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 ⌀ |
|---|---|---|---|---|---|---|
wronskian_add_right (a b c : R[X]) : wronskian a (b + c) = wronskian a b + wronskian a c :=
(wronskianBilin R a).map_add b c | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | wronskian_add_right | null |
wronskian_add_left (a b c : R[X]) : wronskian (a + b) c = wronskian a c + wronskian b c :=
(wronskianBilin R).map_add₂ a b c | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | wronskian_add_left | null |
wronskian_self_eq_zero (a : R[X]) : wronskian a a = 0 := by
rw [wronskian, mul_comm, sub_self] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | wronskian_self_eq_zero | null |
isAlt_wronskianBilin : (wronskianBilin R).IsAlt := wronskian_self_eq_zero | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | isAlt_wronskianBilin | null |
wronskian_neg_eq (a b : R[X]) : -wronskian a b = wronskian b a :=
LinearMap.IsAlt.neg isAlt_wronskianBilin a b | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | wronskian_neg_eq | null |
wronskian_eq_of_sum_zero {a b c : R[X]} (hAdd : a + b + c = 0) :
wronskian a b = wronskian b c := isAlt_wronskianBilin.eq_of_add_add_eq_zero hAdd | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | wronskian_eq_of_sum_zero | null |
degree_wronskian_lt_add {a b : R[X]} (ha : a ≠ 0) (hb : b ≠ 0) :
(wronskian a b).degree < a.degree + b.degree := by
calc
(wronskian a b).degree ≤ max (a * derivative b).degree (derivative a * b).degree :=
Polynomial.degree_sub_le _ _
_ < a.degree + b.degree := by
rw [max_lt_iff]
constructor
case left =>
apply lt_of_le_of_lt
· exact degree_mul_le a (derivative b)
· rw [← Polynomial.degree_ne_bot] at ha
rw [WithBot.add_lt_add_iff_left ha]
exact Polynomial.degree_derivative_lt hb
case right =>
apply lt_of_le_of_lt
· exact degree_mul_le (derivative a) b
· rw [← Polynomial.degree_ne_bot] at hb
rw [WithBot.add_lt_add_iff_right hb]
exact Polynomial.degree_derivative_lt ha | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | degree_wronskian_lt_add | Degree of `W(a,b)` is strictly less than the sum of degrees of `a` and `b` (both nonzero). |
natDegree_wronskian_lt_add {a b : R[X]} (hw : wronskian a b ≠ 0) :
(wronskian a b).natDegree < a.natDegree + b.natDegree := by
have ha : a ≠ 0 := by intro h; subst h; rw [wronskian_zero_left] at hw; exact hw rfl
have hb : b ≠ 0 := by intro h; subst h; rw [wronskian_zero_right] at hw; exact hw rfl
rw [← WithBot.coe_lt_coe, WithBot.coe_add]
convert ← degree_wronskian_lt_add ha hb
· exact Polynomial.degree_eq_natDegree hw
· exact Polynomial.degree_eq_natDegree ha
· exact Polynomial.degree_eq_natDegree hb | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | natDegree_wronskian_lt_add | `natDegree` version of the above theorem.
Note this would be false with just `(ha : a ≠ 0) (hb : b ≠ 0),
as when `a = b = 1` we have `(wronskian a b).natDegree = a.natDegree = b.natDegree = 0`. |
_root_.IsCoprime.wronskian_eq_zero_iff
[NoZeroDivisors R] {a b : R[X]} (hc : IsCoprime a b) :
wronskian a b = 0 ↔ derivative a = 0 ∧ derivative b = 0 where
mp hw := by
rw [wronskian, sub_eq_iff_eq_add, zero_add] at hw
constructor
· rw [← dvd_derivative_iff]
apply hc.dvd_of_dvd_mul_right
rw [← hw]; exact dvd_mul_right _ _
· rw [← dvd_derivative_iff]
apply hc.symm.dvd_of_dvd_mul_left
rw [hw]; exact dvd_mul_left _ _
mpr hdab := by
obtain ⟨hda, hdb⟩ := hdab
rw [wronskian]
rw [hda, hdb]; simp only [mul_zero, zero_mul, sub_self] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Derivative",
"Mathlib.LinearAlgebra.SesquilinearForm",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Polynomial/Wronskian.lean | _root_.IsCoprime.wronskian_eq_zero_iff | For coprime polynomials `a` and `b`, their Wronskian is zero
if and only if their derivatives are zeros. |
@[ext]
PolynomialLaw (R : Type u) [CommSemiring R]
(M : Type*) [AddCommMonoid M] [Module R M] (N : Type*) [AddCommMonoid N] [Module R N] where
/-- The functions `S ⊗[R] M → S ⊗[R] N` underlying a polynomial law -/
toFun' (S : Type u) [CommSemiring S] [Algebra R S] : S ⊗[R] M → S ⊗[R] N
/-- The compatibility relations between the functions underlying a polynomial law -/
isCompat' {S : Type u} [CommSemiring S] [Algebra R S]
{S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') :
φ.toLinearMap.rTensor N ∘ toFun' S = toFun' S' ∘ φ.toLinearMap.rTensor M := by aesop | structure | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | PolynomialLaw | A polynomial law `M →ₚₗ[R] N` between `R`-modules is a functorial family of maps
`S ⊗[R] M → S ⊗[R] N`, for all `R`-algebras `S`.
For universe reasons, `S` has to be restricted to the same universe as `R`. |
id : M →ₚₗ[R] M where
toFun' S _ _ := _root_.id | def | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | id | `M →ₚₗ[R] N` is the type of `R`-polynomial laws from `M` to `N`. -/
notation:25 M " →ₚₗ[" R:25 "] " N:0 => PolynomialLaw R M N
@[local simp]
theorem PolynomialLaw.isCompat_apply'
{R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]
{N : Type*} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N}
{S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S']
(φ : S →ₐ[R] S') (x : S ⊗[R] M) :
(φ.toLinearMap.rTensor N) ((f.toFun' S) x) = (f.toFun' S') (φ.toLinearMap.rTensor M x) := by
simpa only using congr_fun (f.isCompat' φ) x
attribute [local simp] PolynomialLaw.isCompat_apply'
namespace PolynomialLaw
section Module
section CommSemiring
variable {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]
{N : Type*} [AddCommMonoid N] [Module R N] (r a b : R) (f g : M →ₚₗ[R] N)
instance : Zero (M →ₚₗ[R] N) := ⟨{ toFun' _ := 0 }⟩
@[simp]
theorem zero_def (S : Type u) [CommSemiring S] [Algebra R S] :
(0 : PolynomialLaw R M N).toFun' S = 0 := rfl
instance : Inhabited (PolynomialLaw R M N) := ⟨Zero.zero⟩
/-- The identity as a polynomial law |
id_apply' {S : Type u} [CommSemiring S] [Algebra R S] :
(id : M →ₚₗ[R] M).toFun' S = _root_.id := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | id_apply' | null |
noncomputable add : M →ₚₗ[R] N where
toFun' S _ _ := f.toFun' S + g.toFun' S | def | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | add | The sum of two polynomial laws |
@[simp]
add_def (S : Type u) [CommSemiring S] [Algebra R S] :
(f + g).toFun' S = f.toFun' S + g.toFun' S := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | add_def | null |
add_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :
(f + g).toFun' S m = f.toFun' S m + g.toFun' S m := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | add_def_apply | null |
smul : M →ₚₗ[R] N where
toFun' S _ _ := r • f.toFun' S | def | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | smul | External multiplication of a `f : M →ₚₗ[R] N` by `r : R` |
@[simp]
smul_def (S : Type u) [CommSemiring S] [Algebra R S] :
(r • f).toFun' S = r • f.toFun' S := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | smul_def | null |
smul_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :
(r • f).toFun' S m = r • f.toFun' S m := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | smul_def_apply | null |
add_smul : (a + b) • f = a • f + b • f := by
ext; simp only [add_def, smul_def, _root_.add_smul] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | add_smul | null |
zero_smul : (0 : R) • f = 0 := by
ext S; simp only [smul_def, _root_.zero_smul, zero_def, Pi.zero_apply] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | zero_smul | null |
one_smul : (1 : R) • f = f := by
ext S; simp only [smul_def, Pi.smul_apply, _root_.one_smul] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | one_smul | null |
noncomputable neg : M →ₚₗ[R] N where
toFun' S _ _ := (-1 : R) • f.toFun' S | def | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | neg | The opposite of a polynomial law |
@[simp]
neg_def (S : Type u) [CommSemiring S] [Algebra R S] :
(-f).toFun' S = (-1 : R) • f.toFun' S := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | neg_def | null |
ground : M → N := (TensorProduct.lid R N) ∘ (f.toFun' R) ∘ (TensorProduct.lid R M).symm | def | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | ground | The map `M → N` associated with a `f : M →ₚₗ[R] N` (essentially, `f.toFun' R`) |
ground_apply (m : M) : f.ground m = TensorProduct.lid R N (f.toFun' R (1 ⊗ₜ[R] m)) := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | ground_apply | null |
one_tmul_ground_apply' {S : Type u} [CommSemiring S] [Algebra R S] (x : M) :
1 ⊗ₜ (f.ground x) = (f.toFun' S) (1 ⊗ₜ x) := by
rw [ground_apply]
convert f.isCompat_apply' (Algebra.algHom R R S) (1 ⊗ₜ[R] x)
· simp only [includeRight_lid]
· rw [rTensor_tmul, toLinearMap_apply, map_one] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | one_tmul_ground_apply' | null |
lground : (M →ₚₗ[R] N) →ₗ[R] (M → N) where
toFun := ground
map_add' x y := by ext m; simp [ground]
map_smul' r x := by ext m; simp [ground] | def | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | lground | The map ground assigning a function `M → N` to a polynomial map `f : M →ₚₗ[R] N` as a
linear map. |
ground_id : (id : M →ₚₗ[R] M).ground = _root_.id := by
ext; simp [ground_apply, id_apply'] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | ground_id | null |
ground_id_apply (m : M) : (id : M →ₚₗ[R] M).ground m = m := by
rw [ground_id, id_eq] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | ground_id_apply | null |
comp (g : N →ₚₗ[R] P) (f : M →ₚₗ[R] N) : M →ₚₗ[R] P where
toFun' S _ _ := (g.toFun' S).comp (f.toFun' S)
isCompat' φ := by ext; simp only [Function.comp_apply, isCompat_apply'] | def | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | comp | Composition of polynomial maps. |
comp_toFun' (S : Type u) [CommSemiring S] [Algebra R S] :
(g.comp f).toFun' S = (g.toFun' S).comp (f.toFun' S) := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | comp_toFun' | null |
comp_assoc : h.comp (g.comp f) = (h.comp g).comp f := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | comp_assoc | null |
comp_id : g.comp id = g := by ext; rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | comp_id | null |
id_comp : id.comp f = f := by ext; rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/PolynomialLaw/Basic.lean | id_comp | null |
PowerSeries (R : Type*) :=
MvPowerSeries Unit R | abbrev | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries | Formal power series over a coefficient type `R` |
coeff (n : ℕ) : R⟦X⟧ →ₗ[R] R :=
MvPowerSeries.coeff (single () n) | def | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff | `R⟦X⟧` is notation for `PowerSeries R`,
the semiring of formal power series in one variable over a semiring `R`.
-/
scoped notation:9000 R "⟦X⟧" => PowerSeries R
instance [Inhabited R] : Inhabited R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [Zero R] : Zero R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [AddMonoid R] : AddMonoid R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [AddGroup R] : AddGroup R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommMonoid R] : AddCommMonoid R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommGroup R] : AddCommGroup R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [Semiring R] : Semiring R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [CommSemiring R] : CommSemiring R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [Ring R] : Ring R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [CommRing R] : CommRing R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance [Nontrivial R] : Nontrivial R⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]
[IsScalarTower R S A] : IsScalarTower R S A⟦X⟧ :=
Pi.isScalarTower
instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A⟦X⟧ := by
dsimp only [PowerSeries]
infer_instance
end
section Semiring
variable [Semiring R]
/-- The `n`th coefficient of a formal power series. |
monomial (n : ℕ) : R →ₗ[R] R⟦X⟧ :=
MvPowerSeries.monomial (single () n) | def | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | monomial | The `n`th monomial with coefficient `a` as formal power series. |
coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) :
coeff (R := R) n = MvPowerSeries.coeff s := by
rw [coeff, ← h, ← Finsupp.unique_single s] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_def | null |
@[ext]
ext {φ ψ : R⟦X⟧} (h : ∀ n, coeff n φ = coeff n ψ) : φ = ψ :=
MvPowerSeries.ext fun n => by
rw [← coeff_def]
· apply h
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | ext | Two formal power series are equal if all their coefficients are equal. |
forall_coeff_eq_zero (φ : R⟦X⟧) : (∀ n, coeff n φ = 0) ↔ φ = 0 :=
⟨fun h => ext h, fun h => by simp [h]⟩ | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | forall_coeff_eq_zero | null |
mk {R} (f : ℕ → R) : R⟦X⟧ := fun s => f (s ())
@[simp] | def | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | mk | Two formal power series are equal if all their coefficients are equal. -/
add_decl_doc PowerSeries.ext_iff
instance [Subsingleton R] : Subsingleton R⟦X⟧ := by
simp only [subsingleton_iff, PowerSeries.ext_iff]
subsingleton
/-- Constructor for formal power series. |
coeff_mk (n : ℕ) (f : ℕ → R) : coeff n (mk f) = f n :=
congr_arg f Finsupp.single_eq_same | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_mk | null |
coeff_monomial (m n : ℕ) (a : R) : coeff m (monomial n a) = if m = n then a else 0 :=
calc
coeff m (monomial n a) = _ := MvPowerSeries.coeff_monomial _ _ _
_ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_monomial | null |
monomial_eq_mk (n : ℕ) (a : R) : monomial n a = mk fun m => if m = n then a else 0 :=
ext fun m => by rw [coeff_monomial, coeff_mk]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | monomial_eq_mk | null |
coeff_monomial_same (n : ℕ) (a : R) : coeff n (monomial n a) = a :=
MvPowerSeries.coeff_monomial_same _ _
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_monomial_same | null |
coeff_comp_monomial (n : ℕ) : (coeff (R := R) n).comp (monomial n) = LinearMap.id :=
LinearMap.ext <| coeff_monomial_same n | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_comp_monomial | null |
monomial_mul_monomial (m n : ℕ) (a b : R) :
monomial m a * monomial n b = monomial (m + n) (a * b) := by
simpa [monomial] using
MvPowerSeries.monomial_mul_monomial (Finsupp.single () m) (Finsupp.single () n) a b | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | monomial_mul_monomial | null |
constantCoeff : R⟦X⟧ →+* R :=
MvPowerSeries.constantCoeff | def | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff | The constant coefficient of a formal power series. |
C : R →+* R⟦X⟧ :=
MvPowerSeries.C
@[simp] lemma algebraMap_eq {R : Type*} [CommSemiring R] : algebraMap R R⟦X⟧ = C := rfl | def | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | C | The constant formal power series. |
X : R⟦X⟧ :=
MvPowerSeries.X () | def | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X | The variable of the formal power series ring. |
commute_X (φ : R⟦X⟧) : Commute φ X :=
MvPowerSeries.commute_X _ _ | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | commute_X | null |
X_mul {φ : R⟦X⟧} : X * φ = φ * X :=
MvPowerSeries.X_mul | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_mul | null |
commute_X_pow (φ : R⟦X⟧) (n : ℕ) : Commute φ (X ^ n) :=
MvPowerSeries.commute_X_pow _ _ _ | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | commute_X_pow | null |
X_pow_mul {φ : R⟦X⟧} {n : ℕ} : X ^ n * φ = φ * X ^ n :=
MvPowerSeries.X_pow_mul
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_pow_mul | null |
coeff_zero_eq_constantCoeff : ⇑(coeff (R := R) 0) = constantCoeff := by
rw [coeff, Finsupp.single_zero]
rfl | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_zero_eq_constantCoeff | null |
coeff_zero_eq_constantCoeff_apply (φ : R⟦X⟧) : coeff 0 φ = constantCoeff φ := by
rw [coeff_zero_eq_constantCoeff]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_zero_eq_constantCoeff_apply | null |
monomial_zero_eq_C : ⇑(monomial (R := R) 0) = C := by
rw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C]
rfl | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | monomial_zero_eq_C | null |
monomial_zero_eq_C_apply (a : R) : monomial 0 a = C a := by simp | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | monomial_zero_eq_C_apply | null |
coeff_C (n : ℕ) (a : R) : coeff n (C a : R⟦X⟧) = if n = 0 then a else 0 := by
rw [← monomial_zero_eq_C_apply, coeff_monomial]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_C | null |
coeff_zero_C (a : R) : coeff 0 (C a) = a := by
rw [coeff_C, if_pos rfl] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_zero_C | null |
coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff n (C a) = 0 := by
rw [coeff_C, if_neg h]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_ne_zero_C | null |
coeff_succ_C {a : R} {n : ℕ} : coeff (n + 1) (C a) = 0 :=
coeff_ne_zero_C n.succ_ne_zero | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_succ_C | null |
C_injective : Function.Injective (C (R := R)) := by
intro a b H
simp_rw [PowerSeries.ext_iff] at H
simpa only [coeff_zero_C] using H 0 | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | C_injective | null |
protected subsingleton_iff : Subsingleton R⟦X⟧ ↔ Subsingleton R := by
refine ⟨fun h ↦ ?_, fun _ ↦ inferInstance⟩
rw [subsingleton_iff] at h ⊢
exact fun a b ↦ C_injective (h (C a) (C b)) | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | subsingleton_iff | null |
X_eq : (X : R⟦X⟧) = monomial 1 1 :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_eq | null |
coeff_X (n : ℕ) : coeff n (X : R⟦X⟧) = if n = 1 then 1 else 0 := by
rw [X_eq, coeff_monomial]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_X | null |
coeff_zero_X : coeff 0 (X : R⟦X⟧) = 0 := by
rw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_zero_X | null |
coeff_one_X : coeff 1 (X : R⟦X⟧) = 1 := by rw [coeff_X, if_pos rfl]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_one_X | null |
X_ne_zero [Nontrivial R] : (X : R⟦X⟧) ≠ 0 := fun H => by
simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff 1) H | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_ne_zero | null |
X_pow_eq (n : ℕ) : (X : R⟦X⟧) ^ n = monomial n 1 :=
MvPowerSeries.X_pow_eq _ n | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_pow_eq | null |
coeff_X_pow (m n : ℕ) : coeff m ((X : R⟦X⟧) ^ n) = if m = n then 1 else 0 := by
rw [X_pow_eq, coeff_monomial]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_X_pow | null |
coeff_X_pow_self (n : ℕ) : coeff n ((X : R⟦X⟧) ^ n) = 1 := by
rw [coeff_X_pow, if_pos rfl]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_X_pow_self | null |
coeff_one (n : ℕ) : coeff n (1 : R⟦X⟧) = if n = 0 then 1 else 0 :=
coeff_C n 1 | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_one | null |
coeff_zero_one : coeff 0 (1 : R⟦X⟧) = 1 :=
coeff_zero_C 1 | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_zero_one | null |
coeff_mul (n : ℕ) (φ ψ : R⟦X⟧) :
coeff n (φ * ψ) = ∑ p ∈ antidiagonal n, coeff p.1 φ * coeff p.2 ψ := by
refine (MvPowerSeries.coeff_mul _ φ ψ).trans ?_
rw [Finsupp.antidiagonal_single, Finset.sum_map]
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_mul | null |
coeff_mul_C (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff n (φ * C a) = coeff n φ * a :=
MvPowerSeries.coeff_mul_C _ φ a
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_mul_C | null |
coeff_C_mul (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff n (C a * φ) = a * coeff n φ :=
MvPowerSeries.coeff_C_mul _ φ a
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_C_mul | null |
coeff_smul {S : Type*} [Semiring S] [Module R S] (n : ℕ) (φ : PowerSeries S) (a : R) :
coeff n (a • φ) = a • coeff n φ :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_smul | null |
constantCoeff_smul {S : Type*} [Semiring S] [Module R S] (φ : PowerSeries S) (a : R) :
constantCoeff (a • φ) = a • constantCoeff φ :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_smul | null |
smul_eq_C_mul (f : R⟦X⟧) (a : R) : a • f = C a * f := by
ext
simp
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | smul_eq_C_mul | null |
coeff_succ_mul_X (n : ℕ) (φ : R⟦X⟧) : coeff (n + 1) (φ * X) = coeff n φ := by
simp only [coeff, Finsupp.single_add]
convert φ.coeff_add_mul_monomial (single () n) (single () 1) _
rw [mul_one]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_succ_mul_X | null |
coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff (n + 1) (X * φ) = coeff n φ := by
simp only [coeff, Finsupp.single_add, add_comm n 1]
convert φ.coeff_add_monomial_mul (single () 1) (single () n) _
rw [one_mul] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_succ_X_mul | null |
mul_X_cancel {φ ψ : R⟦X⟧} (h : φ * X = ψ * X) : φ = ψ := by
rw [PowerSeries.ext_iff] at h ⊢
intro n
simpa using h (n + 1) | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | mul_X_cancel | null |
mul_X_injective : Function.Injective (· * X : R⟦X⟧ → R⟦X⟧) :=
fun _ _ ↦ mul_X_cancel | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | mul_X_injective | null |
mul_X_inj {φ ψ : R⟦X⟧} : φ * X = ψ * X ↔ φ = ψ :=
mul_X_injective.eq_iff | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | mul_X_inj | null |
X_mul_cancel {φ ψ : R⟦X⟧} (h : X * φ = X * ψ) : φ = ψ := by
rw [PowerSeries.ext_iff] at h ⊢
intro n
simpa using h (n + 1) | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_mul_cancel | null |
X_mul_injective : Function.Injective (X * · : R⟦X⟧ → R⟦X⟧) :=
fun _ _ ↦ X_mul_cancel | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_mul_injective | null |
X_mul_inj {φ ψ : R⟦X⟧} : X * φ = X * ψ ↔ φ = ψ :=
X_mul_injective.eq_iff
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | X_mul_inj | null |
constantCoeff_C (a : R) : constantCoeff (C a) = a :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_C | null |
constantCoeff_comp_C : constantCoeff.comp C = RingHom.id R :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_comp_C | null |
constantCoeff_zero : constantCoeff (R := R) 0 = 0 :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_zero | null |
constantCoeff_one : constantCoeff (R := R) 1 = 1 :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_one | null |
constantCoeff_X : constantCoeff (R := R) X = 0 :=
MvPowerSeries.coeff_zero_X _
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_X | null |
constantCoeff_mk {f : ℕ → R} : constantCoeff (mk f) = f 0 := rfl | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_mk | null |
coeff_zero_mul_X (φ : R⟦X⟧) : coeff 0 (φ * X) = 0 := by simp | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_zero_mul_X | null |
coeff_zero_X_mul (φ : R⟦X⟧) : coeff 0 (X * φ) = 0 := by simp | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_zero_X_mul | null |
constantCoeff_surj : Function.Surjective (constantCoeff (R := R)) :=
fun r => ⟨C r, constantCoeff_C r⟩ | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | constantCoeff_surj | null |
coeff_C_mul_X_pow (x : R) (k n : ℕ) :
coeff n (C x * X ^ k : R⟦X⟧) = if n = k then x else 0 := by
simp [X_pow_eq, coeff_monomial]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_C_mul_X_pow | null |
coeff_mul_X_pow (p : R⟦X⟧) (n d : ℕ) :
coeff (d + n) (p * X ^ n) = coeff d p := by
rw [coeff_mul, Finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one]
· rintro ⟨i, j⟩ h1 h2
rw [coeff_X_pow, if_neg, mul_zero]
rintro rfl
apply h2
rw [mem_antidiagonal, add_right_cancel_iff] at h1
subst h1
rfl
· exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_mul_X_pow | null |
coeff_X_pow_mul (p : R⟦X⟧) (n d : ℕ) :
coeff (d + n) (X ^ n * p) = coeff d p := by
rw [coeff_mul, Finset.sum_eq_single (n, d), coeff_X_pow, if_pos rfl, one_mul]
· rintro ⟨i, j⟩ h1 h2
rw [coeff_X_pow, if_neg, zero_mul]
rintro rfl
apply h2
rw [mem_antidiagonal, add_comm, add_right_cancel_iff] at h1
subst h1
rfl
· rw [add_comm]
exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim | theorem | RingTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Basic",
"Mathlib.RingTheory.MvPowerSeries.Basic",
"Mathlib.Tactic.MoveAdd",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.RingTheory.Ideal.Basic"
] | Mathlib/RingTheory/PowerSeries/Basic.lean | coeff_X_pow_mul | null |
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