statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
num_mul_denom_eq_num_iff_eq' {x y : K} :
y * algebra_map A K (denom A x) = algebra_map A K (num A x) ↔ x = y | ⟨λ h, by simpa only [eq_comm, mk'_num_denom] using eq_mk'_iff_mul_eq.mpr h,
λ h, eq_mk'_iff_mul_eq.mp (by rw [h, mk'_num_denom])⟩ | lemma | is_fraction_ring.num_mul_denom_eq_num_iff_eq' | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"algebra_map",
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_mul_denom_eq_num_mul_denom_iff_eq {x y : K} :
num A y * denom A x = num A x * denom A y ↔ x = y | ⟨λ h, by simpa only [mk'_num_denom] using mk'_eq_of_eq' h,
λ h, by rw h⟩ | lemma | is_fraction_ring.num_mul_denom_eq_num_mul_denom_iff_eq | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_num_eq_zero {x : K} (h : num A x = 0) : x = 0 | num_mul_denom_eq_num_iff_eq'.mp (by rw [zero_mul, h, ring_hom.map_zero]) | lemma | is_fraction_ring.eq_zero_of_num_eq_zero | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"num",
"ring_hom.map_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integer_of_is_unit_denom {x : K} (h : is_unit (denom A x : A)) : is_integer A x | begin
cases h with d hd,
have d_ne_zero : algebra_map A K (denom A x) ≠ 0 :=
is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors (denom A x).2,
use ↑d⁻¹ * num A x,
refine trans _ (mk'_num_denom A x),
rw [map_mul, map_units_inv, hd],
apply mul_left_cancel₀ d_ne_zero,
rw [←mul_assoc, mul_inv_cancel... | lemma | is_fraction_ring.is_integer_of_is_unit_denom | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"algebra_map",
"is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors",
"is_unit",
"map_mul",
"map_units_inv",
"mul_inv_cancel",
"mul_left_cancel₀",
"num",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_denom_of_num_eq_zero {x : K} (h : num A x = 0) : is_unit (denom A x : A) | num_denom_reduced A x (h.symm ▸ dvd_zero _) dvd_rfl | lemma | is_fraction_ring.is_unit_denom_of_num_eq_zero | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"dvd_rfl",
"dvd_zero",
"is_unit",
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule (I : ideal R) : submodule R S | submodule.map (algebra.linear_map R S) I | def | is_localization.coe_submodule | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"algebra.linear_map",
"ideal",
"submodule",
"submodule.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_submodule (I : ideal R) {x : S} :
x ∈ coe_submodule S I ↔ ∃ y : R, y ∈ I ∧ algebra_map R S y = x | iff.rfl | lemma | is_localization.mem_coe_submodule | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"algebra_map",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_mono {I J : ideal R} (h : I ≤ J) :
coe_submodule S I ≤ coe_submodule S J | submodule.map_mono h | lemma | is_localization.coe_submodule_mono | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"submodule.map_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_bot : coe_submodule S (⊥ : ideal R) = ⊥ | by rw [coe_submodule, submodule.map_bot] | lemma | is_localization.coe_submodule_bot | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"submodule.map_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_top : coe_submodule S (⊤ : ideal R) = 1 | by rw [coe_submodule, submodule.map_top, submodule.one_eq_range] | lemma | is_localization.coe_submodule_top | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"submodule.map_top",
"submodule.one_eq_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_sup (I J : ideal R) :
coe_submodule S (I ⊔ J) = coe_submodule S I ⊔ coe_submodule S J | submodule.map_sup _ _ _ | lemma | is_localization.coe_submodule_sup | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"submodule.map_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_mul (I J : ideal R) :
coe_submodule S (I * J) = coe_submodule S I * coe_submodule S J | submodule.map_mul _ _ (algebra.of_id R S) | lemma | is_localization.coe_submodule_mul | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"algebra.of_id",
"ideal",
"submodule.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_fg
(hS : function.injective (algebra_map R S)) (I : ideal R) :
submodule.fg (coe_submodule S I) ↔ submodule.fg I | ⟨submodule.fg_of_fg_map _ (linear_map.ker_eq_bot.mpr hS), submodule.fg.map _⟩ | lemma | is_localization.coe_submodule_fg | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"algebra_map",
"ideal",
"submodule.fg",
"submodule.fg.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_span (s : set R) :
coe_submodule S (ideal.span s) = submodule.span R ((algebra_map R S) '' s) | by { rw [is_localization.coe_submodule, ideal.span, submodule.map_span], refl } | lemma | is_localization.coe_submodule_span | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"algebra_map",
"ideal.span",
"is_localization.coe_submodule",
"submodule.map_span",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_span_singleton (x : R) :
coe_submodule S (ideal.span {x}) = submodule.span R {(algebra_map R S) x} | by rw [coe_submodule_span, set.image_singleton] | lemma | is_localization.coe_submodule_span_singleton | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"algebra_map",
"ideal.span",
"set.image_singleton",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring (h : is_noetherian_ring R) : is_noetherian_ring S | begin
rw [is_noetherian_ring_iff, is_noetherian_iff_well_founded] at h ⊢,
exact order_embedding.well_founded ((is_localization.order_embedding M S).dual) h
end | lemma | is_localization.is_noetherian_ring | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"is_localization.order_embedding",
"is_noetherian_iff_well_founded",
"is_noetherian_ring",
"is_noetherian_ring_iff",
"order_embedding.well_founded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_le_coe_submodule (h : M ≤ non_zero_divisors R)
{I J : ideal R} :
coe_submodule S I ≤ coe_submodule S J ↔ I ≤ J | submodule.map_le_map_iff_of_injective (is_localization.injective _ h) _ _ | lemma | is_localization.coe_submodule_le_coe_submodule | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"is_localization.injective",
"non_zero_divisors",
"submodule.map_le_map_iff_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_strict_mono (h : M ≤ non_zero_divisors R) :
strict_mono (coe_submodule S : ideal R → submodule R S) | strict_mono_of_le_iff_le (λ _ _, (coe_submodule_le_coe_submodule h).symm) | lemma | is_localization.coe_submodule_strict_mono | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"non_zero_divisors",
"strict_mono",
"strict_mono_of_le_iff_le",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_injective (h : M ≤ non_zero_divisors R) :
function.injective (coe_submodule S : ideal R → submodule R S) | injective_of_le_imp_le _ (λ _ _, (coe_submodule_le_coe_submodule h).mp) | lemma | is_localization.coe_submodule_injective | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"injective_of_le_imp_le",
"non_zero_divisors",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_is_principal {I : ideal R} (h : M ≤ non_zero_divisors R) :
(coe_submodule S I).is_principal ↔ I.is_principal | begin
split; unfreezingI { rintros ⟨⟨x, hx⟩⟩ },
{ have x_mem : x ∈ coe_submodule S I := hx.symm ▸ submodule.mem_span_singleton_self x,
obtain ⟨x, x_mem, rfl⟩ := (mem_coe_submodule _ _).mp x_mem,
refine ⟨⟨x, coe_submodule_injective S h _⟩⟩,
rw [ideal.submodule_span_eq, hx, coe_submodule_span_singleton] }... | lemma | is_localization.coe_submodule_is_principal | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"ideal.submodule_span_eq",
"non_zero_divisors",
"submodule.mem_span_singleton_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_iff {N : Type*} [add_comm_group N] [module R N] [module S N] [is_scalar_tower R S N]
{x : N} {a : set N} :
x ∈ submodule.span S a ↔ ∃ (y ∈ submodule.span R a) (z : M), x = mk' S 1 z • y | begin
split, intro h,
{ refine submodule.span_induction h _ _ _ _,
{ rintros x hx,
exact ⟨x, submodule.subset_span hx, 1, by rw [mk'_one, _root_.map_one, one_smul]⟩ },
{ exact ⟨0, submodule.zero_mem _, 1, by rw [mk'_one, _root_.map_one, one_smul]⟩ },
{ rintros _ _ ⟨y, hy, z, rfl⟩ ⟨y', hy', z', rfl... | lemma | is_localization.mem_span_iff | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"add_comm_group",
"is_scalar_tower",
"is_scalar_tower.algebra_map_smul",
"mk'",
"mk'_one",
"mk'_surjective",
"module",
"mul_assoc",
"mul_comm",
"mul_one",
"mul_right_comm",
"one_mul",
"one_smul",
"smul_add",
"smul_smul",
"submodule.add_mem",
"submodule.smul_mem",
"submodule.span",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_map {x : S} {a : set R} :
x ∈ ideal.span (algebra_map R S '' a) ↔
∃ (y ∈ ideal.span a) (z : M), x = mk' S y z | begin
refine (mem_span_iff M).trans _,
split,
{ rw ← coe_submodule_span,
rintros ⟨_, ⟨y, hy, rfl⟩, z, hz⟩,
refine ⟨y, hy, z, _⟩,
rw [hz, algebra.linear_map_apply, smul_eq_mul, mul_comm, mul_mk'_eq_mk'_of_mul, mul_one] },
{ rintros ⟨y, hy, z, hz⟩,
refine ⟨algebra_map R S y, submodule.map_mem_span... | lemma | is_localization.mem_span_map | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"algebra.linear_map_apply",
"algebra_map",
"ideal.span",
"mk'",
"mul_comm",
"mul_one",
"smul_eq_mul",
"submodule.map_mem_span_algebra_map_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_le_coe_submodule
{I J : ideal R} : coe_submodule K I ≤ coe_submodule K J ↔ I ≤ J | is_localization.coe_submodule_le_coe_submodule le_rfl | lemma | is_fraction_ring.coe_submodule_le_coe_submodule | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"is_localization.coe_submodule_le_coe_submodule",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_strict_mono :
strict_mono (coe_submodule K : ideal R → submodule R K) | strict_mono_of_le_iff_le (λ _ _, coe_submodule_le_coe_submodule.symm) | lemma | is_fraction_ring.coe_submodule_strict_mono | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"strict_mono",
"strict_mono_of_le_iff_le",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_injective :
function.injective (coe_submodule K : ideal R → submodule R K) | injective_of_le_imp_le _ (λ _ _, (coe_submodule_le_coe_submodule).mp) | lemma | is_fraction_ring.coe_submodule_injective | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"injective_of_le_imp_le",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_is_principal {I : ideal R} :
(coe_submodule K I).is_principal ↔ I.is_principal | is_localization.coe_submodule_is_principal _ le_rfl | lemma | is_fraction_ring.coe_submodule_is_principal | ring_theory.localization | src/ring_theory/localization/submodule.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.ideal",
"ring_theory.principal_ideal_domain"
] | [
"ideal",
"is_localization.coe_submodule_is_principal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization.away_equiv_adjoin (r : R) : away r ≃ₐ[R] adjoin_root (C r * X - 1) | alg_equiv.of_alg_hom
{ commutes' := is_localization.away.away_map.lift_eq r
(is_unit_of_mul_eq_one _ _ $ root_is_inv r), .. away_lift _ r _ }
(lift_hom _ (is_localization.away.inv_self r) $ by simp only
[map_sub, map_mul, aeval_C, aeval_X, is_localization.away.mul_inv_self, aeval_one, sub_self])
(subsin... | def | localization.away_equiv_adjoin | ring_theory.localization.away | src/ring_theory/localization/away/adjoin_root.lean | [
"ring_theory.adjoin_root",
"ring_theory.localization.away.basic"
] | [
"adjoin_root",
"alg_equiv.of_alg_hom",
"is_localization.away.away_map.lift_eq",
"is_localization.away.inv_self",
"is_localization.away.mul_inv_self",
"is_unit_of_mul_eq_one",
"map_mul"
] | The `R`-`alg_equiv` between the localization of `R` away from `r` and
`R` with an inverse of `r` adjoined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.adjoin_inv (r : R) : is_localization.away r (adjoin_root $ C r * X - 1) | is_localization.is_localization_of_alg_equiv _ (localization.away_equiv_adjoin r) | lemma | is_localization.adjoin_inv | ring_theory.localization.away | src/ring_theory/localization/away/adjoin_root.lean | [
"ring_theory.adjoin_root",
"ring_theory.localization.away.basic"
] | [
"adjoin_root",
"is_localization.away",
"is_localization.is_localization_of_alg_equiv",
"localization.away_equiv_adjoin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.away.finite_presentation (r : R) {S} [comm_ring S] [algebra R S]
[is_localization.away r S] : algebra.finite_presentation R S | (adjoin_root.finite_presentation _).equiv $ (localization.away_equiv_adjoin r).symm.trans $
is_localization.alg_equiv (submonoid.powers r) _ _ | lemma | is_localization.away.finite_presentation | ring_theory.localization.away | src/ring_theory/localization/away/adjoin_root.lean | [
"ring_theory.adjoin_root",
"ring_theory.localization.away.basic"
] | [
"adjoin_root.finite_presentation",
"algebra",
"algebra.finite_presentation",
"comm_ring",
"equiv",
"is_localization.alg_equiv",
"is_localization.away",
"localization.away_equiv_adjoin",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
away (S : Type*) [comm_semiring S] [algebra R S] | is_localization (submonoid.powers x) S | abbreviation | is_localization.away | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra",
"comm_semiring",
"is_localization",
"submonoid.powers"
] | Given `x : R`, the typeclass `is_localization.away x S` states that `S` is
isomorphic to the localization of `R` at the submonoid generated by `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_self : S | mk' S (1 : R) ⟨x, submonoid.mem_powers _⟩ | def | is_localization.away.inv_self | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"mk'",
"submonoid.mem_powers"
] | Given `x : R` and a localization map `F : R →+* S` away from `x`, `inv_self` is `(F x)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_self : algebra_map R S x * inv_self x = 1 | by { convert is_localization.mk'_mul_mk'_eq_one _ 1, symmetry, apply is_localization.mk'_one } | lemma | is_localization.away.mul_inv_self | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra_map",
"is_localization.mk'_mul_mk'_eq_one",
"is_localization.mk'_one",
"mul_inv_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (hg : is_unit (g x)) : S →+* P | is_localization.lift $ λ (y : submonoid.powers x), show is_unit (g y.1),
begin
obtain ⟨n, hn⟩ := y.2,
rw [←hn, g.map_pow],
exact is_unit.map (pow_monoid_hom n : P →* P) hg,
end | def | is_localization.away.lift | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.lift",
"is_unit",
"is_unit.map",
"lift",
"pow_monoid_hom",
"submonoid.powers"
] | Given `x : R`, a localization map `F : R →+* S` away from `x`, and a map of `comm_semiring`s
`g : R →+* P` such that `g x` is invertible, the homomorphism induced from `S` to `P` sending
`z : S` to `g y * (g x)⁻ⁿ`, where `y : R, n : ℕ` are such that `z = F y * (F x)⁻ⁿ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
away_map.lift_eq (hg : is_unit (g x)) (a : R) :
lift x hg ((algebra_map R S) a) = g a | lift_eq _ _ | lemma | is_localization.away.away_map.lift_eq | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra_map",
"is_unit",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
away_map.lift_comp (hg : is_unit (g x)) :
(lift x hg).comp (algebra_map R S) = g | lift_comp _ | lemma | is_localization.away.away_map.lift_comp | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra_map",
"is_unit",
"lift",
"lift_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
away_to_away_right (y : R) [algebra R P] [is_localization.away (x * y) P] :
S →+* P | lift x $ show is_unit ((algebra_map R P) x), from
is_unit_of_mul_eq_one ((algebra_map R P) x) (mk' P y ⟨x * y, submonoid.mem_powers _⟩) $
by rw [mul_mk'_eq_mk'_of_mul, mk'_self] | def | is_localization.away.away_to_away_right | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra",
"algebra_map",
"is_localization.away",
"is_unit",
"is_unit_of_mul_eq_one",
"lift",
"mk'",
"submonoid.mem_powers"
] | Given `x y : R` and localizations `S`, `P` away from `x` and `x * y`
respectively, the homomorphism induced from `S` to `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : R →+* P) (r : R) [is_localization.away r S]
[is_localization.away (f r) Q] : S →+* Q | is_localization.map Q f
(show submonoid.powers r ≤ (submonoid.powers (f r)).comap f,
by { rintros x ⟨n, rfl⟩, use n, simp }) | def | is_localization.away.map | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.away",
"is_localization.map",
"submonoid.powers"
] | Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
at_units (H : ∀ x : M, is_unit (x : R)) : R ≃ₐ[R] S | begin
refine alg_equiv.of_bijective (algebra.of_id R S) ⟨_, _⟩,
{ intros x y hxy,
obtain ⟨c, eq⟩ := (is_localization.eq_iff_exists M S).mp hxy,
obtain ⟨u, hu⟩ := H c,
rwa [← hu, units.mul_right_inj] at eq },
{ intros y,
obtain ⟨⟨x, s⟩, eq⟩ := is_localization.surj M y,
obtain ⟨u, hu⟩ := H s,
... | def | is_localization.at_units | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"alg_equiv.of_bijective",
"alg_hom.coe_mk",
"algebra.of_id",
"is_unit",
"mul_assoc",
"ring_hom.map_mul",
"ring_hom.to_fun_eq_coe",
"units.mul_right_inj"
] | The localization at a module of units is isomorphic to the ring | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
at_unit (x : R) (e : is_unit x) [is_localization.away x S] : R ≃ₐ[R] S | begin
apply at_units R (submonoid.powers x),
rintros ⟨xn, n, hxn⟩,
obtain ⟨u, hu⟩ := e,
rw is_unit_iff_exists_inv,
use u.inv ^ n,
simp[← hxn, ← hu, ← mul_pow]
end | def | is_localization.at_unit | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.away",
"is_unit",
"is_unit_iff_exists_inv",
"mul_pow",
"submonoid.powers"
] | The localization away from a unit is isomorphic to the ring | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
at_one [is_localization.away (1 : R) S] : R ≃ₐ[R] S | @at_unit R _ S _ _ (1 : R) is_unit_one _ | def | is_localization.at_one | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.away",
"is_unit_one"
] | The localization at one is isomorphic to the ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
away_of_is_unit_of_bijective {R : Type*} (S : Type*) [comm_ring R] [comm_ring S]
[algebra R S] {r : R} (hr : is_unit r) (H : function.bijective (algebra_map R S)) :
is_localization.away r S | { map_units := by { rintros ⟨_, n, rfl⟩, exact (algebra_map R S).is_unit_map (hr.pow _) },
surj := λ z, by { obtain ⟨z', rfl⟩ := H.2 z, exact ⟨⟨z', 1⟩, by simp⟩ },
eq_iff_exists := λ x y, begin
erw H.1.eq_iff,
split,
{ rintro rfl, exact ⟨1, rfl⟩ },
{ rintro ⟨⟨_, n, rfl⟩, e⟩, exact (hr.pow _).mul_rig... | lemma | is_localization.away_of_is_unit_of_bijective | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_localization.away",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
away_lift (f : R →+* P) (r : R) (hr : is_unit (f r)) :
localization.away r →+* P | is_localization.away.lift r hr | abbreviation | localization.away_lift | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.away.lift",
"is_unit",
"localization.away"
] | Given a map `f : R →+* S` and an element `r : R`, such that `f r` is invertible,
we may construct a map `Rᵣ →+* S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
away_map (f : R →+* P) (r : R) :
localization.away r →+* localization.away (f r) | is_localization.away.map _ _ f r | abbreviation | localization.away_map | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.away.map",
"localization.away"
] | Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_zpow (m : ℤ) : B | if hm : 0 ≤ m
then algebra_map _ _ x ^ m.nat_abs
else mk' _ (1 : R) (submonoid.pow x m.nat_abs) | def | self_zpow | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra_map",
"mk'",
"submonoid.pow"
] | `self_zpow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_zpow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : self_zpow x B n =
algebra_map R B x ^ n.nat_abs | dif_pos hn | lemma | self_zpow_of_nonneg | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra_map",
"self_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_coe_nat (d : ℕ) : self_zpow x B d = (algebra_map R B x)^d | self_zpow_of_nonneg _ _ (int.coe_nat_nonneg d) | lemma | self_zpow_coe_nat | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra_map",
"int.coe_nat_nonneg",
"self_zpow",
"self_zpow_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_zero : self_zpow x B 0 = 1 | by simp [self_zpow_of_nonneg _ _ le_rfl] | lemma | self_zpow_zero | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"le_rfl",
"self_zpow",
"self_zpow_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_of_neg {n : ℤ} (hn : n < 0) :
self_zpow x B n = mk' _ (1 : R) (submonoid.pow x n.nat_abs) | dif_neg hn.not_le | lemma | self_zpow_of_neg | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"mk'",
"self_zpow",
"submonoid.pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
self_zpow x B n = mk' _ (1 : R) (submonoid.pow x n.nat_abs) | begin
by_cases hn0 : n = 0,
{ simp [hn0, self_zpow_zero, submonoid.pow_apply] },
{ simp [self_zpow_of_neg _ _ (lt_of_le_of_ne hn hn0)] }
end | lemma | self_zpow_of_nonpos | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"mk'",
"self_zpow",
"self_zpow_of_neg",
"self_zpow_zero",
"submonoid.pow",
"submonoid.pow_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_neg_coe_nat (d : ℕ) :
self_zpow x B (-d) = mk' _ (1 : R) (submonoid.pow x d) | by simp [self_zpow_of_nonpos _ _ (neg_nonpos.mpr (int.coe_nat_nonneg d))] | lemma | self_zpow_neg_coe_nat | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"int.coe_nat_nonneg",
"mk'",
"self_zpow",
"self_zpow_of_nonpos",
"submonoid.pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_sub_cast_nat {n m : ℕ} :
self_zpow x B (n - m) = mk' _ (x ^ n) (submonoid.pow x m) | begin
by_cases h : m ≤ n,
{ rw [is_localization.eq_mk'_iff_mul_eq, submonoid.pow_apply, subtype.coe_mk,
← int.coe_nat_sub h, self_zpow_coe_nat, ← map_pow, ← map_mul, ← pow_add,
nat.sub_add_cancel h] },
{ rw [← neg_sub, ← int.coe_nat_sub (le_of_not_le h), self_zpow_neg_coe_nat,
is_localizat... | lemma | self_zpow_sub_cast_nat | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.eq_mk'_iff_mul_eq",
"is_localization.mk'_eq_iff_eq",
"map_mul",
"map_pow",
"mk'",
"pow_add",
"self_zpow",
"self_zpow_coe_nat",
"self_zpow_neg_coe_nat",
"submonoid.pow",
"submonoid.pow_apply",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_add {n m : ℤ} :
self_zpow x B (n + m) = self_zpow x B n * self_zpow x B m | begin
cases le_or_lt 0 n with hn hn; cases le_or_lt 0 m with hm hm,
{ rw [self_zpow_of_nonneg _ _ hn, self_zpow_of_nonneg _ _ hm,
self_zpow_of_nonneg _ _ (add_nonneg hn hm), int.nat_abs_add_nonneg hn hm, pow_add] },
{ have : n + m = n.nat_abs - m.nat_abs,
{ rw [int.nat_abs_of_nonneg hn, int.of_nat_nat... | lemma | self_zpow_add | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.mk'_eq_mul_mk'_one",
"map_pow",
"mul_comm",
"one_mul",
"pow_add",
"self_zpow",
"self_zpow_of_neg",
"self_zpow_of_nonneg",
"self_zpow_sub_cast_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_mul_neg (d : ℤ) : self_zpow x B d * self_zpow x B (-d) = 1 | begin
by_cases hd : d ≤ 0,
{ erw [self_zpow_of_nonpos x B hd, self_zpow_of_nonneg, ← map_pow, int.nat_abs_neg,
is_localization.mk'_spec, map_one],
apply nonneg_of_neg_nonpos,
rwa [neg_neg]},
{ erw [self_zpow_of_nonneg x B (le_of_not_le hd), self_zpow_of_nonpos, ← map_pow, int.nat_abs_neg,
@is_l... | lemma | self_zpow_mul_neg | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"is_localization.mk'_spec",
"is_localization.mk'_spec'",
"map_one",
"map_pow",
"self_zpow",
"self_zpow_of_nonneg",
"self_zpow_of_nonpos",
"submonoid.pow",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_neg_mul (d : ℤ) : self_zpow x B (-d) * self_zpow x B d = 1 | by rw [mul_comm, self_zpow_mul_neg x B d] | lemma | self_zpow_neg_mul | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"mul_comm",
"self_zpow",
"self_zpow_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow_pow_sub (a : R) (b : B) (m d : ℤ) :
(self_zpow x B (m - d)) * mk' B a (1 : submonoid.powers x) = b ↔
(self_zpow x B m) * mk' B a (1 : submonoid.powers x) = (self_zpow x B d) * b | begin
rw [sub_eq_add_neg, self_zpow_add, mul_assoc, mul_comm _ (mk' B a 1), ← mul_assoc],
split,
{ intro h,
have := congr_arg (λ s : B, s * self_zpow x B d) h,
simp only at this,
rwa [mul_assoc, mul_assoc, self_zpow_neg_mul, mul_one, mul_comm b _] at this},
{ intro h,
have := congr_arg (λ s : B,... | lemma | self_zpow_pow_sub | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"mk'",
"mul_assoc",
"mul_comm",
"mul_one",
"self_zpow",
"self_zpow_add",
"self_zpow_mul_neg",
"self_zpow_neg_mul",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : irreducible x) :
∃ (a : R) (n : ℤ), ¬ x ∣ a ∧
self_zpow x B n * algebra_map R B a = b | begin
classical,
obtain ⟨⟨a₀, y⟩, H⟩ := surj (submonoid.powers x) b,
obtain ⟨d, hy⟩ := (submonoid.mem_powers_iff y.1 x).mp y.2,
have ha₀ : a₀ ≠ 0,
{ haveI := @is_domain_of_le_non_zero_divisors B _ R _ _ _ (submonoid.powers x) _
(powers_le_non_zero_divisors_of_no_zero_divisors hx.ne_zero),
simp only ... | theorem | exists_reduced_fraction' | ring_theory.localization.away | src/ring_theory/localization/away/basic.lean | [
"ring_theory.unique_factorization_domain",
"ring_theory.localization.basic"
] | [
"algebra_map",
"irreducible",
"is_localization.injective",
"is_localization.mk'_one",
"is_localization.to_map_ne_zero_of_mem_non_zero_divisors",
"map_mul",
"map_pow",
"mk'_one",
"mul_comm",
"mul_ne_zero",
"pow_ne_zero",
"powers_le_non_zero_divisors_of_no_zero_divisors",
"self_zpow",
"self_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_range_eq_map {R S : Type*} [comm_ring R] [comm_ring S] (p : mv_polynomial σ R)
(f : R →+* S) :
finsupp.map_range f f.map_zero p = map f p | begin
-- `finsupp.map_range_finset_sum` expects `f : R →+ S`
change finsupp.map_range (f : R →+ S) (f : R →+ S).map_zero p = map f p,
rw [p.as_sum, finsupp.map_range_finset_sum, (map f).map_sum],
refine finset.sum_congr rfl (assume n _, _),
rw [map_monomial, ← single_eq_monomial, finsupp.map_range_single, sin... | lemma | mv_polynomial.map_range_eq_map | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"comm_ring",
"finsupp.map_range",
"finsupp.map_range_finset_sum",
"finsupp.map_range_single",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_total_degree : submodule R (mv_polynomial σ R) | finsupp.supported _ _ {n | n.sum (λn e, e) ≤ m } | def | mv_polynomial.restrict_total_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"finsupp.supported",
"mv_polynomial",
"submodule"
] | The submodule of polynomials of total degree less than or equal to `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_degree (m : ℕ) : submodule R (mv_polynomial σ R) | finsupp.supported _ _ {n | ∀i, n i ≤ m } | def | mv_polynomial.restrict_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"finsupp.supported",
"mv_polynomial",
"submodule"
] | The submodule of polynomials such that the degree with respect to each individual variable is
less than or equal to `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_restrict_total_degree (p : mv_polynomial σ R) :
p ∈ restrict_total_degree σ R m ↔ p.total_degree ≤ m | begin
rw [total_degree, finset.sup_le_iff],
refl
end | lemma | mv_polynomial.mem_restrict_total_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"finset.sup_le_iff",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_restrict_degree (p : mv_polynomial σ R) (n : ℕ) :
p ∈ restrict_degree σ R n ↔ (∀s ∈ p.support, ∀i, (s : σ →₀ ℕ) i ≤ n) | begin
rw [restrict_degree, finsupp.mem_supported],
refl
end | lemma | mv_polynomial.mem_restrict_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"finsupp.mem_supported",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_restrict_degree_iff_sup [decidable_eq σ] (p : mv_polynomial σ R) (n : ℕ) :
p ∈ restrict_degree σ R n ↔ ∀i, p.degrees.count i ≤ n | begin
simp only [mem_restrict_degree, degrees_def, multiset.count_finset_sup, finsupp.count_to_multiset,
finset.sup_le_iff],
exact ⟨assume h n s hs, h s hs n, assume h s hs n, h n s hs⟩
end | lemma | mv_polynomial.mem_restrict_degree_iff_sup | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"finset.sup_le_iff",
"finsupp.count_to_multiset",
"multiset.count_finset_sup",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_monomials : basis (σ →₀ ℕ) R (mv_polynomial σ R) | finsupp.basis_single_one | def | mv_polynomial.basis_monomials | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"basis",
"finsupp.basis_single_one",
"mv_polynomial"
] | The monomials form a basis on `mv_polynomial σ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_basis_monomials :
(basis_monomials σ R : (σ →₀ ℕ) → mv_polynomial σ R) = λ s, monomial s 1 | rfl | lemma | mv_polynomial.coe_basis_monomials | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent_X : linear_independent R (X : σ → mv_polynomial σ R) | (basis_monomials σ R).linear_independent.comp
(λ s : σ, finsupp.single s 1) (finsupp.single_left_injective one_ne_zero) | lemma | mv_polynomial.linear_independent_X | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"finsupp.single",
"finsupp.single_left_injective",
"linear_independent",
"linear_independent.comp",
"mv_polynomial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_monomials : basis ℕ R R[X] | basis.of_repr (to_finsupp_iso_alg R).to_linear_equiv | def | polynomial.basis_monomials | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [
"basis"
] | The monomials form a basis on `R[X]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_basis_monomials :
(basis_monomials R : ℕ → R[X]) = λ s, monomial s 1 | _root_.funext $ λ n, of_finsupp_single _ _ | lemma | polynomial.coe_basis_monomials | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/basic.lean | [
"algebra.char_p.basic",
"data.polynomial.algebra_map",
"data.mv_polynomial.variables",
"linear_algebra.finsupp_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homogeneous [comm_semiring R] (φ : mv_polynomial σ R) (n : ℕ) | ∀ ⦃d⦄, coeff d φ ≠ 0 → ∑ i in d.support, d i = n | def | mv_polynomial.is_homogeneous | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"comm_semiring",
"mv_polynomial"
] | A multivariate polynomial `φ` is homogeneous of degree `n`
if all monomials occuring in `φ` have degree `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_submodule [comm_semiring R] (n : ℕ) :
submodule R (mv_polynomial σ R) | { carrier := { x | x.is_homogeneous n },
smul_mem' := λ r a ha c hc, begin
rw coeff_smul at hc,
apply ha,
intro h,
apply hc,
rw h,
exact smul_zero r,
end,
zero_mem' := λ d hd, false.elim (hd $ coeff_zero _),
add_mem' := λ a b ha hb c hc, begin
rw coeff_add at hc,
obtain h|h : coe... | def | mv_polynomial.homogeneous_submodule | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"comm_semiring",
"mv_polynomial",
"smul_zero",
"submodule"
] | The submodule of homogeneous `mv_polynomial`s of degree `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_homogeneous_submodule [comm_semiring R] (n : ℕ) (p : mv_polynomial σ R) :
p ∈ homogeneous_submodule σ R n ↔ p.is_homogeneous n | iff.rfl | lemma | mv_polynomial.mem_homogeneous_submodule | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"comm_semiring",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_submodule_eq_finsupp_supported [comm_semiring R] (n : ℕ) :
homogeneous_submodule σ R n = finsupp.supported _ R {d | ∑ i in d.support, d i = n} | begin
ext,
rw [finsupp.mem_supported, set.subset_def],
simp only [finsupp.mem_support_iff, finset.mem_coe],
refl,
end | lemma | mv_polynomial.homogeneous_submodule_eq_finsupp_supported | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"comm_semiring",
"finset.mem_coe",
"finsupp.mem_support_iff",
"finsupp.mem_supported",
"finsupp.supported",
"set.subset_def"
] | While equal, the former has a convenient definitional reduction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_submodule_mul [comm_semiring R] (m n : ℕ) :
homogeneous_submodule σ R m * homogeneous_submodule σ R n ≤ homogeneous_submodule σ R (m + n) | begin
rw submodule.mul_le,
intros φ hφ ψ hψ c hc,
rw [coeff_mul] at hc,
obtain ⟨⟨d, e⟩, hde, H⟩ := finset.exists_ne_zero_of_sum_ne_zero hc,
have aux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0,
{ contrapose! H,
by_cases h : coeff d φ = 0;
simp only [*, ne.def, not_false_iff, zero_mul, mul_zero] at * },
specia... | lemma | mv_polynomial.homogeneous_submodule_mul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"aux",
"comm_semiring",
"finset.subset_union_left",
"finset.subset_union_right",
"finsupp.mem_antidiagonal",
"finsupp.support_add",
"mul_zero",
"submodule.mul_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homogeneous_monomial (d : σ →₀ ℕ) (r : R) (n : ℕ) (hn : ∑ i in d.support, d i = n) :
is_homogeneous (monomial d r) n | begin
intros c hc,
classical,
rw coeff_monomial at hc,
split_ifs at hc with h,
{ subst c, exact hn },
{ contradiction }
end | lemma | mv_polynomial.is_homogeneous_monomial | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homogeneous_of_total_degree_zero {p : mv_polynomial σ R} (hp : p.total_degree = 0) :
is_homogeneous p 0 | begin
erw [total_degree, finset.sup_eq_bot_iff] at hp,
-- we have to do this in two steps to stop simp changing bot to zero
simp_rw [mem_support_iff] at hp,
exact hp,
end | lemma | mv_polynomial.is_homogeneous_of_total_degree_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.sup_eq_bot_iff",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homogeneous_C (r : R) :
is_homogeneous (C r : mv_polynomial σ R) 0 | begin
apply is_homogeneous_monomial,
simp only [finsupp.zero_apply, finset.sum_const_zero],
end | lemma | mv_polynomial.is_homogeneous_C | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.zero_apply",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homogeneous_zero (n : ℕ) : is_homogeneous (0 : mv_polynomial σ R) n | (homogeneous_submodule σ R n).zero_mem | lemma | mv_polynomial.is_homogeneous_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homogeneous_one : is_homogeneous (1 : mv_polynomial σ R) 0 | is_homogeneous_C _ _ | lemma | mv_polynomial.is_homogeneous_one | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homogeneous_X (i : σ) :
is_homogeneous (X i : mv_polynomial σ R) 1 | begin
apply is_homogeneous_monomial,
simp only [finsupp.support_single_ne_zero _ one_ne_zero, finset.sum_singleton],
exact finsupp.single_eq_same
end | lemma | mv_polynomial.is_homogeneous_X | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.single_eq_same",
"finsupp.support_single_ne_zero",
"mv_polynomial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_zero (hφ : is_homogeneous φ n) (d : σ →₀ ℕ) (hd : ∑ i in d.support, d i ≠ n) :
coeff d φ = 0 | by { have aux := mt (@hφ d) hd, classical, rwa not_not at aux } | lemma | mv_polynomial.is_homogeneous.coeff_eq_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"aux",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_right (hm : is_homogeneous φ m) (hn : is_homogeneous φ n) (hφ : φ ≠ 0) :
m = n | begin
obtain ⟨d, hd⟩ : ∃ d, coeff d φ ≠ 0 := exists_coeff_ne_zero hφ,
rw [← hm hd, ← hn hd]
end | lemma | mv_polynomial.is_homogeneous.inj_right | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add (hφ : is_homogeneous φ n) (hψ : is_homogeneous ψ n) :
is_homogeneous (φ + ψ) n | (homogeneous_submodule σ R n).add_mem hφ hψ | lemma | mv_polynomial.is_homogeneous.add | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum {ι : Type*} (s : finset ι) (φ : ι → mv_polynomial σ R) (n : ℕ)
(h : ∀ i ∈ s, is_homogeneous (φ i) n) :
is_homogeneous (∑ i in s, φ i) n | (homogeneous_submodule σ R n).sum_mem h | lemma | mv_polynomial.is_homogeneous.sum | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul (hφ : is_homogeneous φ m) (hψ : is_homogeneous ψ n) :
is_homogeneous (φ * ψ) (m + n) | homogeneous_submodule_mul m n $ submodule.mul_mem_mul hφ hψ | lemma | mv_polynomial.is_homogeneous.mul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"submodule.mul_mem_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod {ι : Type*} (s : finset ι) (φ : ι → mv_polynomial σ R) (n : ι → ℕ)
(h : ∀ i ∈ s, is_homogeneous (φ i) (n i)) :
is_homogeneous (∏ i in s, φ i) (∑ i in s, n i) | begin
classical,
revert h,
apply finset.induction_on s,
{ intro, simp only [is_homogeneous_one, finset.sum_empty, finset.prod_empty] },
{ intros i s his IH h,
simp only [his, finset.prod_insert, finset.sum_insert, not_false_iff],
apply (h i (finset.mem_insert_self _ _)).mul (IH _),
intros j hjs,
... | lemma | mv_polynomial.is_homogeneous.prod | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_empty",
"finset.prod_insert",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
total_degree (hφ : is_homogeneous φ n) (h : φ ≠ 0) :
total_degree φ = n | begin
rw total_degree,
apply le_antisymm,
{ apply finset.sup_le,
intros d hd,
rw mem_support_iff at hd,
rw [finsupp.sum, hφ hd], },
{ obtain ⟨d, hd⟩ : ∃ d, coeff d φ ≠ 0 := exists_coeff_ne_zero h,
simp only [← hφ hd, finsupp.sum],
replace hd := finsupp.mem_support_iff.mpr hd,
exact finse... | lemma | mv_polynomial.is_homogeneous.total_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_submodule.gcomm_semiring :
set_like.graded_monoid (homogeneous_submodule σ R) | { one_mem := is_homogeneous_one σ R,
mul_mem := λ i j xi xj, is_homogeneous.mul} | instance | mv_polynomial.is_homogeneous.homogeneous_submodule.gcomm_semiring | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"set_like.graded_monoid"
] | The homogeneous submodules form a graded ring. This instance is used by `direct_sum.comm_semiring`
and `direct_sum.algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_component [comm_semiring R] (n : ℕ) :
mv_polynomial σ R →ₗ[R] mv_polynomial σ R | (submodule.subtype _).comp $ finsupp.restrict_dom _ _ {d | ∑ i in d.support, d i = n} | def | mv_polynomial.homogeneous_component | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"comm_semiring",
"finsupp.restrict_dom",
"mv_polynomial",
"submodule.subtype"
] | `homogeneous_component n φ` is the part of `φ` that is homogeneous of degree `n`.
See `sum_homogeneous_component` for the statement that `φ` is equal to the sum
of all its homogeneous components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_homogeneous_component (d : σ →₀ ℕ) :
coeff d (homogeneous_component n φ) = if ∑ i in d.support, d i = n then coeff d φ else 0 | by convert finsupp.filter_apply (λ d : σ →₀ ℕ, ∑ i in d.support, d i = n) φ d | lemma | mv_polynomial.coeff_homogeneous_component | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.filter_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_component_apply :
homogeneous_component n φ =
∑ d in φ.support.filter (λ d, ∑ i in d.support, d i = n), monomial d (coeff d φ) | by convert finsupp.filter_eq_sum (λ d : σ →₀ ℕ, ∑ i in d.support, d i = n) φ | lemma | mv_polynomial.homogeneous_component_apply | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.filter_eq_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_component_is_homogeneous :
(homogeneous_component n φ).is_homogeneous n | begin
intros d hd,
contrapose! hd,
rw [coeff_homogeneous_component, if_neg hd]
end | lemma | mv_polynomial.homogeneous_component_is_homogeneous | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_component_zero : homogeneous_component 0 φ = C (coeff 0 φ) | begin
ext1 d,
rcases em (d = 0) with (rfl|hd),
{ classical,
simp only [coeff_homogeneous_component, sum_eq_zero_iff, finsupp.zero_apply, if_true, coeff_C,
eq_self_iff_true, forall_true_iff] },
{ rw [coeff_homogeneous_component, if_neg, coeff_C, if_neg (ne.symm hd)],
simp only [finsupp.ext_iff, fin... | lemma | mv_polynomial.homogeneous_component_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"em",
"finsupp.ext_iff",
"finsupp.zero_apply",
"forall_true_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_component_C_mul (n : ℕ) (r : R) :
homogeneous_component n (C r * φ) = C r * homogeneous_component n φ | by simp only [C_mul', linear_map.map_smul] | lemma | mv_polynomial.homogeneous_component_C_mul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"linear_map.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_component_eq_zero' (h : ∀ d : σ →₀ ℕ, d ∈ φ.support → ∑ i in d.support, d i ≠ n) :
homogeneous_component n φ = 0 | begin
rw [homogeneous_component_apply, sum_eq_zero],
intros d hd, rw mem_filter at hd,
exfalso, exact h _ hd.1 hd.2
end | lemma | mv_polynomial.homogeneous_component_eq_zero' | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_component_eq_zero (h : φ.total_degree < n) :
homogeneous_component n φ = 0 | begin
apply homogeneous_component_eq_zero',
rw [total_degree, finset.sup_lt_iff] at h,
{ intros d hd, exact ne_of_lt (h d hd), },
{ exact lt_of_le_of_lt (nat.zero_le _) h, }
end | lemma | mv_polynomial.homogeneous_component_eq_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.sup_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_homogeneous_component :
∑ i in range (φ.total_degree + 1), homogeneous_component i φ = φ | begin
ext1 d,
suffices : φ.total_degree < d.support.sum d → 0 = coeff d φ,
by simpa [coeff_sum, coeff_homogeneous_component],
exact λ h, (coeff_eq_zero_of_total_degree_lt h).symm
end | lemma | mv_polynomial.sum_homogeneous_component | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_component_homogeneous_polynomial (m n : ℕ)
(p : mv_polynomial σ R) (h : p ∈ homogeneous_submodule σ R n) :
homogeneous_component m p = if m = n then p else 0 | begin
simp only [mem_homogeneous_submodule] at h,
ext x,
rw coeff_homogeneous_component,
by_cases zero_coeff : coeff x p = 0,
{ split_ifs,
all_goals { simp only [zero_coeff, coeff_zero], }, },
{ rw h zero_coeff,
simp only [show n = m ↔ m = n, from eq_comm],
split_ifs with h1,
{ refl },
{... | lemma | mv_polynomial.homogeneous_component_homogeneous_polynomial | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/homogeneous.lean | [
"algebra.direct_sum.internal",
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_span_monomial_image
{x : mv_polynomial σ R} {s : set (σ →₀ ℕ)} :
x ∈ ideal.span ((λ s, monomial s (1 : R)) '' s) ↔ ∀ xi ∈ x.support, ∃ si ∈ s, si ≤ xi | begin
refine add_monoid_algebra.mem_ideal_span_of'_image.trans _,
simp_rw [le_iff_exists_add, add_comm],
refl,
end | lemma | mv_polynomial.mem_ideal_span_monomial_image | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/ideal.lean | [
"algebra.monoid_algebra.ideal",
"data.mv_polynomial.division"
] | [
"ideal.span",
"mv_polynomial"
] | `x` is in a monomial ideal generated by `s` iff every element of of its support dominates one of
the generators. Note that `si ≤ xi` is analogous to saying that the monomial corresponding to `si`
divides the monomial corresponding to `xi`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ideal_span_monomial_image_iff_dvd {x : mv_polynomial σ R} {s : set (σ →₀ ℕ)} :
x ∈ ideal.span ((λ s, monomial s (1 : R)) '' s) ↔
∀ xi ∈ x.support, ∃ si ∈ s, monomial si 1 ∣ monomial xi (x.coeff xi) | begin
refine mem_ideal_span_monomial_image.trans (forall₂_congr $ λ xi hxi, _),
simp_rw [monomial_dvd_monomial, one_dvd, and_true, mem_support_iff.mp hxi, false_or],
end | lemma | mv_polynomial.mem_ideal_span_monomial_image_iff_dvd | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/ideal.lean | [
"algebra.monoid_algebra.ideal",
"data.mv_polynomial.division"
] | [
"forall₂_congr",
"ideal.span",
"mv_polynomial",
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_span_X_image {x : mv_polynomial σ R} {s : set σ} :
x ∈ ideal.span (mv_polynomial.X '' s : set (mv_polynomial σ R)) ↔
∀ m ∈ x.support, ∃ i ∈ s, (m : σ →₀ ℕ) i ≠ 0 | begin
have := @mem_ideal_span_monomial_image σ R _ _ ((λ i, finsupp.single i 1) '' s),
rw set.image_image at this,
refine this.trans _,
simp [nat.one_le_iff_ne_zero],
end | lemma | mv_polynomial.mem_ideal_span_X_image | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/ideal.lean | [
"algebra.monoid_algebra.ideal",
"data.mv_polynomial.division"
] | [
"finsupp.single",
"ideal.span",
"mv_polynomial",
"mv_polynomial.X",
"nat.one_le_iff_ne_zero",
"set.image_image"
] | `x` is in a monomial ideal generated by variables `X` iff every element of of its support
has a component in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
esymm (s : multiset R) (n : ℕ) : R | ((s.powerset_len n).map multiset.prod).sum | def | multiset.esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"multiset",
"multiset.prod"
] | The `n`th elementary symmetric function evaluated at the elements of `s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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