fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff f = constantCoeff g) : f = g := by
ext n
cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, hc]
| succ n =>
have equ : coeff n (d⁄dX R f) = coeff n (d⁄dX R g) := by rw [hD]
rwa [coeff_derivative, coeff_derivative, ← cast_succ, mul_comm, ← nsmul_eq_mul,
mul_comm, ← nsmul_eq_mul, smul_right_inj n.succ_ne_zero] at equ
@[simp] theorem derivative_inv {R} [CommRing R] (f : R⟦X⟧ˣ) :
d⁄dX R ↑f⁻¹ = -(↑f⁻¹ : R⟦X⟧) ^ 2 * d⁄dX R f := by
apply Derivation.leibniz_of_mul_eq_one
simp
@[simp] theorem derivative_invOf {R} [CommRing R] (f : R⟦X⟧) [Invertible f] :
d⁄dX R ⅟f = -⅟f ^ 2 * d⁄dX R f := by
rw [Derivation.leibniz_invOf, smul_eq_mul]
/-
The following theorem is stated only in the case that `R` is a field. This is because
there is currently no instance of `Inv R⟦X⟧` for more general base rings `R`.
-/
@[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
by_cases h : constantCoeff f = 0
· suffices f⁻¹ = 0 by
rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero]
rwa [MvPowerSeries.inv_eq_zero]
apply Derivation.leibniz_of_mul_eq_one
exact PowerSeries.inv_mul_cancel (h := h) | theorem | RingTheory | [
"Mathlib.RingTheory.PowerSeries.Trunc",
"Mathlib.RingTheory.PowerSeries.Inverse",
"Mathlib.RingTheory.Derivation.Basic"
] | Mathlib/RingTheory/PowerSeries/Derivative.lean | derivative.ext | Abbreviation of `PowerSeries.derivative`, the formal derivative on `R⟦X⟧` -/
scoped notation "d⁄dX" => derivative
variable {R}
@[simp] theorem derivative_C (r : R) : d⁄dX R (C r) = 0 := derivativeFun_C r
theorem coeff_derivative (f : R⟦X⟧) (n : ℕ) :
coeff n (d⁄dX R f) = coeff (n + 1) f * (n + 1) := coeff_derivativeFun f n
theorem derivative_coe (f : R[X]) : d⁄dX R f = Polynomial.derivative f := derivativeFun_coe f
@[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
ext
rw [coeff_derivative, coeff_one, coeff_X, boole_mul]
simp_rw [add_eq_right]
split_ifs with h
· rw [h, cast_zero, zero_add]
· rfl
theorem trunc_derivative (f : R⟦X⟧) (n : ℕ) :
trunc n (d⁄dX R f) = Polynomial.derivative (trunc (n + 1) f) :=
trunc_derivativeFun ..
theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n - 1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by
cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative]
end CommutativeSemiring
/- In the next lemma, we use `smul_right_inj`, which requires not only `NoZeroSMulDivisors ℕ R`, but
also cancellation of addition in `R`. For this reason, the next lemma is stated in the case that `R`
is a `CommRing`. -/
/-- If `f` and `g` have the same constant term and derivative, then they are equal. |
HasEval (a : S) := IsTopologicallyNilpotent a | abbrev | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval | Points at which evaluation of power series is well behaved |
hasEval_def (a : S) : HasEval a ↔ IsTopologicallyNilpotent a := .rfl | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | hasEval_def | null |
hasEval_iff {a : S} :
HasEval a ↔ MvPowerSeries.HasEval (fun (_ : Unit) ↦ a) :=
⟨fun ha ↦ ⟨fun _ ↦ ha, by simp⟩, fun ha ↦ ha.hpow default⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | hasEval_iff | null |
hasEval {a : S} (ha : HasEval a) :
MvPowerSeries.HasEval (fun (_ : Unit) ↦ a) := hasEval_iff.mp ha | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | hasEval | null |
HasEval.mono {S : Type*} [CommRing S] {a : S}
{t u : TopologicalSpace S} (h : t ≤ u) (ha : @HasEval _ _ t a) :
@HasEval _ _ u a := by
simp only [hasEval_iff] at ha ⊢
exact ha.mono h | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval.mono | null |
HasEval.zero : HasEval (0 : S) := by
rw [hasEval_iff]; exact MvPowerSeries.HasEval.zero | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval.zero | null |
HasEval.add [ContinuousAdd S] [IsLinearTopology S S]
{a b : S} (ha : HasEval a) (hb : HasEval b) : HasEval (a + b) := by
simp only [hasEval_iff] at ha hb ⊢
exact ha.add hb | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval.add | null |
HasEval.mul_left [IsLinearTopology S S]
(c : S) {x : S} (hx : HasEval x) : HasEval (c * x) := by
simp only [hasEval_iff] at hx ⊢
exact hx.mul_left _ | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval.mul_left | null |
HasEval.mul_right [IsLinearTopology S S]
(c : S) {x : S} (hx : HasEval x) : HasEval (x * c) := by
simp only [hasEval_iff] at hx ⊢
exact hx.mul_right _ | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval.mul_right | null |
HasEval.map (hφ : Continuous φ) {a : R} (ha : HasEval a) :
HasEval (φ a) := by
simp only [hasEval_iff] at ha ⊢
exact ha.map hφ | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval.map | [Bourbaki, *Algebra*, chap. 4, §4, n°3, Prop. 4 (i) (a & b)][bourbaki1981]. |
protected HasEval.X :
HasEval (X : R⟦X⟧) := by
rw [hasEval_iff]
exact MvPowerSeries.HasEval.X
variable [IsTopologicalRing S] [IsLinearTopology S S] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | HasEval.X | null |
@[simps]
hasEvalIdeal : Ideal S where
carrier := {a | HasEval a}
add_mem' := HasEval.add
zero_mem' := HasEval.zero
smul_mem' := HasEval.mul_left | def | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | hasEvalIdeal | The domain of evaluation of `MvPowerSeries`, as an ideal |
mem_hasEvalIdeal_iff {a : S} :
a ∈ hasEvalIdeal ↔ HasEval a := by
simp [hasEvalIdeal] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | mem_hasEvalIdeal_iff | null |
noncomputable eval₂ : PowerSeries R → S :=
MvPowerSeries.eval₂ φ (fun _ ↦ a)
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | eval₂ | Evaluation of a power series `f` at a point `a`.
It coincides with the evaluation of `f` as a polynomial if `f` is the coercion of a polynomial.
Otherwise, it is only relevant if `φ` is continuous and `a` is topologically nilpotent. |
eval₂_coe (f : Polynomial R) : eval₂ φ a f = f.eval₂ φ a := by
let g : MvPolynomial Unit R := (MvPolynomial.pUnitAlgEquiv R).symm f
have : f = MvPolynomial.pUnitAlgEquiv R g := by
simp only [g, ← AlgEquiv.symm_apply_eq]
simp only [this, PowerSeries.eval₂, MvPolynomial.eval₂_const_pUnitAlgEquiv]
rw [← MvPolynomial.toMvPowerSeries_pUnitAlgEquiv, MvPowerSeries.eval₂_coe]
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | eval₂_coe | null |
eval₂_C (r : R) :
eval₂ φ a (C r) = φ r := by
rw [← Polynomial.coe_C, eval₂_coe, Polynomial.eval₂_C]
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | eval₂_C | null |
eval₂_X :
eval₂ φ a X = a := by
rw [← Polynomial.coe_X, eval₂_coe, Polynomial.eval₂_X]
variable {φ a}
variable [IsUniformAddGroup R] [IsTopologicalSemiring R]
[IsUniformAddGroup S] [T2Space S] [CompleteSpace S]
[IsTopologicalRing S] [IsLinearTopology S S] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | eval₂_X | null |
noncomputable eval₂Hom (hφ : Continuous φ) (ha : HasEval a) :
PowerSeries R →+* S :=
MvPowerSeries.eval₂Hom hφ (hasEval ha) | def | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | eval₂Hom | The evaluation homomorphism at `a` on `PowerSeries`, as a `RingHom`. |
coe_eval₂Hom (hφ : Continuous φ) (ha : HasEval a) :
⇑(eval₂Hom hφ ha) = eval₂ φ a :=
MvPowerSeries.coe_eval₂Hom hφ (hasEval ha) | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | coe_eval₂Hom | null |
uniformContinuous_eval₂ (hφ : Continuous φ) (ha : HasEval a) :
UniformContinuous (eval₂ φ a) :=
MvPowerSeries.uniformContinuous_eval₂ hφ (hasEval ha) | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | uniformContinuous_eval₂ | null |
continuous_eval₂ (hφ : Continuous φ) (ha : HasEval a) :
Continuous (eval₂ φ a : PowerSeries R → S) :=
(uniformContinuous_eval₂ hφ ha).continuous | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | continuous_eval₂ | null |
hasSum_eval₂ (hφ : Continuous φ) (ha : HasEval a) (f : PowerSeries R) :
HasSum (fun (d : ℕ) ↦ φ (coeff d f) * a ^ d) (f.eval₂ φ a) := by
have := MvPowerSeries.hasSum_eval₂ hφ (hasEval ha) f
simp only [PowerSeries.eval₂]
rw [← (Finsupp.single_injective ()).hasSum_iff] at this
· convert this; simp; congr
· intro d hd
exact False.elim (hd ⟨d (), by ext; simp⟩) | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | hasSum_eval₂ | null |
eval₂_eq_tsum (hφ : Continuous φ) (ha : HasEval a) (f : PowerSeries R) :
PowerSeries.eval₂ φ a f =
∑' d : ℕ, φ (coeff d f) * a ^ d :=
(hasSum_eval₂ hφ ha f).tsum_eq.symm | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | eval₂_eq_tsum | null |
eval₂_unique (hφ : Continuous φ) (ha : HasEval a)
{ε : PowerSeries R → S} (hε : Continuous ε)
(h : ∀ p : Polynomial R, ε p = Polynomial.eval₂ φ a p) :
ε = eval₂ φ a := by
apply MvPowerSeries.eval₂_unique hφ (hasEval ha) hε
intro p
rw [MvPolynomial.toMvPowerSeries_pUnitAlgEquiv, h, ← MvPolynomial.eval₂_pUnitAlgEquiv] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | eval₂_unique | null |
comp_eval₂ (hφ : Continuous φ) (ha : HasEval a)
{T : Type*} [UniformSpace T] [CompleteSpace T] [T2Space T]
[CommRing T] [IsTopologicalRing T] [IsLinearTopology T T] [IsUniformAddGroup T]
{ε : S →+* T} (hε : Continuous ε) :
ε ∘ eval₂ φ a = eval₂ (ε.comp φ) (ε a) := by
apply eval₂_unique _ (ha.map hε)
· exact Continuous.comp hε (continuous_eval₂ hφ ha)
· intro p
simp only [Function.comp_apply, eval₂_coe]
exact Polynomial.hom_eval₂ p φ ε a
· simp only [RingHom.coe_comp, Continuous.comp hε hφ]
variable [Algebra R S] [ContinuousSMul R S] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | comp_eval₂ | null |
noncomputable aeval (ha : HasEval a) :
PowerSeries R →ₐ[R] S :=
MvPowerSeries.aeval (hasEval ha) | def | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | aeval | For `HasEval a`,
the evaluation homomorphism at `a` on `PowerSeries`, as an `AlgHom`. |
coe_aeval (ha : HasEval a) :
↑(aeval ha) = eval₂ (algebraMap R S) a :=
MvPowerSeries.coe_aeval (hasEval ha) | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | coe_aeval | null |
continuous_aeval (ha : HasEval a) :
Continuous (aeval ha : PowerSeries R → S) :=
MvPowerSeries.continuous_aeval (hasEval ha)
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | continuous_aeval | null |
aeval_coe (ha : HasEval a) (p : Polynomial R) :
aeval ha (p : PowerSeries R) = Polynomial.aeval a p := by
rw [coe_aeval, Polynomial.aeval_def, eval₂_coe] | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | aeval_coe | null |
aeval_unique {ε : PowerSeries R →ₐ[R] S} (hε : Continuous ε) :
aeval (HasEval.X.map hε) = ε :=
MvPowerSeries.aeval_unique hε | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | aeval_unique | null |
hasSum_aeval (ha : HasEval a) (f : PowerSeries R) :
HasSum (fun d ↦ coeff d f • a ^ d) (f.aeval ha) := by
simp_rw [coe_aeval, ← algebraMap_smul (R := R) S, smul_eq_mul]
exact hasSum_eval₂ (continuous_algebraMap R S) ha f | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | hasSum_aeval | null |
aeval_eq_sum (ha : HasEval a) (f : PowerSeries R) :
aeval ha f = tsum fun d ↦ coeff d f • a ^ d :=
(hasSum_aeval ha f).tsum_eq.symm | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | aeval_eq_sum | null |
comp_aeval (ha : HasEval a)
{T : Type*} [CommRing T] [UniformSpace T] [IsUniformAddGroup T]
[IsTopologicalRing T] [IsLinearTopology T T]
[T2Space T] [Algebra R T] [ContinuousSMul R T] [CompleteSpace T]
{ε : S →ₐ[R] T} (hε : Continuous ε) :
ε.comp (aeval ha) = aeval (ha.map hε) :=
MvPowerSeries.comp_aeval (hasEval ha) hε | theorem | RingTheory | [
"Mathlib.RingTheory.MvPowerSeries.Evaluation",
"Mathlib.RingTheory.PowerSeries.PiTopology",
"Mathlib.Algebra.MvPolynomial.Equiv"
] | Mathlib/RingTheory/PowerSeries/Evaluation.lean | comp_aeval | null |
noncomputable gaussNorm : ℝ := ⨆ i : ℕ, v (f.coeff i) * c ^ i | def | RingTheory | [
"Mathlib.Data.Real.Archimedean",
"Mathlib.RingTheory.PowerSeries.Order"
] | Mathlib/RingTheory/PowerSeries/GaussNorm.lean | gaussNorm | Given a power series `f` in `R⟦X⟧`, a function `v : R → ℝ` and a real number `c`, the Gauss norm
is defined as the supremum of the set of all values of `v (f.coeff i) * c ^ i` for all `i : ℕ`. |
le_gaussNorm (hbd : BddAbove {x | ∃ i, v (f.coeff i) * c ^ i = x}) (i : ℕ) :
v (f.coeff i) * c ^ i ≤ f.gaussNorm v c := le_ciSup hbd i
@[simp] | lemma | RingTheory | [
"Mathlib.Data.Real.Archimedean",
"Mathlib.RingTheory.PowerSeries.Order"
] | Mathlib/RingTheory/PowerSeries/GaussNorm.lean | le_gaussNorm | null |
gaussNorm_zero [ZeroHomClass F R ℝ] : gaussNorm v c 0 = 0 := by simp [gaussNorm] | theorem | RingTheory | [
"Mathlib.Data.Real.Archimedean",
"Mathlib.RingTheory.PowerSeries.Order"
] | Mathlib/RingTheory/PowerSeries/GaussNorm.lean | gaussNorm_zero | null |
gaussNorm_nonneg [NonnegHomClass F R ℝ] : 0 ≤ f.gaussNorm v c := by
by_cases h : BddAbove (Set.range fun i ↦ v (f.coeff i) * c ^ i)
· calc
0 ≤ v (f.coeff 0) * c ^ 0 :=
mul_nonneg (apply_nonneg v (f.coeff 0)) <| pow_nonneg (le_of_lt (zero_lt_one)) 0
_ ≤ f.gaussNorm v c := le_ciSup h 0
· simp [gaussNorm, h]
@[simp] | theorem | RingTheory | [
"Mathlib.Data.Real.Archimedean",
"Mathlib.RingTheory.PowerSeries.Order"
] | Mathlib/RingTheory/PowerSeries/GaussNorm.lean | gaussNorm_nonneg | null |
gaussNorm_eq_zero_iff [ZeroHomClass F R ℝ] [NonnegHomClass F R ℝ] {v : F}
(h_eq_zero : ∀ x : R, v x = 0 → x = 0) {f : R⟦X⟧} {c : ℝ} (hc : 0 < c)
(hbd : BddAbove (Set.range fun i ↦ v (f.coeff i) * c ^ i)) :
f.gaussNorm v c = 0 ↔ f = 0 := by
refine ⟨?_, fun hf ↦ by simp [hf]⟩
contrapose!
intro hf
apply ne_of_gt
obtain ⟨n, hn⟩ := exists_coeff_ne_zero_iff_ne_zero.mpr hf
calc
0 < v (f.coeff n) * c ^ n := by
have := fun h ↦ hn (h_eq_zero (coeff n f) h)
positivity
_ ≤ gaussNorm v c f := le_ciSup hbd n | theorem | RingTheory | [
"Mathlib.Data.Real.Archimedean",
"Mathlib.RingTheory.PowerSeries.Order"
] | Mathlib/RingTheory/PowerSeries/GaussNorm.lean | gaussNorm_eq_zero_iff | null |
protected inv.aux : R → R⟦X⟧ → R⟦X⟧ :=
MvPowerSeries.inv.aux | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | inv.aux | Auxiliary function used for computing inverse of a power series |
coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) :
coeff n (inv.aux a φ) =
if n = 0 then a
else
-a *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff x.1 φ * coeff x.2 (inv.aux a φ) else 0 := by
rw [coeff, inv.aux, MvPowerSeries.coeff_inv_aux]
simp only [Finsupp.single_eq_zero]
split_ifs; · rfl
congr 1
symm
apply Finset.sum_nbij' (fun (a, b) ↦ (single () a, single () b))
fun (f, g) ↦ (f (), g ())
· aesop
· aesop
· aesop
· aesop
· rintro ⟨i, j⟩ _hij
obtain H | H := le_or_gt n j
· aesop
rw [if_pos H, if_pos]
· rfl
refine ⟨?_, fun hh ↦ H.not_ge ?_⟩
· rintro ⟨⟩
simpa [Finsupp.single_eq_same] using le_of_lt H
· simpa [Finsupp.single_eq_same] using hh () | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | coeff_inv_aux | null |
invOfUnit (φ : R⟦X⟧) (u : Rˣ) : R⟦X⟧ :=
MvPowerSeries.invOfUnit φ u | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | invOfUnit | A formal power series is invertible if the constant coefficient is invertible. |
coeff_invOfUnit (n : ℕ) (φ : R⟦X⟧) (u : Rˣ) :
coeff n (invOfUnit φ u) =
if n = 0 then ↑u⁻¹
else
-↑u⁻¹ *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff x.1 φ * coeff x.2 (invOfUnit φ u) else 0 :=
coeff_inv_aux n (↑u⁻¹ : R) φ
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | coeff_invOfUnit | null |
constantCoeff_invOfUnit (φ : R⟦X⟧) (u : Rˣ) :
constantCoeff (invOfUnit φ u) = ↑u⁻¹ := by
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | constantCoeff_invOfUnit | null |
mul_invOfUnit (φ : R⟦X⟧) (u : Rˣ) (h : constantCoeff φ = u) :
φ * invOfUnit φ u = 1 :=
MvPowerSeries.mul_invOfUnit φ u <| h
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | mul_invOfUnit | null |
invOfUnit_mul (φ : R⟦X⟧) (u : Rˣ) (h : constantCoeff φ = u) :
invOfUnit φ u * φ = 1 :=
MvPowerSeries.invOfUnit_mul φ u h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | invOfUnit_mul | null |
isUnit_iff_constantCoeff {φ : R⟦X⟧} :
IsUnit φ ↔ IsUnit (constantCoeff φ) :=
MvPowerSeries.isUnit_iff_constantCoeff | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | isUnit_iff_constantCoeff | null |
sub_const_eq_shift_mul_X (φ : R⟦X⟧) :
φ - C (constantCoeff φ) = (mk fun p ↦ coeff (p + 1) φ) * X :=
sub_eq_iff_eq_add.mpr (eq_shift_mul_X_add_const φ) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | sub_const_eq_shift_mul_X | Two ways of removing the constant coefficient of a power series are the same. |
sub_const_eq_X_mul_shift (φ : R⟦X⟧) :
φ - C (constantCoeff φ) = X * mk fun p ↦ coeff (p + 1) φ :=
sub_eq_iff_eq_add.mpr (eq_X_mul_shift_add_const φ) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | sub_const_eq_X_mul_shift | null |
protected inv : k⟦X⟧ → k⟦X⟧ :=
MvPowerSeries.inv | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | inv | The inverse 1/f of a power series f defined over a field |
inv_eq_inv_aux (φ : k⟦X⟧) : φ⁻¹ = inv.aux (constantCoeff φ)⁻¹ φ :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | inv_eq_inv_aux | null |
coeff_inv (n) (φ : k⟦X⟧) :
coeff n φ⁻¹ =
if n = 0 then (constantCoeff φ)⁻¹
else
-(constantCoeff φ)⁻¹ *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff x.1 φ * coeff x.2 φ⁻¹ else 0 := by
rw [inv_eq_inv_aux, coeff_inv_aux n (constantCoeff φ)⁻¹ φ]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | coeff_inv | null |
constantCoeff_inv (φ : k⟦X⟧) : constantCoeff φ⁻¹ = (constantCoeff φ)⁻¹ :=
MvPowerSeries.constantCoeff_inv φ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | constantCoeff_inv | null |
inv_eq_zero {φ : k⟦X⟧} : φ⁻¹ = 0 ↔ constantCoeff φ = 0 :=
MvPowerSeries.inv_eq_zero | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | inv_eq_zero | null |
zero_inv : (0 : k⟦X⟧)⁻¹ = 0 :=
MvPowerSeries.zero_inv
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | zero_inv | null |
invOfUnit_eq (φ : k⟦X⟧) (h : constantCoeff φ ≠ 0) :
invOfUnit φ (Units.mk0 _ h) = φ⁻¹ :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | invOfUnit_eq | null |
invOfUnit_eq' (φ : k⟦X⟧) (u : Units k) (h : constantCoeff φ = u) :
invOfUnit φ u = φ⁻¹ :=
MvPowerSeries.invOfUnit_eq' φ _ h
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | invOfUnit_eq' | null |
protected mul_inv_cancel (φ : k⟦X⟧) (h : constantCoeff φ ≠ 0) : φ * φ⁻¹ = 1 :=
MvPowerSeries.mul_inv_cancel φ h
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | mul_inv_cancel | null |
protected inv_mul_cancel (φ : k⟦X⟧) (h : constantCoeff φ ≠ 0) : φ⁻¹ * φ = 1 :=
MvPowerSeries.inv_mul_cancel φ h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | inv_mul_cancel | null |
eq_mul_inv_iff_mul_eq {φ₁ φ₂ φ₃ : k⟦X⟧} (h : constantCoeff φ₃ ≠ 0) :
φ₁ = φ₂ * φ₃⁻¹ ↔ φ₁ * φ₃ = φ₂ :=
MvPowerSeries.eq_mul_inv_iff_mul_eq h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | eq_mul_inv_iff_mul_eq | null |
eq_inv_iff_mul_eq_one {φ ψ : k⟦X⟧} (h : constantCoeff ψ ≠ 0) :
φ = ψ⁻¹ ↔ φ * ψ = 1 :=
MvPowerSeries.eq_inv_iff_mul_eq_one h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | eq_inv_iff_mul_eq_one | null |
inv_eq_iff_mul_eq_one {φ ψ : k⟦X⟧} (h : constantCoeff ψ ≠ 0) :
ψ⁻¹ = φ ↔ φ * ψ = 1 :=
MvPowerSeries.inv_eq_iff_mul_eq_one h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | inv_eq_iff_mul_eq_one | null |
protected mul_inv_rev (φ ψ : k⟦X⟧) : (φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹ :=
MvPowerSeries.mul_inv_rev _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | mul_inv_rev | null |
@[simp]
C_inv (r : k) : (C r)⁻¹ = C r⁻¹ :=
MvPowerSeries.C_inv _
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | C_inv | null |
X_inv : (X : k⟦X⟧)⁻¹ = 0 :=
MvPowerSeries.X_inv _ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | X_inv | null |
smul_inv (r : k) (φ : k⟦X⟧) : (r • φ)⁻¹ = r⁻¹ • φ⁻¹ :=
MvPowerSeries.smul_inv _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | smul_inv | null |
firstUnitCoeff {f : k⟦X⟧} (hf : f ≠ 0) : kˣ :=
have : Invertible (constantCoeff (divXPowOrder f)) := by
apply invertibleOfNonzero
simpa [constantCoeff_divXPowOrder_eq_zero_iff.not]
unitOfInvertible (constantCoeff (divXPowOrder f)) | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | firstUnitCoeff | `firstUnitCoeff` is the non-zero coefficient whose index is `f.order`, seen as a unit of the
field. It is obtained using `divided_by_X_pow_order`, defined in `PowerSeries.Order`. |
Inv_divided_by_X_pow_order {f : k⟦X⟧} (hf : f ≠ 0) : k⟦X⟧ :=
invOfUnit (divXPowOrder f) (firstUnitCoeff hf)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | Inv_divided_by_X_pow_order | `Inv_divided_by_X_pow_order` is the inverse of the element obtained by diving a non-zero power
series by the largest power of `X` dividing it. Useful to create a term of type `Units`, done in
`Unit_divided_by_X_pow_order` |
Inv_divided_by_X_pow_order_rightInv {f : k⟦X⟧} (hf : f ≠ 0) :
divXPowOrder f * Inv_divided_by_X_pow_order hf = 1 :=
mul_invOfUnit (divXPowOrder f) (firstUnitCoeff hf) rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | Inv_divided_by_X_pow_order_rightInv | null |
Inv_divided_by_X_pow_order_leftInv {f : k⟦X⟧} (hf : f ≠ 0) :
Inv_divided_by_X_pow_order hf * divXPowOrder f = 1 := by
rw [mul_comm]
exact mul_invOfUnit (divXPowOrder f) (firstUnitCoeff hf) rfl
open scoped Classical in | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | Inv_divided_by_X_pow_order_leftInv | null |
Unit_of_divided_by_X_pow_order (f : k⟦X⟧) : k⟦X⟧ˣ :=
if hf : f = 0 then 1
else
{ val := divXPowOrder f
inv := Inv_divided_by_X_pow_order hf
val_inv := Inv_divided_by_X_pow_order_rightInv hf
inv_val := Inv_divided_by_X_pow_order_leftInv hf } | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | Unit_of_divided_by_X_pow_order | `Unit_of_divided_by_X_pow_order` is the unit power series obtained by dividing a non-zero
power series by the largest power of `X` that divides it. |
isUnit_divided_by_X_pow_order {f : k⟦X⟧} (hf : f ≠ 0) :
IsUnit (divXPowOrder f) :=
⟨Unit_of_divided_by_X_pow_order f,
by simp only [Unit_of_divided_by_X_pow_order, dif_neg hf, Units.val_mk]⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | isUnit_divided_by_X_pow_order | null |
Unit_of_divided_by_X_pow_order_nonzero {f : k⟦X⟧} (hf : f ≠ 0) :
↑(Unit_of_divided_by_X_pow_order f) = divXPowOrder f := by
simp only [Unit_of_divided_by_X_pow_order, dif_neg hf, Units.val_mk]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | Unit_of_divided_by_X_pow_order_nonzero | null |
Unit_of_divided_by_X_pow_order_zero : Unit_of_divided_by_X_pow_order (0 : k⟦X⟧) = 1 := by
simp only [Unit_of_divided_by_X_pow_order, dif_pos] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | Unit_of_divided_by_X_pow_order_zero | null |
eq_divided_by_X_pow_order_Iff_Unit {f : k⟦X⟧} (hf : f ≠ 0) :
f = divXPowOrder f ↔ IsUnit f :=
⟨fun h ↦ by rw [h]; exact isUnit_divided_by_X_pow_order hf, fun h ↦ by
have : f.order = 0 := by
simp [order_zero_of_unit h]
conv_lhs => rw [← X_pow_order_mul_divXPowOrder (f := f), this, ENat.toNat_zero,
pow_zero, one_mul]⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | eq_divided_by_X_pow_order_Iff_Unit | null |
@[instance]
map.isLocalHom : IsLocalHom (map f) :=
MvPowerSeries.map.isLocalHom f
variable [IsLocalRing R] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | map.isLocalHom | null |
hasUnitMulPowIrreducibleFactorization :
HasUnitMulPowIrreducibleFactorization k⟦X⟧ :=
⟨X, And.intro X_irreducible
(by
intro f hf
use f.order.toNat
use Unit_of_divided_by_X_pow_order f
simp only [Unit_of_divided_by_X_pow_order_nonzero hf]
exact X_pow_order_mul_divXPowOrder)⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | hasUnitMulPowIrreducibleFactorization | null |
isNoetherianRing : IsNoetherianRing k⟦X⟧ :=
PrincipalIdealRing.isNoetherianRing | instance | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | isNoetherianRing | null |
maximalIdeal_eq_span_X : IsLocalRing.maximalIdeal (k⟦X⟧) = Ideal.span {X} := by
have hX : (Ideal.span {(X : k⟦X⟧)}).IsMaximal := by
rw [Ideal.isMaximal_iff]
constructor
· rw [Ideal.mem_span_singleton]
exact Prime.not_dvd_one X_prime
· intro I f hI hfX hfI
rw [Ideal.mem_span_singleton, X_dvd_iff] at hfX
have hfI0 : C (f 0) ∈ I := by
have : C (f 0) = f - (f - C (f 0)) := by rw [sub_sub_cancel]
rw [this]
apply Ideal.sub_mem I hfI
apply hI
rw [Ideal.mem_span_singleton, X_dvd_iff, map_sub, constantCoeff_C, ←
coeff_zero_eq_constantCoeff_apply, sub_eq_zero, coeff_zero_eq_constantCoeff]
rfl
rw [← Ideal.eq_top_iff_one]
apply Ideal.eq_top_of_isUnit_mem I hfI0 (IsUnit.map C (Ne.isUnit hfX))
rw [IsLocalRing.eq_maximalIdeal hX] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | maximalIdeal_eq_span_X | The maximal ideal of `k⟦X⟧` is generated by `X`. |
normUnit_X : normUnit (X : k⟦X⟧) = 1 := by
simp [normUnit, ← Units.val_eq_one, Unit_of_divided_by_X_pow_order_nonzero] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | normUnit_X | null |
X_eq_normalizeX : (X : k⟦X⟧) = normalize X := by
simp only [normalize_apply, normUnit_X, Units.val_one, mul_one]
open UniqueFactorizationMonoid
open scoped Classical in | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | X_eq_normalizeX | null |
normalized_count_X_eq_of_coe {P : k[X]} (hP : P ≠ 0) :
Multiset.count PowerSeries.X (normalizedFactors (P : k⟦X⟧)) =
Multiset.count Polynomial.X (normalizedFactors P) := by
apply eq_of_forall_le_iff
simp only [← Nat.cast_le (α := ℕ∞)]
rw [X_eq_normalize, PowerSeries.X_eq_normalizeX, ← emultiplicity_eq_count_normalizedFactors
irreducible_X hP, ← emultiplicity_eq_count_normalizedFactors X_irreducible] <;>
simp only [← pow_dvd_iff_le_emultiplicity, Polynomial.X_pow_dvd_iff,
PowerSeries.X_pow_dvd_iff, Polynomial.coeff_coe P, implies_true, ne_eq, coe_eq_zero_iff, hP,
not_false_eq_true]
open IsLocalRing | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | normalized_count_X_eq_of_coe | null |
ker_coeff_eq_max_ideal : RingHom.ker (constantCoeff (R := k)) = maximalIdeal _ :=
Ideal.ext fun _ ↦ by
rw [RingHom.mem_ker, maximalIdeal_eq_span_X, Ideal.mem_span_singleton, X_dvd_iff] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | ker_coeff_eq_max_ideal | null |
residueFieldOfPowerSeries : ResidueField k⟦X⟧ ≃+* k :=
(Ideal.quotEquivOfEq (ker_coeff_eq_max_ideal).symm).trans
(RingHom.quotientKerEquivOfSurjective constantCoeff_surj) | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.MvPowerSeries.Inverse",
"Mathlib.RingTheory.PowerSeries.NoZeroDivisors",
"Mathlib.RingTheory.LocalRing.ResidueField.Defs",
"Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity",
"Mathl... | Mathlib/RingTheory/PowerSeries/Inverse.lean | residueFieldOfPowerSeries | The ring isomorphism between the residue field of the ring of power series valued in a field `K`
and `K` itself. |
span_X_isPrime : (Ideal.span ({X} : Set R⟦X⟧)).IsPrime := by
suffices Ideal.span ({X} : Set R⟦X⟧) = RingHom.ker constantCoeff by
rw [this]
exact RingHom.ker_isPrime _
apply Ideal.ext
intro φ
rw [RingHom.mem_ker, Ideal.mem_span_singleton, X_dvd_iff] | theorem | RingTheory | [
"Mathlib.RingTheory.PowerSeries.Order",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/PowerSeries/NoZeroDivisors.lean | span_X_isPrime | The ideal spanned by the variable in the power series ring
over an integral domain is a prime ideal. |
X_prime : Prime (X : R⟦X⟧) := by
rw [← Ideal.span_singleton_prime]
· exact span_X_isPrime
· intro h
simpa [map_zero (coeff 1)] using congr_arg (coeff 1) h | theorem | RingTheory | [
"Mathlib.RingTheory.PowerSeries.Order",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/PowerSeries/NoZeroDivisors.lean | X_prime | The variable of the power series ring over an integral domain is prime. |
X_irreducible : Irreducible (X : R⟦X⟧) := X_prime.irreducible | theorem | RingTheory | [
"Mathlib.RingTheory.PowerSeries.Order",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/PowerSeries/NoZeroDivisors.lean | X_irreducible | The variable of the power series ring over an integral domain is irreducible. |
rescale_injective {a : R} (ha : a ≠ 0) : Function.Injective (rescale a) := by
intro p q h
rw [PowerSeries.ext_iff] at *
intro n
specialize h n
rwa [coeff_rescale, coeff_rescale, mul_right_inj' <| pow_ne_zero _ ha] at h | theorem | RingTheory | [
"Mathlib.RingTheory.PowerSeries.Order",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/PowerSeries/NoZeroDivisors.lean | rescale_injective | null |
exists_coeff_ne_zero_iff_ne_zero : (∃ n : ℕ, coeff n φ ≠ 0) ↔ φ ≠ 0 := by
refine not_iff_not.mp ?_
push_neg
simp | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | exists_coeff_ne_zero_iff_ne_zero | null |
order (φ : R⟦X⟧) : ℕ∞ :=
letI := Classical.decEq R
letI := Classical.decEq R⟦X⟧
if h : φ = 0 then ⊤ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h) | def | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | order | The order of a formal power series `φ` is the greatest `n : PartENat`
such that `X^n` divides `φ`. The order is `⊤` if and only if `φ = 0`. |
@[simp]
order_zero : order (0 : R⟦X⟧) = ⊤ :=
dif_pos rfl | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | order_zero | The order of the `0` power series is infinite. |
order_finite_iff_ne_zero : (order φ < ⊤) ↔ φ ≠ 0 := by
simp only [order]
split_ifs with h <;> simpa | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | order_finite_iff_ne_zero | null |
@[simp]
order_eq_top {φ : R⟦X⟧} : φ.order = ⊤ ↔ φ = 0 := by
simpa using order_finite_iff_ne_zero.not_left | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | order_eq_top | The `0` power series is the unique power series with infinite order. |
coe_toNat_order {φ : R⟦X⟧} (hf : φ ≠ 0) : φ.order.toNat = φ.order := by
rw [ENat.coe_toNat_eq_self.mpr (order_eq_top.not.mpr hf)] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | coe_toNat_order | null |
coeff_order (h : φ ≠ 0) : coeff φ.order.toNat φ ≠ 0 := by
classical
simp only [order, h, not_false_iff, dif_neg]
generalize_proofs h
exact Nat.find_spec h | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | coeff_order | If the order of a formal power series is finite,
then the coefficient indexed by the order is nonzero. |
order_le (n : ℕ) (h : coeff n φ ≠ 0) : order φ ≤ n := by
classical
rw [order, dif_neg]
· simpa using ⟨n, le_rfl, h⟩
· exact exists_coeff_ne_zero_iff_ne_zero.mp ⟨n, h⟩ | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | order_le | If the `n`th coefficient of a formal power series is nonzero,
then the order of the power series is less than or equal to `n`. |
coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff n φ = 0 := by
contrapose! h
exact order_le _ h | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | coeff_of_lt_order | The `n`th coefficient of a formal power series is `0` if `n` is strictly
smaller than the order of the power series. |
coeff_of_lt_order_toNat (n : ℕ) (h : n < φ.order.toNat) : coeff n φ = 0 := by
by_cases h' : φ = 0
· simp [h']
· refine coeff_of_lt_order _ ?_
rwa [← coe_toNat_order h', ENat.coe_lt_coe] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | coeff_of_lt_order_toNat | null |
nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff i φ = 0) : ↑n ≤ order φ := by
classical
simp only [order]
split_ifs
· simp
· simpa [Nat.le_find_iff] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | nat_le_order | The order of a formal power series is at least `n` if
the `i`th coefficient is `0` for all `i < n`. |
le_order (φ : R⟦X⟧) (n : ℕ∞) (h : ∀ i : ℕ, ↑i < n → coeff i φ = 0) :
n ≤ order φ := by
cases n with
| top => simpa using ext (by simpa using h)
| coe n =>
convert nat_le_order φ n _
simpa using h | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.RingTheory.PowerSeries.Basic"
] | Mathlib/RingTheory/PowerSeries/Order.lean | le_order | The order of a formal power series is at least `n` if
the `i`th coefficient is `0` for all `i < n`. |
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