mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: case refine_2.h
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
i : ι
x : G i
⊢ (V i).toLinearMap x ∈ hV.linearIsometry.range

Tactic:
use lp.single 2 i x

Goal state after:
case h
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
i : ι
x : G i
⊢ hV.linearIsometry.toLinearMap (lp.single 2 i x) = (V i).toLinearMap x
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
n : ℤ
hn : n ≠ 0
hT : 0 < T
x : AddCircle T
⊢ ↑(toCircle ↑(n • (T / 2 / ↑n))) = -1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
x : X
n : ℕ
⊢ (D.P n) x =
    ((fun s ↦
          ∑ b ∈ s,
            ({ toFun := coeff b, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (‖succSub D.P b‖ / ‖D.e b‖) ⋯) x •
              D.e b) ∘
        Finset.range)
      n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
⊢ hV.linearIsometry.range = (⨆ i, (V i).range).topologicalClosure

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
x : X
⊢ HasSum
    (fun i ↦ ({ toFun := coeff i, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (‖succSub D.P i‖ / ‖D.e i‖) ⋯) x • D.e i)
    x (SummationFilter.conditional ℕ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
n : ℤ
hn : n ≠ 0
hT : 0 < T
x : AddCircle T
⊢ (fourier n) (x + ↑(T / 2 / ↑n)) = -(fourier n) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
inst✝ : ∀ (i : ι), CompleteSpace (G i)
hVortho : OrthogonalFamily 𝕜 G V
hVtotal : ⊤ ≤ (⨆ i, (V i).range).topologicalClosure
⊢ Function.Surjective ⇑hVortho.linearIsometry

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
⊢ ∀ {a : C(AddCircle T, ℂ)}, a ∈ (adjoin ℂ (range fourier)).carrier → star a ∈ (adjoin ℂ (range fourier)).carrier

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
F : ι → Submodule 𝕜 E
inst✝ : ∀ (i : ι), CompleteSpace ↥(F i)
hFortho : OrthogonalFamily 𝕜 (fun i ↦ ↥(F i)) fun i ↦ (F i).subtypeₗᵢ
hFtotal : ⊤ ≤ (⨆ i, F i).topologicalClosure
⊢ ⊤ ≤ (⨆ i, (F i).subtypeₗᵢ.range).topologicalClosure

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
D : RankOneDecomposition 𝕜 X
⊢ D.basis.proj = D.P

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : IsHilbertSum 𝕜 G V
w : ↥(lp G 2)
⊢ hV.linearIsometryEquiv.symm w = ∑' (i : ι), (V i) (↑w i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: case refine_1
T : ℝ
x : C(AddCircle T, ℂ)
hx : x ∈ ↑(Submonoid.closure (range fourier))
z : AddCircle T
⊢ (fourier 0) z = 1 z

Tactic:
exact fourier_zero

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : IsHilbertSum 𝕜 G V
w : ↥(lp G 2)
⊢ HasSum (fun i ↦ (V i) (↑w i)) (hV.linearIsometryEquiv.symm w)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
inst✝ : DecidableEq ι
hV : IsHilbertSum 𝕜 G V
i : ι
x : G i
⊢ hV.linearIsometryEquiv.symm (lp.single 2 i x) = (V i) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
⊢ Subalgebra.toSubmodule fourierSubalgebra.toSubalgebra = span ℂ (range fourier)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝⁴ : (i : ι) → NormedAddCommGroup (G i)
inst✝³ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝² : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
hV : IsHilbertSum 𝕜 G V
W₀ : Π₀ (i : ι), G i
⊢ hV.linearIsometryEquiv.symm (W₀.sum (lp.single 2)) = W₀.sum fun i ↦ ⇑(V i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
⊢ fourierSubalgebra.SeparatesPoints

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
⊢ (span ℂ (range fourier)).topologicalClosure = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝⁴ : (i : ι) → NormedAddCommGroup (G i)
inst✝³ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝² : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
hV : IsHilbertSum 𝕜 G V
W₀ : Π₀ (i : ι), G i
⊢ ↑(W₀.sum fun a b ↦ hV.linearIsometryEquiv ((V a) b)) = ⇑W₀

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
hsp : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure
⊢ ⊤ ≤ (⨆ i, (LinearIsometry.toSpanSingleton 𝕜 E ⋯).range).topologicalClosure

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
⊢ (span ℂ (range (fourierLp p))).topologicalClosure = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
K : Submodule 𝕜 E
hK : CompleteSpace ↥K
⊢ ∀ (b : Bool), CompleteSpace ↥(bif b then K else Kᗮ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
i j : ℤ
h : ¬i = j
⊢ j + -i ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
⊢ Orthonormal ℂ (fourierLp 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
⊢ ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
K : Submodule 𝕜 E
hK : CompleteSpace ↥K
⊢ IsHilbertSum 𝕜 (fun b ↦ ↥(bif b then K else Kᗮ)) fun b ↦ (bif b then K else Kᗮ).subtypeₗᵢ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
b₁ b₂ : E ≃ₗᵢ[𝕜] ↥(lp (fun x ↦ 𝕜) 2)
h : (fun b i ↦ b.repr.symm (lp.single 2 i 1)) { repr := b₁ } = (fun b i ↦ b.repr.symm (lp.single 2 i 1)) { repr := b₂ }
⊢ { repr := b₁ } = { repr := b₂ }

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : DecidableEq ι
b : HilbertBasis ι 𝕜 E
i : ι
⊢ b.repr.symm (lp.single 2 i 1) = b i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : DecidableEq ι
b : HilbertBasis ι 𝕜 E
i : ι
⊢ b.repr (b i) = lp.single 2 i 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
⊢ fourierCoeff f n = (1 / T) • ∫ (x : ℝ) in a..a + T, (fourier (-n)) ↑x • f ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
v : E
i : ι
⊢ ↑(b.repr v) i = ⟪b i, v⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
⊢ Orthonormal 𝕜 ⇑b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
f : ↥(lp (fun x ↦ 𝕜) 2)
H : (fun i ↦ ↑f i • b i) = fun b_1 ↦ ↑b.repr.symm ((fun i ↦ lp.single 2 i (↑f i)) b_1)
⊢ HasSum (fun i ↦ ↑f i • b i) (b.repr.symm f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
hf : Integrable f haarAddCircle
n : ℤ
⊢ Integrable (fun t ↦ (fourier n) t • f t) haarAddCircle

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
f : ↥(lp (fun x ↦ 𝕜) 2)
i : ι
⊢ NormedSpace 𝕜 ↥(lp (fun _i ↦ 𝕜) 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : AddCircle T → E
hf : Integrable f haarAddCircle
hg : Integrable g haarAddCircle
⊢ fourierCoeff (f + g) = fourierCoeff f + fourierCoeff g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
ι : Type u_2
f : ι → AddCircle T → E
a : ι
s : Finset ι
ha : a ∉ s
iha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)
hf : ∀ i ∈ insert a s, Integrable (f i) haarAddCircle
⊢ ?m.70

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
f : ↥(lp (fun x ↦ 𝕜) 2)
⊢ HasSum (fun i ↦ ↑f i • b i) (b.repr.symm f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
x : E
⊢ HasSum (fun i ↦ ↑(b.repr x) i • b i) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
ι : Type u_2
s : Finset ι
f : ι → AddCircle T → E
hf : ∀ i ∈ s, Integrable (f i) haarAddCircle
⊢ fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
⊢ (span 𝕜 (Set.range ⇑b)).topologicalClosure = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
c : ℂ
n : ℤ
⊢ fourierCoeff (c • f) n = c • fourierCoeff f n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
hab : a < b
f : ℝ → E
n : ℤ
⊢ 0 < b - a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
n : ℤ
hab : a < b
⊢ 0 < b - a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
x y : E
⊢ HasSum (fun i ↦ ⟪x, b i⟫ * ⟪b i, y⟫) ⟪x, y⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : Fintype ι
b : HilbertBasis ι 𝕜 E
⊢ ⊤ ≤ span 𝕜 (Set.range ⇑b)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
b : HilbertBasis ι 𝕜 E
i j : ι
⊢ ((innerSL 𝕜) (b i)) (b j) = Pi.single j 1 i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
b : HilbertBasis ι 𝕜 E
x : E
⊢ HasSum (fun i ↦ ((innerSL 𝕜) (b i)) x • b i) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
inst✝ : CompleteSpace ↥U
b : HilbertBasis ι 𝕜 ↥U
x : E
⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : DecidableEq E
b : HilbertBasis ι 𝕜 E
⊢ (span 𝕜 (Set.range ⇑b)).topologicalClosure ≤ (⨆ J, span 𝕜 ↑(Finset.image (⇑b) J)).topologicalClosure

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
n : ℤ
hab : a < b
⊢ fourierCoeffOn hab f n = (1 / (b - a)) • ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x • f x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
c : ℂ
n : ℤ
hab : a < b
⊢ 0 < b - a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
c : ℂ
n : ℤ
hab : a < b
⊢ fourierCoeffOn hab (c • f) n = c • fourierCoeffOn hab f n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
inst✝ : DecidableEq ι
h : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure
i : ι
⊢ ⋯.linearIsometryEquiv.symm (lp.single 2 i 1) = v i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
hsp : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure
⊢ ⇑(HilbertBasis.mk hv hsp) = v

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
hsp : (span 𝕜 (Set.range v))ᗮ = ⊥
⊢ ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
⊢ ⊤ ≤ (span 𝕜 (Set.range ⇑b)).topologicalClosure

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
hab : a < b
f g : ℝ → E
h : f =ᵐ[volume.restrict (Ioc a b)] g
⊢ fourierCoeffOn hab f = fourierCoeffOn hab g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
s : Set E
hs : Orthonormal 𝕜 Subtype.val
w : Set E
hws : w ⊇ s
hw_ortho : Orthonormal 𝕜 Subtype.val
hw_max : ∀ u ⊇ w, Orthonormal 𝕜 Subtype.val → u = w
⊢ (span 𝕜 (Set.range Subtype.val))ᗮ = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
a : ℝ
f : ℝ → ℂ
n : ℤ
⊢ fourierCoeff (liftIoc T a f) n = fourierCoeffOn ⋯ f n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
a : ℝ
f : ℝ → ℂ
n : ℤ
⊢ fourierCoeff (liftIco T a f) n = fourierCoeffOn ⋯ f n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
⊢ 2 ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
i : ℤ
⊢ ↑(fourierBasis.repr f) i = fourierCoeff (↑↑f) i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
⊢ HasSum (fun i ↦ fourierCoeff (↑↑f) i • fourierLp 2 i) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
⊢ HasSum (fun i ↦ ‖↑(fourierBasis.repr f) i‖ ^ 2) (‖fourierBasis.repr f‖ ^ 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
H₁ : HasSum (fun i ↦ ‖↑(fourierBasis.repr f) i‖ ^ 2) (‖fourierBasis.repr f‖ ^ 2)
⊢ ‖fourierBasis.repr f‖ ^ 2 = ‖f‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
⊢ HasSum (fun i ↦ ‖fourierCoeff (↑↑f) i‖ ^ 2) (∫ (t : AddCircle T), ‖↑↑f t‖ ^ 2 ∂haarAddCircle)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: a b : ℝ
f : ℝ → ℂ
hab : a < b
hL2 : MemLp f 2 (volume.restrict (Ioc a b))
⊢ 0 < b - a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: a b : ℝ
f : ℝ → ℂ
hab : a < b
hL2 : MemLp f 2 (volume.restrict (Ioc a b))
⊢ HasSum (fun i ↦ ‖fourierCoeffOn hab f i‖ ^ 2) ((b - a)⁻¹ • ∫ (x : ℝ) in a..b, ‖f x‖ ^ 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
n : ℤ
⊢ ∀ (x : AddCircle T), (fourier (-n)) x = (fourier (-n)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
h : Summable (fourierCoeff ⇑f)
⊢ HasSum (fun i ↦ fourierCoeff (⇑f) i • fourier i) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
h : Summable (fourierCoeff ⇑f)
x : AddCircle T
⊢ HasSum (fun i ↦ fourierCoeff (⇑f) i • (fourier i) x) (f x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
n : ℤ
⊢ fourierCoeff ⇑(fourier n) = Pi.single n 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
n : ℤ
x : ℝ
⊢ HasDerivAt (fun y ↦ (fourier n) ↑y) (2 * ↑π * I * ↑n / ↑T * (fourier n) ↑x) x

Tactic:
fourier_coe_apply

Goal state after:
T : ℝ
n : ℤ
x : ℝ
⊢ HasDerivAt (fun y ↦ cexp (2 * ↑π * I * ↑n * ↑y / ↑T)) (2 * ↑π * I * ↑n / ↑T * cexp (2 * ↑π * I * ↑n * ↑x / ↑T)) x