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
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|---|---|---|---|---|---|---|---|---|---|---|
coeff_trunc_fun (m : σ →₀ ℕ) (φ : mv_power_series σ R) :
(trunc_fun n φ).coeff m = if m < n then coeff R m φ else 0 | by simp [trunc_fun, mv_polynomial.coeff_sum] | lemma | mv_power_series.coeff_trunc_fun | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial.coeff_sum",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc : mv_power_series σ R →+ mv_polynomial σ R | { to_fun := trunc_fun n,
map_zero' := by { ext, simp [coeff_trunc_fun] },
map_add' := by { intros, ext, simp [coeff_trunc_fun, ite_add], split_ifs; refl } } | def | mv_power_series.trunc | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series",
"trunc"
] | The `n`th truncation of a multivariate formal power series to a multivariate polynomial | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_trunc (m : σ →₀ ℕ) (φ : mv_power_series σ R) :
(trunc R n φ).coeff m = if m < n then coeff R m φ else 0 | by simp [trunc, coeff_trunc_fun] | lemma | mv_power_series.coeff_trunc | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc_one (hnn : n ≠ 0) : trunc R n 1 = 1 | mv_polynomial.ext _ _ $ λ m,
begin
rw [coeff_trunc, coeff_one],
split_ifs with H H' H',
{ subst m, simp },
{ symmetry, rw mv_polynomial.coeff_one, exact if_neg (ne.symm H'), },
{ symmetry, rw mv_polynomial.coeff_one, refine if_neg _,
rintro rfl, apply H, exact ne.bot_lt hnn, }
end | lemma | mv_power_series.trunc_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial.coeff_one",
"mv_polynomial.ext",
"ne.bot_lt",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc_C (hnn : n ≠ 0) (a : R) : trunc R n (C σ R a) = mv_polynomial.C a | mv_polynomial.ext _ _ $ λ m,
begin
rw [coeff_trunc, coeff_C, mv_polynomial.coeff_C],
split_ifs with H; refl <|> try {simp * at *},
exfalso, apply H, subst m, exact ne.bot_lt hnn,
end | lemma | mv_power_series.trunc_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial.C",
"mv_polynomial.coeff_C",
"mv_polynomial.ext",
"ne.bot_lt",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_dvd_iff {s : σ} {n : ℕ} {φ : mv_power_series σ R} :
(X s : mv_power_series σ R)^n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 | begin
split,
{ rintros ⟨φ, rfl⟩ m h,
rw [coeff_mul, finset.sum_eq_zero],
rintros ⟨i,j⟩ hij, rw [coeff_X_pow, if_neg, zero_mul],
contrapose! h, subst i, rw finsupp.mem_antidiagonal at hij,
rw [← hij, finsupp.add_apply, finsupp.single_eq_same], exact nat.le_add_right n _ },
{ intro h, refine ⟨λ m, c... | lemma | mv_power_series.X_pow_dvd_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"add_tsub_cancel_left",
"finsupp.add_apply",
"finsupp.mem_antidiagonal",
"finsupp.single_apply",
"finsupp.single_eq_same",
"finsupp.tsub_apply",
"mv_power_series",
"one_mul",
"prod.mk.inj_iff",
"tsub_add_cancel_of_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_dvd_iff {s : σ} {φ : mv_power_series σ R} :
(X s : mv_power_series σ R) ∣ φ ↔ ∀ m : σ →₀ ℕ, m s = 0 → coeff R m φ = 0 | begin
rw [← pow_one (X s : mv_power_series σ R), X_pow_dvd_iff],
split; intros h m hm,
{ exact h m (hm.symm ▸ zero_lt_one) },
{ exact h m (nat.eq_zero_of_le_zero $ nat.le_of_succ_le_succ hm) }
end | lemma | mv_power_series.X_dvd_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series",
"pow_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv.aux (a : R) (φ : mv_power_series σ R) : mv_power_series σ R | | n := if n = 0 then a else
- a * ∑ x in n.antidiagonal,
if h : x.2 < n then coeff R x.1 φ * inv.aux x.2 else 0
using_well_founded
{ rel_tac := λ _ _, `[exact ⟨_, finsupp.lt_wf σ⟩],
dec_tac := tactic.assumption } | def | mv_power_series.inv.aux | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.lt_wf",
"mv_power_series"
] | Auxiliary definition that unifies
the totalised inverse formal power series `(_)⁻¹` and
the inverse formal power series that depends on
an inverse of the constant coefficient `inv_of_unit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_inv_aux [decidable_eq σ] (n : σ →₀ ℕ) (a : R) (φ : mv_power_series σ R) :
coeff R n (inv.aux a φ) = if n = 0 then a else
- a * ∑ x in n.antidiagonal,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 | show inv.aux a φ n = _,
begin
rw inv.aux,
convert rfl -- unify `decidable` instances
end | lemma | mv_power_series.coeff_inv_aux | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_unit (φ : mv_power_series σ R) (u : Rˣ) : mv_power_series σ R | inv.aux (↑u⁻¹) φ | def | mv_power_series.inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | A multivariate formal power series is invertible if the constant coefficient is invertible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_inv_of_unit [decidable_eq σ] (n : σ →₀ ℕ) (φ : mv_power_series σ R) (u : Rˣ) :
coeff R n (inv_of_unit φ u) = if n = 0 then ↑u⁻¹ else
- ↑u⁻¹ * ∑ x in n.antidiagonal,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv_of_unit φ u) else 0 | coeff_inv_aux n (↑u⁻¹) φ | lemma | mv_power_series.coeff_inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_inv_of_unit (φ : mv_power_series σ R) (u : Rˣ) :
constant_coeff σ R (inv_of_unit φ u) = ↑u⁻¹ | by rw [← coeff_zero_eq_constant_coeff_apply, coeff_inv_of_unit, if_pos rfl] | lemma | mv_power_series.constant_coeff_inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_of_unit (φ : mv_power_series σ R) (u : Rˣ) (h : constant_coeff σ R φ = u) :
φ * inv_of_unit φ u = 1 | ext $ λ n, if H : n = 0 then by { rw H, simp [coeff_mul, support_single_ne_zero, h], }
else
begin
have : ((0 : σ →₀ ℕ), n) ∈ n.antidiagonal,
{ rw [finsupp.mem_antidiagonal, zero_add] },
rw [coeff_one, if_neg H, coeff_mul,
← finset.insert_erase this, finset.sum_insert (finset.not_mem_erase _ _),
coeff_zero... | lemma | mv_power_series.mul_inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.insert_erase",
"finset.mem_erase",
"finset.not_mem_erase",
"finsupp.mem_antidiagonal",
"le_rfl",
"mul_neg",
"mv_power_series",
"neg_mul",
"units.mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map.is_local_ring_hom : is_local_ring_hom (map σ f) | ⟨begin
rintros φ ⟨ψ, h⟩,
replace h := congr_arg (constant_coeff σ S) h,
rw constant_coeff_map at h,
have : is_unit (constant_coeff σ S ↑ψ) := @is_unit_constant_coeff σ S _ (↑ψ) ψ.is_unit,
rw h at this,
rcases is_unit_of_map_unit f _ this with ⟨c, hc⟩,
exact is_unit_of_mul_eq_one φ (inv_of_unit φ c) (mul_i... | instance | mv_power_series.map.is_local_ring_hom | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"is_local_ring_hom",
"is_unit",
"is_unit_of_map_unit",
"is_unit_of_mul_eq_one"
] | The map `A[[X]] → B[[X]]` induced by a local ring hom `A → B` is local | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv (φ : mv_power_series σ k) : mv_power_series σ k | inv.aux (constant_coeff σ k φ)⁻¹ φ | def | mv_power_series.inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | The inverse `1/f` of a multivariable power series `f` over a field | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_inv [decidable_eq σ] (n : σ →₀ ℕ) (φ : mv_power_series σ k) :
coeff k n (φ⁻¹) = if n = 0 then (constant_coeff σ k φ)⁻¹ else
- (constant_coeff σ k φ)⁻¹ * ∑ x in n.antidiagonal,
if x.2 < n then coeff k x.1 φ * coeff k x.2 (φ⁻¹) else 0 | coeff_inv_aux n _ φ | lemma | mv_power_series.coeff_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_inv (φ : mv_power_series σ k) :
constant_coeff σ k (φ⁻¹) = (constant_coeff σ k φ)⁻¹ | by rw [← coeff_zero_eq_constant_coeff_apply, coeff_inv, if_pos rfl] | lemma | mv_power_series.constant_coeff_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_zero {φ : mv_power_series σ k} :
φ⁻¹ = 0 ↔ constant_coeff σ k φ = 0 | ⟨λ h, by simpa using congr_arg (constant_coeff σ k) h,
λ h, ext $ λ n, by { rw coeff_inv, split_ifs;
simp only [h, mv_power_series.coeff_zero, zero_mul, inv_zero, neg_zero] }⟩ | lemma | mv_power_series.inv_eq_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_eq_zero",
"inv_zero",
"mv_power_series",
"mv_power_series.coeff_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_inv : (0 : mv_power_series σ k)⁻¹ = 0 | by rw [inv_eq_zero, constant_coeff_zero] | lemma | mv_power_series.zero_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_eq_zero",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_unit_eq (φ : mv_power_series σ k) (h : constant_coeff σ k φ ≠ 0) :
inv_of_unit φ (units.mk0 _ h) = φ⁻¹ | rfl | lemma | mv_power_series.inv_of_unit_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_unit_eq' (φ : mv_power_series σ k) (u : units k) (h : constant_coeff σ k φ = u) :
inv_of_unit φ u = φ⁻¹ | begin
rw ← inv_of_unit_eq φ (h.symm ▸ u.ne_zero),
congr' 1, rw [units.ext_iff], exact h.symm,
end | lemma | mv_power_series.inv_of_unit_eq' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series",
"units",
"units.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel (φ : mv_power_series σ k) (h : constant_coeff σ k φ ≠ 0) :
φ * φ⁻¹ = 1 | by rw [← inv_of_unit_eq φ h, mul_inv_of_unit φ (units.mk0 _ h) rfl] | lemma | mv_power_series.mul_inv_cancel | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mul_inv_cancel",
"mv_power_series",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel (φ : mv_power_series σ k) (h : constant_coeff σ k φ ≠ 0) :
φ⁻¹ * φ = 1 | by rw [mul_comm, φ.mul_inv_cancel h] | lemma | mv_power_series.inv_mul_cancel | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_mul_cancel",
"mul_comm",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_inv_iff_mul_eq {φ₁ φ₂ φ₃ : mv_power_series σ k}
(h : constant_coeff σ k φ₃ ≠ 0) :
φ₁ = φ₂ * φ₃⁻¹ ↔ φ₁ * φ₃ = φ₂ | ⟨λ k, by simp [k, mul_assoc, mv_power_series.inv_mul_cancel _ h],
λ k, by simp [← k, mul_assoc, mv_power_series.mul_inv_cancel _ h]⟩ | lemma | mv_power_series.eq_mul_inv_iff_mul_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"eq_mul_inv_iff_mul_eq",
"mul_assoc",
"mv_power_series",
"mv_power_series.inv_mul_cancel",
"mv_power_series.mul_inv_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_iff_mul_eq_one {φ ψ : mv_power_series σ k} (h : constant_coeff σ k ψ ≠ 0) :
φ = ψ⁻¹ ↔ φ * ψ = 1 | by rw [← mv_power_series.eq_mul_inv_iff_mul_eq h, one_mul] | lemma | mv_power_series.eq_inv_iff_mul_eq_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"eq_inv_iff_mul_eq_one",
"mv_power_series",
"mv_power_series.eq_mul_inv_iff_mul_eq",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_iff_mul_eq_one {φ ψ : mv_power_series σ k} (h : constant_coeff σ k ψ ≠ 0) :
ψ⁻¹ = φ ↔ φ * ψ = 1 | by rw [eq_comm, mv_power_series.eq_inv_iff_mul_eq_one h] | lemma | mv_power_series.inv_eq_iff_mul_eq_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_eq_iff_mul_eq_one",
"mv_power_series",
"mv_power_series.eq_inv_iff_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_rev (φ ψ : mv_power_series σ k) :
(φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹ | begin
by_cases h : constant_coeff σ k (φ * ψ) = 0,
{ rw inv_eq_zero.mpr h,
simp only [map_mul, mul_eq_zero] at h,
-- we don't have `no_zero_divisors (mw_power_series σ k)` yet,
cases h;
simp [inv_eq_zero.mpr h] },
{ rw [mv_power_series.inv_eq_iff_mul_eq_one h],
simp only [not_or_distrib, map_m... | lemma | mv_power_series.mul_inv_rev | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"map_mul",
"mul_assoc",
"mul_eq_zero",
"mul_inv_rev",
"mul_one",
"mv_power_series",
"mv_power_series.inv_eq_iff_mul_eq_one",
"mv_power_series.inv_mul_cancel",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_inv (r : k) : (C σ k r)⁻¹ = C σ k r⁻¹ | begin
rcases eq_or_ne r 0 with rfl|hr,
{ simp },
rw [mv_power_series.inv_eq_iff_mul_eq_one, ←map_mul, inv_mul_cancel hr, map_one],
simpa using hr
end | lemma | mv_power_series.C_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"eq_or_ne",
"inv_mul_cancel",
"map_one",
"mv_power_series.inv_eq_iff_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_inv (s : σ) : (X s : mv_power_series σ k)⁻¹ = 0 | by rw [inv_eq_zero, constant_coeff_X] | lemma | mv_power_series.X_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_eq_zero",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_inv (r : k) (φ : mv_power_series σ k) :
(r • φ)⁻¹ = r⁻¹ • φ⁻¹ | by simp [smul_eq_C_mul, mul_comm] | lemma | mv_power_series.smul_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mul_comm",
"mv_power_series",
"smul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mv_power_series : has_coe (mv_polynomial σ R) (mv_power_series σ R) | ⟨λ φ n, coeff n φ⟩ | instance | mv_polynomial.coe_to_mv_power_series | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | The natural inclusion from multivariate polynomials into multivariate formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_def : (φ : mv_power_series σ R) = λ n, coeff n φ | rfl | lemma | mv_polynomial.coe_def | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_coe (n : σ →₀ ℕ) :
mv_power_series.coeff R n ↑φ = coeff n φ | rfl | lemma | mv_polynomial.coeff_coe | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monomial (n : σ →₀ ℕ) (a : R) :
(monomial n a : mv_power_series σ R) = mv_power_series.monomial R n a | mv_power_series.ext $ λ m,
begin
rw [coeff_coe, coeff_monomial, mv_power_series.coeff_monomial],
split_ifs with h₁ h₂; refl <|> subst m; contradiction
end | lemma | mv_polynomial.coe_monomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series",
"mv_power_series.coeff_monomial",
"mv_power_series.ext",
"mv_power_series.monomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : mv_polynomial σ R) : mv_power_series σ R) = 0 | rfl | lemma | mv_polynomial.coe_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : mv_polynomial σ R) : mv_power_series σ R) = 1 | coe_monomial _ _ | lemma | mv_polynomial.coe_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add :
((φ + ψ : mv_polynomial σ R) : mv_power_series σ R) = φ + ψ | rfl | lemma | mv_polynomial.coe_add | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul :
((φ * ψ : mv_polynomial σ R) : mv_power_series σ R) = φ * ψ | mv_power_series.ext $ λ n,
by simp only [coeff_coe, mv_power_series.coeff_mul, coeff_mul] | lemma | mv_polynomial.coe_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series",
"mv_power_series.coeff_mul",
"mv_power_series.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_C (a : R) :
((C a : mv_polynomial σ R) : mv_power_series σ R) = mv_power_series.C σ R a | coe_monomial _ _ | lemma | mv_polynomial.coe_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series",
"mv_power_series.C"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit0 :
((bit0 φ : mv_polynomial σ R) : mv_power_series σ R) = bit0 (φ : mv_power_series σ R) | coe_add _ _ | lemma | mv_polynomial.coe_bit0 | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit1 :
((bit1 φ : mv_polynomial σ R) : mv_power_series σ R) = bit1 (φ : mv_power_series σ R) | by rw [bit1, bit1, coe_add, coe_one, coe_bit0] | lemma | mv_polynomial.coe_bit1 | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_X (s : σ) :
((X s : mv_polynomial σ R) : mv_power_series σ R) = mv_power_series.X s | coe_monomial _ _ | lemma | mv_polynomial.coe_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series",
"mv_power_series.X"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : mv_polynomial σ R → mv_power_series σ R) | λ x y h, by { ext, simp_rw [←coeff_coe, h] } | lemma | mv_polynomial.coe_injective | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj : (φ : mv_power_series σ R) = ψ ↔ φ = ψ | (coe_injective σ R).eq_iff | lemma | mv_polynomial.coe_inj | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero_iff : (φ : mv_power_series σ R) = 0 ↔ φ = 0 | by rw [←coe_zero, coe_inj] | lemma | mv_polynomial.coe_eq_zero_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_one_iff : (φ : mv_power_series σ R) = 1 ↔ φ = 1 | by rw [←coe_one, coe_inj] | lemma | mv_polynomial.coe_eq_one_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mv_power_series.ring_hom : mv_polynomial σ R →+* mv_power_series σ R | { to_fun := (coe : mv_polynomial σ R → mv_power_series σ R),
map_zero' := coe_zero,
map_one' := coe_one,
map_add' := coe_add,
map_mul' := coe_mul } | def | mv_polynomial.coe_to_mv_power_series.ring_hom | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | The coercion from multivariable polynomials to multivariable power series
as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pow (n : ℕ) :
((φ ^ n : mv_polynomial σ R) : mv_power_series σ R) = (φ : mv_power_series σ R) ^ n | coe_to_mv_power_series.ring_hom.map_pow _ _ | lemma | mv_polynomial.coe_pow | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mv_power_series.ring_hom_apply : coe_to_mv_power_series.ring_hom φ = φ | rfl | lemma | mv_polynomial.coe_to_mv_power_series.ring_hom_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mv_power_series.alg_hom : mv_polynomial σ R →ₐ[R] mv_power_series σ A | { commutes' := λ r, by simp [algebra_map_apply, mv_power_series.algebra_map_apply],
..(mv_power_series.map σ (algebra_map R A)).comp coe_to_mv_power_series.ring_hom} | def | mv_polynomial.coe_to_mv_power_series.alg_hom | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"algebra_map_apply",
"mv_polynomial",
"mv_power_series",
"mv_power_series.algebra_map_apply",
"mv_power_series.map"
] | The coercion from multivariable polynomials to multivariable power series
as an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_mv_power_series.alg_hom_apply : (coe_to_mv_power_series.alg_hom A φ) =
mv_power_series.map σ (algebra_map R A) ↑φ | rfl | lemma | mv_polynomial.coe_to_mv_power_series.alg_hom_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"mv_power_series.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_mv_polynomial : algebra (mv_polynomial σ R) (mv_power_series σ A) | ring_hom.to_algebra (mv_polynomial.coe_to_mv_power_series.alg_hom A).to_ring_hom | instance | mv_power_series.algebra_mv_polynomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra",
"mv_polynomial",
"mv_polynomial.coe_to_mv_power_series.alg_hom",
"mv_power_series",
"ring_hom.to_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_mv_power_series : algebra (mv_power_series σ R) (mv_power_series σ A) | (map σ (algebra_map R A)).to_algebra | instance | mv_power_series.algebra_mv_power_series | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra",
"algebra_map",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply' (p : mv_polynomial σ R):
algebra_map (mv_polynomial σ R) (mv_power_series σ A) p = map σ (algebra_map R A) p | rfl | lemma | mv_power_series.algebra_map_apply' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"mv_polynomial",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply'' :
algebra_map (mv_power_series σ R) (mv_power_series σ A) f = map σ (algebra_map R A) f | rfl | lemma | mv_power_series.algebra_map_apply'' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_series (R : Type*) | mv_power_series unit R | def | power_series | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series"
] | Formal power series over the coefficient ring `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff (n : ℕ) : power_series R →ₗ[R] R | mv_power_series.coeff R (single () n) | def | power_series.coeff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.coeff",
"power_series"
] | The `n`th coefficient of a formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monomial (n : ℕ) : R →ₗ[R] power_series R | mv_power_series.monomial R (single () n) | def | power_series.monomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.monomial",
"power_series"
] | The `n`th monomial with coefficient `a` as formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_def {s : unit →₀ ℕ} {n : ℕ} (h : s () = n) :
coeff R n = mv_power_series.coeff R s | by erw [coeff, ← h, ← finsupp.unique_single s] | lemma | power_series.coeff_def | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.unique_single",
"mv_power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {φ ψ : power_series R} (h : ∀ n, coeff R n φ = coeff R n ψ) :
φ = ψ | mv_power_series.ext $ λ n,
by { rw ← coeff_def, { apply h }, refl } | lemma | power_series.ext | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.ext",
"power_series"
] | Two formal power series are equal if all their coefficients are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff {φ ψ : power_series R} : φ = ψ ↔ (∀ n, coeff R n φ = coeff R n ψ) | ⟨λ h n, congr_arg (coeff R n) h, ext⟩ | lemma | power_series.ext_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | Two formal power series are equal if all their coefficients are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {R} (f : ℕ → R) : power_series R | λ s, f (s ()) | def | power_series.mk | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | Constructor for formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_mk (n : ℕ) (f : ℕ → R) : coeff R n (mk f) = f n | congr_arg f finsupp.single_eq_same | lemma | power_series.coeff_mk | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_eq_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_monomial (m n : ℕ) (a : R) :
coeff R m (monomial R n a) = if m = n then a else 0 | calc coeff R m (monomial R n a) = _ : mv_power_series.coeff_monomial _ _ _
... = if m = n then a else 0 :
by simp only [finsupp.unique_single_eq_iff] | lemma | power_series.coeff_monomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.unique_single_eq_iff",
"mv_power_series.coeff_monomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomial_eq_mk (n : ℕ) (a : R) :
monomial R n a = mk (λ m, if m = n then a else 0) | ext $ λ m, by { rw [coeff_monomial, coeff_mk] } | lemma | power_series.monomial_eq_mk | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_monomial_same (n : ℕ) (a : R) :
coeff R n (monomial R n a) = a | mv_power_series.coeff_monomial_same _ _ | lemma | power_series.coeff_monomial_same | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.coeff_monomial_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_comp_monomial (n : ℕ) :
(coeff R n).comp (monomial R n) = linear_map.id | linear_map.ext $ coeff_monomial_same n | lemma | power_series.coeff_comp_monomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"linear_map.ext",
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff : power_series R →+* R | mv_power_series.constant_coeff unit R | def | power_series.constant_coeff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.constant_coeff",
"power_series"
] | The constant coefficient of a formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
C : R →+* power_series R | mv_power_series.C unit R | def | power_series.C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.C",
"power_series"
] | The constant formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X : power_series R | mv_power_series.X () | def | power_series.X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.X",
"power_series"
] | The variable of the formal power series ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute_X (φ : power_series R) : commute φ X | φ.commute_X _ | lemma | power_series.commute_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"commute",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_eq_constant_coeff :
⇑(coeff R 0) = constant_coeff R | by { rw [coeff, finsupp.single_zero], refl } | lemma | power_series.coeff_zero_eq_constant_coeff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_eq_constant_coeff_apply (φ : power_series R) :
coeff R 0 φ = constant_coeff R φ | by rw [coeff_zero_eq_constant_coeff]; refl | lemma | power_series.coeff_zero_eq_constant_coeff_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomial_zero_eq_C : ⇑(monomial R 0) = C R | by rw [monomial, finsupp.single_zero, mv_power_series.monomial_zero_eq_C, C] | lemma | power_series.monomial_zero_eq_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_zero",
"mv_power_series.monomial_zero_eq_C"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a | by simp | lemma | power_series.monomial_zero_eq_C_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_C (n : ℕ) (a : R) :
coeff R n (C R a : power_series R) = if n = 0 then a else 0 | by rw [← monomial_zero_eq_C_apply, coeff_monomial] | lemma | power_series.coeff_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_C (a : R) : coeff R 0 (C R a) = a | by rw [← monomial_zero_eq_C_apply, coeff_monomial_same 0 a] | lemma | power_series.coeff_zero_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_eq : (X : power_series R) = monomial R 1 1 | rfl | lemma | power_series.X_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_X (n : ℕ) :
coeff R n (X : power_series R) = if n = 1 then 1 else 0 | by rw [X_eq, coeff_monomial] | lemma | power_series.coeff_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_X : coeff R 0 (X : power_series R) = 0 | by rw [coeff, finsupp.single_zero, X, mv_power_series.coeff_zero_X] | lemma | power_series.coeff_zero_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_zero",
"mv_power_series.coeff_zero_X",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_one_X : coeff R 1 (X : power_series R) = 1 | by rw [coeff_X, if_pos rfl] | lemma | power_series.coeff_one_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_ne_zero [nontrivial R] : (X : power_series R) ≠ 0 | λ H, by simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H | lemma | power_series.X_ne_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"nontrivial",
"one_ne_zero",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_eq (n : ℕ) : (X : power_series R)^n = monomial R n 1 | mv_power_series.X_pow_eq _ n | lemma | power_series.X_pow_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.X_pow_eq",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_X_pow (m n : ℕ) :
coeff R m ((X : power_series R)^n) = if m = n then 1 else 0 | by rw [X_pow_eq, coeff_monomial] | lemma | power_series.coeff_X_pow | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_X_pow_self (n : ℕ) :
coeff R n ((X : power_series R)^n) = 1 | by rw [coeff_X_pow, if_pos rfl] | lemma | power_series.coeff_X_pow_self | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_one (n : ℕ) :
coeff R n (1 : power_series R) = if n = 0 then 1 else 0 | coeff_C n 1 | lemma | power_series.coeff_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_one : coeff R 0 (1 : power_series R) = 1 | coeff_zero_C 1 | lemma | power_series.coeff_zero_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul (n : ℕ) (φ ψ : power_series R) :
coeff R n (φ * ψ) = ∑ p in finset.nat.antidiagonal n, coeff R p.1 φ * coeff R p.2 ψ | begin
symmetry,
apply finset.sum_bij (λ (p : ℕ × ℕ) h, (single () p.1, single () p.2)),
{ rintros ⟨i,j⟩ hij, rw finset.nat.mem_antidiagonal at hij,
rw [finsupp.mem_antidiagonal, ← finsupp.single_add, hij], },
{ rintros ⟨i,j⟩ hij, refl },
{ rintros ⟨i,j⟩ ⟨k,l⟩ hij hkl,
simpa only [prod.mk.inj_iff, fins... | lemma | power_series.coeff_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.antidiagonal",
"finset.nat.mem_antidiagonal",
"finsupp.add_apply",
"finsupp.mem_antidiagonal",
"finsupp.single_add",
"finsupp.single_eq_same",
"finsupp.unique_single",
"finsupp.unique_single_eq_iff",
"power_series",
"prod.mk.inj_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul_C (n : ℕ) (φ : power_series R) (a : R) :
coeff R n (φ * C R a) = coeff R n φ * a | mv_power_series.coeff_mul_C _ φ a | lemma | power_series.coeff_mul_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.coeff_mul_C",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_C_mul (n : ℕ) (φ : power_series R) (a : R) :
coeff R n (C R a * φ) = a * coeff R n φ | mv_power_series.coeff_C_mul _ φ a | lemma | power_series.coeff_C_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.coeff_C_mul",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_smul {S : Type*} [semiring S] [module R S]
(n : ℕ) (φ : power_series S) (a : R) : coeff S n (a • φ) = a • coeff S n φ | rfl | lemma | power_series.coeff_smul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"module",
"power_series",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_C_mul (f : power_series R) (a : R) :
a • f = C R a * f | by { ext, simp } | lemma | power_series.smul_eq_C_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_succ_mul_X (n : ℕ) (φ : power_series R) :
coeff R (n+1) (φ * X) = coeff R n φ | begin
simp only [coeff, finsupp.single_add],
convert φ.coeff_add_mul_monomial (single () n) (single () 1) _,
rw mul_one
end | lemma | power_series.coeff_succ_mul_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_add",
"mul_one",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_succ_X_mul (n : ℕ) (φ : power_series R) :
coeff R (n + 1) (X * φ) = coeff R n φ | begin
simp only [coeff, finsupp.single_add, add_comm n 1],
convert φ.coeff_add_monomial_mul (single () 1) (single () n) _,
rw one_mul,
end | lemma | power_series.coeff_succ_X_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_add",
"one_mul",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_C (a : R) : constant_coeff R (C R a) = a | rfl | lemma | power_series.constant_coeff_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_comp_C :
(constant_coeff R).comp (C R) = ring_hom.id R | rfl | lemma | power_series.constant_coeff_comp_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_zero : constant_coeff R 0 = 0 | rfl | lemma | power_series.constant_coeff_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_one : constant_coeff R 1 = 1 | rfl | lemma | power_series.constant_coeff_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_X : constant_coeff R X = 0 | mv_power_series.coeff_zero_X _ | lemma | power_series.constant_coeff_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.coeff_zero_X"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_mul_X (φ : power_series R) : coeff R 0 (φ * X) = 0 | by simp | lemma | power_series.coeff_zero_mul_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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