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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • x₀ + b • T x₀ = 0
⊢ T x₀ = T.rayleighQuotient x₀ • x₀

Tactic:
by_cases hx₀ : x₀ = 0

Goal state after:
case pos
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • x₀ + b • T x₀ = 0
hx₀ : x₀ = 0
⊢ T x₀ = T.rayleighQuotient x₀ • x₀

case neg
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • x₀ + b • T x₀ = 0
hx₀ : ¬x₀ = 0
⊢ T x₀ = T.rayleighQuotient x₀ • x₀
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : MeasurableSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
p : ℝ≥0
hp : ↑p ≠ 0
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
hp_pos : 0 < ↑p
l : ℕ
hl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ
k : ℕ := ⌈↑l / ↑p⌉₊
hlk : ↑l ≤ ↑k * ↑p
this : ∫⁻ (a : E), ‖(1 + ‖a‖) ^ (-(↑k * ↑p))‖ₑ ∂μ < ∞
⊢ eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) (↑p) μ < ∞

Tactic:
eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top hp ENNReal.coe_ne_top

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : MeasurableSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
p : ℝ≥0
hp : ↑p ≠ 0
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
hp_pos : 0 < ↑p
l : ℕ
hl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ
k : ℕ := ⌈↑l / ↑p⌉₊
hlk : ↑l ≤ ↑k * ↑p
this : ∫⁻ (a : E), ‖(1 + ‖a‖) ^ (-(↑k * ↑p))‖ₑ ∂μ < ∞
⊢ ∫⁻ (a : E), ‖(1 + ‖a‖) ^ (-↑k)‖ₑ ^ (↑p).toReal ∂μ < ∞
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
⊢ ((Multiset.map (⇑RCLike.re) A.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) = List.ofFn hA.eigenvalues₀

Tactic:
simp_rw [hA.roots_charpoly_eq_eigenvalues₀, Fin.univ_val_map, Multiset.map_coe, List.map_ofFn,
    Function.comp_def, RCLike.ofReal_re, Multiset.coe_sort]
  apply List.mergeSort_of_pairwise
  simp_rw [decide_eq_true_eq, ← List.sortedGE_iff_pairwise]
  exact (eigenvalues₀_antitone hA).sortedGE_ofFn

Goal state after:
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
h : A = 0
⊢ hA.eigenvalues = 0

Tactic:
by ext; simp [h, eigenvalues_eq]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ (hT.unsortedEigenvectorBasis hn) i ∈ eigenspace T μ

Tactic:
unsortedEigenvectorBasis

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ (DirectSum.IsInternal.subordinateOrthonormalBasis hn ⋯ ⋯) i ∈ eigenspace T μ
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
μ : 𝕜
⊢ {i | ↑((hT.unsortedEigenvalues hn ∘ ⇑(Tuple.sort (hT.unsortedEigenvalues hn)) ∘ ⇑Fin.revPerm) i) = μ}.card =
    {i | ↑(hT.unsortedEigenvalues hn i) = μ}.card

Tactic:
apply Finset.card_equiv (Fin.revPerm.trans (Tuple.sort (hT.unsortedEigenvalues hn)))

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
μ : 𝕜
⊢ ∀ (i : Fin n),
    i ∈ {i | ↑((hT.unsortedEigenvalues hn ∘ ⇑(Tuple.sort (hT.unsortedEigenvalues hn)) ∘ ⇑Fin.revPerm) i) = μ} ↔
      (Equiv.trans Fin.revPerm (Tuple.sort (hT.unsortedEigenvalues hn))) i ∈ {i | ↑(hT.unsortedEigenvalues hn i) = μ}
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: case e_f
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)

Tactic:
←
    ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) zero_le_two,

Goal state after:
case e_f
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ENNReal.ofReal ‖↑↑f x‖ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2)
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
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.OuterRegular
hp : p ≠ ∞
s u : Set α
s_closed : IsClosed[inst✝⁶] s
u_open : IsOpen[inst✝⁶] u
hsu : s ⊆ u
hs : μ s ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
η : ℝ≥0
η_pos : 0 < η
hη : ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun _x ↦ c) p μ ≤ ε
ηpos : 0 < ↑η
⊢ ∃ f,
    Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧
      eLpNorm (fun x ↦ f x - s.indicator (fun _y ↦ c) x) p μ ≤ ε ∧
        (∀ (x : α), ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ MemLp f p μ

Tactic:
obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α), V ⊇ s ∧ IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η :=
    s_closed.measurableSet.exists_isOpen_sdiff_lt hs ηpos.ne'

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
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.OuterRegular
hp : p ≠ ∞
s u : Set α
s_closed : IsClosed[inst✝⁶] s
u_open : IsOpen[inst✝⁶] u
hsu : s ⊆ u
hs : μ s ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
η : ℝ≥0
η_pos : 0 < η
hη : ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun _x ↦ c) p μ ≤ ε
ηpos : 0 < ↑η
V : Set α
sV : V ⊇ s
V_open : IsOpen[inst✝⁶] V
h'V : μ V < ∞
hV : μ (V \ s) < ↑η
⊢ ∃ f,
    Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧
      eLpNorm (fun x ↦ f x - s.indicator (fun _y ↦ c) x) p μ ≤ ε ∧
        (∀ (x : α), ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ MemLp f p μ
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∀ (f g : α → E),
    Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ MemLp f p μ ∧ HasCompactSupport f →
      Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ HasCompactSupport g →
        Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f + g) ∧
          MemLp (f + g) p μ ∧ HasCompactSupport (f + g)

Tactic:
rintro f g ⟨f_cont, f_mem, hf⟩ ⟨g_cont, g_mem, hg⟩
    exact ⟨f_cont.add g_cont, f_mem.add g_mem, hf.add hg⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f✝ : α → E
hf : MemLp f✝ p μ
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → E
_f_cont : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
f_mem : MemLp f p μ
_hf : HasCompactSupport f
⊢ AEStronglyMeasurable f μ

Tactic:
exact f_mem.aestronglyMeasurable

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

Goal state before:
pp: case e_A
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
b : SchauderBasis 𝕜 X
n m : ℕ
x : X
a✝ : ℕ
⊢ a✝ < n ∧ a✝ < m ↔ a✝ < min n m

Tactic:
omega

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
X : Type u_2
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
b : SchauderBasis 𝕜 X
inst✝ : CompleteSpace X
⊢ BddAbove (Set.range fun n ↦ ‖b.proj n‖₊)

Tactic:
by
  obtain ⟨C, hC⟩ := b.exists_norm_proj_le
  have hCpos : 0 ≤ C := by simpa [proj_zero] using hC 0
  refine ⟨C.toNNReal, ?_⟩
  rintro _ ⟨n, rfl⟩
  rw [← NNReal.coe_le_coe, Real.coe_toNNReal C hCpos, coe_nnnorm]
  exact hC n

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
P : ℕ → X →L[𝕜] X
hrank : ∀ (n : ℕ), Module.finrank 𝕜 ↥(↑(P n)).range = n
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
n : ℕ
U : Submodule 𝕜 X := (↑(succSub P n)).range
V : Submodule 𝕜 X := (↑(P n)).range
W : Submodule 𝕜 X := (↑(P (n + 1))).range
hV : V ≤ W
hUW : U ≤ W
hW : W = U ⊔ V
⊢ Module.finrank 𝕜 ↥(↑(succSub P n)).range = 1

Tactic:
have hdisj : U ⊓ V = ⊥ := eq_bot_iff.mpr fun x ⟨⟨y, hy⟩, ⟨z, hz⟩⟩ ↦ by
    simp only [Submodule.mem_bot]
    calc x = (P n) x := by rw [← hz, ContinuousLinearMap.coe_coe, hcomp, min_self]
         _ = 0       := by rw [← hy, ContinuousLinearMap.coe_coe]; simp [succSub, map_sub, hcomp]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
P : ℕ → X →L[𝕜] X
hrank : ∀ (n : ℕ), Module.finrank 𝕜 ↥(↑(P n)).range = n
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
n : ℕ
U : Submodule 𝕜 X := (↑(succSub P n)).range
V : Submodule 𝕜 X := (↑(P n)).range
W : Submodule 𝕜 X := (↑(P (n + 1))).range
hV : V ≤ W
hUW : U ≤ W
hW : W = U ⊔ V
hdisj : U ⊓ V = ⊥
⊢ Module.finrank 𝕜 ↥(↑(succSub P n)).range = 1
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
n : ℤ
x : ℝ
⊢ Complex.exp (↑2 * ↑π / ↑T * (↑↑n * ↑x) * Complex.I) = Complex.exp (2 * ↑π * Complex.I * ↑n * ↑x / ↑T)

Tactic:
Complex.ofReal_intCast

Goal state after:
T : ℝ
n : ℤ
x : ℝ
⊢ Complex.exp (↑2 * ↑π / ↑T * (↑n * ↑x) * Complex.I) = Complex.exp (2 * ↑π * Complex.I * ↑n * ↑x / ↑T)
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:
rw [orthonormal_iff_ite]
  intro i j
  rw [← b.repr.inner_map_map (b i) (b j), b.repr_self, b.repr_self, lp.inner_single_left,
    lp.single_apply, Pi.single_apply]
  simp

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) • fourierSMulRight L f v) w' =
    ((ContinuousLinearMap.toSpanSingleton ℝ (2 * ↑π * I * ↑(𝐞 (-(L v) w))) ∘SL (-L v)).smulRight (f v)) w'

Tactic:
simp_rw [fourierSMulRight, smul_apply, ContinuousLinearMap.smulRight_apply]

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
⊢ 𝐞 (-(L v) w) • -(2 * ↑π * I) • (L v) w' • f v =
    (ContinuousLinearMap.toSpanSingleton ℝ (2 * ↑π * I * ↑(𝐞 (-(L v) w))) ∘SL (-L v)) w' • f v
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
h't : μ t ≠ ∞
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝⁴] 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 a)
h : α → E := g - t.indicator fun x ↦ a
A : Tendsto (fun i ↦ ∫ (x : α) in s, φ i x • h x ∂μ + (∫ (x : α) in t, φ i x ∂μ) • a) l (𝓝 a)
i : ι
hi : IntegrableOn (fun x ↦ φ i x • g x) s μ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < 1
⊢ Integrable (t.indicator fun a_1 ↦ φ i a_1 • a) (μ.restrict s)

Tactic:
integrable_indicator_iff ht

Goal state after:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
h't : μ t ≠ ∞
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝⁴] 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 a)
h : α → E := g - t.indicator fun x ↦ a
A : Tendsto (fun i ↦ ∫ (x : α) in s, φ i x • h x ∂μ + (∫ (x : α) in t, φ i x ∂μ) • a) l (𝓝 a)
i : ι
hi : IntegrableOn (fun x ↦ φ i x • g x) s μ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < 1
⊢ IntegrableOn (fun a_1 ↦ φ i a_1 • a) t (μ.restrict s)
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: case h₁
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 : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
m : ℕ × ℕ
k n : ℕ
hk : k ≤ m.1
hn : n ≤ m.2
f : 𝓢(E, F)
x : E
hk' : Finset.range (k + 1) ⊆ Finset.range (m.1 + 1)
i : ℕ
hi : i ∈ Finset.range (m.1 + 1)
⊢ (SchwartzMap.seminorm 𝕜 i n) f ≤ ((Finset.Iic m).sup fun m ↦ SchwartzMap.seminorm 𝕜 m.1 m.2) f

Tactic:
apply Seminorm.le_def.1

Goal state after:
case h₁
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 : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
m : ℕ × ℕ
k n : ℕ
hk : k ≤ m.1
hn : n ≤ m.2
f : 𝓢(E, F)
x : E
hk' : Finset.range (k + 1) ⊆ Finset.range (m.1 + 1)
i : ℕ
hi : i ∈ Finset.range (m.1 + 1)
⊢ SchwartzMap.seminorm 𝕜 i n ≤ (Finset.Iic m).sup fun m ↦ SchwartzMap.seminorm 𝕜 m.1 m.2
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 := ⋯
m : ℕ
this : ‖T‖ ≤ 1
⊢ ↑((Fintype.card (Fin n)).descFactorial m) * ‖T‖ * ‖fun x ↦ L v‖ ^ (Fintype.card (Fin n) - m) ≤
    ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)

Tactic:
simp only [Fintype.card_fin]

Goal state after:
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
m : ℕ
this : ‖T‖ ≤ 1
⊢ ↑(n.descFactorial m) * ‖T‖ * ‖fun x ↦ L v‖ ^ (n - m) ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
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:
gcongr; exacts [le_add_self, Finset.mem_range_succ_iff.mp hi]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : CommRing 𝕜
V : Type u_2
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module 𝕜 V
inst✝⁹ : MeasurableSpace V
W : Type u_3
inst✝⁸ : AddCommGroup W
inst✝⁷ : Module 𝕜 W
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace V
inst✝¹ : BorelSpace V
inst✝ : TopologicalSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
he : Continuous[inst✝⁴, _] ⇑e
hL : Continuous[instTopologicalSpaceProd, inst✝⁴] fun p ↦ (L p.1) p.2
f g : V → E
hf : Integrable f μ
hg : Integrable g μ
w : W
⊢ fourierIntegral e μ L (f + g) w = (fourierIntegral e μ L f + fourierIntegral e μ L g) w

Tactic:
dsimp only [Pi.add_apply, fourierIntegral]

Goal state after:
𝕜 : Type u_1
inst✝¹² : CommRing 𝕜
V : Type u_2
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module 𝕜 V
inst✝⁹ : MeasurableSpace V
W : Type u_3
inst✝⁸ : AddCommGroup W
inst✝⁷ : Module 𝕜 W
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace V
inst✝¹ : BorelSpace V
inst✝ : TopologicalSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
he : Continuous[inst✝⁴, _] ⇑e
hL : Continuous[instTopologicalSpaceProd, inst✝⁴] fun p ↦ (L p.1) p.2
f g : V → E
hf : Integrable f μ
hg : Integrable g μ
w : W
⊢ ∫ (v : V), e (-(L v) w) • (f v + g v) ∂μ = ∫ (v : V), e (-(L v) w) • f v ∂μ + ∫ (v : V), e (-(L v) w) • g v ∂μ
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : MeasurableSpace E
inst✝² : OpensMeasurableSpace E
inst✝¹ : SecondCountableTopologyEither E F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝ : μ.IsOpenPosMeasure
f g : 𝓢(E, F)
⊢ (fun f ↦ f.toLp p μ) f = (fun f ↦ f.toLp p μ) g → f = g

Tactic:
by simpa [toLp] using (Continuous.ae_eq_iff_eq μ f.continuous g.continuous).mp

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

Goal state before:
pp: c : ℝ
⊢ ∫ (x : ℝ) in Iic c, rexp x = rexp c

Tactic:
refine
    tendsto_nhds_unique
      (intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) ?_
  simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
  exact tendsto_exp_atBot.const_sub _

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: case neg
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : ¬IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
this : ¬Integrable (fun x ↦ f |x|) volume
⊢ ∫ (x : ℝ), f |x| = 2 * 0

Tactic:
integral_undef this,

Goal state after:
case neg
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : ¬IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
this : ¬Integrable (fun x ↦ f |x|) volume
⊢ 0 = 2 * 0
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s : ℝ
⊢ ¬IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume

Tactic:
intro h
  rcases le_or_gt s (-1) with hs | hs
  · have : IntegrableOn (fun x ↦ x ^ s) (Ioo (0 : ℝ) 1) := h.mono Ioo_subset_Ioi_self le_rfl
    rw [integrableOn_Ioo_rpow_iff zero_lt_one] at this
    exact hs.not_gt this
  · have : IntegrableOn (fun x ↦ x ^ s) (Ioi (1 : ℝ)) := h.mono (Ioi_subset_Ioi zero_le_one) le_rfl
    rw [integrableOn_Ioi_rpow_iff zero_lt_one] at this
    exact hs.not_gt this

Goal state after:
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
h : ↑↑(c • f) x = (c • ↑↑f) x
⊢ 𝐞 (-⟪x, x✝⟫) • (c • ↑↑f) x = (RingHom.id ℂ) c • 𝐞 (-⟪x, x✝⟫) • ↑↑f x

Tactic:
smul_comm

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 = 𝐞 (-⟪x, x✝⟫) • (RingHom.id ℂ) c • ↑↑f x
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
hf' : Integrable (fun x ↦ x • f x) volume
w : ℝ
hf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume
L : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := (ContinuousLinearMap.mul ℝ ℝ).flip
⊢ Integrable (fun v ↦ (L v).smulRight (f v)) volume

Tactic:
convert! ((ContinuousLinearMap.toSpanSingletonLIE ℝ
      E).toContinuousLinearEquiv.toContinuousLinearMap).integrable_comp hf' using 2 with _ v

Goal state after:
case e'_6
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
hf' : Integrable (fun x ↦ x • f x) volume
w : ℝ
hf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume
L : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := (ContinuousLinearMap.mul ℝ ℝ).flip
v : ℝ
⊢ (L v).smulRight (f v) = ↑↑(ContinuousLinearMap.toSpanSingletonLIE ℝ E) (v • f v)