mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra ℝ 𝕜
inst✝ : NormedSpace 𝕜 F
g₁ g₂ : E → 𝕜
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.SchwartzSpace.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra ℝ 𝕜
inst✝ : NormedSpace 𝕜 F
g : E → 𝕜
hg : Function.HasTemperateGrowth g
⊢ smulLeftCLM F (-g) = -smulLeftCLM F g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra ℝ 𝕜
inst✝ : NormedSpace 𝕜 F
g : ι → E → 𝕜
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.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
x : E
⊢ Function.HasTemperateGrowth fun x ↦ ↑(g x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
x : E
⊢ Function.HasTemperateGrowth g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
⊢ (smulLeftCLM F fun x ↦ ↑(g x)) f = (smulLeftCLM F g) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → 𝕜'
hg : Function.HasTemperateGrowth g
c : ℝ
⊢ smulLeftCLM F (c • g) = c • smulLeftCLM F g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra ℝ 𝕜
inst✝ : NormedSpace 𝕜 F
g : E → 𝕜
f : 𝓢(E, F)
⊢ tsupport ⇑((smulLeftCLM F g) f) ⊆ tsupport ⇑f ∩ tsupport g

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y ↦ T fun x ↦ L y
⊢ ∑ i ∈ Finset.range (k + 1),
      ↑(k.choose i) * ‖iteratedFDeriv ℝ i (fun y ↦ T fun x ↦ L y) v‖ * ‖iteratedFDeriv ℝ (k - i) f v‖ ≤
    ∑ i ∈ Finset.range (k + 1),
      ↑(k.choose i) * (↑(n.descFactorial i) * ‖L‖ ^ n * ‖v‖ ^ (n - i)) * ‖iteratedFDeriv ℝ (k - i) f v‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
inst✝⁶ : NormedAddCommGroup D
inst✝⁵ : NormedSpace ℝ D
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
x✝¹ x✝ : 𝓢(D, E)
⊢ bilinLeftCLM B.flip ⋯ = bilinLeftCLM B.flip ⋯ + bilinLeftCLM B.flip ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
inst✝⁶ : NormedAddCommGroup D
inst✝⁵ : NormedSpace ℝ D
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
x✝¹ : 𝕜
x✝ : 𝓢(D, E)
⊢ bilinLeftCLM B.flip ⋯ = (RingHom.id 𝕜) x✝¹ • bilinLeftCLM B.flip ⋯

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y ↦ T fun x ↦ L y
⊢ ∑ i ∈ Finset.range (k + 1),
      ↑(k.choose i) * (↑(n.descFactorial i) * ‖L‖ ^ n * ‖v‖ ^ (n - i)) * ‖iteratedFDeriv ℝ (k - i) f v‖ =
    ∑ i ∈ Finset.range (k + 1),
      ↑(k.choose i) * ↑(n.descFactorial i) * ‖L‖ ^ n * (‖v‖ ^ (n - i) * ‖iteratedFDeriv ℝ (k - i) f v‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
⊢ ∀ (f g : 𝓢(E, F)) (x : E), (L x).smulRight ((f + g) x) = (L x).smulRight (f x) + (L x).smulRight (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
⊢ ∀ (a : 𝕜) (f : 𝓢(E, F)) (x : E), (L x).smulRight ((a • f) x) = (RingHom.id 𝕜) a • (L x).smulRight (f x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
⊢ ∀ (f : 𝓢(E, F)), ContDiff ℝ ∞ fun x ↦ (L x).smulRight (f x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
⊢ 0 ≤ 2 * ‖L‖ * ↑(max 1 n)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y ↦ T fun x ↦ L y
i : ℕ
hi : i ∈ Finset.range (k + 1)
⊢ n ^ i ≤ (n + 1) ^ i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ContDiff ℝ ∞ ⇑L

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ (k, n).1 * ‖iteratedFDeriv ℝ (k, n).2 (fun x ↦ (L x).smulRight (f x)) x‖ ≤
    ‖x‖ ^ k *
      ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑L) x‖ * ‖iteratedFDeriv ℝ (n - i) (⇑f) x‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
J : ∀ (n : ℕ), 0 ≤ᵐ[μ.restrict s] fun x ↦ c x ^ n
P : ∀ (n : ℕ), 0 < ∫ (x : α) in s, c x ^ n ∂μ
hiφ : ∀ (n : ℕ), ∫ (x : α) in s, φ n x ∂μ = 1
u : Set α
u_open : IsOpen[inst✝⁶] u
x₀u : x₀ ∈ u
t : ℝ
t_pos : 0 ≤ t
tx₀ : t < c x₀
ht : ∀ x ∈ s \ u, c x ≤ t
t' : ℝ
tt' : t < t'
t'x₀ : t' < c x₀
t'_pos : 0 < t'
v : Set α
v_open : IsOpen[inst✝⁶] v
x₀_v : x₀ ∈ v
hv : v ∩ s ⊆ c ⁻¹' Ioi t'
⊢ ∀ (n : ℕ), ∀ x ∈ s \ u, φ n x ≤ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y ↦ T fun x ↦ L y
i : ℕ
hi : i ∈ Finset.range (k + 1)
⊢ (n + 1) ^ i ≤ (n + 1) ^ k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k : ℕ
f : 𝓢(E, F)
x : E
n : ℕ
⊢ ∑ k ∈ Finset.range n,
      ↑((n + 1).choose (k + 1 + 1)) * ‖iteratedFDeriv ℝ (k + 1 + 1) (⇑L) x‖ *
        ‖iteratedFDeriv ℝ (n + 1 - (k + 1 + 1)) (⇑f) x‖ =
    0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
J : ∀ (n : ℕ), 0 ≤ᵐ[μ.restrict s] fun x ↦ c x ^ n
P : ∀ (n : ℕ), 0 < ∫ (x : α) in s, c x ^ n ∂μ
hiφ : ∀ (n : ℕ), ∫ (x : α) in s, φ n x ∂μ = 1
u : Set α
u_open : IsOpen[inst✝⁶] u
x₀u : x₀ ∈ u
t : ℝ
t_pos : 0 ≤ t
tx₀ : t < c x₀
ht : ∀ x ∈ s \ u, c x ≤ t
t' : ℝ
tt' : t < t'
t'x₀ : t' < c x₀
t'_pos : 0 < t'
v : Set α
v_open : IsOpen[inst✝⁶] v
x₀_v : x₀ ∈ v
hv : v ∩ s ⊆ c ⁻¹' Ioi t'
M : ∀ (n : ℕ), ∀ x ∈ s \ u, φ n x ≤ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n
⊢ Tendsto (fun n ↦ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n) atTop (𝓝 ((μ.real (v ∩ s))⁻¹ * 0))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ k *
      ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑L) x‖ * ‖iteratedFDeriv ℝ (n - i) (⇑f) x‖ ≤
    ‖x‖ ^ k * (‖L x‖ * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ↑n * ‖L‖ * ‖iteratedFDeriv ℝ (n - 1) (⇑f) x‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ k * (‖L x‖ * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ↑n * ‖L‖ * ‖iteratedFDeriv ℝ (n - 1) (⇑f) x‖) =
    ‖x‖ ^ k * ‖L x‖ * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ‖x‖ ^ k * ↑n * ‖L‖ * ‖iteratedFDeriv ℝ (n - 1) (⇑f) x‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y ↦ T fun x ↦ L y
i : ℕ
hi : i ∈ Finset.range (k + 1)
⊢ k - i ≤ k

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y ↦ T fun x ↦ L y
i : ℕ
hi : i ∈ Finset.range (k + 1)
⊢ n - i ≤ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ k * ‖L x‖ * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ‖x‖ ^ k * ↑n * ‖L‖ * ‖iteratedFDeriv ℝ (n - 1) (⇑f) x‖ ≤
    ‖L‖ * 1 * (SchwartzMap.seminorm 𝕜 (k + 1) n) f + ‖L‖ * ↑n * (SchwartzMap.seminorm 𝕜 k (n - 1)) f

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y ↦ T fun x ↦ L y
⊢ ∑ i ∈ Finset.range (k + 1),
      ↑(k.choose i) * ↑(n.descFactorial i) * ‖L‖ ^ n * (‖v‖ ^ (n - i) * ‖iteratedFDeriv ℝ (k - i) f v‖) ≤
    ∑ i ∈ Finset.range (k + 1), ↑(k.choose i) * ↑(n + 1) ^ k * ‖L‖ ^ n * C

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖L‖ * 1 * (SchwartzMap.seminorm 𝕜 (k + 1) n) f + ‖L‖ * ↑n * (SchwartzMap.seminorm 𝕜 k (n - 1)) f ≤
    ‖L‖ * ↑(max 1 n) * max ((SchwartzMap.seminorm 𝕜 (k + 1) n) f) ((SchwartzMap.seminorm 𝕜 k (n - 1)) f) +
      ‖L‖ * ↑(max 1 n) * max ((SchwartzMap.seminorm 𝕜 (k + 1) n) f) ((SchwartzMap.seminorm 𝕜 k (n - 1)) f)

Tactic:
by

Goal state after: