mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
w : Multiplicative W
v : V
⊢ ((charMonoidHom he hL) w) v = ↑(e ((L v) (Multiplicative.toAdd w)))

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝¹ : Fintype d
inst✝ : DecidableEq d
z : d → AddCircle 1
i : d
⊢ (mFourier (Pi.single i 1)) z = (fourier 1) (z i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : CompleteSpace F
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
f : ↥(Lp F p μ)
hf : (toTemperedDistributionCLM F μ p) f = 0
g : E → ℝ
g_smooth : ContDiff ℝ ∞ g
g_cpt : HasCompactSupport g
hg₁ : HasCompactSupport (⇑Complex.ofRealCLM ∘ g)
hg₂ : ContDiff ℝ ∞ (⇑Complex.ofRealCLM ∘ g)
⊢ ∫ (x : E), g x • ↑↑f x ∂μ = ((toTemperedDistributionCLM F μ p) f) (hg₁.toSchwartzMap hg₂)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
w : AddMonoidAlgebra ℂ W
v : V
⊢ ((charAlgHom he hL) w) v = w.coeff.sum fun a z ↦ z • ↑(e ((L v) a))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
f g : 𝓢'(E, F)
hf : MemSobolev s p f
hg : MemSobolev s p g
⊢ MemSobolev s p (f - g)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : CompleteSpace F
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
f : ↥(Lp F p μ)
hf : (toTemperedDistributionCLM F μ p) f = 0
g : E → ℝ
g_smooth : ContDiff ℝ ∞ g
g_cpt : HasCompactSupport g
hg₁ : HasCompactSupport (⇑Complex.ofRealCLM ∘ g)
hg₂ : ContDiff ℝ ∞ (⇑Complex.ofRealCLM ∘ g)
⊢ ((toTemperedDistributionCLM F μ p) f) (hg₁.toSchwartzMap hg₂) = 0

Tactic:
by

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

Goal state before:
pp: d✝ : Type u_1
inst✝¹ : Fintype d✝
d : Type u_2
inst✝ : Fintype d
⊢ ∀ {a : C(UnitAddTorus d, ℂ)}, a ∈ (adjoin ℂ (range mFourier)).carrier → star a ∈ (adjoin ℂ (range mFourier)).carrier

Tactic:
by

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

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
x : V →ᵇ ℂ
hx : x ∈ (charAlgHom he hL).range
⊢ star x ∈ (charAlgHom he hL).range

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
f : 𝓢'(E, F)
hf : MemSobolev s p f
⊢ MemSobolev s p (-f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : CompleteSpace F
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
⊢ (↑(toTemperedDistributionCLM F μ p)).ker = ⊥

Tactic:
by

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

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
f : V →ᵇ ℂ
⊢ f ∈ charPoly he hL ↔ ∃ w, ⇑f = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : E → ℂ
f : 𝓢'(E, F)
f' : 𝓢(E, ℂ)
⊢ ((smulLeftCLM F g) f) f' = f ((SchwartzMap.smulLeftCLM ℂ g) f')

Tactic:
by

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

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
w : W
⊢ (charAlgHom he hL).toRingHom (AddMonoidAlgebra.single w 1) = char he hL w

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
⊢ Subalgebra.toSubmodule (mFourierSubalgebra d).toSubalgebra = span ℂ (range mFourier)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
c : ℂ
f : 𝓢'(E, F)
⊢ (smulLeftCLM F fun x ↦ c) f = c • f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
f : 𝓢'(E, F)
⊢ (smulLeftCLM F g₂) ((smulLeftCLM F g₁) f) = (smulLeftCLM F (g₁ * g₂)) f

Tactic:
by

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

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL : Continuous fun p ↦ (L p.1) p.2
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
v v' : V
hvv' : v ≠ v'
⊢ ∃ w, (char he hL w) v ≠ (char he hL w) v'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
c : ℂ
f : 𝓢'(E, F)
hf : MemSobolev s p f
⊢ MemSobolev s p (c • f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ smulLeftCLM F g₂ ∘SL smulLeftCLM F g₁ = smulLeftCLM F (g₁ * g₂)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL : Continuous fun p ↦ (L p.1) p.2
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
⊢ (StarSubalgebra.map (toContinuousMapStarₐ ℂ) (charPoly he hL)).SeparatesPoints

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : E → ℂ
hg : Function.HasTemperateGrowth g
c : ℂ
⊢ smulLeftCLM F (c • g) = c • smulLeftCLM F g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
⊢ MemSobolev s p 0

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
⊢ (mFourierSubalgebra d).SeparatesPoints

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
s r : ℝ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
f : 𝓢'(E, F)
⊢ MemSobolev s p ((besselPotential E F r) f) ↔ MemSobolev (r + s) p f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ smulLeftCLM F (g₁ + g₂) = smulLeftCLM F g₁ + smulLeftCLM F g₂

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
⊢ (span ℂ (range mFourier)).topologicalClosure = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ smulLeftCLM F (g₁ - g₂) = smulLeftCLM F g₁ - smulLeftCLM F g₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : E → ℂ
hg : Function.HasTemperateGrowth g
⊢ smulLeftCLM F (-g) = -smulLeftCLM F g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
f : 𝓢(E, F)
⊢ Function.HasTemperateGrowth fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
a : d → ℝ
⊢ MeasurePreserving (⇑(measurableEquivPiIoc a)) volume (Measure.comap Subtype.val volume)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
g : ι → E → ℂ
s : Finset ι
hg : ∀ i ∈ s, Function.HasTemperateGrowth (g i)
⊢ (smulLeftCLM F fun x ↦ ∑ i ∈ s, g i x) = ∑ i ∈ s, smulLeftCLM F (g i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
p : ℝ≥0∞
hp : Fact (1 ≤ p)
f : 𝓢(E, F)
⊢ MemSobolev s p ((SchwartzMap.toTemperedDistributionCLM E F volume) f)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedAddCommGroup F₂
inst✝⁷ : NormedAddCommGroup F₃
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : NormedSpace 𝕜 F₁
inst✝¹ : NormedSpace 𝕜 F₂
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
f₁ : E → F₁
f₂ : E → F₂
hf₁ : Integrable f₁ volume
hf₂ : Integrable f₂ volume
⊢ Integrable (fun p ↦ ‖B‖ * (‖f₁ (p.1 - p.2)‖ * ‖f₂ p.2‖)) (volume.prod volume)

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
f : UnitAddTorus d → ℝ≥0∞
a : d → ℝ
⊢ ∫⁻ (x : UnitAddTorus d), f x = ∫⁻ (x : d → ℝ) in {x | ∀ (i : d), x i ∈ Ioc (a i) (a i + 1)}, f fun i ↦ ↑(x i)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F₁
inst✝⁹ : NormedAddCommGroup F₂
inst✝⁸ : NormedAddCommGroup F₃
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : NormedSpace 𝕜 F₁
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℂ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
f₁ : E → F₁
f₂ : E → F₂
hf₁ : Integrable f₁ volume
hf₂ : Integrable f₂ volume
ξ : E
⊢ 𝓕 (f₁ ⋆[B, volume] f₂) ξ = 𝓕 (f₂ ⋆[B.flip, volume] f₁) ξ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F₁
inst✝⁹ : NormedAddCommGroup F₂
inst✝⁸ : NormedAddCommGroup F₃
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : NormedSpace 𝕜 F₁
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℂ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
f₁ : E → F₁
f₂ : E → F₂
hf₁ : Integrable f₁ volume
hf₂ : Integrable f₂ volume
ξ : E
⊢ 𝓕 (f₂ ⋆[B.flip, volume] f₁) ξ = ∫ (x : E), 𝐞 (-inner ℝ x ξ) • ∫ (y : E), (B (f₁ (x - y))) (f₂ y)

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝² : Fintype d
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : UnitAddTorus d → E
a : d → ℝ
⊢ ∫ (x : UnitAddTorus d), f x = ∫ (x : d → ℝ) in {x | ∀ (i : d), x i ∈ Ioc (a i) (a i + 1)}, f fun i ↦ ↑(x i)

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝¹ : Fintype d
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
⊢ (span ℂ (range (mFourierLp p))).topologicalClosure = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
inst✝⁵ : CompleteSpace F
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
p q r : ℝ≥0∞
inst✝² : p.HolderTriple q r
inst✝¹ : Fact (1 ≤ q)
inst✝ : Fact (1 ≤ r)
g : E → ℂ
hg₁ : Function.HasTemperateGrowth g
hg₂ : MemLp g p μ
f : ↥(Lp F q μ)
⊢ Lp.toTemperedDistribution (MemLp.toLp g hg₂ • f) = (smulLeftCLM F g) (Lp.toTemperedDistribution f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: 𝕜 : Type u_2
F : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : NormedSpace 𝕜 F
f : 𝓢(ℝ, F)
⊢ (derivCLM F) ((toTemperedDistributionCLM ℝ F volume) f) =
    (toTemperedDistributionCLM ℝ F volume) ((SchwartzMap.derivCLM 𝕜 F) f)

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
⊢ Orthonormal ℂ (mFourierLp 2)

Tactic:
by

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
⊢ 2 ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
F₁ : Type u_5
F₂ : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
x y : E
f : 𝓢'(E, F)
⊢ ∂_{x + y} f = ∂_{x} f + ∂_{y} f

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F₁
inst✝⁹ : NormedAddCommGroup F₂
inst✝⁸ : NormedAddCommGroup F₃
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : NormedSpace 𝕜 F₁
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℂ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
f₁ : E → F₁
f₂ : E → F₂
hf₁ : Integrable f₁ volume
hf₂ : Integrable f₂ volume
ξ : E
⊢ ∫ (x : E), 𝐞 (-inner ℝ x ξ) • ∫ (y : E), (B (f₁ (x - y))) (f₂ y) =
    ∫ (x : E) (y : E), 𝐞 (-inner ℝ x ξ) • (B (f₁ (x - y))) (f₂ y)

Tactic:
congr
    ext x
    simp_rw [Circle.smul_def, integral_smul]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F₁
inst✝⁹ : NormedAddCommGroup F₂
inst✝⁸ : NormedAddCommGroup F₃
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : NormedSpace 𝕜 F₁
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℂ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
f₁ : E → F₁
f₂ : E → F₂
hf₁ : Integrable f₁ volume
hf₂ : Integrable f₂ volume
ξ : E
⊢ ∫ (x : E), 𝐞 (-inner ℝ x ξ) • ∫ (y : E), (B (f₁ (x - y))) (f₂ y) =
    ∫ (x : E) (y : E), 𝐞 (-inner ℝ x ξ) • (B (f₁ (x - y))) (f₂ y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
F₁ : Type u_5
F₂ : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
r : ℝ
x : E
f : 𝓢'(E, F)
⊢ ∂_{r • x} f = r • ∂_{x} f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : 𝓢(E, F)
m : E
⊢ ∂_{m} ((toTemperedDistributionCLM E F μ) f) = (toTemperedDistributionCLM E F μ) (∂_{m} f)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F₁
inst✝⁹ : NormedAddCommGroup F₂
inst✝⁸ : NormedAddCommGroup F₃
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : NormedSpace 𝕜 F₁
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℂ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
f₁ : E → F₁
f₂ : E → F₂
hf₁ : Integrable f₁ volume
hf₂ : Integrable f₂ volume
ξ : E
hB : Integrable (fun p ↦ (B.flip (f₂ p.2)) (f₁ (p.1 - p.2))) (volume.prod volume)
⊢ Continuous fun a ↦ -inner ℝ a.1 ξ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
f : 𝓢'(E, F)
⊢ (laplacianCLM ℂ E 𝓢'(E, F)) f = Δ f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢'(E, F)
u : 𝓢(E, ℂ)
⊢ (Δ f) u = f (Δ u)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F₁
inst✝⁹ : NormedAddCommGroup F₂
inst✝⁸ : NormedAddCommGroup F₃
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : NormedSpace 𝕜 F₁
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℂ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
f₁ : E → F₁
f₂ : E → F₂
hf₁ : Integrable f₁ volume
hf₂ : Integrable f₂ volume
ξ : E
hB : Integrable (fun p ↦ (B.flip (f₂ p.2)) (f₁ (p.1 - p.2))) (volume.prod volume)
x y : E
⊢ ‖Function.uncurry (fun x y ↦ 𝐞 (-inner ℝ x ξ) • (B (f₁ (x - y))) (f₂ y)) (x, y)‖ ≤
    ‖(B.flip (f₂ (x, y).2)) (f₁ ((x, y).1 - (x, y).2))‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : 𝓢(E, F)
⊢ Δ ((toTemperedDistributionCLM E F volume) f) = (toTemperedDistributionCLM E F volume) (Δ f)

Tactic:
by

Goal state after: