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 |
|---|---|---|---|---|---|---|---|---|---|---|
algebraic_independent.aeval_equiv (hx : algebraic_independent R x) :
(mv_polynomial ι R) ≃ₐ[R] algebra.adjoin R (range x) | begin
apply alg_equiv.of_bijective
(alg_hom.cod_restrict (@aeval R A ι _ _ _ x) (algebra.adjoin R (range x)) _),
swap,
{ intros x,
rw [adjoin_range_eq_range_aeval],
exact alg_hom.mem_range_self _ _ },
{ split,
{ exact (alg_hom.injective_cod_restrict _ _ _).2 hx },
{ rintros ⟨x, hx⟩,
rw... | def | algebraic_independent.aeval_equiv | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_equiv.of_bijective",
"alg_hom.cod_restrict",
"alg_hom.injective_cod_restrict",
"alg_hom.mem_range_self",
"algebra.adjoin",
"algebraic_independent",
"mv_polynomial"
] | Canonical isomorphism between polynomials and the subalgebra generated by
algebraically independent elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebraic_independent.algebra_map_aeval_equiv (hx : algebraic_independent R x)
(p : mv_polynomial ι R) : algebra_map (algebra.adjoin R (range x)) A (hx.aeval_equiv p) =
aeval x p | rfl | lemma | algebraic_independent.algebra_map_aeval_equiv | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra.adjoin",
"algebra_map",
"algebraic_independent",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.repr (hx : algebraic_independent R x) :
algebra.adjoin R (range x) →ₐ[R] mv_polynomial ι R | hx.aeval_equiv.symm | def | algebraic_independent.repr | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra.adjoin",
"algebraic_independent",
"mv_polynomial"
] | The canonical map from the subalgebra generated by an algebraic independent family
into the polynomial ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebraic_independent.aeval_repr (p) : aeval x (hx.repr p) = p | subtype.ext_iff.1 (alg_equiv.apply_symm_apply hx.aeval_equiv p) | lemma | algebraic_independent.aeval_repr | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_equiv.apply_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.aeval_comp_repr :
(aeval x).comp hx.repr = subalgebra.val _ | alg_hom.ext $ hx.aeval_repr | lemma | algebraic_independent.aeval_comp_repr | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_hom.ext",
"subalgebra.val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.repr_ker :
(hx.repr : adjoin R (range x) →+* mv_polynomial ι R).ker = ⊥ | (ring_hom.injective_iff_ker_eq_bot _).1 (alg_equiv.injective _) | lemma | algebraic_independent.repr_ker | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_equiv.injective",
"mv_polynomial",
"ring_hom.injective_iff_ker_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin
(hx : algebraic_independent R x) :
mv_polynomial (option ι) R ≃+* polynomial (adjoin R (set.range x)) | (mv_polynomial.option_equiv_left _ _).to_ring_equiv.trans
(polynomial.map_equiv hx.aeval_equiv.to_ring_equiv) | def | algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"mv_polynomial",
"mv_polynomial.option_equiv_left",
"polynomial",
"polynomial.map_equiv",
"set.range"
] | The isomorphism between `mv_polynomial (option ι) R` and the polynomial ring over
the algebra generated by an algebraically independent family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply
(hx : algebraic_independent R x) (y) :
hx.mv_polynomial_option_equiv_polynomial_adjoin y =
polynomial.map (hx.aeval_equiv : mv_polynomial ι R →+* adjoin R (range x))
(aeval (λ (o : option ι), o.elim polynomial.X (λ (s : ι), polynomia... | rfl | lemma | algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"mv_polynomial",
"polynomial.C",
"polynomial.X",
"polynomial.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_C
(hx : algebraic_independent R x) (r) :
hx.mv_polynomial_option_equiv_polynomial_adjoin (C r) =
polynomial.C (algebra_map _ _ r) | begin
rw [algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply, aeval_C,
is_scalar_tower.algebra_map_apply R (mv_polynomial ι R), ← polynomial.C_eq_algebra_map,
polynomial.map_C, ring_hom.coe_coe, alg_equiv.commutes]
end | lemma | algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_C | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_equiv.commutes",
"algebra_map",
"algebraic_independent",
"algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply",
"is_scalar_tower.algebra_map_apply",
"mv_polynomial",
"polynomial.C",
"polynomial.C_eq_algebra_map",
"polynomial.map_C",
"ring_hom.coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_X_none
(hx : algebraic_independent R x) :
hx.mv_polynomial_option_equiv_polynomial_adjoin (X none) = polynomial.X | by rw [algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply, aeval_X,
option.elim, polynomial.map_X] | lemma | algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_X_none | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply",
"option.elim",
"polynomial.X",
"polynomial.map_X"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_X_some
(hx : algebraic_independent R x) (i) :
hx.mv_polynomial_option_equiv_polynomial_adjoin (X (some i)) =
polynomial.C (hx.aeval_equiv (X i)) | by rw [algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply, aeval_X,
option.elim, polynomial.map_C, ring_hom.coe_coe] | lemma | algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_X_some | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent.mv_polynomial_option_equiv_polynomial_adjoin_apply",
"option.elim",
"polynomial.C",
"polynomial.map_C",
"ring_hom.coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.aeval_comp_mv_polynomial_option_equiv_polynomial_adjoin
(hx : algebraic_independent R x) (a : A) :
(ring_hom.comp (↑(polynomial.aeval a : polynomial (adjoin R (set.range x)) →ₐ[_] A) :
polynomial (adjoin R (set.range x)) →+* A)
hx.mv_polynomial_option_equiv_polynomial_adjoin.to_ring_... | begin
refine mv_polynomial.ring_hom_ext _ _;
simp only [ring_hom.comp_apply, ring_equiv.to_ring_hom_eq_coe, ring_equiv.coe_to_ring_hom,
alg_hom.coe_to_ring_hom, alg_hom.coe_to_ring_hom],
{ intro r,
rw [hx.mv_polynomial_option_equiv_polynomial_adjoin_C,
aeval_C, polynomial.aeval_C, is_scalar_... | lemma | algebraic_independent.aeval_comp_mv_polynomial_option_equiv_polynomial_adjoin | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_hom.coe_to_ring_hom",
"algebraic_independent",
"is_scalar_tower.algebra_map_apply",
"mv_polynomial",
"mv_polynomial.aeval",
"mv_polynomial.ring_hom_ext",
"option.elim",
"polynomial",
"polynomial.aeval",
"polynomial.aeval_C",
"polynomial.aeval_X",
"ring_equiv.coe_to_ring_hom",
"ring_equi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.option_iff (hx : algebraic_independent R x) (a : A) :
(algebraic_independent R (λ o : option ι, o.elim a x)) ↔
¬ is_algebraic (adjoin R (set.range x)) a | by erw [algebraic_independent_iff_injective_aeval, is_algebraic_iff_not_injective, not_not,
← alg_hom.coe_to_ring_hom,
← hx.aeval_comp_mv_polynomial_option_equiv_polynomial_adjoin,
ring_hom.coe_comp, injective.of_comp_iff' _ (ring_equiv.bijective _), alg_hom.coe_to_ring_hom] | theorem | algebraic_independent.option_iff | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_hom.coe_to_ring_hom",
"algebraic_independent",
"algebraic_independent_iff_injective_aeval",
"is_algebraic",
"is_algebraic_iff_not_injective",
"not_not",
"ring_equiv.bijective",
"ring_hom.coe_comp",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transcendence_basis (x : ι → A) : Prop | algebraic_independent R x ∧
∀ (s : set A) (i' : algebraic_independent R (coe : s → A)) (h : range x ≤ s), range x = s | def | is_transcendence_basis | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | A family is a transcendence basis if it is a maximal algebraically independent subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_transcendence_basis (h : injective (algebra_map R A)) :
∃ s : set A, is_transcendence_basis R (coe : s → A) | begin
cases exists_maximal_algebraic_independent (∅ : set A) set.univ
(set.subset_univ _) ((algebraic_independent_empty_iff R A).2 h) with s hs,
use [s, hs.1],
intros t ht hr,
simp only [subtype.range_coe_subtype, set_of_mem_eq] at *,
exact eq.symm (hs.2.2.2 t ht hr (set.subset_univ _))
end | lemma | exists_is_transcendence_basis | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra_map",
"algebraic_independent_empty_iff",
"exists_maximal_algebraic_independent",
"is_transcendence_basis",
"set.subset_univ",
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.is_transcendence_basis_iff
{ι : Type w} {R : Type u} [comm_ring R] [nontrivial R]
{A : Type v} [comm_ring A] [algebra R A] {x : ι → A} (i : algebraic_independent R x) :
is_transcendence_basis R x ↔ ∀ (κ : Type v) (w : κ → A) (i' : algebraic_independent R w)
(j : ι → κ) (h : w ∘ j = x), s... | begin
fsplit,
{ rintros p κ w i' j rfl,
have p := p.2 (range w) i'.coe_range (range_comp_subset_range _ _),
rw [range_comp, ←@image_univ _ _ w] at p,
exact range_iff_surjective.mp (image_injective.mpr i'.injective p) },
{ intros p,
use i,
intros w i' h,
specialize p w (coe : w → A) i'
... | lemma | algebraic_independent.is_transcendence_basis_iff | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra",
"algebraic_independent",
"comm_ring",
"is_transcendence_basis",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transcendence_basis.is_algebraic [nontrivial R]
(hx : is_transcendence_basis R x) : is_algebraic (adjoin R (range x)) A | begin
intro a,
rw [← not_iff_comm.1 (hx.1.option_iff _).symm],
intro ai,
have h₁ : range x ⊆ range (λ o : option ι, o.elim a x),
{ rintros x ⟨y, rfl⟩, exact ⟨some y, rfl⟩ },
have h₂ : range x ≠ range (λ o : option ι, o.elim a x),
{ intro h,
have : a ∈ range x, { rw h, exact ⟨none, rfl⟩ },
rcases t... | lemma | is_transcendence_basis.is_algebraic | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent_subtype_range",
"is_algebraic",
"is_transcendence_basis",
"nontrivial",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_empty_type [is_empty ι] [nontrivial A] :
algebraic_independent K x | begin
rw [algebraic_independent_empty_type_iff],
exact ring_hom.injective _,
end | lemma | algebraic_independent_empty_type | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_empty_type_iff",
"is_empty",
"nontrivial",
"ring_hom.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_empty [nontrivial A] :
algebraic_independent K (coe : ((∅ : set A) → A)) | algebraic_independent_empty_type | lemma | algebraic_independent_empty | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_empty_type",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible.algebra_tower (r : R) [invertible (algebra_map R S r)] :
invertible (algebra_map R A r) | invertible.copy (invertible.map (algebra_map S A) (algebra_map R S r)) (algebra_map R A r)
(is_scalar_tower.algebra_map_apply R S A r) | def | is_scalar_tower.invertible.algebra_tower | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"algebra_map",
"invertible",
"invertible.copy",
"invertible.map",
"is_scalar_tower.algebra_map_apply"
] | Suppose that `R -> S -> A` is a tower of algebras.
If an element `r : R` is invertible in `S`, then it is invertible in `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible_algebra_coe_nat (n : ℕ) [inv : invertible (n : R)] :
invertible (n : A) | by { haveI : invertible (algebra_map ℕ R n) := inv, exact invertible.algebra_tower ℕ R A n } | def | is_scalar_tower.invertible_algebra_coe_nat | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"algebra_map",
"invertible"
] | A natural number that is invertible when coerced to `R` is also invertible
when coerced to any `R`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis.algebra_map_coeffs : basis ι A M | b.map_coeffs (ring_equiv.of_bijective _ h) (λ c x, by simp) | def | basis.algebra_map_coeffs | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"basis",
"ring_equiv.of_bijective"
] | If `R` and `A` have a bijective `algebra_map R A` and act identically on `M`,
then a basis for `M` as `R`-module is also a basis for `M` as `R'`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis.algebra_map_coeffs_apply (i : ι) : b.algebra_map_coeffs A h i = b i | b.map_coeffs_apply _ _ _ | lemma | basis.algebra_map_coeffs_apply | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.coe_algebra_map_coeffs : (b.algebra_map_coeffs A h : ι → M) = b | b.coe_map_coeffs _ _ | lemma | basis.coe_algebra_map_coeffs | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent_smul {ι : Type v₁} {b : ι → S} {ι' : Type w₁} {c : ι' → A}
(hb : linear_independent R b) (hc : linear_independent S c) :
linear_independent R (λ p : ι × ι', b p.1 • c p.2) | begin
rw linear_independent_iff' at hb hc, rw linear_independent_iff'', rintros s g hg hsg ⟨i, k⟩,
by_cases hik : (i, k) ∈ s,
{ have h1 : ∑ i in s.image prod.fst ×ˢ s.image prod.snd, g i • b i.1 • c i.2 = 0,
{ rw ← hsg, exact (finset.sum_subset finset.subset_product $ λ p _ hp,
show g p • b p.1 • c p.... | theorem | linear_independent_smul | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"finset.mem_image_of_mem",
"finset.subset_product",
"finset.sum_smul",
"linear_independent",
"linear_independent_iff'",
"linear_independent_iff''",
"smul_assoc",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.smul {ι : Type v₁} {ι' : Type w₁}
(b : basis ι R S) (c : basis ι' S A) : basis (ι × ι') R A | basis.of_repr ((c.repr.restrict_scalars R) ≪≫ₗ
((finsupp.lcongr (equiv.refl _) b.repr) ≪≫ₗ
((finsupp_prod_lequiv R).symm ≪≫ₗ
((finsupp.lcongr (equiv.prod_comm ι' ι) (linear_equiv.refl _ _)))))) | def | basis.smul | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"basis",
"equiv.prod_comm",
"equiv.refl",
"finsupp.lcongr",
"linear_equiv.refl"
] | `basis.smul (b : basis ι R S) (c : basis ι S A)` is the `R`-basis on `A`
where the `(i, j)`th basis vector is `b i • c j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis.smul_repr {ι : Type v₁} {ι' : Type w₁}
(b : basis ι R S) (c : basis ι' S A) (x ij):
(b.smul c).repr x ij = b.repr (c.repr x ij.2) ij.1 | by simp [basis.smul] | theorem | basis.smul_repr | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"basis",
"basis.smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.smul_repr_mk {ι : Type v₁} {ι' : Type w₁}
(b : basis ι R S) (c : basis ι' S A) (x i j):
(b.smul c).repr x (i, j) = b.repr (c.repr x j) i | b.smul_repr c x (i, j) | theorem | basis.smul_repr_mk | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.smul_apply {ι : Type v₁} {ι' : Type w₁}
(b : basis ι R S) (c : basis ι' S A) (ij) :
(b.smul c) ij = b ij.1 • c ij.2 | begin
obtain ⟨i, j⟩ := ij,
rw basis.apply_eq_iff,
ext ⟨i', j'⟩,
rw [basis.smul_repr, linear_equiv.map_smul, basis.repr_self, finsupp.smul_apply,
finsupp.single_apply],
dsimp only,
split_ifs with hi,
{ simp [hi, finsupp.single_apply] },
{ simp [hi] },
end | theorem | basis.smul_apply | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"basis",
"basis.apply_eq_iff",
"basis.repr_self",
"basis.smul_repr",
"finsupp.single_apply",
"finsupp.smul_apply",
"linear_equiv.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.algebra_map_injective {ι : Type*} [no_zero_divisors R] [nontrivial S]
(b : basis ι R S) :
function.injective (algebra_map R S) | have no_zero_smul_divisors R S := b.no_zero_smul_divisors,
by exactI no_zero_smul_divisors.algebra_map_injective R S | lemma | basis.algebra_map_injective | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"algebra_map",
"basis",
"no_zero_divisors",
"no_zero_smul_divisors",
"no_zero_smul_divisors.algebra_map_injective",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom.restrict_domain : B →ₐ[A] D | f.comp (is_scalar_tower.to_alg_hom A B C) | def | alg_hom.restrict_domain | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"is_scalar_tower.to_alg_hom"
] | Restrict the domain of an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom.extend_scalars : @alg_hom B C D _ _ _ _ (f.restrict_domain B).to_ring_hom.to_algebra | { commutes' := λ _, rfl .. f } | def | alg_hom.extend_scalars | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"alg_hom"
] | Extend the scalars of an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom_equiv_sigma :
(C →ₐ[A] D) ≃ Σ (f : B →ₐ[A] D), @alg_hom B C D _ _ _ _ f.to_ring_hom.to_algebra | { to_fun := λ f, ⟨f.restrict_domain B, f.extend_scalars B⟩,
inv_fun := λ fg,
let alg := fg.1.to_ring_hom.to_algebra in by exactI fg.2.restrict_scalars A,
left_inv := λ f, by { dsimp only, ext, refl },
right_inv :=
begin
rintros ⟨⟨f, _, _, _, _, _⟩, g, _, _, _, _, hg⟩,
obtain rfl : f = λ x, g (algebr... | def | alg_hom_equiv_sigma | ring_theory | src/ring_theory/algebra_tower.lean | [
"algebra.algebra.tower",
"algebra.invertible",
"algebra.module.big_operators",
"linear_algebra.basis"
] | [
"alg_hom",
"algebra_map",
"inv_fun"
] | `alg_hom`s from the top of a tower are equivalent to a pair of `alg_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_artinian (R M) [semiring R] [add_comm_monoid M] [module R M] : Prop | (well_founded_submodule_lt [] : well_founded ((<) : submodule R M → submodule R M → Prop)) | class | is_artinian | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_monoid",
"module",
"semiring",
"submodule"
] | `is_artinian R M` is the proposition that `M` is an Artinian `R`-module,
implemented as the well-foundedness of submodule inclusion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_artinian_of_injective (f : M →ₗ[R] P) (h : function.injective f)
[is_artinian R P] : is_artinian R M | ⟨subrelation.wf
(λ A B hAB, show A.map f < B.map f,
from submodule.map_strict_mono_of_injective h hAB)
(inv_image.wf (submodule.map f) (is_artinian.well_founded_submodule_lt R P))⟩ | theorem | is_artinian_of_injective | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"submodule.map",
"submodule.map_strict_mono_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_submodule' [is_artinian R M] (N : submodule R M) : is_artinian R N | is_artinian_of_injective N.subtype subtype.val_injective | instance | is_artinian_submodule' | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"is_artinian_of_injective",
"submodule",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_le {s t : submodule R M} [ht : is_artinian R t]
(h : s ≤ t) : is_artinian R s | is_artinian_of_injective (submodule.of_le h) (submodule.of_le_injective h) | lemma | is_artinian_of_le | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"is_artinian_of_injective",
"submodule",
"submodule.of_le",
"submodule.of_le_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_surjective (f : M →ₗ[R] P) (hf : function.surjective f)
[is_artinian R M] : is_artinian R P | ⟨subrelation.wf
(λ A B hAB, show A.comap f < B.comap f,
from submodule.comap_strict_mono_of_surjective hf hAB)
(inv_image.wf (submodule.comap f) (is_artinian.well_founded_submodule_lt _ _))⟩ | theorem | is_artinian_of_surjective | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"submodule.comap",
"submodule.comap_strict_mono_of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_linear_equiv (f : M ≃ₗ[R] P)
[is_artinian R M] : is_artinian R P | is_artinian_of_surjective _ f.to_linear_map f.to_equiv.surjective | theorem | is_artinian_of_linear_equiv | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"is_artinian_of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_range_eq_ker
[is_artinian R M] [is_artinian R P]
(f : M →ₗ[R] N) (g : N →ₗ[R] P)
(hf : function.injective f)
(hg : function.surjective g)
(h : f.range = g.ker) :
is_artinian R N | ⟨well_founded_lt_exact_sequence
(is_artinian.well_founded_submodule_lt _ _)
(is_artinian.well_founded_submodule_lt _ _)
f.range
(submodule.map f)
(submodule.comap f)
(submodule.comap g)
(submodule.map g)
(submodule.gci_map_comap hf)
(submodule.gi_map_comap hg)
(by simp [submodule.map_comap_eq, inf_c... | theorem | is_artinian_of_range_eq_ker | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"inf_comm",
"is_artinian",
"submodule.comap",
"submodule.comap_map_eq",
"submodule.gci_map_comap",
"submodule.gi_map_comap",
"submodule.map",
"submodule.map_comap_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_prod [is_artinian R M]
[is_artinian R P] : is_artinian R (M × P) | is_artinian_of_range_eq_ker
(linear_map.inl R M P)
(linear_map.snd R M P)
linear_map.inl_injective
linear_map.snd_surjective
(linear_map.range_inl R M P) | instance | is_artinian_prod | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"is_artinian_of_range_eq_ker",
"linear_map.inl",
"linear_map.inl_injective",
"linear_map.range_inl",
"linear_map.snd",
"linear_map.snd_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_finite [finite M] : is_artinian R M | ⟨finite.well_founded_of_trans_of_irrefl _⟩ | instance | is_artinian_of_finite | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"finite",
"is_artinian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_pi {R ι : Type*} [finite ι] : Π {M : ι → Type*} [ring R]
[Π i, add_comm_group (M i)], by exactI Π [Π i, module R (M i)],
by exactI Π [∀ i, is_artinian R (M i)], is_artinian R (Π i, M i) | finite.induction_empty_option
(begin
introsI α β e hα M _ _ _ _,
exact is_artinian_of_linear_equiv
(linear_equiv.Pi_congr_left R M e)
end)
(by { introsI M _ _ _ _, apply_instance })
(begin
introsI α _ ih M _ _ _ _,
exact is_artinian_of_linear_equiv
(linear_equiv.pi_option_equiv_p... | instance | is_artinian_pi | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"finite",
"finite.induction_empty_option",
"ih",
"is_artinian",
"is_artinian_of_linear_equiv",
"linear_equiv.Pi_congr_left",
"linear_equiv.pi_option_equiv_prod",
"module",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_pi' {R ι M : Type*} [ring R] [add_comm_group M] [module R M] [finite ι]
[is_artinian R M] : is_artinian R (ι → M) | is_artinian_pi | instance | is_artinian_pi' | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"finite",
"is_artinian",
"is_artinian_pi",
"module",
"ring"
] | A version of `is_artinian_pi` for non-dependent functions. We need this instance because
sometimes Lean fails to apply the dependent version in non-dependent settings (e.g., it fails to
prove that `ι → ℝ` is finite dimensional over `ℝ`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_artinian_iff_well_founded :
is_artinian R M ↔ well_founded ((<) : submodule R M → submodule R M → Prop) | ⟨λ h, h.1, is_artinian.mk⟩ | theorem | is_artinian_iff_well_founded | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian.finite_of_linear_independent [nontrivial R] [is_artinian R M]
{s : set M} (hs : linear_independent R (coe : s → M)) : s.finite | begin
refine classical.by_contradiction (λ hf, (rel_embedding.well_founded_iff_no_descending_seq.1
(well_founded_submodule_lt R M)).elim' _),
have f : ℕ ↪ s, from set.infinite.nat_embedding s hf,
have : ∀ n, (coe ∘ f) '' {m | n ≤ m} ⊆ s,
{ rintros n x ⟨y, hy₁, rfl⟩, exact (f y).2 },
have : ∀ a b : ℕ, a ≤ ... | lemma | is_artinian.finite_of_linear_independent | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"le_rfl",
"linear_independent",
"nontrivial",
"set.image_subset_image_iff",
"set.infinite.nat_embedding",
"set.subset_def",
"span_le_span_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_has_minimal_iff_artinian :
(∀ a : set $ submodule R M, a.nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬ I < M') ↔ is_artinian R M | by rw [is_artinian_iff_well_founded, well_founded.well_founded_iff_has_min] | theorem | set_has_minimal_iff_artinian | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"is_artinian_iff_well_founded",
"submodule",
"well_founded.well_founded_iff_has_min"
] | A module is Artinian iff every nonempty set of submodules has a minimal submodule among them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_artinian.set_has_minimal [is_artinian R M] (a : set $ submodule R M) (ha : a.nonempty) :
∃ M' ∈ a, ∀ I ∈ a, ¬ I < M' | set_has_minimal_iff_artinian.mpr ‹_› a ha | theorem | is_artinian.set_has_minimal | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_stabilizes_iff_artinian :
(∀ (f : ℕ →o (submodule R M)ᵒᵈ), ∃ n, ∀ m, n ≤ m → f n = f m) ↔ is_artinian R M | by { rw is_artinian_iff_well_founded, exact well_founded.monotone_chain_condition.symm } | theorem | monotone_stabilizes_iff_artinian | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"is_artinian_iff_well_founded",
"submodule"
] | A module is Artinian iff every decreasing chain of submodules stabilizes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_stabilizes (f : ℕ →o (submodule R M)ᵒᵈ) : ∃ n, ∀ m, n ≤ m → f n = f m | monotone_stabilizes_iff_artinian.mpr ‹_› f | theorem | is_artinian.monotone_stabilizes | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction {P : submodule R M → Prop} (hgt : ∀ I, (∀ J < I, P J) → P I) (I : submodule R M) :
P I | (well_founded_submodule_lt R M).recursion I hgt | lemma | is_artinian.induction | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"submodule"
] | If `∀ I > J, P I` implies `P J`, then `P` holds for all submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_endomorphism_iterate_ker_sup_range_eq_top (f : M →ₗ[R] M) :
∃ n : ℕ, n ≠ 0 ∧ (f ^ n).ker ⊔ (f ^ n).range = ⊤ | begin
obtain ⟨n, w⟩ := monotone_stabilizes (f.iterate_range.comp ⟨λ n, n+1, λ n m w, by linarith⟩),
specialize w ((n + 1) + n) (by linarith),
dsimp at w,
refine ⟨n + 1, nat.succ_ne_zero _, _⟩,
simp_rw [eq_top_iff', mem_sup],
intro x,
have : (f^(n + 1)) x ∈ (f ^ ((n + 1) + n + 1)).range,
{ rw ← w, exact ... | theorem | is_artinian.exists_endomorphism_iterate_ker_sup_range_eq_top | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"linear_map.map_sub",
"linear_map.mem_ker",
"pow_add"
] | For any endomorphism of a Artinian module, there is some nontrivial iterate
with disjoint kernel and range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_of_injective_endomorphism (f : M →ₗ[R] M) (s : injective f) : surjective f | begin
obtain ⟨n, ne, w⟩ := exists_endomorphism_iterate_ker_sup_range_eq_top f,
rw [linear_map.ker_eq_bot.mpr (linear_map.iterate_injective s n), bot_sup_eq,
linear_map.range_eq_top] at w,
exact linear_map.surjective_of_iterate_surjective ne w,
end | theorem | is_artinian.surjective_of_injective_endomorphism | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"bot_sup_eq",
"linear_map.iterate_injective",
"linear_map.range_eq_top",
"linear_map.surjective_of_iterate_surjective"
] | Any injective endomorphism of an Artinian module is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bijective_of_injective_endomorphism (f : M →ₗ[R] M) (s : injective f) : bijective f | ⟨s, surjective_of_injective_endomorphism f s⟩ | theorem | is_artinian.bijective_of_injective_endomorphism | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [] | Any injective endomorphism of an Artinian module is bijective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint_partial_infs_eventually_top (f : ℕ → submodule R M)
(h : ∀ n, disjoint (partial_sups (order_dual.to_dual ∘ f) n) (order_dual.to_dual (f (n+1)))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊤ | begin
-- A little off-by-one cleanup first:
rsuffices ⟨n, w⟩ : ∃ n : ℕ, ∀ m, n ≤ m → order_dual.to_dual f (m+1) = ⊤,
{ use n+1,
rintros (_|m) p,
{ cases p, },
{ apply w,
exact nat.succ_le_succ_iff.mp p }, },
obtain ⟨n, w⟩ := monotone_stabilizes (partial_sups (order_dual.to_dual ∘ f)),
refin... | lemma | is_artinian.disjoint_partial_infs_eventually_top | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"disjoint",
"order_dual.to_dual",
"partial_sups",
"submodule"
] | A sequence `f` of submodules of a artinian module,
with the supremum `f (n+1)` and the infinum of `f 0`, ..., `f n` being ⊤,
is eventually ⊤. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_smul_pow_stabilizes (r : R) : ∃ n : ℕ, ∀ m, n ≤ m →
(r^n • linear_map.id : M →ₗ[R] M).range = (r^m • linear_map.id : M →ₗ[R] M).range | monotone_stabilizes
⟨λ n, (r^n • linear_map.id : M →ₗ[R] M).range,
λ n m h x ⟨y, hy⟩, ⟨r ^ (m - n) • y,
by { dsimp at ⊢ hy, rw [←smul_assoc, smul_eq_mul, ←pow_add, ←hy, add_tsub_cancel_of_le h] }⟩⟩ | lemma | is_artinian.range_smul_pow_stabilizes | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_tsub_cancel_of_le",
"linear_map.id",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pow_succ_smul_dvd (r : R) (x : M) :
∃ (n : ℕ) (y : M), r ^ n.succ • y = r ^ n • x | begin
obtain ⟨n, hn⟩ := is_artinian.range_smul_pow_stabilizes M r,
simp_rw [set_like.ext_iff] at hn,
exact ⟨n, by simpa using hn n.succ n.le_succ (r ^ n • x)⟩,
end | lemma | is_artinian.exists_pow_succ_smul_dvd | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian.range_smul_pow_stabilizes",
"set_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_ring (R) [ring R] | is_artinian R R | def | is_artinian_ring | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"ring"
] | A ring is Artinian if it is Artinian as a module over itself.
Strictly speaking, this should be called `is_left_artinian_ring` but we omit the `left_` for
convenience in the commutative case. For a right Artinian ring, use `is_artinian Rᵐᵒᵖ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_artinian_ring_iff {R} [ring R] : is_artinian_ring R ↔ is_artinian R R | iff.rfl | theorem | is_artinian_ring_iff | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian",
"is_artinian_ring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.is_artinian_of_zero_eq_one {R} [ring R] (h01 : (0 : R) = 1) : is_artinian_ring R | have _ := subsingleton_of_zero_eq_one h01, by exactI infer_instance | theorem | ring.is_artinian_of_zero_eq_one | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian_ring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_submodule_of_artinian (R M) [ring R] [add_comm_group M] [module R M]
(N : submodule R M) (h : is_artinian R M) : is_artinian R N | by apply_instance | theorem | is_artinian_of_submodule_of_artinian | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"is_artinian",
"module",
"ring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_quotient_of_artinian (R) [ring R] (M) [add_comm_group M] [module R M]
(N : submodule R M) (h : is_artinian R M) : is_artinian R (M ⧸ N) | is_artinian_of_surjective M (submodule.mkq N) (submodule.quotient.mk_surjective N) | theorem | is_artinian_of_quotient_of_artinian | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"is_artinian",
"is_artinian_of_surjective",
"module",
"ring",
"submodule",
"submodule.mkq",
"submodule.quotient.mk_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_tower (R) {S M} [comm_ring R] [ring S]
[add_comm_group M] [algebra R S] [module S M] [module R M] [is_scalar_tower R S M]
(h : is_artinian R M) : is_artinian S M | begin
rw is_artinian_iff_well_founded at h ⊢,
refine (submodule.restrict_scalars_embedding R S M).well_founded h
end | theorem | is_artinian_of_tower | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"algebra",
"comm_ring",
"is_artinian",
"is_artinian_iff_well_founded",
"is_scalar_tower",
"module",
"ring",
"submodule.restrict_scalars_embedding"
] | If `M / S / R` is a scalar tower, and `M / R` is Artinian, then `M / S` is
also Artinian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_artinian_of_fg_of_artinian {R M} [ring R] [add_comm_group M] [module R M]
(N : submodule R M) [is_artinian_ring R] (hN : N.fg) : is_artinian R N | let ⟨s, hs⟩ := hN in
begin
haveI := classical.dec_eq M,
haveI := classical.dec_eq R,
have : ∀ x ∈ s, x ∈ N, from λ x hx, hs ▸ submodule.subset_span hx,
refine @@is_artinian_of_surjective ((↑s : set M) → R) _ _ _ (pi.module _ _ _)
_ _ _ is_artinian_pi,
{ fapply linear_map.mk,
{ exact λ f, ⟨∑ i in s.att... | theorem | is_artinian_of_fg_of_artinian | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"add_smul",
"classical.dec_eq",
"finsupp.mem_span_image_iff_total",
"finsupp.total_apply",
"is_artinian",
"is_artinian_of_surjective",
"is_artinian_pi",
"is_artinian_ring",
"module",
"pi.module",
"ring",
"set.image_id",
"smul_eq_mul",
"submodule",
"submodule.subset_sp... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_of_fg_of_artinian' {R M} [ring R] [add_comm_group M] [module R M]
[is_artinian_ring R] (h : (⊤ : submodule R M).fg) : is_artinian R M | have is_artinian R (⊤ : submodule R M), from is_artinian_of_fg_of_artinian _ h,
by exactI is_artinian_of_linear_equiv (linear_equiv.of_top (⊤ : submodule R M) rfl) | lemma | is_artinian_of_fg_of_artinian' | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"is_artinian",
"is_artinian_of_fg_of_artinian",
"is_artinian_of_linear_equiv",
"is_artinian_ring",
"linear_equiv.of_top",
"module",
"ring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_span_of_finite (R) {M} [ring R] [add_comm_group M] [module R M]
[is_artinian_ring R] {A : set M} (hA : A.finite) : is_artinian R (submodule.span R A) | is_artinian_of_fg_of_artinian _ (submodule.fg_def.mpr ⟨A, hA, rfl⟩) | theorem | is_artinian_span_of_finite | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"add_comm_group",
"is_artinian",
"is_artinian_of_fg_of_artinian",
"is_artinian_ring",
"module",
"ring",
"submodule.span"
] | In a module over a artinian ring, the submodule generated by finitely many vectors is
artinian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.is_artinian_ring {R} [ring R] {S} [ring S] {F} [ring_hom_class F R S]
{f : F} (hf : function.surjective f) [H : is_artinian_ring R] : is_artinian_ring S | begin
rw [is_artinian_ring_iff, is_artinian_iff_well_founded] at H ⊢,
exact (ideal.order_embedding_of_surjective f hf).well_founded H,
end | theorem | function.surjective.is_artinian_ring | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"ideal.order_embedding_of_surjective",
"is_artinian_iff_well_founded",
"is_artinian_ring",
"is_artinian_ring_iff",
"ring",
"ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_artinian_ring_range {R} [ring R] {S} [ring S] (f : R →+* S) [is_artinian_ring R] :
is_artinian_ring f.range | f.range_restrict_surjective.is_artinian_ring | instance | is_artinian_ring_range | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian_ring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_jacobson_bot : is_nilpotent (ideal.jacobson (⊥ : ideal R)) | begin
let Jac := ideal.jacobson (⊥ : ideal R),
let f : ℕ →o (ideal R)ᵒᵈ := ⟨λ n, Jac ^ n, λ _ _ h, ideal.pow_le_pow h⟩,
obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := is_artinian.monotone_stabilizes f,
refine ⟨n, _⟩,
let J : ideal R := annihilator (Jac ^ n),
suffices : J = ⊤,
{ have hJ : J • Jac ... | lemma | is_artinian_ring.is_nilpotent_jacobson_bot | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"by_contradiction",
"eq_of_le_of_not_lt",
"ideal",
"ideal.jacobson",
"ideal.pow_le_pow",
"ideal.span",
"ideal.zero_eq_bot",
"is_artinian.monotone_stabilizes",
"is_artinian.set_has_minimal",
"is_nilpotent",
"le_rfl",
"le_sup_left",
"le_sup_right",
"mul_assoc",
"mul_comm",
"mul_le_mul",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_surjective : function.surjective (algebra_map R L) | begin
intro r',
obtain ⟨r₁, s, rfl⟩ := is_localization.mk'_surjective S r',
obtain ⟨r₂, h⟩ : ∃ r : R, is_localization.mk' L 1 s = algebra_map R L r,
swap, { exact ⟨r₁ * r₂, by rw [is_localization.mk'_eq_mul_mk'_one, map_mul, h]⟩ },
obtain ⟨n, r, hr⟩ := is_artinian.exists_pow_succ_smul_dvd (s : R) (1 : R),
u... | theorem | is_artinian_ring.localization_surjective | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"algebra_map",
"is_artinian.exists_pow_succ_smul_dvd",
"is_localization.mk'",
"is_localization.mk'_eq_iff_eq",
"is_localization.mk'_eq_mul_mk'_one",
"is_localization.mk'_surjective",
"map_mul",
"mul_assoc",
"mul_left_cancel",
"mul_one",
"pow_succ'",
"smul_eq_mul",
"submonoid.coe_one"
] | Localizing an artinian ring can only reduce the amount of elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_artinian : is_artinian_ring L | (localization_surjective S L).is_artinian_ring | lemma | is_artinian_ring.localization_artinian | ring_theory | src/ring_theory/artinian.lean | [
"ring_theory.nakayama",
"data.set_like.fintype"
] | [
"is_artinian_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bezout : Prop | (is_principal_of_fg : ∀ I : ideal R, I.fg → I.is_principal) | class | is_bezout | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal"
] | A Bézout ring is a ring whose finitely generated ideals are principal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_pair_is_principal [is_bezout R] (x y : R) :
(ideal.span {x, y} : ideal R).is_principal | by { classical, exact is_principal_of_fg (ideal.span {x, y}) ⟨{x, y}, by simp⟩ } | instance | is_bezout.span_pair_is_principal | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal",
"ideal.span",
"is_bezout"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_span_pair_is_principal :
is_bezout R ↔ (∀ x y : R, (ideal.span {x, y} : ideal R).is_principal) | begin
classical,
split,
{ introsI H x y, apply_instance },
{ intro H,
constructor,
apply submodule.fg_induction,
{ exact λ _, ⟨⟨_, rfl⟩⟩ },
{ rintro _ _ ⟨⟨x, rfl⟩⟩ ⟨⟨y, rfl⟩⟩, rw ← submodule.span_insert, exact H _ _ } },
end | lemma | is_bezout.iff_span_pair_is_principal | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal",
"ideal.span",
"is_bezout",
"submodule.fg_induction",
"submodule.span_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd (x y : R) : R | submodule.is_principal.generator (ideal.span {x, y}) | def | is_bezout.gcd | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal.span",
"submodule.is_principal.generator"
] | The gcd of two elements in a bezout domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_gcd (x y : R) : (ideal.span {gcd x y} : ideal R) = ideal.span {x, y} | ideal.span_singleton_generator _ | lemma | is_bezout.span_gcd | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal",
"ideal.span",
"ideal.span_singleton_generator",
"span_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_left (x y : R) : gcd x y ∣ x | (submodule.is_principal.mem_iff_generator_dvd _).mp (ideal.subset_span (by simp)) | lemma | is_bezout.gcd_dvd_left | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal.subset_span",
"submodule.is_principal.mem_iff_generator_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_right (x y : R) : gcd x y ∣ y | (submodule.is_principal.mem_iff_generator_dvd _).mp (ideal.subset_span (by simp)) | lemma | is_bezout.gcd_dvd_right | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal.subset_span",
"submodule.is_principal.mem_iff_generator_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_gcd {x y z : R} (hx : z ∣ x) (hy : z ∣ y) : z ∣ gcd x y | begin
rw [← ideal.span_singleton_le_span_singleton] at hx hy ⊢,
rw [span_gcd, ideal.span_insert, sup_le_iff],
exact ⟨hx, hy⟩
end | lemma | is_bezout.dvd_gcd | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal.span_insert",
"ideal.span_singleton_le_span_singleton",
"span_gcd",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_sum (x y : R) : ∃ a b : R, a * x + b * y = gcd x y | ideal.mem_span_pair.mp (by { rw ← span_gcd, apply ideal.subset_span, simp }) | lemma | is_bezout.gcd_eq_sum | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal.subset_span",
"span_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_gcd_domain [is_domain R] [decidable_eq R] :
gcd_monoid R | gcd_monoid_of_gcd gcd gcd_dvd_left gcd_dvd_right
(λ _ _ _, dvd_gcd) | def | is_bezout.to_gcd_domain | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"gcd_monoid",
"gcd_monoid_of_gcd",
"is_domain"
] | Any bezout domain is a GCD domain. This is not an instance since `gcd_monoid` contains data,
and this might not be how we would like to construct it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.function.surjective.is_bezout {S : Type v} [comm_ring S] (f : R →+* S)
(hf : function.surjective f) [is_bezout R] : is_bezout S | begin
rw iff_span_pair_is_principal,
intros x y,
obtain ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩ := ⟨hf x, hf y⟩,
use f (gcd x y),
transitivity ideal.map f (ideal.span {gcd x y}),
{ rw [span_gcd, ideal.map_span, set.image_insert_eq, set.image_singleton] },
{ rw [ideal.map_span, set.image_singleton], refl }
end | lemma | function.surjective.is_bezout | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"comm_ring",
"ideal.map",
"ideal.map_span",
"ideal.span",
"is_bezout",
"set.image_insert_eq",
"set.image_singleton",
"span_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_principal_ideal_ring [is_principal_ideal_ring R] : is_bezout R | ⟨λ I _, is_principal_ideal_ring.principal I⟩ | instance | is_bezout.of_is_principal_ideal_ring | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"is_bezout",
"is_principal_ideal_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tfae [is_bezout R] [is_domain R] :
tfae [is_noetherian_ring R,
is_principal_ideal_ring R,
unique_factorization_monoid R,
wf_dvd_monoid R] | begin
classical,
tfae_have : 1 → 2,
{ introI H, exact ⟨λ I, is_principal_of_fg _ (is_noetherian.noetherian _)⟩ },
tfae_have : 2 → 3,
{ introI _, apply_instance },
tfae_have : 3 → 4,
{ introI _, apply_instance },
tfae_have : 4 → 1,
{ rintro ⟨h⟩,
rw [is_noetherian_ring_iff, is_noetherian_iff_fg_well... | lemma | is_bezout.tfae | ring_theory | src/ring_theory/bezout.lean | [
"ring_theory.principal_ideal_domain",
"algebra.gcd_monoid.integrally_closed"
] | [
"ideal",
"ideal.span",
"ideal.span_singleton_lt_span_singleton",
"is_bezout",
"is_domain",
"is_noetherian_iff_fg_well_founded",
"is_noetherian_ring",
"is_noetherian_ring_iff",
"is_principal_ideal_ring",
"rel_embedding.well_founded",
"unique_factorization_monoid",
"wf_dvd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates.is_atom_iff {p : associates M} (h₁ : p ≠ 0) :
is_atom p ↔ irreducible p | ⟨λ hp, ⟨by simpa only [associates.is_unit_iff_eq_one] using hp.1,
λ a b h, (hp.le_iff.mp ⟨_, h⟩).cases_on
(λ ha, or.inl (a.is_unit_iff_eq_one.mpr ha))
(λ ha, or.inr (show is_unit b, by {rw ha at h, apply is_unit_of_associated_mul
(show associated (p * b) p, by conv_rhs {rw h}) h₁ }... | lemma | associates.is_atom_iff | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associated",
"associates",
"associates.bot_eq_one",
"associates.is_unit_iff_eq_one",
"irreducible",
"is_atom",
"is_unit",
"is_unit.mul_coe_inv",
"is_unit_of_associated_mul",
"mul_assoc",
"units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_chain_of_prime_pow {p : associates M} {n : ℕ} (hn : n ≠ 0) (hp : prime p) :
∃ c : fin (n + 1) → associates M,
c 1 = p ∧ strict_mono c ∧
∀ {r : associates M}, r ≤ p^n ↔ ∃ i, r = c i | begin
refine ⟨λ i, p^(i : ℕ), _, λ n m h, _, λ y, ⟨λ h, _, _⟩⟩,
{ rw [fin.coe_one', nat.mod_eq_of_lt, pow_one],
exact nat.lt_succ_of_le (nat.one_le_iff_ne_zero.mpr hn) },
{ exact associates.dvd_not_unit_iff_lt.mp ⟨pow_ne_zero n hp.ne_zero, p^(m - n : ℕ),
not_is_unit_of_not_is_unit_dvd hp.not_unit (dvd_p... | lemma | divisor_chain.exists_chain_of_prime_pow | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associated_iff_eq",
"associates",
"dvd_pow",
"dvd_prime_pow",
"dvd_rfl",
"fin.coe_one'",
"not_is_unit_of_not_is_unit_dvd",
"pow_mul_pow_sub",
"pow_one",
"prime",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
element_of_chain_not_is_unit_of_index_ne_zero {n : ℕ} {i : fin (n + 1)} (i_pos : i ≠ 0)
{c : fin (n + 1) → associates M} (h₁ : strict_mono c) :
¬ is_unit (c i) | dvd_not_unit.not_unit (associates.dvd_not_unit_iff_lt.2
(h₁ $ show (0 : fin (n + 1)) < i, from i.pos_iff_ne_zero.mpr i_pos)) | lemma | divisor_chain.element_of_chain_not_is_unit_of_index_ne_zero | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"dvd_not_unit.not_unit",
"is_unit",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
first_of_chain_is_unit {q : associates M} {n : ℕ} {c : fin (n + 1) → associates M}
(h₁ : strict_mono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) : is_unit (c 0) | begin
obtain ⟨i, hr⟩ := h₂.mp associates.one_le,
rw [associates.is_unit_iff_eq_one, ← associates.le_one_iff, hr],
exact h₁.monotone (fin.zero_le i)
end | lemma | divisor_chain.first_of_chain_is_unit | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"associates.is_unit_iff_eq_one",
"associates.le_one_iff",
"associates.one_le",
"fin.zero_le",
"is_unit",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_of_chain_is_irreducible {q : associates M} {n : ℕ} (hn : n ≠ 0)
{c : fin (n + 1) → associates M} (h₁ : strict_mono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
(hq : q ≠ 0) : irreducible (c 1) | begin
cases n, { contradiction },
refine (associates.is_atom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨_, λ b hb, _⟩,
{ exact ne_bot_of_gt (h₁ (show (0 : fin (n + 2)) < 1, from fin.one_pos)) },
obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩)),
cases i,
{ exact (associates.is_unit_iff_e... | lemma | divisor_chain.second_of_chain_is_irreducible | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"associates.is_atom_iff",
"associates.is_unit_iff_eq_one",
"fin.lt_iff_coe_lt_coe",
"fin.one_pos",
"irreducible",
"ne_bot_of_gt",
"ne_zero_of_dvd_ne_zero",
"strict_mono"
] | The second element of a chain is irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_second_of_chain_of_prime_dvd {p q r : associates M} {n : ℕ} (hn : n ≠ 0)
{c : fin (n + 1) → associates M} (h₁ : strict_mono c)
(h₂ : ∀ {r : associates M}, r ≤ q ↔ ∃ i, r = c i) (hp : prime p) (hr : r ∣ q) (hp' : p ∣ r) :
p = c 1 | begin
cases n,
{ contradiction },
obtain ⟨i, rfl⟩ := h₂.1 (dvd_trans hp' hr),
refine congr_arg c (eq_of_ge_of_not_gt _ $ λ hi, _),
{ rw [fin.le_iff_coe_le_coe, fin.coe_one, nat.succ_le_iff, ← fin.coe_zero (n.succ + 1),
← fin.lt_iff_coe_lt_coe, fin.pos_iff_ne_zero],
rintro rfl,
exact hp.not_uni... | lemma | divisor_chain.eq_second_of_chain_of_prime_dvd | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"dvd_not_unit.not_unit",
"dvd_trans",
"eq_of_ge_of_not_gt",
"fin.coe_eq_cast_succ",
"fin.coe_one",
"fin.coe_zero",
"fin.le_iff_coe_le_coe",
"fin.lt_iff_coe_lt_coe",
"fin.lt_succ",
"fin.pos_iff_ne_zero",
"fin.succ_lt_succ_iff",
"nat.succ_le_iff",
"not_irreducible_of_not_unit_d... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_subset_divisors_le_length_of_chain {q : associates M}
{n : ℕ} {c : fin (n + 1) → associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
{m : finset (associates M)} (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 | begin
classical,
have mem_image : ∀ (r : associates M), r ≤ q → r ∈ finset.univ.image c,
{ intros r hr,
obtain ⟨i, hi⟩ := h₂.1 hr,
exact finset.mem_image.2 ⟨i, finset.mem_univ _, hi.symm⟩ },
rw ←finset.card_fin (n + 1),
exact (finset.card_le_of_subset $ λ x hx, mem_image x $ hm x hx).trans finset.card... | lemma | divisor_chain.card_subset_divisors_le_length_of_chain | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"finset",
"finset.card_image_le",
"finset.card_le_of_subset",
"finset.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
element_of_chain_eq_pow_second_of_chain {q r : associates M} {n : ℕ} (hn : n ≠ 0)
{c : fin (n + 1) → associates M} (h₁ : strict_mono c)
(h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) (hr : r ∣ q)
(hq : q ≠ 0) : ∃ (i : fin (n + 1)), r = (c 1) ^ (i : ℕ) | begin
classical,
let i := (normalized_factors r).card,
have hi : normalized_factors r = multiset.replicate i (c 1),
{ apply multiset.eq_replicate_of_mem,
intros b hb,
refine eq_second_of_chain_of_prime_dvd hn h₁ (λ r', h₂) (prime_of_normalized_factor b hb) hr
(dvd_of_mem_normalized_factors hb) },
... | lemma | divisor_chain.element_of_chain_eq_pow_second_of_chain | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associated_iff_eq",
"associates",
"dvd_trans",
"fin.ext",
"finset.card_fin",
"finset.card_image_iff",
"finset.mem_image",
"irreducible.ne_zero",
"multiset.prod_replicate",
"multiset.replicate",
"nat.succ_le_iff",
"ne_zero_of_dvd_ne_zero",
"pow_injective_of_not_unit",
"pow_mul_pow_sub",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pow_second_of_chain_of_has_chain {q : associates M} {n : ℕ} (hn : n ≠ 0)
{c : fin (n + 1) → associates M} (h₁ : strict_mono c)
(h₂ : ∀ {r : associates M}, r ≤ q ↔ ∃ i, r = c i) (hq : q ≠ 0) : q = (c 1)^n | begin
classical,
obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (λ r, h₂) (dvd_refl q) hq,
convert hi',
refine (nat.lt_succ_iff.1 i.prop).antisymm' (nat.le_of_succ_le_succ _),
calc n + 1 = (finset.univ : finset (fin (n + 1))).card : (finset.card_fin _).symm
... = (finset.univ.image ... | lemma | divisor_chain.eq_pow_second_of_chain_of_has_chain | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"antisymm'",
"associated_iff_eq",
"associates",
"dvd_prime_pow",
"dvd_refl",
"fin.coe_coe_of_lt",
"finset",
"finset.card_fin",
"finset.card_image_le",
"finset.card_le_of_subset",
"finset.mem_univ",
"finset.univ",
"prime",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_pow_of_has_chain {q : associates M} {n : ℕ} (hn : n ≠ 0)
{c : fin (n + 1) → associates M} (h₁ : strict_mono c)
(h₂ : ∀ {r : associates M}, r ≤ q ↔ ∃ i, r = c i) (hq : q ≠ 0) : is_prime_pow q | ⟨c 1, n, irreducible_iff_prime.mp (second_of_chain_is_irreducible hn h₁ @h₂ hq),
zero_lt_iff.mpr hn, (eq_pow_second_of_chain_of_has_chain hn h₁ @h₂ hq).symm⟩ | lemma | divisor_chain.is_prime_pow_of_has_chain | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"is_prime_pow",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor_order_iso_map_one_eq_bot {m : associates M} {n : associates N}
(d : {l : associates M // l ≤ m} ≃o {l : associates N // l ≤ n}) :
(d ⟨1, one_dvd m⟩ : associates N) = 1 | begin
letI : order_bot {l : associates M // l ≤ m} := subtype.order_bot bot_le,
letI : order_bot {l : associates N // l ≤ n} := subtype.order_bot bot_le,
simp [←associates.bot_eq_one]
end | lemma | factor_order_iso_map_one_eq_bot | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"bot_le",
"one_dvd",
"order_bot",
"subtype.order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_factor_order_iso_map_eq_one_iff {m u : associates M} {n : associates N}
(hu' : u ≤ m) (d : set.Iic m ≃o set.Iic n) :
(d ⟨u, hu'⟩ : associates N) = 1 ↔ u = 1 | ⟨λ hu, by { rw (show u = ↑(d.symm ⟨↑(d ⟨u, hu'⟩), (d ⟨u, hu'⟩).prop⟩), by simp only
[subtype.coe_eta, order_iso.symm_apply_apply, subtype.coe_mk]),
convert factor_order_iso_map_one_eq_bot d.symm },
λ hu, by {simp_rw hu, convert factor_order_iso_map_one_eq_bot d } ⟩ | lemma | coe_factor_order_iso_map_eq_one_iff | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"factor_order_iso_map_one_eq_bot",
"order_iso.symm_apply_apply",
"set.Iic",
"subtype.coe_eta",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_image_of_prime_by_factor_order_iso_dvd [decidable_eq (associates M)] {m p : associates M}
{n : associates N} (hn : n ≠ 0) (hp : p ∈ normalized_factors m)
(d : set.Iic m ≃o set.Iic n) {s : ℕ} (hs' : p^s ≤ m) :
(d ⟨p, dvd_of_mem_normalized_factors hp⟩ : associates N)^s ≤ n | begin
by_cases hs : s = 0,
{ simp [hs], },
suffices : (d ⟨p, dvd_of_mem_normalized_factors hp⟩ : associates N)^s = ↑(d ⟨p^s, hs'⟩),
{ rw this,
apply subtype.prop (d ⟨p^s, hs'⟩) },
obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp),
set c₂ : fin (s + 1) → ass... | lemma | pow_image_of_prime_by_factor_order_iso_dvd | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"ne_zero_of_dvd_ne_zero",
"set.Iic",
"subtype.coe_eta",
"subtype.coe_le_coe",
"subtype.coe_lt_coe",
"subtype.coe_mk",
"subtype.mk_le_mk",
"subtype.mk_lt_mk",
"subtype.prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prime_of_factor_order_iso [decidable_eq (associates M)]
{m p : associates M} {n : associates N} (hn : n ≠ 0) (hp : p ∈ normalized_factors m)
(d : set.Iic m ≃o set.Iic n) : prime (d ⟨p, dvd_of_mem_normalized_factors hp⟩ : associates N) | begin
rw ← irreducible_iff_prime,
refine (associates.is_atom_iff $ ne_zero_of_dvd_ne_zero hn (d ⟨p, _⟩).prop).mp ⟨_, λ b hb, _⟩,
{ rw [ne.def, ← associates.is_unit_iff_eq_bot, associates.is_unit_iff_eq_one,
coe_factor_order_iso_map_eq_one_iff _ d],
rintro rfl,
exact (prime_of_normalized_factor 1 hp)... | lemma | map_prime_of_factor_order_iso | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"associates.is_atom_iff",
"associates.is_unit_iff_eq_bot",
"associates.is_unit_iff_eq_one",
"bot_le",
"coe_factor_order_iso_map_eq_one_iff",
"is_unit_one",
"ne_zero_of_dvd_ne_zero",
"order_bot",
"order_iso.map_bot",
"prime",
"prime.ne_zero",
"set.Iic",
"subtype.coe_lt_coe",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_normalized_factors_factor_order_iso_of_mem_normalized_factors
[decidable_eq (associates M)] [decidable_eq (associates N)] {m p : associates M}
{n : associates N} (hn : n ≠ 0) (hp : p ∈ normalized_factors m)
(d : set.Iic m ≃o set.Iic n) :
↑(d ⟨p, dvd_of_mem_normalized_factors hp⟩) ∈ normalized_factors n | begin
obtain ⟨q, hq, hq'⟩ := exists_mem_normalized_factors_of_dvd hn
(map_prime_of_factor_order_iso hn hp d).irreducible
(d ⟨p, dvd_of_mem_normalized_factors hp⟩).prop,
rw associated_iff_eq at hq',
rwa hq'
end | lemma | mem_normalized_factors_factor_order_iso_of_mem_normalized_factors | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associated_iff_eq",
"associates",
"irreducible",
"map_prime_of_factor_order_iso",
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_prime_le_multiplicity_image_by_factor_order_iso [decidable_eq (associates M)]
{m p : associates M} {n : associates N} (hp : p ∈ normalized_factors m)
(d : set.Iic m ≃o set.Iic n) :
multiplicity p m ≤ multiplicity ↑(d ⟨p, dvd_of_mem_normalized_factors hp⟩) n | begin
by_cases hn : n = 0,
{ simp [hn], },
by_cases hm : m = 0,
{ simpa [hm] using hp, },
rw [← part_enat.coe_get
(finite_iff_dom.1 $ finite_prime_left (prime_of_normalized_factor p hp) hm),
← pow_dvd_iff_le_multiplicity],
exact pow_image_of_prime_by_factor_order_iso_dvd hn hp d (pow_multiplicity_dv... | lemma | multiplicity_prime_le_multiplicity_image_by_factor_order_iso | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"multiplicity",
"part_enat.coe_get",
"pow_image_of_prime_by_factor_order_iso_dvd",
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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