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 |
|---|---|---|---|---|---|---|---|---|---|---|
degree_le_eq_span_X_pow {n : ℕ} :
degree_le R n = submodule.span R ↑((finset.range (n+1)).image (λ n, (X : R[X])^n)) | begin
apply le_antisymm,
{ intros p hp, replace hp := mem_degree_le.1 hp,
rw [← polynomial.sum_monomial_eq p, polynomial.sum],
refine submodule.sum_mem _ (λ k hk, _),
show monomial _ _ ∈ _,
have := with_bot.coe_le_coe.1 (finset.sup_le_iff.1 hp k hk),
rw [← C_mul_X_pow_eq_monomial, C_mul'],
r... | theorem | polynomial.degree_le_eq_span_X_pow | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finset.coe_image",
"finset.range",
"polynomial.sum",
"polynomial.sum_monomial_eq",
"set.image_subset_iff",
"submodule.smul_mem",
"submodule.span",
"submodule.span_le",
"submodule.subset_span",
"submodule.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_degree_lt {n : ℕ} {f : R[X]} :
f ∈ degree_lt R n ↔ degree f < n | by { simp_rw [degree_lt, submodule.mem_infi, linear_map.mem_ker, degree, finset.max_eq_sup_coe,
finset.sup_lt_iff (with_bot.bot_lt_coe n), mem_support_iff,
with_bot.coe_lt_coe, lt_iff_not_le, ne, not_imp_not], refl } | theorem | polynomial.mem_degree_lt | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finset.max_eq_sup_coe",
"finset.sup_lt_iff",
"linear_map.mem_ker",
"lt_iff_not_le",
"not_imp_not",
"submodule.mem_infi",
"with_bot.bot_lt_coe",
"with_bot.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_lt_mono {m n : ℕ} (H : m ≤ n) :
degree_lt R m ≤ degree_lt R n | λ f hf, mem_degree_lt.2 (lt_of_lt_of_le (mem_degree_lt.1 hf) $ with_bot.coe_le_coe.2 H) | theorem | polynomial.degree_lt_mono | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_lt_eq_span_X_pow {n : ℕ} :
degree_lt R n = submodule.span R ↑((finset.range n).image (λ n, X^n) : finset R[X]) | begin
apply le_antisymm,
{ intros p hp, replace hp := mem_degree_lt.1 hp,
rw [← polynomial.sum_monomial_eq p, polynomial.sum],
refine submodule.sum_mem _ (λ k hk, _),
show monomial _ _ ∈ _,
have := with_bot.coe_lt_coe.1 ((finset.sup_lt_iff $ with_bot.bot_lt_coe n).1 hp k hk),
rw [← C_mul_X_pow_e... | theorem | polynomial.degree_lt_eq_span_X_pow | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finset",
"finset.coe_image",
"finset.range",
"finset.sup_lt_iff",
"polynomial.sum",
"polynomial.sum_monomial_eq",
"set.image_subset_iff",
"submodule.smul_mem",
"submodule.span",
"submodule.span_le",
"submodule.subset_span",
"submodule.sum_mem",
"with_bot.bot_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_lt_equiv (R) [semiring R] (n : ℕ) : degree_lt R n ≃ₗ[R] (fin n → R) | { to_fun := λ p n, (↑p : R[X]).coeff n,
inv_fun := λ f, ⟨∑ i : fin n, monomial i (f i),
(degree_lt R n).sum_mem (λ i _, mem_degree_lt.mpr (lt_of_le_of_lt
(degree_monomial_le i (f i)) (with_bot.coe_lt_coe.mpr i.is_lt)))⟩,
map_add' := λ p q, by { ext, rw [submodule.coe_add, coeff_add], refl },
map_smul' :... | def | polynomial.degree_lt_equiv | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"fin.ext_iff",
"finset.mem_univ",
"inv_fun",
"linear_map.map_zero",
"semiring",
"submodule.coe_add",
"submodule.coe_mk",
"submodule.coe_smul",
"with_bot.coe_lt_coe"
] | The first `n` coefficients on `degree_lt n` form a linear equivalence with `fin n → R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_lt_equiv_eq_zero_iff_eq_zero {n : ℕ} {p : R[X]} (hp : p ∈ degree_lt R n) :
degree_lt_equiv _ _ ⟨p, hp⟩ = 0 ↔ p = 0 | by rw [linear_equiv.map_eq_zero_iff, submodule.mk_eq_zero] | theorem | polynomial.degree_lt_equiv_eq_zero_iff_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"linear_equiv.map_eq_zero_iff",
"submodule.mk_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_eq_sum_degree_lt_equiv {n : ℕ} {p : R[X]} (hp : p ∈ degree_lt R n) (x : R) :
p.eval x = ∑ i, degree_lt_equiv _ _ ⟨p, hp⟩ i * (x ^ (i : ℕ)) | begin
simp_rw [eval_eq_sum],
exact (sum_fin _ (by simp_rw [zero_mul, forall_const]) (mem_degree_lt.mp hp)).symm
end | theorem | polynomial.eval_eq_sum_degree_lt_equiv | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"forall_const",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frange (p : R[X]) : finset R | finset.image (λ n, p.coeff n) p.support | def | polynomial.frange | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finset",
"finset.image"
] | The finset of nonzero coefficients of a polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frange_zero : frange (0 : R[X]) = ∅ | rfl | lemma | polynomial.frange_zero | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_frange_iff {p : R[X]} {c : R} :
c ∈ p.frange ↔ ∃ n ∈ p.support, c = p.coeff n | by simp [frange, eq_comm] | lemma | polynomial.mem_frange_iff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frange_one : frange (1 : R[X]) ⊆ {1} | begin
simp [frange, finset.image_subset_iff],
simp only [← C_1, coeff_C],
assume n hn,
simp only [exists_prop, ite_eq_right_iff, not_forall] at hn,
simp [hn],
end | lemma | polynomial.frange_one | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"exists_prop",
"finset.image_subset_iff",
"ite_eq_right_iff",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mem_frange (p : R[X]) (n : ℕ) (h : p.coeff n ≠ 0) :
p.coeff n ∈ p.frange | begin
simp only [frange, exists_prop, mem_support_iff, finset.mem_image, ne.def],
exact ⟨n, h, rfl⟩,
end | lemma | polynomial.coeff_mem_frange | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"exists_prop",
"finset.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_X_comp_X_add_one_eq_sum (n : ℕ) :
(∑ i in range n, (X : R[X]) ^ i).comp (X + 1) =
(finset.range n).sum (λ (i : ℕ), (n.choose (i + 1) : R[X]) * X ^ i) | begin
ext i,
transitivity (n.choose (i + 1) : R), swap,
{ simp only [finset_sum_coeff, ← C_eq_nat_cast, coeff_C_mul_X_pow],
rw [finset.sum_eq_single i, if_pos rfl],
{ simp only [@eq_comm _ i, if_false, eq_self_iff_true, implies_true_iff] {contextual := tt}, },
{ simp only [nat.lt_add_one_iff, nat.choo... | lemma | polynomial.geom_sum_X_comp_X_add_one_eq_sum | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finset.mem_range",
"finset.range",
"geom_sum_succ'",
"geom_sum_zero",
"ih",
"nat.cast_add",
"nat.cast_zero",
"nat.choose_eq_zero_of_lt",
"nat.choose_succ_succ",
"nat.choose_zero_succ",
"nat.lt_add_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic.geom_sum {P : R[X]}
(hP : P.monic) (hdeg : 0 < P.nat_degree) {n : ℕ} (hn : n ≠ 0) : (∑ i in range n, P ^ i).monic | begin
nontriviality R,
cases n, { exact (hn rfl).elim },
rw [geom_sum_succ'],
refine (hP.pow _).add_of_left _,
refine lt_of_le_of_lt (degree_sum_le _ _) _,
rw [finset.sup_lt_iff],
{ simp only [finset.mem_range, degree_eq_nat_degree (hP.pow _).ne_zero,
with_bot.coe_lt_coe, hP.nat_degree_pow],
int... | lemma | polynomial.monic.geom_sum | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"bot_lt_iff_ne_bot",
"finset.mem_range",
"finset.sup_lt_iff",
"geom_sum_succ'",
"ne_zero",
"with_bot.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic.geom_sum' {P : R[X]}
(hP : P.monic) (hdeg : 0 < P.degree) {n : ℕ} (hn : n ≠ 0) : (∑ i in range n, P ^ i).monic | hP.geom_sum (nat_degree_pos_iff_degree_pos.2 hdeg) hn | lemma | polynomial.monic.geom_sum' | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_geom_sum_X {n : ℕ} (hn : n ≠ 0) :
(∑ i in range n, (X : R[X]) ^ i).monic | begin
nontriviality R,
apply monic_X.geom_sum _ hn,
simpa only [nat_degree_X] using zero_lt_one
end | lemma | polynomial.monic_geom_sum_X | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restriction (p : R[X]) : polynomial (subring.closure (↑p.frange : set R)) | ∑ i in p.support, monomial i (⟨p.coeff i,
if H : p.coeff i = 0 then H.symm ▸ (subring.closure _).zero_mem
else subring.subset_closure (p.coeff_mem_frange _ H)⟩ : (subring.closure (↑p.frange : set R))) | def | polynomial.restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"polynomial",
"subring.closure",
"subring.subset_closure"
] | Given a polynomial, return the polynomial whose coefficients are in
the ring closure of the original coefficients. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_restriction {p : R[X]} {n : ℕ} :
↑(coeff (restriction p) n) = coeff p n | begin
simp only [restriction, coeff_monomial, finset_sum_coeff, mem_support_iff, finset.sum_ite_eq',
ne.def, ite_not],
split_ifs,
{ rw h, refl },
{ refl }
end | theorem | polynomial.coeff_restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ite_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_restriction' {p : R[X]} {n : ℕ} :
(coeff (restriction p) n).1 = coeff p n | coeff_restriction | theorem | polynomial.coeff_restriction' | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_restriction (p : R[X]) :
support (restriction p) = support p | begin
ext i,
simp only [mem_support_iff, not_iff_not, ne.def],
conv_rhs { rw [← coeff_restriction] },
exact ⟨λ H, by { rw H, refl }, λ H, subtype.coe_injective H⟩
end | lemma | polynomial.support_restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"not_iff_not",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_restriction {R : Type u} [comm_ring R]
(p : R[X]) : p.restriction.map (algebra_map _ _) = p | ext $ λ n, by rw [coeff_map, algebra.algebra_map_of_subring_apply, coeff_restriction] | theorem | polynomial.map_restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"algebra.algebra_map_of_subring_apply",
"algebra_map",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_restriction {p : R[X]} : (restriction p).degree = p.degree | by simp [degree] | theorem | polynomial.degree_restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_restriction {p : R[X]} :
(restriction p).nat_degree = p.nat_degree | by simp [nat_degree] | theorem | polynomial.nat_degree_restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_restriction {p : R[X]} : monic (restriction p) ↔ monic p | begin
simp only [monic, leading_coeff, nat_degree_restriction],
rw [←@coeff_restriction _ _ p],
exact ⟨λ H, by { rw H, refl }, λ H, subtype.coe_injective H⟩
end | theorem | polynomial.monic_restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restriction_zero : restriction (0 : R[X]) = 0 | by simp only [restriction, finset.sum_empty, support_zero] | theorem | polynomial.restriction_zero | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restriction_one : restriction (1 : R[X]) = 1 | ext $ λ i, subtype.eq $ by rw [coeff_restriction', coeff_one, coeff_one]; split_ifs; refl | theorem | polynomial.restriction_one | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval₂_restriction {p : R[X]} :
eval₂ f x p =
eval₂ (f.comp (subring.subtype (subring.closure (p.frange : set R)))) x p.restriction | begin
simp only [eval₂_eq_sum, sum, support_restriction, ←@coeff_restriction _ _ p],
refl,
end | theorem | polynomial.eval₂_restriction | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"subring.closure",
"subring.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subring (hp : (↑p.frange : set R) ⊆ T) : T[X] | ∑ i in p.support, monomial i (⟨p.coeff i,
if H : p.coeff i = 0 then H.symm ▸ T.zero_mem
else hp (p.coeff_mem_frange _ H)⟩ : T) | def | polynomial.to_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | Given a polynomial `p` and a subring `T` that contains the coefficients of `p`,
return the corresponding polynomial whose coefficients are in `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_to_subring {n : ℕ} : ↑(coeff (to_subring p T hp) n) = coeff p n | begin
simp only [to_subring, coeff_monomial, finset_sum_coeff, mem_support_iff, finset.sum_ite_eq',
ne.def, ite_not],
split_ifs,
{ rw h, refl },
{ refl }
end | theorem | polynomial.coeff_to_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ite_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_to_subring' {n : ℕ} : (coeff (to_subring p T hp) n).1 = coeff p n | coeff_to_subring _ _ hp | theorem | polynomial.coeff_to_subring' | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_to_subring :
support (to_subring p T hp) = support p | begin
ext i,
simp only [mem_support_iff, not_iff_not, ne.def],
conv_rhs { rw [← coeff_to_subring p T hp] },
exact ⟨λ H, by { rw H, refl }, λ H, subtype.coe_injective H⟩
end | lemma | polynomial.support_to_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"not_iff_not",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_to_subring : (to_subring p T hp).degree = p.degree | by simp [degree] | theorem | polynomial.degree_to_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_to_subring : (to_subring p T hp).nat_degree = p.nat_degree | by simp [nat_degree] | theorem | polynomial.nat_degree_to_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_to_subring : monic (to_subring p T hp) ↔ monic p | begin
simp_rw [monic, leading_coeff, nat_degree_to_subring, ← coeff_to_subring p T hp],
exact ⟨λ H, by { rw H, refl }, λ H, subtype.coe_injective H⟩
end | theorem | polynomial.monic_to_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subring_zero : to_subring (0 : R[X]) T (by simp [frange_zero]) = 0 | by { ext i, simp } | theorem | polynomial.to_subring_zero | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subring_one : to_subring (1 : R[X]) T
(set.subset.trans frange_one $finset.singleton_subset_set_iff.2 T.one_mem) = 1 | ext $ λ i, subtype.eq $ by rw [coeff_to_subring', coeff_one, coeff_one]; split_ifs; refl | theorem | polynomial.to_subring_one | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_to_subring : (p.to_subring T hp).map (subring.subtype T) = p | by { ext n, simp [coeff_map] } | theorem | polynomial.map_to_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"subring.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subring (p : T[X]) : R[X] | ∑ i in p.support, monomial i (p.coeff i : R) | def | polynomial.of_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | Given a polynomial whose coefficients are in some subring, return
the corresponding polynomial whose coefficients are in the ambient ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_of_subring (p : T[X]) (n : ℕ) :
coeff (of_subring T p) n = (coeff p n : T) | begin
simp only [of_subring, coeff_monomial, finset_sum_coeff, mem_support_iff, finset.sum_ite_eq',
ite_eq_right_iff, ne.def, ite_not, not_not, ite_eq_left_iff],
assume h,
rw h,
refl
end | lemma | polynomial.coeff_of_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ite_eq_left_iff",
"ite_eq_right_iff",
"ite_not",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frange_of_subring {p : T[X]} :
(↑(p.of_subring T).frange : set R) ⊆ T | begin
assume i hi,
simp only [frange, set.mem_image, mem_support_iff, ne.def, finset.mem_coe, finset.coe_image]
at hi,
rcases hi with ⟨n, hn, h'n⟩,
rw [← h'n, coeff_of_subring],
exact subtype.mem (coeff p n : T)
end | theorem | polynomial.frange_of_subring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finset.coe_image",
"finset.mem_coe",
"set.mem_image",
"subtype.mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ker_mod_by_monic (hq : q.monic) {p : R[X]} :
p ∈ (mod_by_monic_hom q).ker ↔ q ∣ p | linear_map.mem_ker.trans (dvd_iff_mod_by_monic_eq_zero hq) | lemma | polynomial.mem_ker_mod_by_monic | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_mod_by_monic_hom (hq : q.monic) :
(polynomial.mod_by_monic_hom q).ker = (ideal.span {q}).restrict_scalars R | submodule.ext (λ f, (mem_ker_mod_by_monic hq).trans ideal.mem_span_singleton.symm) | lemma | polynomial.ker_mod_by_monic_hom | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal.span",
"polynomial.mod_by_monic_hom",
"restrict_scalars",
"submodule.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_polynomial (I : ideal R[X]) : submodule R R[X] | { carrier := I.carrier,
zero_mem' := I.zero_mem,
add_mem' := λ _ _, I.add_mem,
smul_mem' := λ c x H, by { rw [← C_mul'], exact I.mul_mem_left _ H } } | def | ideal.of_polynomial | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"submodule"
] | Transport an ideal of `R[X]` to an `R`-submodule of `R[X]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_of_polynomial (x) : x ∈ I.of_polynomial ↔ x ∈ I | iff.rfl | theorem | ideal.mem_of_polynomial | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_le (n : with_bot ℕ) : submodule R R[X] | degree_le R n ⊓ I.of_polynomial | def | ideal.degree_le | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"submodule",
"with_bot"
] | Given an ideal `I` of `R[X]`, make the `R`-submodule of `I`
consisting of polynomials of degree ≤ `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
leading_coeff_nth (n : ℕ) : ideal R | (I.degree_le n).map $ lcoeff R n | def | ideal.leading_coeff_nth | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal"
] | Given an ideal `I` of `R[X]`, make the ideal in `R` of
leading coefficients of polynomials in `I` with degree ≤ `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
leading_coeff : ideal R | ⨆ n : ℕ, I.leading_coeff_nth n | def | ideal.leading_coeff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal"
] | Given an ideal `I` in `R[X]`, make the ideal in `R` of the
leading coefficients in `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_mem_ideal_of_coeff_mem_ideal (I : ideal R[X]) (p : R[X])
(hp : ∀ (n : ℕ), (p.coeff n) ∈ I.comap (C : R →+* R[X])) : p ∈ I | sum_C_mul_X_pow_eq p ▸ submodule.sum_mem I (λ n hn, I.mul_mem_right _ (hp n)) | lemma | ideal.polynomial_mem_ideal_of_coeff_mem_ideal | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"submodule.sum_mem"
] | If every coefficient of a polynomial is in an ideal `I`, then so is the polynomial itself | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map_C_iff {I : ideal R} {f : R[X]} :
f ∈ (ideal.map (C : R →+* R[X]) I : ideal R[X]) ↔ ∀ n : ℕ, f.coeff n ∈ I | begin
split,
{ intros hf,
apply submodule.span_induction hf,
{ intros f hf n,
cases (set.mem_image _ _ _).mp hf with x hx,
rw [← hx.right, coeff_C],
by_cases (n = 0),
{ simpa [h] using hx.left },
{ simp [h] } },
{ simp },
{ exact λ f g hf hg n, by simp [I.add_mem (hf n)... | theorem | ideal.mem_map_C_iff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"ideal.map",
"mul_comm",
"set.mem_image",
"smul_eq_mul",
"submodule.span_induction"
] | The push-forward of an ideal `I` of `R` to `R[X]` via inclusion
is exactly the set of polynomials whose coefficients are in `I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.polynomial.ker_map_ring_hom (f : R →+* S) :
(polynomial.map_ring_hom f).ker = f.ker.map (C : R →+* R[X]) | begin
ext,
rw [mem_map_C_iff, ring_hom.mem_ker, polynomial.ext_iff],
simp_rw [coe_map_ring_hom, coeff_map, coeff_zero, ring_hom.mem_ker],
end | lemma | polynomial.ker_map_ring_hom | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"polynomial.ext_iff",
"polynomial.map_ring_hom",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_leading_coeff_nth (n : ℕ) (x) :
x ∈ I.leading_coeff_nth n ↔ ∃ p ∈ I, degree p ≤ n ∧ p.leading_coeff = x | begin
simp only [leading_coeff_nth, degree_le, submodule.mem_map, lcoeff_apply, submodule.mem_inf,
mem_degree_le],
split,
{ rintro ⟨p, ⟨hpdeg, hpI⟩, rfl⟩,
cases lt_or_eq_of_le hpdeg with hpdeg hpdeg,
{ refine ⟨0, I.zero_mem, bot_le, _⟩,
rw [leading_coeff_zero, eq_comm],
exact coeff_eq_zero... | theorem | ideal.mem_leading_coeff_nth | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"add_tsub_cancel_of_le",
"bot_le",
"le_rfl",
"polynomial.leading_coeff",
"submodule.mem_inf",
"submodule.mem_map",
"with_bot.coe_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_leading_coeff_nth_zero (x) :
x ∈ I.leading_coeff_nth 0 ↔ C x ∈ I | (mem_leading_coeff_nth _ _ _).trans
⟨λ ⟨p, hpI, hpdeg, hpx⟩, by rwa [← hpx, polynomial.leading_coeff,
nat.eq_zero_of_le_zero (nat_degree_le_of_degree_le hpdeg),
← eq_C_of_degree_le_zero hpdeg],
λ hx, ⟨C x, hx, degree_C_le, leading_coeff_C x⟩⟩ | theorem | ideal.mem_leading_coeff_nth_zero | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"polynomial.leading_coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_nth_mono {m n : ℕ} (H : m ≤ n) :
I.leading_coeff_nth m ≤ I.leading_coeff_nth n | begin
intros r hr,
simp only [set_like.mem_coe, mem_leading_coeff_nth] at hr ⊢,
rcases hr with ⟨p, hpI, hpdeg, rfl⟩,
refine ⟨p * X ^ (n - m), I.mul_mem_right _ hpI, _, leading_coeff_mul_X_pow⟩,
refine le_trans (degree_mul_le _ _) _,
refine le_trans (add_le_add hpdeg (degree_X_pow_le _)) _,
rw [← with_bot.... | theorem | ideal.leading_coeff_nth_mono | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"add_tsub_cancel_of_le",
"le_rfl",
"set_like.mem_coe",
"with_bot.coe_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_leading_coeff (x) :
x ∈ I.leading_coeff ↔ ∃ p ∈ I, polynomial.leading_coeff p = x | begin
rw [leading_coeff, submodule.mem_supr_of_directed],
simp only [mem_leading_coeff_nth],
{ split, { rintro ⟨i, p, hpI, hpdeg, rfl⟩, exact ⟨p, hpI, rfl⟩ },
rintro ⟨p, hpI, rfl⟩, exact ⟨nat_degree p, p, hpI, degree_le_nat_degree, rfl⟩ },
intros i j, exact ⟨i + j, I.leading_coeff_nth_mono (nat.le_add_right... | theorem | ideal.mem_leading_coeff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"polynomial.leading_coeff",
"submodule.mem_supr_of_directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.polynomial.coeff_prod_mem_ideal_pow_tsub {ι : Type*} (s : finset ι) (f : ι → R[X])
(I : ideal R) (n : ι → ℕ) (h : ∀ (i ∈ s) k, (f i).coeff k ∈ I ^ (n i - k)) (k : ℕ) :
(s.prod f).coeff k ∈ I ^ (s.sum n - k) | begin
classical,
induction s using finset.induction with a s ha hs generalizing k,
{ rw [sum_empty, prod_empty, coeff_one, zero_tsub, pow_zero, ideal.one_eq_top],
exact submodule.mem_top },
{ rw [sum_insert ha, prod_insert ha, coeff_mul],
apply sum_mem,
rintro ⟨i, j⟩ e,
obtain rfl : i + j = k :=... | lemma | polynomial.coeff_prod_mem_ideal_pow_tsub | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"add_tsub_add_le_tsub_add_tsub",
"finset",
"finset.induction",
"ideal",
"ideal.mul_mem_mul",
"ideal.one_eq_top",
"ideal.pow_le_pow",
"pow_add",
"pow_zero",
"submodule.mem_top",
"zero_tsub"
] | If `I` is an ideal, and `pᵢ` is a finite family of polynomials each satisfying
`∀ k, (pᵢ)ₖ ∈ Iⁿⁱ⁻ᵏ` for some `nᵢ`, then `p = ∏ pᵢ` also satisfies `∀ k, pₖ ∈ Iⁿ⁻ᵏ` with `n = ∑ nᵢ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_not_is_field : ¬ is_field R[X] | begin
nontriviality R,
intro hR,
obtain ⟨p, hp⟩ := hR.mul_inv_cancel X_ne_zero,
have hp0 : p ≠ 0,
{ rintro rfl,
rw [mul_zero] at hp,
exact zero_ne_one hp },
have := degree_lt_degree_mul_X hp0,
rw [←X_mul, congr_arg degree hp, degree_one, nat.with_bot.lt_zero_iff, degree_eq_bot] at this,
exact hp... | lemma | ideal.polynomial_not_is_field | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"is_field",
"mul_zero",
"nat.with_bot.lt_zero_iff",
"zero_ne_one"
] | `R[X]` is never a field for any ring `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_constant_mem_of_maximal (hR : is_field R)
(I : ideal R[X]) [hI : I.is_maximal] (x : R) (hx : C x ∈ I) : x = 0 | begin
refine classical.by_contradiction (λ hx0, hI.ne_top ((eq_top_iff_one I).2 _)),
obtain ⟨y, hy⟩ := hR.mul_inv_cancel hx0,
convert I.mul_mem_left (C y) hx,
rw [← C.map_mul, hR.mul_comm y x, hy, ring_hom.map_one],
end | lemma | ideal.eq_zero_of_constant_mem_of_maximal | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"is_field",
"ring_hom.map_one"
] | The only constant in a maximal ideal over a field is `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime_map_C_iff_is_prime (P : ideal R) :
is_prime (map (C : R →+* R[X]) P : ideal R[X]) ↔ is_prime P | begin
-- Porting note: the following proof avoids quotient rings
-- It can be golfed substantially by using something like
-- `(quotient.is_domain_iff_prime (map C P : ideal R[X]))`
split,
{ intro H,
have := @comap_is_prime R R[X] (R →+* R[X]) _ _ _ C (map C P) H,
convert this using 1,
ext x,
... | lemma | ideal.is_prime_map_C_iff_is_prime | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finset.insert_erase",
"finset.mem_erase",
"finset.nat.mem_antidiagonal",
"finset.not_mem_erase",
"ideal",
"mul_comm",
"not_and_distrib",
"not_forall",
"not_or_distrib",
"or_iff_not_imp_left",
"prod.mk.inj_iff",
"submodule.zero_mem"
] | If `P` is a prime ideal of `R`, then `P.R[x]` is a prime ideal of `R[x]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime_map_C_of_is_prime {P : ideal R} (H : is_prime P) :
is_prime (map (C : R →+* R[X]) P : ideal R[X]) | (is_prime_map_C_iff_is_prime P).mpr H | lemma | ideal.is_prime_map_C_of_is_prime | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal"
] | If `P` is a prime ideal of `R`, then `P.R[x]` is a prime ideal of `R[x]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fg_degree_le [is_noetherian_ring R] (I : ideal R[X]) (n : ℕ) :
submodule.fg (I.degree_le n) | is_noetherian_submodule_left.1 (is_noetherian_of_fg_of_noetherian _
⟨_, degree_le_eq_span_X_pow.symm⟩) _ | theorem | ideal.is_fg_degree_le | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"is_noetherian_of_fg_of_noetherian",
"is_noetherian_ring",
"submodule.fg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_C_iff : prime (C r) ↔ prime r | ⟨ comap_prime C (eval_ring_hom (0 : R)) (λ r, eval_C),
λ hr, by { have := hr.1,
rw ← ideal.span_singleton_prime at hr ⊢,
{ convert ideal.is_prime_map_C_of_is_prime hr using 1,
rw [ideal.map_span, set.image_singleton] },
exacts [λ h, this (C_eq_zero.1 h), this] } ⟩ | lemma | polynomial.prime_C_iff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"comap_prime",
"ideal.is_prime_map_C_of_is_prime",
"ideal.map_span",
"ideal.span_singleton_prime",
"prime",
"set.image_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_C_iff_of_fintype [fintype σ] : prime (C r : mv_polynomial σ R) ↔ prime r | begin
rw (rename_equiv R (fintype.equiv_fin σ)).to_mul_equiv.prime_iff,
convert_to prime (C r) ↔ _, { congr, apply rename_C },
{ symmetry, induction fintype.card σ with d hd,
{ exact (is_empty_alg_equiv R (fin 0)).to_mul_equiv.symm.prime_iff },
{ rw [hd, ← polynomial.prime_C_iff],
convert (fin_succ_... | lemma | mv_polynomial.prime_C_iff_of_fintype | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"fin_succ_equiv",
"fintype",
"fintype.card",
"fintype.equiv_fin",
"mv_polynomial",
"polynomial.prime_C_iff",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_C_iff : prime (C r : mv_polynomial σ R) ↔ prime r | ⟨ comap_prime C constant_coeff (constant_coeff_C _),
λ hr, ⟨ λ h, hr.1 $ by { rw [← C_inj, h], simp },
λ h, hr.2.1 $ by { rw ← constant_coeff_C _ r, exact h.map _ },
λ a b hd, begin
obtain ⟨s,a',b',rfl,rfl⟩ := exists_finset_rename₂ a b,
rw ← algebra_map_eq at hd, have : algebra_map R _ r ∣ a' * b'... | lemma | mv_polynomial.prime_C_iff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"algebra_map",
"comap_prime",
"mv_polynomial",
"prime",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_rename_iff (s : set σ) {p : mv_polynomial s R} :
prime (rename (coe : s → σ) p) ↔ prime p | begin
classical, symmetry, let eqv := (sum_alg_equiv R _ _).symm.trans
(rename_equiv R $ (equiv.sum_comm ↥sᶜ s).trans $ equiv.set.sum_compl s),
rw [← prime_C_iff ↥sᶜ, eqv.to_mul_equiv.prime_iff], convert iff.rfl,
suffices : (rename coe).to_ring_hom = eqv.to_alg_hom.to_ring_hom.comp C,
{ apply ring_hom.congr... | lemma | mv_polynomial.prime_rename_iff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"equiv.set.sum_compl",
"equiv.sum_comm",
"mv_polynomial",
"prime",
"ring_hom.congr_fun",
"ring_hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial.is_noetherian_ring [is_noetherian_ring R] :
is_noetherian_ring R[X] | is_noetherian_ring_iff.2 ⟨assume I : ideal R[X],
let M := well_founded.min (is_noetherian_iff_well_founded.1 (by apply_instance))
(set.range I.leading_coeff_nth) ⟨_, ⟨0, rfl⟩⟩ in
have hm : M ∈ set.range I.leading_coeff_nth := well_founded.min_mem _ _ _,
let ⟨N, HN⟩ := hm, ⟨s, hs⟩ := I.is_fg_degree_le N in
have hm2 : ... | theorem | polynomial.is_noetherian_ring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"add_tsub_cancel_of_le",
"ideal",
"ideal.add_mem",
"ideal.mul_mem_left",
"ideal.span",
"ideal.subset_span",
"ideal.zero_mem",
"ih",
"is_noetherian_ring",
"mul_one",
"mul_zero",
"nontrivial",
"polynomial.X",
"polynomial.degree_X_pow",
"polynomial.degree_eq_nat_degree",
"polynomial.degre... | Hilbert basis theorem: a polynomial ring over a noetherian ring is a noetherian ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_irreducible_of_degree_pos
{R : Type u} [comm_ring R] [is_domain R] [wf_dvd_monoid R]
{f : R[X]} (hf : 0 < f.degree) : ∃ g, irreducible g ∧ g ∣ f | wf_dvd_monoid.exists_irreducible_factor
(λ huf, ne_of_gt hf $ degree_eq_zero_of_is_unit huf)
(λ hf0, not_lt_of_lt hf $ hf0.symm ▸ (@degree_zero R _).symm ▸ with_bot.bot_lt_coe _) | theorem | polynomial.exists_irreducible_of_degree_pos | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"irreducible",
"is_domain",
"wf_dvd_monoid",
"wf_dvd_monoid.exists_irreducible_factor",
"with_bot.bot_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_irreducible_of_nat_degree_pos
{R : Type u} [comm_ring R] [is_domain R] [wf_dvd_monoid R]
{f : R[X]} (hf : 0 < f.nat_degree) : ∃ g, irreducible g ∧ g ∣ f | exists_irreducible_of_degree_pos $ by { contrapose! hf, exact nat_degree_le_of_degree_le hf } | theorem | polynomial.exists_irreducible_of_nat_degree_pos | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"irreducible",
"is_domain",
"wf_dvd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_irreducible_of_nat_degree_ne_zero
{R : Type u} [comm_ring R] [is_domain R] [wf_dvd_monoid R]
{f : R[X]} (hf : f.nat_degree ≠ 0) : ∃ g, irreducible g ∧ g ∣ f | exists_irreducible_of_nat_degree_pos $ nat.pos_of_ne_zero hf | theorem | polynomial.exists_irreducible_of_nat_degree_ne_zero | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"irreducible",
"is_domain",
"wf_dvd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent_powers_iff_aeval
(f : M →ₗ[R] M) (v : M) :
linear_independent R (λ n : ℕ, (f ^ n) v)
↔ ∀ (p : R[X]), aeval f p v = 0 → p = 0 | begin
rw linear_independent_iff,
simp only [finsupp.total_apply, aeval_endomorphism, forall_iff_forall_finsupp, sum, support,
coeff, of_finsupp_eq_zero],
exact iff.rfl,
end | lemma | polynomial.linear_independent_powers_iff_aeval | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finsupp.total_apply",
"linear_independent",
"linear_independent_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_ker_aeval_of_coprime
(f : M →ₗ[R] M) {p q : R[X]} (hpq : is_coprime p q) :
disjoint (aeval f p).ker (aeval f q).ker | begin
rw disjoint_iff_inf_le,
intros v hv,
rcases hpq with ⟨p', q', hpq'⟩,
simpa [linear_map.mem_ker.1 (submodule.mem_inf.1 hv).1,
linear_map.mem_ker.1 (submodule.mem_inf.1 hv).2]
using congr_arg (λ p : R[X], aeval f p v) hpq'.symm,
end | lemma | polynomial.disjoint_ker_aeval_of_coprime | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"disjoint",
"disjoint_iff_inf_le",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_aeval_range_eq_top_of_coprime
(f : M →ₗ[R] M) {p q : R[X]} (hpq : is_coprime p q) :
(aeval f p).range ⊔ (aeval f q).range = ⊤ | begin
rw eq_top_iff,
intros v hv,
rw submodule.mem_sup,
rcases hpq with ⟨p', q', hpq'⟩,
use aeval f (p * p') v,
use linear_map.mem_range.2 ⟨aeval f p' v, by simp only [linear_map.mul_apply, aeval_mul]⟩,
use aeval f (q * q') v,
use linear_map.mem_range.2 ⟨aeval f q' v, by simp only [linear_map.mul_apply,... | lemma | polynomial.sup_aeval_range_eq_top_of_coprime | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"eq_top_iff",
"is_coprime",
"linear_map.mul_apply",
"mul_comm",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_ker_aeval_le_ker_aeval_mul {f : M →ₗ[R] M} {p q : R[X]} :
(aeval f p).ker ⊔ (aeval f q).ker ≤ (aeval f (p * q)).ker | begin
intros v hv,
rcases submodule.mem_sup.1 hv with ⟨x, hx, y, hy, hxy⟩,
have h_eval_x : aeval f (p * q) x = 0,
{ rw [mul_comm, aeval_mul, linear_map.mul_apply, linear_map.mem_ker.1 hx, linear_map.map_zero] },
have h_eval_y : aeval f (p * q) y = 0,
{ rw [aeval_mul, linear_map.mul_apply, linear_map.mem_ker... | lemma | polynomial.sup_ker_aeval_le_ker_aeval_mul | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"linear_map.map_add",
"linear_map.map_zero",
"linear_map.mem_ker",
"linear_map.mul_apply",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_ker_aeval_eq_ker_aeval_mul_of_coprime
(f : M →ₗ[R] M) {p q : R[X]} (hpq : is_coprime p q) :
(aeval f p).ker ⊔ (aeval f q).ker = (aeval f (p * q)).ker | begin
apply le_antisymm sup_ker_aeval_le_ker_aeval_mul,
intros v hv,
rw submodule.mem_sup,
rcases hpq with ⟨p', q', hpq'⟩,
have h_eval₂_qpp' := calc
aeval f (q * (p * p')) v = aeval f (p' * (p * q)) v :
by rw [mul_comm, mul_assoc, mul_comm, mul_assoc, mul_comm q p]
... = 0 :
by rw [aeval_m... | lemma | polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"is_coprime",
"linear_map.map_zero",
"linear_map.mul_apply",
"map_mul",
"mul_assoc",
"mul_comm",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring_fin_0 [is_noetherian_ring R] :
is_noetherian_ring (mv_polynomial (fin 0) R) | is_noetherian_ring_of_ring_equiv R
((mv_polynomial.is_empty_ring_equiv R pempty).symm.trans
(rename_equiv R fin_zero_equiv'.symm).to_ring_equiv) | lemma | mv_polynomial.is_noetherian_ring_fin_0 | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"is_noetherian_ring",
"is_noetherian_ring_of_ring_equiv",
"mv_polynomial",
"mv_polynomial.is_empty_ring_equiv",
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring_fin [is_noetherian_ring R] :
∀ {n : ℕ}, is_noetherian_ring (mv_polynomial (fin n) R) | | 0 := is_noetherian_ring_fin_0
| (n+1) :=
@is_noetherian_ring_of_ring_equiv (polynomial (mv_polynomial (fin n) R)) _ _ _
(mv_polynomial.fin_succ_equiv _ n).to_ring_equiv.symm
(@polynomial.is_noetherian_ring (mv_polynomial (fin n) R) _ (is_noetherian_ring_fin)) | theorem | mv_polynomial.is_noetherian_ring_fin | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"is_noetherian_ring",
"is_noetherian_ring_of_ring_equiv",
"mv_polynomial",
"mv_polynomial.fin_succ_equiv",
"polynomial",
"polynomial.is_noetherian_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring [finite σ] [is_noetherian_ring R] :
is_noetherian_ring (mv_polynomial σ R) | by casesI nonempty_fintype σ; exact
@is_noetherian_ring_of_ring_equiv (mv_polynomial (fin (fintype.card σ)) R) _ _ _
(rename_equiv R (fintype.equiv_fin σ).symm).to_ring_equiv is_noetherian_ring_fin | instance | mv_polynomial.is_noetherian_ring | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"finite",
"fintype.card",
"fintype.equiv_fin",
"is_noetherian_ring",
"is_noetherian_ring_of_ring_equiv",
"mv_polynomial",
"nonempty_fintype"
] | The multivariate polynomial ring in finitely many variables over a noetherian ring
is itself a noetherian ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
no_zero_divisors_fin (R : Type u) [comm_semiring R] [no_zero_divisors R] :
∀ (n : ℕ), no_zero_divisors (mv_polynomial (fin n) R) | | 0 := (mv_polynomial.is_empty_alg_equiv R _).injective.no_zero_divisors _ (map_zero _) (map_mul _)
| (n+1) := begin
haveI := no_zero_divisors_fin n,
exact (mv_polynomial.fin_succ_equiv R n).injective.no_zero_divisors _ (map_zero _) (map_mul _)
end | lemma | mv_polynomial.no_zero_divisors_fin | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"comm_semiring",
"map_mul",
"mv_polynomial",
"mv_polynomial.fin_succ_equiv",
"mv_polynomial.is_empty_alg_equiv",
"no_zero_divisors"
] | Auxiliary lemma:
Multivariate polynomials over an integral domain
with variables indexed by `fin n` form an integral domain.
This fact is proven inductively,
and then used to prove the general case without any finiteness hypotheses.
See `mv_polynomial.no_zero_divisors` for the general case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
no_zero_divisors_of_finite (R : Type u) (σ : Type v) [comm_semiring R] [finite σ]
[no_zero_divisors R] : no_zero_divisors (mv_polynomial σ R) | begin
casesI nonempty_fintype σ,
haveI := no_zero_divisors_fin R (fintype.card σ),
exact (rename_equiv R (fintype.equiv_fin σ)).injective.no_zero_divisors _ (map_zero _) (map_mul _)
end | lemma | mv_polynomial.no_zero_divisors_of_finite | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"comm_semiring",
"finite",
"fintype.card",
"fintype.equiv_fin",
"map_mul",
"mv_polynomial",
"no_zero_divisors",
"nonempty_fintype"
] | Auxiliary definition:
Multivariate polynomials in finitely many variables over an integral domain form an integral domain.
This fact is proven by transport of structure from the `mv_polynomial.no_zero_divisors_fin`,
and then used to prove the general case without finiteness hypotheses.
See `mv_polynomial.no_zero_diviso... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mv_polynomial_eq_eval₂ {S : Type*} [comm_ring S] [finite σ]
(ϕ : mv_polynomial σ R →+* S) (p : mv_polynomial σ R) :
ϕ p = mv_polynomial.eval₂ (ϕ.comp mv_polynomial.C) (λ s, ϕ (mv_polynomial.X s)) p | begin
casesI nonempty_fintype σ,
refine trans (congr_arg ϕ (mv_polynomial.as_sum p)) _,
rw [mv_polynomial.eval₂_eq', ϕ.map_sum],
congr,
ext,
simp only [monomial_eq, ϕ.map_pow, ϕ.map_prod, ϕ.comp_apply, ϕ.map_mul, finsupp.prod_pow],
end | lemma | mv_polynomial.map_mv_polynomial_eq_eval₂ | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"finite",
"finsupp.prod_pow",
"mv_polynomial",
"mv_polynomial.C",
"mv_polynomial.X",
"mv_polynomial.as_sum",
"mv_polynomial.eval₂",
"mv_polynomial.eval₂_eq'",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_of_coeff_mem_ideal (I : ideal (mv_polynomial σ R)) (p : mv_polynomial σ R)
(hcoe : ∀ (m : σ →₀ ℕ), p.coeff m ∈ I.comap (C : R →+* mv_polynomial σ R)) : p ∈ I | begin
rw as_sum p,
suffices : ∀ m ∈ p.support, monomial m (mv_polynomial.coeff m p) ∈ I,
{ exact submodule.sum_mem I this },
intros m hm,
rw [← mul_one (coeff m p), ← C_mul_monomial],
suffices : C (coeff m p) ∈ I,
{ exact I.mul_mem_right (monomial m 1) this },
simpa [ideal.mem_comap] using hcoe m
end | lemma | mv_polynomial.mem_ideal_of_coeff_mem_ideal | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"ideal.mem_comap",
"mul_one",
"mv_polynomial",
"mv_polynomial.coeff",
"submodule.sum_mem"
] | If every coefficient of a polynomial is in an ideal `I`, then so is the polynomial itself,
multivariate version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map_C_iff {I : ideal R} {f : mv_polynomial σ R} :
f ∈ (ideal.map (C : R →+* mv_polynomial σ R) I :
ideal (mv_polynomial σ R)) ↔ ∀ (m : σ →₀ ℕ), f.coeff m ∈ I | begin
split,
{ intros hf,
apply submodule.span_induction hf,
{ intros f hf n,
cases (set.mem_image _ _ _).mp hf with x hx,
rw [← hx.right, coeff_C],
by_cases (n = 0),
{ simpa [h] using hx.left },
{ simp [ne.symm h] } },
{ simp },
{ exact λ f g hf hg n, by simp [I.add_me... | theorem | mv_polynomial.mem_map_C_iff | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"ideal.map",
"ideal.mem_map_of_mem",
"ideal.mul_mem_right",
"mul_one",
"mv_polynomial",
"set.mem_image",
"smul_eq_mul",
"submodule.span_induction",
"submodule.sum_mem"
] | The push-forward of an ideal `I` of `R` to `mv_polynomial σ R` via inclusion
is exactly the set of polynomials whose coefficients are in `I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_map (f : R →+* S) :
(map f : mv_polynomial σ R →+* mv_polynomial σ S).ker = f.ker.map (C : R →+* mv_polynomial σ R) | begin
ext,
rw [mv_polynomial.mem_map_C_iff, ring_hom.mem_ker, mv_polynomial.ext_iff],
simp_rw [coeff_map, coeff_zero, ring_hom.mem_ker],
end | lemma | mv_polynomial.ker_map | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"mv_polynomial",
"mv_polynomial.ext_iff",
"mv_polynomial.mem_map_C_iff",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_factorization_monoid : unique_factorization_monoid D[X] | begin
haveI := arbitrary (normalization_monoid D),
haveI := to_normalized_gcd_monoid D,
exact ufm_of_gcd_of_wf_dvd_monoid
end | instance | polynomial.unique_factorization_monoid | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"normalization_monoid",
"ufm_of_gcd_of_wf_dvd_monoid",
"unique_factorization_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_factorization_monoid_of_fintype [fintype σ] :
unique_factorization_monoid (mv_polynomial σ D) | (rename_equiv D (fintype.equiv_fin σ)).to_mul_equiv.symm.unique_factorization_monoid $
begin
induction fintype.card σ with d hd,
{ apply (is_empty_alg_equiv D (fin 0)).to_mul_equiv.symm.unique_factorization_monoid,
apply_instance },
{ apply (fin_succ_equiv D d).to_mul_equiv.symm.unique_factorization_monoid,
... | lemma | mv_polynomial.unique_factorization_monoid_of_fintype | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"fin_succ_equiv",
"fintype",
"fintype.card",
"fintype.equiv_fin",
"mv_polynomial",
"polynomial.unique_factorization_monoid",
"unique_factorization_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bernstein_polynomial (n ν : ℕ) : R[X] | choose n ν * X^ν * (1 - X)^(n - ν) | def | bernstein_polynomial | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [] | `bernstein_polynomial R n ν` is `(choose n ν) * X^ν * (1 - X)^(n - ν)`.
Although the coefficients are integers, it is convenient to work over an arbitrary commutative ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_lt {n ν : ℕ} (h : n < ν) : bernstein_polynomial R n ν = 0 | by simp [bernstein_polynomial, nat.choose_eq_zero_of_lt h] | lemma | bernstein_polynomial.eq_zero_of_lt | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"nat.choose_eq_zero_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : R →+* S) (n ν : ℕ) :
(bernstein_polynomial R n ν).map f = bernstein_polynomial S n ν | by simp [bernstein_polynomial] | lemma | bernstein_polynomial.map | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip (n ν : ℕ) (h : ν ≤ n) :
(bernstein_polynomial R n ν).comp (1-X) = bernstein_polynomial R n (n-ν) | by simp [bernstein_polynomial, h, tsub_tsub_assoc, mul_right_comm] | lemma | bernstein_polynomial.flip | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"mul_right_comm",
"tsub_tsub_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip' (n ν : ℕ) (h : ν ≤ n) :
bernstein_polynomial R n ν = (bernstein_polynomial R n (n-ν)).comp (1-X) | by simp [←flip _ _ _ h, polynomial.comp_assoc] | lemma | bernstein_polynomial.flip' | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"polynomial.comp_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_at_0 (n ν : ℕ) : (bernstein_polynomial R n ν).eval 0 = if ν = 0 then 1 else 0 | begin
rw [bernstein_polynomial],
split_ifs,
{ subst h, simp, },
{ simp [zero_pow (nat.pos_of_ne_zero h)], },
end | lemma | bernstein_polynomial.eval_at_0 | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_at_1 (n ν : ℕ) : (bernstein_polynomial R n ν).eval 1 = if ν = n then 1 else 0 | begin
rw [bernstein_polynomial],
split_ifs,
{ subst h, simp, },
{ obtain w | w := (n - ν).eq_zero_or_pos,
{ simp [nat.choose_eq_zero_of_lt ((tsub_eq_zero_iff_le.mp w).lt_of_ne (ne.symm h))] },
{ simp [zero_pow w] } },
end. | lemma | bernstein_polynomial.eval_at_1 | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"nat.choose_eq_zero_of_lt",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivative_succ_aux (n ν : ℕ) :
(bernstein_polynomial R (n+1) (ν+1)).derivative =
(n+1) * (bernstein_polynomial R n ν - bernstein_polynomial R n (ν + 1)) | begin
rw [bernstein_polynomial],
suffices :
↑((n + 1).choose (ν + 1)) * (↑(ν + 1) * X ^ ν) * (1 - X) ^ (n - ν)
-(↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) * (↑(n - ν) * (1 - X) ^ (n - ν - 1))) =
↑(n + 1) * (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν) -
↑(n.choose (ν + 1)) * X ^ (ν + 1) * (1 - X) ... | lemma | bernstein_polynomial.derivative_succ_aux | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"algebra_map.coe_one",
"bernstein_polynomial",
"mul_assoc",
"mul_comm",
"mul_neg",
"mul_one",
"nat.cast_add",
"nat.cast_succ",
"nat.choose_mul_succ_eq",
"nat.succ_mul_choose_eq",
"polynomial.C_eq_nat_cast",
"polynomial.derivative_X",
"polynomial.derivative_mul",
"polynomial.derivative_nat_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivative_succ (n ν : ℕ) :
(bernstein_polynomial R n (ν+1)).derivative =
n * (bernstein_polynomial R (n-1) ν - bernstein_polynomial R (n-1) (ν+1)) | begin
cases n,
{ simp [bernstein_polynomial], },
{ rw nat.cast_succ, apply derivative_succ_aux, }
end | lemma | bernstein_polynomial.derivative_succ | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"nat.cast_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivative_zero (n : ℕ) :
(bernstein_polynomial R n 0).derivative = -n * bernstein_polynomial R (n-1) 0 | by simp [bernstein_polynomial, polynomial.derivative_pow] | lemma | bernstein_polynomial.derivative_zero | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"polynomial.derivative_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_derivative_at_0_eq_zero_of_lt (n : ℕ) {ν k : ℕ} :
k < ν → (polynomial.derivative^[k] (bernstein_polynomial R n ν)).eval 0 = 0 | begin
cases ν,
{ rintro ⟨⟩, },
{ rw nat.lt_succ_iff,
induction k with k ih generalizing n ν,
{ simp [eval_at_0], },
{ simp only [derivative_succ, int.coe_nat_eq_zero, mul_eq_zero,
function.comp_app, function.iterate_succ,
polynomial.iterate_derivative_sub, polynomial.iterate_derivative... | lemma | bernstein_polynomial.iterate_derivative_at_0_eq_zero_of_lt | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"function.iterate_succ",
"ih",
"int.coe_nat_eq_zero",
"mul_eq_zero",
"mul_eq_zero_of_right",
"nat.lt_succ_iff",
"polynomial.derivative",
"polynomial.eval_mul",
"polynomial.eval_nat_cast",
"polynomial.eval_sub",
"polynomial.iterate_derivative_nat_cast_mul",
"polynomial... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_derivative_succ_at_0_eq_zero (n ν : ℕ) :
(polynomial.derivative^[ν] (bernstein_polynomial R n (ν+1))).eval 0 = 0 | iterate_derivative_at_0_eq_zero_of_lt R n (lt_add_one ν) | lemma | bernstein_polynomial.iterate_derivative_succ_at_0_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"lt_add_one",
"polynomial.derivative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_derivative_at_0 (n ν : ℕ) :
(polynomial.derivative^[ν] (bernstein_polynomial R n ν)).eval 0 =
(pochhammer R ν).eval (n - (ν - 1) : ℕ) | begin
by_cases h : ν ≤ n,
{ induction ν with ν ih generalizing n h,
{ simp [eval_at_0], },
{ have h' : ν ≤ n-1 := le_tsub_of_add_le_right h,
simp only [derivative_succ, ih (n-1) h', iterate_derivative_succ_at_0_eq_zero,
nat.succ_sub_succ_eq_sub, tsub_zero, sub_zero,
iterate_derivative_... | lemma | bernstein_polynomial.iterate_derivative_at_0 | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"function.iterate_succ",
"ih",
"le_tsub_of_add_le_right",
"nat.le_pred_of_lt",
"pochhammer",
"pochhammer_eval_succ",
"pochhammer_succ_left",
"polynomial.derivative",
"pos_of_gt",
"tsub_add_cancel_of_le",
"tsub_add_eq_tsub_tsub",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_derivative_at_0_ne_zero [char_zero R] (n ν : ℕ) (h : ν ≤ n) :
(polynomial.derivative^[ν] (bernstein_polynomial R n ν)).eval 0 ≠ 0 | begin
simp only [int.coe_nat_eq_zero, bernstein_polynomial.iterate_derivative_at_0, ne.def,
nat.cast_eq_zero],
simp only [←pochhammer_eval_cast],
norm_cast,
apply ne_of_gt,
obtain rfl|h' := nat.eq_zero_or_pos ν,
{ simp, },
{ rw ← nat.succ_pred_eq_of_pos h' at h,
exact pochhammer_pos _ _ (tsub_pos_... | lemma | bernstein_polynomial.iterate_derivative_at_0_ne_zero | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"bernstein_polynomial.iterate_derivative_at_0",
"char_zero",
"int.coe_nat_eq_zero",
"nat.cast_eq_zero",
"pochhammer_pos",
"polynomial.derivative",
"tsub_pos_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_derivative_at_1_eq_zero_of_lt (n : ℕ) {ν k : ℕ} :
k < n - ν → (polynomial.derivative^[k] (bernstein_polynomial R n ν)).eval 1 = 0 | begin
intro w,
rw flip' _ _ _ (tsub_pos_iff_lt.mp (pos_of_gt w)).le,
simp [polynomial.eval_comp, iterate_derivative_at_0_eq_zero_of_lt R n w],
end | lemma | bernstein_polynomial.iterate_derivative_at_1_eq_zero_of_lt | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"polynomial.derivative",
"polynomial.eval_comp",
"pos_of_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_derivative_at_1 (n ν : ℕ) (h : ν ≤ n) :
(polynomial.derivative^[n-ν] (bernstein_polynomial R n ν)).eval 1 =
(-1)^(n-ν) * (pochhammer R (n - ν)).eval (ν + 1) | begin
rw flip' _ _ _ h,
simp [polynomial.eval_comp, h],
obtain rfl | h' := h.eq_or_lt,
{ simp, },
{ congr,
norm_cast,
rw [← tsub_add_eq_tsub_tsub, tsub_tsub_cancel_of_le (nat.succ_le_iff.mpr h')] },
end | lemma | bernstein_polynomial.iterate_derivative_at_1 | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"pochhammer",
"polynomial.derivative",
"polynomial.eval_comp",
"tsub_add_eq_tsub_tsub",
"tsub_tsub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.