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[Merged by Bors] - refactor(Analysis/Calculus/FormalMultilinearSeries): generalize to additive monoids #9467

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[Merged by Bors] - refactor(Analysis/Calculus/FormalMultilinearSeries): generalize to additive monoids #9467

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refactor(Analysis/Calculus/FormalMultilinearSeries): generalize to ad…
eric-wieser Jan 5, 2024
2003ead
regeneralise
YaelDillies Feb 18, 2025
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more
YaelDillies Feb 18, 2025
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more!
YaelDillies Feb 18, 2025
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4 changes: 2 additions & 2 deletions Mathlib/Analysis/Analytic/Basic.lean
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Expand Up @@ -79,9 +79,9 @@ open Topology NNReal Filter ENNReal Set Asymptotics

namespace FormalMultilinearSeries

variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [Semiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [IsTopologicalAddGroup E] [IsTopologicalAddGroup F]
variable [ContinuousAdd E] [ContinuousAdd F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]

/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
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56 changes: 33 additions & 23 deletions Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
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Expand Up @@ -33,23 +33,23 @@ variable {𝕜 : Type u} {𝕜' : Type u'} {E : Type v} {F : Type w} {G : Type x

section

variable [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [IsTopologicalAddGroup E]
[ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F]
[IsTopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [AddCommGroup G] [Module 𝕜 G]
[TopologicalSpace G] [IsTopologicalAddGroup G] [ContinuousConstSMul 𝕜 G]
variable [Semiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousAdd E]
[ContinuousConstSMul 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F]
[ContinuousAdd F] [ContinuousConstSMul 𝕜 F] [AddCommMonoid G] [Module 𝕜 G]
[TopologicalSpace G] [ContinuousAdd G] [ContinuousConstSMul 𝕜 G]

/-- A formal multilinear series over a field `𝕜`, from `E` to `F`, is given by a family of
multilinear maps from `E^n` to `F` for all `n`. -/
@[nolint unusedArguments]
def FormalMultilinearSeries (𝕜 : Type*) (E : Type*) (F : Type*) [Ring 𝕜] [AddCommGroup E]
[Module 𝕜 E] [TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E]
[AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [IsTopologicalAddGroup F]
def FormalMultilinearSeries (𝕜 : Type*) (E : Type*) (F : Type*) [Semiring 𝕜] [AddCommMonoid E]
[Module 𝕜 E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousConstSMul 𝕜 E]
[AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F] [ContinuousAdd F]
[ContinuousConstSMul 𝕜 F] :=
∀ n : ℕ, E[×n]→L[𝕜] F

-- Porting note: was `deriving`
instance : AddCommGroup (FormalMultilinearSeries 𝕜 E F) :=
inferInstanceAs <| AddCommGroup <| ∀ n : ℕ, E[×n]→L[𝕜] F
instance : AddCommMonoid (FormalMultilinearSeries 𝕜 E F) :=
inferInstanceAs <| AddCommMonoid <| ∀ n : ℕ, E[×n]→L[𝕜] F

instance : Inhabited (FormalMultilinearSeries 𝕜 E F) :=
⟨0⟩
Expand All @@ -69,15 +69,9 @@ the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`. -/
@[simp, nolint simpNF]
theorem zero_apply (n : ℕ) : (0 : FormalMultilinearSeries 𝕜 E F) n = 0 := rfl

@[simp]
theorem neg_apply (f : FormalMultilinearSeries 𝕜 E F) (n : ℕ) : (-f) n = - f n := rfl

@[simp]
theorem add_apply (p q : FormalMultilinearSeries 𝕜 E F) (n : ℕ) : (p + q) n = p n + q n := rfl

@[simp]
theorem sub_apply (p q : FormalMultilinearSeries 𝕜 E F) (n : ℕ) : (p - q) n = p n - q n := rfl

@[simp]
theorem smul_apply [Semiring 𝕜'] [Module 𝕜' F] [ContinuousConstSMul 𝕜' F] [SMulCommClass 𝕜 𝕜' F]
(f : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (a : 𝕜') : (a • f) n = a • f n := rfl
Expand Down Expand Up @@ -146,7 +140,7 @@ theorem compContinuousLinearMap_apply (p : FormalMultilinearSeries 𝕜 F G) (u
(v : Fin n → E) : (p.compContinuousLinearMap u) n v = p n (u ∘ v) :=
rfl

variable (𝕜) [Ring 𝕜'] [SMul 𝕜 𝕜']
variable (𝕜) [Semiring 𝕜'] [SMul 𝕜 𝕜']
variable [Module 𝕜' E] [ContinuousConstSMul 𝕜' E] [IsScalarTower 𝕜 𝕜' E]
variable [Module 𝕜' F] [ContinuousConstSMul 𝕜' F] [IsScalarTower 𝕜 𝕜' F]

Expand All @@ -159,6 +153,22 @@ end FormalMultilinearSeries

end

namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [IsTopologicalAddGroup E]
[ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F]
[IsTopologicalAddGroup F] [ContinuousConstSMul 𝕜 F]

instance : AddCommGroup (FormalMultilinearSeries 𝕜 E F) :=
inferInstanceAs <| AddCommGroup <| ∀ n : ℕ, E[×n]→L[𝕜] F

@[simp]
theorem neg_apply (f : FormalMultilinearSeries 𝕜 E F) (n : ℕ) : (-f) n = - f n := rfl

@[simp]
theorem sub_apply (f g : FormalMultilinearSeries 𝕜 E F) (n : ℕ) : (f - g) n = f n - g n := rfl

end FormalMultilinearSeries

namespace FormalMultilinearSeries

variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
Expand Down Expand Up @@ -190,10 +200,10 @@ end FormalMultilinearSeries

section

variable [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [IsTopologicalAddGroup E]
[ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F]
[IsTopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [AddCommGroup G] [Module 𝕜 G]
[TopologicalSpace G] [IsTopologicalAddGroup G] [ContinuousConstSMul 𝕜 G]
variable [Semiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousAdd E]
[ContinuousConstSMul 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F]
[ContinuousAdd F] [ContinuousConstSMul 𝕜 F] [AddCommMonoid G] [Module 𝕜 G]
[TopologicalSpace G] [ContinuousAdd G] [ContinuousConstSMul 𝕜 G]

namespace ContinuousLinearMap

Expand Down Expand Up @@ -235,9 +245,9 @@ namespace FormalMultilinearSeries

section Order

variable [Ring 𝕜] {n : ℕ} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
[IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F]
[TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousConstSMul 𝕜 F]
variable [Semiring 𝕜] {n : ℕ} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
[ContinuousAdd E] [ContinuousConstSMul 𝕜 E] [AddCommMonoid F] [Module 𝕜 F]
[TopologicalSpace F] [ContinuousAdd F] [ContinuousConstSMul 𝕜 F]
{p : FormalMultilinearSeries 𝕜 E F}

/-- The index of the first non-zero coefficient in `p` (or `0` if all coefficients are zero). This
Expand Down