mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
p : ℝ
hp : 0 < p
f : α → E
hf : MemLp f (ENNReal.ofReal p) μ
ε : ℝ
hε : 0 < ε
I : 0 < ε ^ (1 / p)
A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0
B : ENNReal.ofReal p ≠ 0
g : α →ᵇ E
g_mem : MemLp (⇑g) (↑p.toNNReal) μ
hg : (∫ (a : α), ‖(f - ⇑g) a‖ ^ (ENNReal.ofReal p).toReal ∂μ) ^ (ENNReal.ofReal p).toReal⁻¹ ≤ ε ^ p⁻¹
⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε

Tactic:
ENNReal.toReal_ofReal hp.le,

Goal state after:
α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
p : ℝ
hp : 0 < p
f : α → E
hf : MemLp f (ENNReal.ofReal p) μ
ε : ℝ
hε : 0 < ε
I : 0 < ε ^ (1 / p)
A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0
B : ENNReal.ofReal p ≠ 0
g : α →ᵇ E
g_mem : MemLp (⇑g) (↑p.toNNReal) μ
hg : (∫ (a : α), ‖(f - ⇑g) a‖ ^ p ∂μ) ^ p⁻¹ ≤ ε ^ p⁻¹
⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε
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
n : ℕ
x : X
⊢ ‖{ toFun := coeff n, map_add' := ⋯, map_smul' := ⋯ } x‖ * ‖D.e n‖ ≤ ‖succSub D.P n‖ * ‖x‖

Tactic:
calc ‖coeff n x‖ * ‖D.e n‖ = ‖coeff n x • D.e n‖ := (norm_smul _ _).symm
          _ = ‖(succSub D.P n) x‖ := by rw [hcoeff]
          _ ≤ ‖succSub D.P n‖ * ‖x‖ := ContinuousLinearMap.le_opNorm _ _

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:
simpa only [innerSL_apply_apply, ← b.repr_apply_apply] using b.hasSum_repr x

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

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝³] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε
δpos : 0 < δ
δone : δ < 1
u : Set α
u_open : IsOpen[inst✝³] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball 0 δ
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < δ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < δ
hφpos : ∀ x ∈ s, 0 ≤ φ i x
h''i : IntegrableOn (fun x ↦ φ i x • g x) s μ
I : IntegrableOn (φ i) t μ
⊢ IntegrableOn (fun x ↦ ‖φ i x‖ * δ) (s ∩ u) μ

Tactic:
exact IntegrableOn.mono_set (I.norm.mul_const _) ut

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

Goal state before:
pp: case e'_12
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
v : V
w : W
ha : HasFDerivAt (fun w' ↦ (L v) w') (L v) w
w' : W
⊢ (𝐞 (-(L v) w) • -(2 * ↑π * I)) • (L v) w' • f v = (-(L v) w' • (2 * ↑π * I * ↑(𝐞 (-(L v) w)))) • f v

Tactic:
smul_comm,

Goal state after:
case e'_12
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
v : V
w : W
ha : HasFDerivAt (fun w' ↦ (L v) w') (L v) w
w' : W
⊢ (L v) w' • (𝐞 (-(L v) w) • -(2 * ↑π * I)) • f v = (-(L v) w' • (2 * ↑π * I * ↑(𝐞 (-(L v) w)))) • f v
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : 𝓢(E, F)
k n : ℕ
⊢ ∃ C, 0 < C ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C

Tactic:
by
  rcases f.decay' k n with ⟨C, hC⟩
  exact ⟨max C 1, by positivity, fun x => (hC x).trans (le_max_left _ _)⟩

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

Goal state before:
pp: case h₁
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
g : D → F
hg : Function.HasTemperateGrowth g
k n l : ℕ
C : ℝ
hC : 0 ≤ C
f : 𝓢(D, E)
x : D
hxk : 0 ≤ ‖x‖ ^ k
hnorm_mul :
  ‖iteratedFDeriv ℝ n (fun y ↦ ((ContinuousLinearMap.bilinearRestrictScalars ℝ B) (f y)) (g y)) x‖ ≤
    ‖ContinuousLinearMap.bilinearRestrictScalars ℝ B‖ *
      ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑f) x‖ * ‖iteratedFDeriv ℝ (n - i) g x‖
this : ∑ _x ∈ Finset.range (n + 1), 1 = ↑n + 1
i : ℕ
hgrowth : ‖iteratedFDeriv ℝ (n - i) g x‖ ≤ C * (1 + ‖x‖) ^ l
hi : i ∈ Finset.range (n + 1)
⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ i (⇑f) x‖ * ‖iteratedFDeriv ℝ (n - i) g x‖ ≤
    C * 2 ^ (l + k) * ((Finset.Iic (l + k, n)).sup (schwartzSeminormFamily 𝕜 D E)) f

Tactic:
grw [hgrowth]

Goal state after:
case h₁
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
g : D → F
hg : Function.HasTemperateGrowth g
k n l : ℕ
C : ℝ
hC : 0 ≤ C
f : 𝓢(D, E)
x : D
hxk : 0 ≤ ‖x‖ ^ k
hnorm_mul :
  ‖iteratedFDeriv ℝ n (fun y ↦ ((ContinuousLinearMap.bilinearRestrictScalars ℝ B) (f y)) (g y)) x‖ ≤
    ‖ContinuousLinearMap.bilinearRestrictScalars ℝ B‖ *
      ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑f) x‖ * ‖iteratedFDeriv ℝ (n - i) g x‖
this : ∑ _x ∈ Finset.range (n + 1), 1 = ↑n + 1
i : ℕ
hgrowth : ‖iteratedFDeriv ℝ (n - i) g x‖ ≤ C * (1 + ‖x‖) ^ l
hi : i ∈ Finset.range (n + 1)
⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ i (⇑f) x‖ * (C * (1 + ‖x‖) ^ l) ≤
    C * 2 ^ (l + k) * ((Finset.Iic (l + k, n)).sup (schwartzSeminormFamily 𝕜 D E)) f
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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup D
inst✝¹ : NormedSpace ℝ D
inst✝ : NormedSpace 𝕜 F
g : D → E
hg : Function.HasTemperateGrowth g
k n l : ℕ
C : ℝ
hC : 0 ≤ C
hgrowth : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N g x‖ ≤ C * (1 + ‖x‖) ^ l
kg : ℕ
Cg : ℝ
hg_upper' : ∀ (x : D), ‖x‖ ≤ Cg * (1 + ‖g x‖) ^ kg
hCg : 1 ≤ 1 + Cg
k' : ℕ := kg * (k + l * n)
f : 𝓢(E, F)
x : D
seminorm_f : ℝ := ((Finset.Iic (k', n)).sup (schwartzSeminormFamily 𝕜 E F)) f
hg_upper'' : (1 + ‖x‖) ^ (k + l * n) ≤ (1 + Cg) ^ (k + l * n) * (1 + ‖g x‖) ^ k'
hbound : ∀ i ≤ n, ‖iteratedFDeriv ℝ i (⇑f) (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k'
hgrowth' : ∀ (N : ℕ), 1 ≤ N → N ≤ n → ‖iteratedFDeriv ℝ N g x‖ ≤ ((C + 1) * (1 + ‖x‖) ^ l) ^ N
this : ‖iteratedFDeriv ℝ n (⇑f ∘ g) x‖ ≤ ↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) * (1 + ‖x‖) ^ l) ^ n
hxk : ‖x‖ ^ k ≤ (1 + ‖x‖) ^ k
⊢ (1 + ‖x‖) ^ k * (↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) * (1 + ‖x‖) ^ l) ^ n) =
    (1 + ‖x‖) ^ (k + l * n) / (1 + ‖g x‖) ^ k' * ((C + 1) ^ n * ↑n ! * 2 ^ k' * seminorm_f)

Tactic:
mul_pow,

Goal state after:
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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup D
inst✝¹ : NormedSpace ℝ D
inst✝ : NormedSpace 𝕜 F
g : D → E
hg : Function.HasTemperateGrowth g
k n l : ℕ
C : ℝ
hC : 0 ≤ C
hgrowth : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N g x‖ ≤ C * (1 + ‖x‖) ^ l
kg : ℕ
Cg : ℝ
hg_upper' : ∀ (x : D), ‖x‖ ≤ Cg * (1 + ‖g x‖) ^ kg
hCg : 1 ≤ 1 + Cg
k' : ℕ := kg * (k + l * n)
f : 𝓢(E, F)
x : D
seminorm_f : ℝ := ((Finset.Iic (k', n)).sup (schwartzSeminormFamily 𝕜 E F)) f
hg_upper'' : (1 + ‖x‖) ^ (k + l * n) ≤ (1 + Cg) ^ (k + l * n) * (1 + ‖g x‖) ^ k'
hbound : ∀ i ≤ n, ‖iteratedFDeriv ℝ i (⇑f) (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k'
hgrowth' : ∀ (N : ℕ), 1 ≤ N → N ≤ n → ‖iteratedFDeriv ℝ N g x‖ ≤ ((C + 1) * (1 + ‖x‖) ^ l) ^ N
this : ‖iteratedFDeriv ℝ n (⇑f ∘ g) x‖ ≤ ↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) * (1 + ‖x‖) ^ l) ^ n
hxk : ‖x‖ ^ k ≤ (1 + ‖x‖) ^ k
⊢ (1 + ‖x‖) ^ k * (↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) ^ n * ((1 + ‖x‖) ^ l) ^ n)) =
    (1 + ‖x‖) ^ (k + l * n) / (1 + ‖g x‖) ^ k' * ((C + 1) ^ n * ↑n ! * 2 ^ k' * seminorm_f)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
⊢ ∫ (x : ℝ), f |x| = 2 * ∫ (x : ℝ) in Ioi 0, f x

Tactic:
by
  have eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x := by
    refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
    rw [abs_eq_self.mpr (le_of_lt (by exact hx))]
  by_cases hf : IntegrableOn (fun x => f |x|) (Ioi 0)
  · have int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) := by
      rw [← Measure.map_neg_eq_self (volume : Measure ℝ)]
      let m : MeasurableEmbedding fun x : ℝ => -x := (Homeomorph.neg ℝ).measurableEmbedding
      rw [m.integrableOn_map_iff]
      simp_rw [Function.comp_def, abs_neg, neg_preimage, neg_Iic, neg_zero]
      exact Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi hf
    calc
      _ = (∫ x in Iic 0, f |x|) + ∫ x in Ioi 0, f |x| := by
        rw [← setIntegral_union (Iic_disjoint_Ioi le_rfl) measurableSet_Ioi int_Iic hf,
          Iic_union_Ioi, restrict_univ]
      _ = 2 * ∫ x in Ioi 0, f x := by
        rw [two_mul, eq]
        congr! 1
        rw [← neg_zero, ← integral_comp_neg_Iic, neg_zero]
        refine setIntegral_congr_fun measurableSet_Iic (fun _ hx => ?_)
        rw [abs_eq_neg_self.mpr (by exact hx)]
  · have : ¬ Integrable (fun x => f |x|) := by
      contrapose hf
      exact hf.integrableOn
    rw [← eq, integral_undef hf, integral_undef this, mul_zero]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
f : α → E
g : α → F
a : α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F
inst✝² : LinearOrder α
inst✝¹ : CompactIccSpace α
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrableOn f (Ici a) μ
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
⊢ StronglyMeasurableAtFilter f atTop μ

Tactic:
exact ⟨Ici a, Ici_mem_atTop a, hf.aestronglyMeasurable⟩

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
K N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) volume
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
w : V
this :
  (‖w‖ * ‖w‖) ^ n *
      ‖iteratedFDeriv ℝ k (fourierIntegral 𝐞 volume (ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)) f) w‖ ≤
    ‖w‖ ^ n *
      ((2 * (π * ‖innerSL ℝ‖)) ^ k *
        ((2 * ↑k + 2) ^ n *
          ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1),
            ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂volume))
⊢ ‖w‖ ^ n * (‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖) ≤
    ‖w‖ ^ n *
      ((2 * (π * ‖innerSL ℝ‖)) ^ k *
        ((2 * ↑k + 2) ^ n *
          ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖))

Tactic:
mul_pow,

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
K N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) volume
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
w : V
this :
  ‖w‖ ^ n * ‖w‖ ^ n *
      ‖iteratedFDeriv ℝ k (fourierIntegral 𝐞 volume (ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)) f) w‖ ≤
    ‖w‖ ^ n *
      ((2 * (π * ‖innerSL ℝ‖)) ^ k *
        ((2 * ↑k + 2) ^ n *
          ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1),
            ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂volume))
⊢ ‖w‖ ^ n * (‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖) ≤
    ‖w‖ ^ n *
      ((2 * (π * ‖innerSL ℝ‖)) ^ k *
        ((2 * ↑k + 2) ^ n *
          ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖))
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: V : Type u_1
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
f : V → E
y : V
⊢ 𝓕⁻ f y = 𝓕 (fun x ↦ f (-x)) y

Tactic:
fourierInv_eq_fourier_neg

Goal state after:
V : Type u_1
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
f : V → E
y : V
⊢ 𝓕 f (-y) = 𝓕 (fun x ↦ f (-x)) y
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: V : Type u_1
W : Type u_2
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℂ E
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : InnerProductSpace ℝ V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : InnerProductSpace ℝ W
inst✝³ : MeasurableSpace W
inst✝² : BorelSpace W
inst✝¹ : FiniteDimensional ℝ W
inst✝ : FiniteDimensional ℝ V
c : ℂ
f : ↥(Lp E 1 volume)
x✝ x : V
⊢ ↑↑(c • f) x = (c • ↑↑f) x → 𝐞 (-⟪x, x✝⟫) • ↑↑(c • f) x = (RingHom.id ℂ) c • 𝐞 (-⟪x, x✝⟫) • ↑↑f x

Tactic:
h

Goal state after:
V : Type u_1
W : Type u_2
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℂ E
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : InnerProductSpace ℝ V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : InnerProductSpace ℝ W
inst✝³ : MeasurableSpace W
inst✝² : BorelSpace W
inst✝¹ : FiniteDimensional ℝ W
inst✝ : FiniteDimensional ℝ V
c : ℂ
f : ↥(Lp E 1 volume)
x✝ x : V
h : ↑↑(c • f) x = (c • ↑↑f) x
⊢ 𝐞 (-⟪x, x✝⟫) • ↑↑(c • f) x = (RingHom.id ℂ) c • 𝐞 (-⟪x, x✝⟫) • ↑↑f x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ MeasurableSet (Ioc 0 Y)

Tactic:
exact measurableSet_Ioc

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.snd ℝ ℝ ℝ).ker
h : LinearMap.snd ℝ ℝ ℝ = 0
⊢ (LinearMap.snd ℝ ℝ ℝ) (0, 1) = 0 (0, 1)

Tactic:
h

Goal state after:
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.snd ℝ ℝ ℝ).ker
h : LinearMap.snd ℝ ℝ ℝ = 0
⊢ 0 (0, 1) = 0 (0, 1)
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: ι : Type u_2
inst✝³ : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
f : X i → E
hf : AEStronglyMeasurable f (μ i)
⊢ ∫ (x : (i : ι) → X i), f (x i) ∂Measure.pi μ = ∫ (x : X i), f x ∂μ i

Tactic:
rw [← (measurePreserving_eval μ i).map_eq, integral_map]

Goal state after:
case hφ
ι : Type u_2
inst✝³ : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
f : X i → E
hf : AEStronglyMeasurable f (μ i)
⊢ AEMeasurable (Function.eval i) (Measure.pi μ)

case hfm
ι : Type u_2
inst✝³ : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
f : X i → E
hf : AEStronglyMeasurable f (μ i)
⊢ AEStronglyMeasurable f (map (Function.eval i) (Measure.pi μ))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target,
      p.1 • cexp (-b * ((↑p.1 * Complex.cos ↑p.2) ^ 2 + (↑p.1 * Complex.sin ↑p.2) ^ 2)) =
    ∫ (z : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, ↑z.1 * cexp (-b * ↑z.1 ^ 2) * 1 ∂volume.prod volume

Tactic:
congr with p : 1

Goal state after:
case e_f
b : ℂ
hb : 0 < b.re
p : ℝ × ℝ
⊢ p.1 • cexp (-b * ((↑p.1 * Complex.cos ↑p.2) ^ 2 + (↑p.1 * Complex.sin ↑p.2) ^ 2)) = ↑p.1 * cexp (-b * ↑p.1 ^ 2) * 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
aux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
x : ℝ
hx : 0 < x
⊢ ↑x ^ (a - 1) * cexp (-(↑r * ↑x)) = 1 ^ (a - 1) * ↑x ^ (a - 1) * cexp (-(↑r * ↑x))

Tactic:
one_cpow,

Goal state after:
a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
aux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
x : ℝ
hx : 0 < x
⊢ ↑x ^ (a - 1) * cexp (-(↑r * ↑x)) = 1 * ↑x ^ (a - 1) * cexp (-(↑r * ↑x))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
c : ℂ
hc : c ∈ {c | 0 < c.re}
⊢ ¬(↑π / c = 0 ∧ 1 / 2 ≠ 0)

Tactic:
not_and_or

Goal state after:
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
c : ℂ
hc : c ∈ {c | 0 < c.re}
⊢ ¬↑π / c = 0 ∨ ¬1 / 2 ≠ 0
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: case h
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := ⋯
⊢ Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ V * φ x) (cobounded V) (𝓝 0)

Tactic:
have A : Tendsto (fun (w : V) ↦ π ^ 2 * ‖w‖ ^ 2) (cobounded V) atTop := by
        rw [tendsto_const_mul_atTop_of_pos (by positivity)]
        apply (tendsto_pow_atTop two_ne_zero).comp tendsto_norm_cobounded_atTop

Goal state after:
case h
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := fun w ↦ π ^ (↑(finrank ℝ V) / 2) * rexp (-π ^ 2 * ‖w‖ ^ 2)
A : Tendsto (fun w ↦ π ^ 2 * ‖w‖ ^ 2) (cobounded V) atTop
⊢ Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ V * φ x) (cobounded V) (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T ↦ ∫ (y : ℝ) in 0..c, cexp (-b * (↑T + ↑y * I) ^ 2)
I₅ : ℝ → ℂ := fun T ↦ ∫ (y : ℝ) in 0..c, cexp (-b * (-↑T + ↑y * I) ^ 2)
T : ℝ
C : I₂ T - I₁ T + I * I₄ T = I * I₅ T
⊢ I₁ T =
    I₂ T + I * ((∫ (x : ℝ) in 0..c, cexp (-b * (↑T + ↑x * I) ^ 2)) - ∫ (x : ℝ) in 0..c, cexp (-b * (-↑T + ↑x * I) ^ 2))

Tactic:
change I₁ T = I₂ T + I * (I₄ T - I₅ T)

Goal state after:
b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T ↦ ∫ (y : ℝ) in 0..c, cexp (-b * (↑T + ↑y * I) ^ 2)
I₅ : ℝ → ℂ := fun T ↦ ∫ (y : ℝ) in 0..c, cexp (-b * (-↑T + ↑y * I) ^ 2)
T : ℝ
C : I₂ T - I₁ T + I * I₄ T = I * I₅ T
⊢ I₁ T = I₂ T + I * (I₄ T - I₅ T)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
t : ℂ
⊢ (↑π / b) ^ (1 / 2) * cexp ((I * t) ^ 2 / (4 * b)) = (↑π / b) ^ (1 / 2) * cexp (-t ^ 2 / (4 * b))

Tactic:
mul_pow,

Goal state after:
b : ℂ
hb : 0 < b.re
t : ℂ
⊢ (↑π / b) ^ (1 / 2) * cexp (I ^ 2 * t ^ 2 / (4 * b)) = (↑π / b) ^ (1 / 2) * cexp (-t ^ 2 / (4 * b))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
hb : 0 < b.re
c : ℂ
w : V
⊢ Integrable (fun v ↦ cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫)) volume

Tactic:
by
  let e := (stdOrthonormalBasis ℝ V).repr.symm
  rw [← e.measurePreserving.integrable_comp_emb e.toHomeomorph.measurableEmbedding]
  convert! integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace hb c (e.symm w) with v
  simp only [neg_mul, Function.comp_apply, LinearIsometryEquiv.norm_map,
    LinearIsometryEquiv.symm_symm,
    LinearIsometryEquiv.inner_map_eq_flip]

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

Goal state before:
pp: case e_f.e_a.e_a
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := fun w ↦ π ^ (↑(finrank ℝ V) / 2) * rexp (-π ^ 2 * ‖w‖ ^ 2)
A : Tendsto (fun c ↦ ∫ (w : V), (c ^ finrank ℝ V * φ (c • (v - w))) • f w) atTop (𝓝 (f v))
B : Tendsto (fun c ↦ ∫ (w : V), ((c ^ (1 / 2)) ^ finrank ℝ V * φ (c ^ (1 / 2) • (v - w))) • f w) atTop (𝓝 (f v))
c : ℝ
hc : 0 < c
w : V
⊢ ↑(-π ^ 2 * ‖c ^ (1 / 2) • (v - w)‖ ^ 2) = -↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2

Tactic:
· norm_cast
    simp only [one_div, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs, neg_mul, neg_inj,
      ← rpow_natCast, ← rpow_mul hc.le, mul_assoc]
    simp

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)
g : 𝓢(ℝ, ℂ)
⊢ ((derivCLM F) ((toTemperedDistributionCLM ℝ F volume) f)) g =
    ((toTemperedDistributionCLM ℝ F volume) ((SchwartzMap.derivCLM 𝕜 F) f)) g

Tactic:
simp [integral_smul_deriv_right_eq_neg_left, integral_neg]

Goal state after: