finset.sum_ite_eq', mem_support_iff, ne.def, not_not, ite_eq_right_iff],
intros h₁ h₂,
exact (h₂ h₁).rec _,
end | lemma | normalize_scale_roots_support | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"ite_eq_right_iff",
"normalize_scale_roots",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_scale_roots_degree :
(normalize_scale_roots p).degree = p.degree | begin
apply le_antisymm,
{ exact finset.sup_mono (normalize_scale_roots_support p) },
{ rw [← degree_scale_roots, ← leading_coeff_smul_normalize_scale_roots],
exact degree_smul_le _ _ }
end | lemma | normalize_scale_roots_degree | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"finset.sup_mono",
"leading_coeff_smul_normalize_scale_roots",
"normalize_scale_roots",
"normalize_scale_roots_support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_scale_roots_eval₂_leading_coeff_mul (h : 1 ≤ p.nat_degree) (f : R →+* S) (x : S) :
(normalize_scale_roots p).eval₂ f (f p.leading_coeff * x) =
f p.leading_coeff ^ (p.nat_degree - 1) * (p.eval₂ f x) | begin
rw [eval₂_eq_sum_range, eval₂_eq_sum_range, finset.mul_sum],
apply finset.sum_congr,
{ rw nat_degree_eq_of_degree_eq (normalize_scale_roots_degree p) },
intros n hn,
rw [mul_pow, ← mul_assoc, ← f.map_pow, ← f.map_mul,
normalize_scale_roots_coeff_mul_leading_coeff_pow _ _ h, f.map_mul, f.map_pow],
... | lemma | normalize_scale_roots_eval₂_leading_coeff_mul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"finset.mul_sum",
"mul_assoc",
"mul_pow",
"normalize_scale_roots",
"normalize_scale_roots_coeff_mul_leading_coeff_pow",
"normalize_scale_roots_degree",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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