mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module 𝕜₁ M₁
inst✝⁴ : Module 𝕜₂ M₂
inst✝³ : IsTopologicalAddGroup M₁
inst✝² : ContinuousConstSMul 𝕜₁ M₁
inst✝¹ : IsTopologicalAddGroup M₂
inst✝ : ContinuousSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
U : Set M₂
hU : U ∈ 𝓝 0
K : Set M₂
hK : IsCompact K
hKf : ⇑f ⁻¹' K ∈ 𝓝 0
r : ℝ
hr : r > 0
hrU : ∀ (c : 𝕜₂), r ≤ ‖c‖ → K ⊆ c • U
c : 𝕜₁
hc : r < ‖c‖
hcnz : c ≠ 0
this✝ : σ₁₂ c⁻¹ • K ⊆ U
this : IsUnit c⁻¹
⊢ ⇑f ⁻¹' (σ₁₂ c⁻¹ • K) ∈ 𝓝 0

Tactic:
this.preimage_smul_setₛₗ σ₁₂,

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module 𝕜₁ M₁
inst✝⁴ : Module 𝕜₂ M₂
inst✝³ : IsTopologicalAddGroup M₁
inst✝² : ContinuousConstSMul 𝕜₁ M₁
inst✝¹ : IsTopologicalAddGroup M₂
inst✝ : ContinuousSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
U : Set M₂
hU : U ∈ 𝓝 0
K : Set M₂
hK : IsCompact K
hKf : ⇑f ⁻¹' K ∈ 𝓝 0
r : ℝ
hr : r > 0
hrU : ∀ (c : 𝕜₂), r ≤ ‖c‖ → K ⊆ c • U
c : 𝕜₁
hc : r < ‖c‖
hcnz : c ≠ 0
this✝ : σ₁₂ c⁻¹ • K ⊆ U
this : IsUnit c⁻¹
⊢ c⁻¹ • ⇑f ⁻¹' K ∈ 𝓝 0
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case a
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := ⋯
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
hx' : 1 / 2 < x
hx'' : 0 < x
h✝ : r - ↑n < 0
h : 0 < ↑n - r
⊢ Real.log 2 * (↑n - r) ≤ Real.log (3 / 2) * ↑k

Tactic:
apply le_trans _ hk₂

Goal state after:
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := ⋯
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
hx' : 1 / 2 < x
hx'' : 0 < x
h✝ : r - ↑n < 0
h : 0 < ↑n - r
⊢ Real.log 2 * (↑n - r) ≤ Real.log 2 * (↑N - r)
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
hK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖
C : 𝕜
hC : 1 < ‖C‖
ε : ℝ
hε : ε > 0
x : X
hx : ‖(T - μ • 1) x‖ < ε * ‖x‖
⊢ x ≠ 0

Tactic:
aesop

Goal state after:
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: n : ℕ
σ : Equiv.Perm (Fin n)
⊢ Monotone (⇑σ ∘ ⇑σ⁻¹)

pp: n : ℕ
σ : Equiv.Perm (Fin n)
⊢ ∀ (i j : Fin n), i < j → σ (σ⁻¹ i) = σ (σ⁻¹ j) → σ⁻¹ i < σ⁻¹ j

Tactic:
· simpa using monotone_id

Goal state after:
n : ℕ
σ : Equiv.Perm (Fin n)
⊢ ∀ (i j : Fin n), i < j → σ (σ⁻¹ i) = σ (σ⁻¹ j) → σ⁻¹ i < σ⁻¹ j
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
⊢ A.det = ∏ i, ↑(hA.eigenvalues i)

Tactic:
by
  simp [det_eq_prod_roots_charpoly_of_splits hA.splits_charpoly, hA.roots_charpoly_eq_eigenvalues]

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
x : 𝕜
| x ∈ spectrum 𝕜 A

Tactic:
rw [hA.spectral_theorem]

Goal state after:
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
x : 𝕜
| x ∈ spectrum 𝕜 (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues)))
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
μ : 𝕜
hμ : HasEigenvalue T μ
v : E
hv₁ : v ∈ (genEigenspace T μ) 1
hv₂ : v ≠ 0
⊢ (starRingEnd 𝕜) μ = μ

Tactic:
rw [mem_eigenspace_iff] at hv₁

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
μ : 𝕜
hμ : HasEigenvalue T μ
v : E
hv₁ : T v = μ • v
hv₂ : v ≠ 0
⊢ (starRingEnd 𝕜) μ = μ
Module: Mathlib.Analysis.Matrix.HermitianFunctionalCalculus

Goal state before:
pp: n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
| A

Tactic:
rw [hA.spectral_theorem]

Goal state after:
n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
| ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues))
Module: Mathlib.Analysis.Matrix.PosDef

Goal state before:
pp: n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
| A.PosSemidef

Tactic:
rw [hA.spectral_theorem]

Goal state after:
n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
| (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues))).PosSemidef
Module: Mathlib.Analysis.Matrix.PosDef

Goal state before:
pp: n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
| A.PosDef

Tactic:
rw [hA.spectral_theorem]

Goal state after:
n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
| (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues))).PosDef
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
μ : 𝕜
p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
⊢ eigenspace (T.restrict ⋯) μ = ⊥

Tactic:
refine eigenspace_restrict_eq_bot hT.orthogonalComplement_iSup_eigenspaces_invariant ?_

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
μ : 𝕜
p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
⊢ Disjoint (eigenspace T μ) (⨆ μ, eigenspace T μ)ᗮ
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
hμ : HasEigenvalue T μ
⊢ ∃ i, ↑(hT.eigenvalues hn i) = μ

Tactic:
obtain ⟨i, hi⟩ := hT.exists_unsortedEigenvalues_eq hn hμ
  use ((Tuple.sort (hT.unsortedEigenvalues hn)).symm i).revPerm
  simp [eigenvalues_def, hi]

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
⊢ (hA.cfc fun x ↦ ‖x‖).det = ↑‖A.det‖

Tactic:
simp [IsHermitian.cfc, -Unitary.conjStarAlgAut_apply, hA.det_eq_prod_eigenvalues]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b : A
h₁ : Commute a b
h₂ : Commute a (star b)
hab : Commute (abs a) (abs b)
⊢ abs a * abs b * (abs a * abs b) = star (a * b) * (a * b)

Tactic:
hab.eq,

Goal state after:
A : Type u_2
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b : A
h₁ : Commute a b
h₂ : Commute a (star b)
hab : Commute (abs a) (abs b)
⊢ abs b * abs a * (abs b * abs a) = star (a * b) * (a * b)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Module ℝ A
inst✝¹² : SMulCommClass ℝ A A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁹ : PartialOrder A
inst✝⁸ : StarOrderedRing A
inst✝⁷ : NonnegSpectrumClass ℝ A
inst✝⁶ : IsTopologicalRing A
inst✝⁵ : T2Space A
inst✝⁴ : StarModule ℝ A
R : Type u_3
inst✝³ : Semiring R
inst✝² : SMulWithZero R ℝ≥0
inst✝¹ : SMul R A
inst✝ : IsScalarTower R ℝ≥0 A
r✝ : R
a : A
r : ℝ≥0
⊢ r • abs a * r • abs a = star (r • a) * r • a

Tactic:
simp [mul_smul_comm, smul_mul_assoc, abs_mul_abs]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: 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 g : ↥(Lp E 2 μ)
x : α
⊢ ‖↑↑f x‖ * ‖↑↑g x‖ ≤ 2 * ‖↑↑f x‖ * ‖↑↑g x‖

Tactic:
by
        gcongr
        exact le_mul_of_one_le_left (norm_nonneg _) one_le_two

Goal state after:
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.WeaklyRegular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
H :
  ∃ g,
    eLpNorm (f - g) p μ ≤ ε ∧
      Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ IsBounded (range g)
⊢ ∃ g, eLpNorm (f - ⇑g) p μ ≤ ε ∧ MemLp (⇑g) p μ

Tactic:
rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩
    exact ⟨⟨⟨g, g_cont⟩, Metric.isBounded_range_iff.1 g_bd⟩, hg, g_mem⟩

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

Goal state before:

Tactic:
conj _ = conj (∑' i, ⟪g i, f i⟫) := by congr
        _ = ∑' i, conj ⟪g i, f i⟫ := RCLike.conjCLE.map_tsum
        _ = ∑' i, ⟪f i, g i⟫ := by simp only [inner_conj_symm]
        _ = _ := by congr

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

Goal state before:
pp: case pos
ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : DecidableEq ι
f : ↥(lp G 2)
j : ι
a : G j
⊢ ⟪Pi.single j a j, ↑f j⟫ = ⟪a, ↑f j⟫

Tactic:
Pi.single_eq_same

Goal state after:
case pos
ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : DecidableEq ι
f : ↥(lp G 2)
j : ι
a : G j
⊢ ⟪a, ↑f j⟫ = ⟪a, ↑f j⟫
Module: Mathlib.Analysis.Normed.Module.Bases

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

Tactic:
simp only [Function.comp, LinearMap.coe_mk, AddHom.coe_mk,
                   LinearMap.mkContinuous_apply, ← hcoeff]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
x : X
n : ℕ
⊢ (D.P n) x = ∑ b ∈ Finset.range n, (succSub D.P b) x
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
i : ℤ
⊢ ∫ (a : AddCircle T), inner ℂ (↑↑(fourierBasis i) a) (↑↑f a) ∂haarAddCircle =
    ∫ (t : AddCircle T), (starRingEnd ℂ) (↑↑(fourierLp 2 i) t) * ↑↑f t ∂haarAddCircle

Tactic:
coe_fourierBasis

Goal state after:
T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
i : ℤ
⊢ ∫ (a : AddCircle T), inner ℂ (↑↑(fourierLp 2 i) a) (↑↑f a) ∂haarAddCircle =
    ∫ (t : AddCircle T), (starRingEnd ℂ) (↑↑(fourierLp 2 i) t) * ↑↑f t ∂haarAddCircle
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
n m : ℤ
⊢ inner ℂ (fourierBasis m) (fourierLp 2 n) = Pi.single n 1 m

Tactic:
coe_fourierBasis

Goal state after:
T : ℝ
hT : Fact (0 < T)
n m : ℤ
⊢ inner ℂ (fourierLp 2 m) (fourierLp 2 n) = Pi.single n 1 m
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: x : ℝ
⊢ HasDerivAt (fun x ↦ ↑(𝐞 x)) (2 * ↑π * I * ↑(𝐞 x)) x

Tactic:
by
  have h1 (y : ℝ) : 𝐞 y = fourier 1 (y : UnitAddCircle) := by
    rw [fourierChar_apply, fourier_coe_apply]
    push_cast
    ring_nf
  simpa only [h1, Int.cast_one, ofReal_one, div_one, mul_one] using hasDerivAt_fourier 1 1 x

Goal state after:
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
d : ℝ
left✝ : 0 < d
hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d
x : E
hx : x ≠ 0
⊢ ‖f x‖ ≤ d * ‖x‖ ^ (-↑k)

Tactic:
zpow_neg,

Goal state after:
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
d : ℝ
left✝ : 0 < d
hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d
x : E
hx : x ≠ 0
⊢ ‖f x‖ ≤ d * (‖x‖ ^ ↑k)⁻¹
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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
h₁ : HasCompactSupport f
h₂ : ContDiff ℝ ∞ f
k n : ℕ
g : E → ℝ := fun x ↦ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖
hg₁ : Continuous g
hg₂ : HasCompactSupport g
x₀ : E
hx₀ : ∀ (y : E), g y ≤ g x₀
⊢ ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ C

Tactic:
exact ⟨g x₀, hx₀⟩

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

Goal state before:
pp: E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
H :
  ((fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fderiv ℝ (iteratedFDeriv ℝ n f)) w) (m 0)) (Fin.tail m) =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ x, -(L (m x)) w) • ∫ (v : V), 𝐞 (-(L v) w) • f v ∂μ
⊢ fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L)
      (fun x ↦ ((fderiv ℝ (iteratedFDeriv ℝ n f) x) (m 0)) (Fin.tail m)) w =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ i, -(L (m i)) w) • fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f w

Tactic:
← fourierIntegral_continuousMultilinearMap_apply' ((J.apply_continuousLinearMap _)),

Goal state after:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
H :
  ((fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fderiv ℝ (iteratedFDeriv ℝ n f)) w) (m 0)) (Fin.tail m) =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ x, -(L (m x)) w) • ∫ (v : V), 𝐞 (-(L v) w) • f v ∂μ
⊢ (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun a ↦ (fderiv ℝ (iteratedFDeriv ℝ n f) a) (m 0)) w)
      (Fin.tail m) =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ i, -(L (m i)) w) • fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f w
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℂ
ha : a.re < 0
c : ℝ
⊢ a ≠ 0

Tactic:
aesop

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

Goal state before:
pp: E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
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‖) μ
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
v : V
w : W
⊢ (2 * π) ^ n * (|(L v) w| ^ n * ‖iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w‖) =
    ‖(fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)) w
          n)
        fun x ↦ v‖

Tactic:
simp [norm_smul, abs_of_nonneg pi_nonneg]

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✝⁸ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝⁷ : TopologicalSpace α
inst✝⁶ : SecondCountableTopology α
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : NormedAddCommGroup F
inst✝³ : LinearOrder α
inst✝² : CompactIccSpace α
g' : α → F
inst✝¹ : atBot.IsMeasurablyGenerated
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[atBot] g
hg : IntegrableAtFilter g atBot μ
ho' : f =O[atTop] g'
hg' : IntegrableAtFilter g' atTop μ
⊢ Integrable f μ

Tactic:
by
  refine integrable_iff_integrableAtFilter_atBot_atTop.mpr
    ⟨⟨ho.integrableAtFilter ?_ hg, ho'.integrableAtFilter ?_ hg'⟩, hf⟩
  all_goals exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter

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

Goal state before:
pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
⊢ Tendsto (fun x ↦ ↑x ^ (a + 1)) atTop (𝓝 0)

Tactic:
rw [tendsto_zero_iff_norm_tendsto_zero]

Goal state after:
a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
⊢ Tendsto (fun x ↦ ‖↑x ^ (a + 1)‖) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
this : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
⊢ Tendsto (fun X ↦ -↑X ^ s * ↑(rexp (-X))) atTop (𝓝 0)

Tactic:
rw [tendsto_zero_iff_norm_tendsto_zero]

Goal state after:
s : ℂ
hs : 0 < s.re
this : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
⊢ Tendsto (fun x ↦ ‖-↑x ^ s * ↑(rexp (-x))‖) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: case inr
b : ℝ
hb : 0 < b
⊢ (∫ (x : ℝ), ↑(rexp (-b * x ^ 2))) ^ 2 = ↑(π / b)

Tactic:
ofReal_div

Goal state after:
case inr
b : ℝ
hb : 0 < b
⊢ (∫ (x : ℝ), ↑(rexp (-b * x ^ 2))) ^ 2 = ↑π / ↑b
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
i0 : 1 - s.re - 1 < ↑⌊1 - s.re⌋₊
⊢ -s.re < ↑⌊1 - s.re⌋₊ + ↑k

Tactic:
simp only [sub_sub_cancel_left] at i0

Goal state after:
case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
i0 : -s.re < ↑⌊1 - s.re⌋₊
⊢ -s.re < ↑⌊1 - s.re⌋₊ + ↑k
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: case hx
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ 0 ≤ π / 1

Tactic:
rw [div_one]; exact pi_pos.le

Goal state after:
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}
⊢ (∫ (x : ℝ), cexp (-c * ↑x ^ 2)) ^ 2 = ((↑π / c) ^ (1 / 2)) ^ 2

Tactic:
integral_gaussian_sq_complex hc,

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 = ((↑π / c) ^ (1 / 2)) ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (verticalIntegral b c) atTop (𝓝 0)

Tactic:
rw [tendsto_zero_iff_norm_tendsto_zero]

Goal state after:
b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun x ↦ ‖verticalIntegral b c x‖) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case hf
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ 0 ≤ᵐ[volume.restrict (Ioi 0)] fun x ↦ rexp (-x) * x ^ (s - 1)

Tactic:
refine eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi) ?_

Goal state after:
case hf
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ ∀ x ∈ Ioi 0, 0 x ≤ (fun x ↦ rexp (-x) * x ^ (s - 1)) x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ ↑(∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * rexp (-(r * t))) = ↑(1 / r) ^ ↑a * Complex.Gamma ↑a

Tactic:
ofReal_div

Goal state after:
a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ ↑(∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * rexp (-(r * t))) = (↑1 / ↑r) ^ ↑a * Complex.Gamma ↑a
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s + 1 = -↑m
⊢ s = -↑(1 + m)

Tactic:
← eq_sub_iff_add_eq

Goal state after:
case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ s = -↑(1 + m)
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_11
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
inst✝ : FiniteDimensional ℝ E
x : E
hx : x ∈ Ω
s : Set E
hxs : x ∈ s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
t : Set E := s ∩ ↑Ω
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
htx : x ∈ t
u : E → ℝ
h₁ : tsupport u ⊆ t
h₂ : HasCompactSupport u
h₃ : ContDiff ℝ (↑n) u
h₄ : u x = 1
⊢ tsupport u ⊆ ↑Ω

Tactic:
aesop

Goal state after: