mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
H1 : ¬BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
n : ℕ
⊢ g =ᵐ[volume.restrict (Ioc 0 ↑n)] 0

Tactic:
rcases not_bddAbove_iff.1 H1 n with ⟨s, hs, ns⟩

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
H1 : ¬BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
n : ℕ
s : ℝ
hs : s ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
ns : ↑n < s
⊢ g =ᵐ[volume.restrict (Ioc 0 ↑n)] 0
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
t : ℝ
ht : g t = G t
a : α
ha : f a = F a
⊢ {a | t ≤ f a} a = {a | t ≤ F a} a

Tactic:
by simp [Set.ofPred, ha]

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

Goal state before:
pp: 𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
F' : Type u_5
inst✝¹¹ : NormedAddCommGroup F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
toFun : 𝓓^{n}(Ω, F) → V
T : (K : Compacts E) → ↑K ⊆ ↑Ω → 𝓓^{n}_{K}(E, F) →L[𝕜] V
toFun_eq_T :
  ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω) (f : 𝓓^{n}_{K}(E, F)), toFun (ofSupportedIn K_sub_Ω f) = (T K K_sub_Ω) f
f g : 𝓓^{n}(Ω, F)
⊢ toFun (f + g) = toFun f + toFun g

Tactic:
set K : Compacts E := ⟨tsupport f ∪ tsupport g, .union f.hasCompactSupport g.hasCompactSupport⟩

Goal state after:
𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
F' : Type u_5
inst✝¹¹ : NormedAddCommGroup F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
toFun : 𝓓^{n}(Ω, F) → V
T : (K : Compacts E) → ↑K ⊆ ↑Ω → 𝓓^{n}_{K}(E, F) →L[𝕜] V
toFun_eq_T :
  ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω) (f : 𝓓^{n}_{K}(E, F)), toFun (ofSupportedIn K_sub_Ω f) = (T K K_sub_Ω) f
f g : 𝓓^{n}(Ω, F)
K : Compacts E := { carrier := tsupport ⇑f ∪ tsupport ⇑g, isCompact' := ⋯ }
⊢ toFun (f + g) = toFun f + toFun g
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_6
F₂ : Type u_7
F₃ : Type u_8
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : LocallyIntegrableOn φ (↑Ω) μ
f : 𝓓^{n}(Ω, F₁)
this : IntegrableOn (fun x ↦ (B (f x)) (φ x)) (tsupport ⇑f) μ
⊢ (support fun x ↦ (B (f x)) (φ x)) ⊆ tsupport ⇑f

Tactic:
refine subset_trans ?_ (subset_tsupport f)

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_6
F₂ : Type u_7
F₃ : Type u_8
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : LocallyIntegrableOn φ (↑Ω) μ
f : 𝓓^{n}(Ω, F₁)
this : IntegrableOn (fun x ↦ (B (f x)) (φ x)) (tsupport ⇑f) μ
⊢ (support fun x ↦ (B (f x)) (φ x)) ⊆ support ⇑f
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

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✝ : Nontrivial E
hT' : (algebraMap ℝ 𝕜) ‖T‖ ∈ resolventSet 𝕜 T
hT'' : -(algebraMap ℝ 𝕜) ‖T‖ ∈ resolventSet 𝕜 T
⊢ ∃ ε > 0, ∀ (x : E), |T.rayleighQuotient x| ≤ ‖T‖ - ε

Tactic:
obtain ⟨ε, hε0, hε⟩ := T.rayleighQuotient_le_of_norm_mem_resolventSet hT'

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✝ : Nontrivial E
hT' : (algebraMap ℝ 𝕜) ‖T‖ ∈ resolventSet 𝕜 T
hT'' : -(algebraMap ℝ 𝕜) ‖T‖ ∈ resolventSet 𝕜 T
ε : ℝ
hε0 : ε > 0
hε : ∀ (x : E), T.rayleighQuotient x ≤ ‖T‖ - ε
⊢ ∃ ε > 0, ∀ (x : E), |T.rayleighQuotient x| ≤ ‖T‖ - ε
Module: Mathlib.LinearAlgebra.Eigenspace.Charpoly

Goal state before:
pp: K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
f : End K V
h : (charpoly f).Splits
⊢ LinearMap.det f = (charpoly f).roots.prod

Tactic:
rw [← det_toMatrix (Module.Free.chooseBasis K V),
    Matrix.det_eq_prod_roots_charpoly_of_splits (by simpa using h), charpoly_toMatrix]

Goal state after:
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
r : ℝ
⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary)
      (diagonal (RCLike.ofReal ∘ ⇑((algebraMap ℝ C(↑(spectrum ℝ A), ℝ)) r) ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩)) =
    (algebraMap ℝ (Matrix n n 𝕜)) r

Tactic:
by
    simp only [Function.comp_def, algebraMap_apply, smul_eq_mul, mul_one]
    rw [← mul_one (algebraMap _ _ _), ← coe_mul_star_self hA.eigenvectorUnitary,
      ← Algebra.left_comm, coe_star, ← mul_assoc, conjStarAlgAut_apply]
    rfl

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.PosSemidef
⊢ (CFC.sqrt A)⁻¹ ^ 2 = A⁻¹

Tactic:
inv_pow',

Goal state after:
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.PosSemidef
⊢ (CFC.sqrt A ^ 2)⁻¹ = A⁻¹
Module: Mathlib.Analysis.InnerProductSpace.GramMatrix

Goal state before:
pp: E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
v : n → E
f : E →L[𝕜] F
c : n →₀ 𝕜
⊢ 0 ≤
    ∑ x ∈ c.support, ∑ x_1 ∈ c.support, star (c x) * ‖f‖ ^ 2 • gram 𝕜 v x x_1 * c x_1 -
      ∑ x ∈ c.support, ∑ x_1 ∈ c.support, star (c x) * gram 𝕜 (⇑f ∘ v) x x_1 * c x_1

Tactic:
sub_nonneg

Goal state after:
E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
v : n → E
f : E →L[𝕜] F
c : n →₀ 𝕜
⊢ ∑ x ∈ c.support, ∑ x_1 ∈ c.support, star (c x) * gram 𝕜 (⇑f ∘ v) x x_1 * c x_1 ≤
    ∑ x ∈ c.support, ∑ x_1 ∈ c.support, star (c x) * ‖f‖ ^ 2 • gram 𝕜 v x x_1 * c x_1
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
F : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
hf : AEStronglyMeasurable f μ
⊢ MemLp f 2 μ ↔ Integrable (fun x ↦ ‖f x‖ ^ 2) μ

Tactic:
rw [← memLp_one_iff_integrable]
  convert! (memLp_norm_rpow_iff hf two_ne_zero ENNReal.ofNat_ne_top).symm
  · simp
  · rw [div_eq_mul_inv, ENNReal.mul_inv_cancel two_ne_zero ENNReal.ofNat_ne_top]

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
s : Set α
f : ↥(Lp E 2 μ)
hs : MeasurableSet s
c : E
hμs : μ s ≠ ∞
x : α
hx : ↑↑(indicatorConstLp 2 hs hμs c) x = s.indicator (fun x ↦ c) x
⊢ ⟪↑↑(indicatorConstLp 2 hs hμs c) x, ↑↑f x⟫ = s.indicator (fun x ↦ ⟪c, ↑↑f x⟫) x

Tactic:
by_cases hxs : x ∈ s <;> simp [hx, hxs]

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

Goal state before:
pp: case neg.inl
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
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
i j : ℕ
x : X
h : ¬i = j
h' : i < j
⊢ (P (min i j + 1)) x - (P (min i (j + 1))) x - ((P (min (i + 1) j)) x - (P (min i j)) x) = 0

Tactic:
min_eq_left_of_lt h',

Goal state after:
case neg.inl
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
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
i j : ℕ
x : X
h : ¬i = j
h' : i < j
⊢ (P (i + 1)) x - (P (min i (j + 1))) x - ((P (min (i + 1) j)) x - (P i) x) = 0
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
K : Submodule 𝕜 E
hK : CompleteSpace ↥K
this : ∀ (b : Bool), CompleteSpace ↥(bif b then K else Kᗮ)
⊢ ⊤ ≤ (bif true then K else Kᗮ) ⊔ bif false then K else Kᗮ

Tactic:
cond,

Goal state after:
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
K : Submodule 𝕜 E
hK : CompleteSpace ↥K
this : ∀ (b : Bool), CompleteSpace ↥(bif b then K else Kᗮ)
⊢ ⊤ ≤ K ⊔ bif false then K else Kᗮ
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
x : E
s : Finset ι
⊢ ∀ c ∈ s, ↑(b.repr x) c • b c ∈ (span 𝕜 (Set.range ⇑b)).toAddSubmonoid

Tactic:
rintro i -

Goal state after:
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
x : E
s : Finset ι
i : ι
⊢ ↑(b.repr x) i • b i ∈ (span 𝕜 (Set.range ⇑b)).toAddSubmonoid
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : Fintype ι
b : HilbertBasis ι 𝕜 E
⊢ ⊤ ≤ span 𝕜 (Set.range ⇑b)

Tactic:
refine Eq.ge ?_
      classical
      have := (span 𝕜 (Finset.univ.image b : Set E)).closed_of_finiteDimensional
      simpa only [Finset.coe_image, Finset.coe_univ, Set.image_univ, HilbertBasis.dense_span] using
        this.submodule_topologicalClosure_eq.symm

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

Tactic:
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.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'
i : ℕ
hi : i ≤ n
⊢ ‖iteratedFDeriv ℝ i (⇑f) (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k'

Tactic:
have hpos : 0 < (1 + ‖g x‖) ^ k' := by positivity

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'
i : ℕ
hi : i ≤ n
hpos : 0 < (1 + ‖g x‖) ^ k'
⊢ ‖iteratedFDeriv ℝ i (⇑f) (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k'
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁷ : TopologicalSpace α
inst✝⁶ : BorelSpace α
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝³ : CompleteSpace E
inst✝² : MetrizableSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
hs : IsCompact 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 ∈ closure[inst✝⁷] (interior s)
hmg : ContinuousOn g s
⊢ x₀ ∈ s

Tactic:
by rw [← hs.isClosed.closure_eq]; exact closure_mono interior_subset h₀

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

Goal state before:
pp: case hg
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
⊢ Integrable (fun v ↦ e (-(L v) w) • g v) μ

Tactic:
exact (fourierIntegral_convergent_iff he hL w).2 hg

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

Goal state before:
pp: c y : ℝ
x✝ : y ∈ Iic 0
⊢ 0 ≤ rexp (id y)

Tactic:
exact (exp_pos _).le

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
V : Type u_5
W : Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → F →L[ℝ] E
a : F
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ (fourierIntegral e μ (ContinuousLinearMap.toLinearMap₁₂ L) f w) a =
    fourierIntegral e μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun x ↦ (f x) a) w

Tactic:
by
  rw [fourierIntegral, ContinuousLinearMap.integral_apply]
  · rfl
  · apply (fourierIntegral_convergent_iff he _ _).2 hf
    exact L.continuous₂

Goal state after:
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))
⊢ HasFiniteIntegral (fun a ↦ ‖s‖ * (‖↑(rexp (-a))‖ * ‖↑a ^ (s - 1)‖)) (volume.restrict (Ioc 0 Y))

Tactic:
refine (((Real.GammaIntegral_convergent hs).mono_set
    Ioc_subset_Ioi_self).hasFiniteIntegral.congr ?_).const_mul _

Goal state after:
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))
⊢ (fun x ↦ rexp (-x) * x ^ (s.re - 1)) =ᵐ[volume.restrict (Ioc 0 Y)] fun a ↦ ‖↑(rexp (-a))‖ * ‖↑a ^ (s - 1)‖
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ⊢ polarCoord.source =ᵐ[volume] univ

Tactic:
have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_ofPred_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    rw [Ne, LinearMap.ker_eq_top]
    intro h
    have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by rw [h]
    simp at this
  simp only [ae_eq_univ]
  exact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] (f : (i : Fin n) → E i → 𝕜),
    ∫ (x : (i : Fin n) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
⊢ ∫ (x : (i : Fin (n + 1)) → E i), ∏ i, f i (x i) ∂Measure.pi μ =
    ∫ (x : E 0 × ((i : Fin n) → E i.succ)), f 0 x.1 * ∏ i, f i.succ (x.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ)

Tactic:
rw [← ((measurePreserving_piFinSuccAbove μ 0).symm).integral_comp']

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] (f : (i : Fin n) → E i → 𝕜),
    ∫ (x : (i : Fin n) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
⊢ ∫ (x : E 0 × ((j : Fin n) → E (Fin.succAbove 0 j))),
      ∏ i, f i ((MeasurableEquiv.piFinSuccAbove E 0).symm x i) ∂(μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)) =
    ∫ (x : E 0 × ((i : Fin n) → E i.succ)), f 0 x.1 * ∏ i, f i.succ (x.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ)
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : RCLike 𝕜
E : Type u_3
f : E → 𝕜
mE : MeasurableSpace E
μ : Measure E
inst✝ : SigmaFinite μ
⊢ ∏ i, ∫ (x : E), f x ∂μ = (∫ (x : E), f x ∂μ) ^ card ι

Tactic:
Finset.prod_const,

Goal state after:
𝕜 : Type u_1
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : RCLike 𝕜
E : Type u_3
f : E → 𝕜
mE : MeasurableSpace E
μ : Measure E
inst✝ : SigmaFinite μ
⊢ (∫ (x : E), f x ∂μ) ^ Finset.univ.card = (∫ (x : E), f x ∂μ) ^ card ι
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: case e'_3.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
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
c : ℝ
w : V
⊢ 2 * ↑π * I * ↑⟪w, v⟫ = ↑(2 * π * ⟪w, v⟫) * I

Tactic:
simp only [ofReal_mul, ofReal_ofNat]

Goal state after:
case e'_3.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
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
c : ℝ
w : V
⊢ 2 * ↑π * I * ↑⟪w, v⟫ = 2 * ↑π * ↑⟪w, v⟫ * I
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ι → ℂ
⊢ (∏ x, (↑π / b) ^ 2⁻¹) * ∏ x, cexp (c x ^ 2 / (4 * b)) =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp ((∑ i, c i ^ 2) / (4 * b))

Tactic:
Finset.prod_const,

Goal state after:
b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ι → ℂ
⊢ ((↑π / b) ^ 2⁻¹) ^ Finset.univ.card * ∏ x, cexp (c x ^ 2 / (4 * b)) =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp ((∑ i, c i ^ 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
x w : V
⊢ ∫ (v : V), cexp (-(2 * ↑π * ↑⟪w, v⟫ * I) + (-(b * ↑‖v‖ ^ 2) + 2 * ↑π * I * ↑⟪x, v⟫)) =
    (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp (-(↑π ^ 2 * ↑‖x - w‖ ^ 2) / b)

Tactic:
convert! integral_cexp_neg_mul_sq_norm_add hb (2 * π * Complex.I) (x - w) using 3 with v

Goal state after:
case e'_2.e'_7
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
x w v : V
⊢ cexp (-(2 * ↑π * ↑⟪w, v⟫ * I) + (-(b * ↑‖v‖ ^ 2) + 2 * ↑π * I * ↑⟪x, v⟫)) =
    cexp (-b * ↑‖v‖ ^ 2 + 2 * ↑π * I * ↑⟪x - w, v⟫)

case e'_3
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
x w : V
⊢ -(↑π ^ 2 * ↑‖x - w‖ ^ 2) / b = (2 * ↑π * I) ^ 2 * ↑‖x - w‖ ^ 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
w : V
⊢ 𝓕 (fun v ↦ cexp (-b * ↑‖v‖ ^ 2)) w = (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖w‖ ^ 2 / b)

Tactic:
by
  simpa using fourier_gaussian_innerProductSpace' hb 0 w

Goal state after: