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
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
x : ℝ
hx : x ∈ Ioc 0 s
⊢ ∞ * ENNReal.ofReal (g x) ≤ μ {a | x ≤ f a} * ENNReal.ofReal (g x)

Tactic:
rw [← h's]

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
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
x : ℝ
hx : x ∈ Ioc 0 s
⊢ μ {a | s < f a} * ENNReal.ofReal (g x) ≤ μ {a | x ≤ f a} * ENNReal.ofReal (g x)
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case hf
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : μ.HasTemperateGrowth
f : E → F
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : E), ‖f x‖ ≤ C₁
h'f : ∀ (x : E), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
h''f : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable (fun x ↦ ‖x‖ ^ k * ‖f x‖) μ

pp: case h
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : μ.HasTemperateGrowth
f : E → F
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : E), ‖f x‖ ≤ C₁
h'f : ∀ (x : E), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
h''f : AEStronglyMeasurable f μ
⊢ ∀ᵐ (a : E) ∂μ, ‖‖a‖ ^ k * ‖f a‖‖ ≤ 2 ^ μ.integrablePower * (C₁ + C₂) * (1 + ‖a‖) ^ (-↑μ.integrablePower)

Tactic:
· exact AEStronglyMeasurable.mul (aestronglyMeasurable_id.norm.pow _) h''f.norm

Goal state after:
case h
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : μ.HasTemperateGrowth
f : E → F
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : E), ‖f x‖ ≤ C₁
h'f : ∀ (x : E), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
h''f : AEStronglyMeasurable f μ
⊢ ∀ᵐ (a : E) ∂μ, ‖‖a‖ ^ k * ‖f a‖‖ ≤ 2 ^ μ.integrablePower * (C₁ + C₂) * (1 + ‖a‖) ^ (-↑μ.integrablePower)
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
inst✝ : CompleteSpace X
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h₁ : ¬HasEigenvalue (↑T) μ
S : X →L[𝕜] X := T - μ • 1
K✝ : NNReal
hK✝ : AntilipschitzWith K✝ ⇑S
h₂ : ¬Function.Surjective ⇑S
c : 𝕜
hc : 1 < ‖c‖
f : ℕ → X
hf_norm_lower : ∀ (n : ℕ), 1 ≤ ‖f n‖
hf_norm_upper : ∀ (n : ℕ), ‖f n‖ ≤ ‖c‖ + 1
hf_far : ∀ (n : ℕ), ∀ y ∈ LinearMap.range (↑S ^ (n + 1)), 1 ≤ ‖f n - y‖
hf_mem : ∀ {n m : ℕ}, m ≤ n → f n ∈ (↑S ^ m).range
hf_mem' : ∀ {n m : ℕ}, m ≤ n → S (f n) ∈ (↑S ^ (m + 1)).range
hp : Pairwise fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 (‖c‖ + 1) ⊆ K
y : X
hyK : y ∈ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hψy : CauchySeq ((fun n ↦ ↑T (f n)) ∘ ψ)
⊢ False

Tactic:
rw [Metric.cauchySeq_iff'] at hψy

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
inst✝ : CompleteSpace X
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h₁ : ¬HasEigenvalue (↑T) μ
S : X →L[𝕜] X := T - μ • 1
K✝ : NNReal
hK✝ : AntilipschitzWith K✝ ⇑S
h₂ : ¬Function.Surjective ⇑S
c : 𝕜
hc : 1 < ‖c‖
f : ℕ → X
hf_norm_lower : ∀ (n : ℕ), 1 ≤ ‖f n‖
hf_norm_upper : ∀ (n : ℕ), ‖f n‖ ≤ ‖c‖ + 1
hf_far : ∀ (n : ℕ), ∀ y ∈ LinearMap.range (↑S ^ (n + 1)), 1 ≤ ‖f n - y‖
hf_mem : ∀ {n m : ℕ}, m ≤ n → f n ∈ (↑S ^ m).range
hf_mem' : ∀ {n m : ℕ}, m ≤ n → S (f n) ∈ (↑S ^ (m + 1)).range
hp : Pairwise fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 (‖c‖ + 1) ⊆ K
y : X
hyK : y ∈ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hψy : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (((fun n ↦ ↑T (f n)) ∘ ψ) n) (((fun n ↦ ↑T (f n)) ∘ ψ) N) < ε
⊢ False
Module: Mathlib.Geometry.Manifold.SmoothApprox

Goal state before:
pp: E : Type u_1
F : Type u_2
H : Type u_3
M : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
f : M → F
ε : M → ℝ
n : ℕ∞
f_cont : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
ε_cont : Continuous[inst✝⁴, _] ε
ε_pos : ∀ (x : M), 0 < ε x
S U : Set M
hS : IsClosed[inst✝⁴] S
hU : U ∈ 𝓝ˢ S
hfU : ContMDiffOn I 𝓘(ℝ, F) (↑n) f U
⊢ ∃ g, (∀ (x : M), dist (g x) (f x) < ε x) ∧ EqOn (⇑g) f S ∧ support ⇑g ⊆ support f

Tactic:
have dist_f_f : ∀ x, dist (f x) (f x) < ε x := by simpa only [dist_self] using ε_pos

Goal state after:
E : Type u_1
F : Type u_2
H : Type u_3
M : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
f : M → F
ε : M → ℝ
n : ℕ∞
f_cont : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
ε_cont : Continuous[inst✝⁴, _] ε
ε_pos : ∀ (x : M), 0 < ε x
S U : Set M
hS : IsClosed[inst✝⁴] S
hU : U ∈ 𝓝ˢ S
hfU : ContMDiffOn I 𝓘(ℝ, F) (↑n) f U
dist_f_f : ∀ (x : M), dist (f x) (f x) < ε x
⊢ ∃ g, (∀ (x : M), dist (g x) (f x) < ε x) ∧ EqOn (⇑g) f S ∧ support ⇑g ⊆ support f
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 f' g : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑(f + f') a, ↑↑g a⟫ ∂μ = ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫ ∂μ + ∫ (a : α), ⟪↑↑f' a, ↑↑g a⟫ ∂μ

Tactic:
← integral_add (integrable_inner (𝕜 := 𝕜) f g) (integrable_inner f' g),

Goal state after:
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 f' g : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑(f + f') a, ↑↑g a⟫ ∂μ = ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫ + ⟪↑↑f' a, ↑↑g a⟫ ∂μ
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
hp : p ≠ ∞
hp₂ : 1 ≤ p
f : E → F
hf : MemLp f p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε

Tactic:
have hε₂ : 0 < ε / 2 := by positivity
  have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
  obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
  obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
  refine ⟨g', hg'₁, hg'₂, ?_⟩
  have : f - g' = (f - g) - (g' - g) := by simp
  grw [this, eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂, hg₂,
    eLpNorm_sub_comm, hg'₃, ← ENNReal.ofReal_add hε₂.le hε₂.le, add_halves]

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 (i + 1)) x - (P (min i (j + 1))) x - ((P i.succ) x - (P i) x) = 0

Tactic:
min_eq_left_of_lt (Nat.lt_succ_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 i) x - ((P i.succ) x - (P i) x) = 0
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Goal state before:
pp: E : Type u_5
F : Type u_8
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ E
n : ℕ
m : Fin n → E
f : 𝓢(E, F)
⊢ tsupport ⇑(∂^{m} f) ⊆ tsupport ⇑f

Tactic:
induction n with
  | zero => simp
  | succ n IH =>
    rw [iteratedLineDerivOp_succ_left]
    exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m)

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

Goal state before:
pp: case neg
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
i : ι
a : G i
f : ↥(lp G 2)
j : ι
h : ¬j = i
⊢ ⟪Pi.single i a j, ↑f j⟫ = 0

Tactic:
· simp [Pi.single_eq_of_ne h]

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
x y : AddCircle T
hxy : ↑x.toCircle = ↑y.toCircle
⊢ x = y

Tactic:
Subtype.coe_inj

Goal state after:
T : ℝ
hT : Fact (0 < T)
x y : AddCircle T
hxy : x.toCircle = y.toCircle
⊢ x = y
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: x y : ℝ
⊢ cexp (2 * ↑π * ↑y * I) = cexp (2 * ↑π * I * 1 * ↑y / 1)

Tactic:
ring_nf

Goal state after:
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✝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
B : ‖∫ (x : α) in s ∩ u, φ i x • g x ∂μ‖ ≤ 2 * δ
C : ‖∫ (x : α) in s \ u, φ i x • g x ∂μ‖ ≤ δ * ∫ (x : α) in s, ‖g x‖ ∂μ
⊢ ‖∫ (x : α) in s \ u ∪ s ∩ u, φ i x • g x ∂μ‖ =
    ‖∫ (x : α) in s \ u, φ i x • g x ∂μ + ∫ (x : α) in s ∩ u, φ i x • g x ∂μ‖

Tactic:
rw [setIntegral_union disjoint_sdiff_inter (hs.inter u_open.measurableSet)
          (h''i.mono_set sdiff_subset) (h''i.mono_set inter_subset_left)]

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
⊢ (↑((L v) w') * (↑(𝐞 (-(L v) w)) * -(2 * ↑π * I))) • f v = (-↑((L v) w') * (2 * ↑π * I * ↑(𝐞 (-(L v) w)))) • f v

Tactic:
ring_nf

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

Goal state before:
pp: case hcg
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 := ⋯
⊢ Tendsto h (𝓝[s] x₀) (𝓝 0)

Tactic:
rw [← sub_self a]

Goal state after:
case hcg
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
⊢ Tendsto h (𝓝[s] x₀) (𝓝 (a - a))
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
J : ∀ (n : ℕ), 0 ≤ᵐ[μ.restrict s] fun x ↦ c x ^ n
P : ∀ (n : ℕ), 0 < ∫ (x : α) in s, c x ^ n ∂μ
n : ℕ
⊢ ∫ (x : α) in s, φ n x ∂μ = 1

Tactic:
integral_const_mul,

Goal state after:
α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
J : ∀ (n : ℕ), 0 ≤ᵐ[μ.restrict s] fun x ↦ c x ^ n
P : ∀ (n : ℕ), 0 < ∫ (x : α) in s, c x ^ n ∂μ
n : ℕ
⊢ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * ∫ (a : α) in s, c a ^ n ∂μ = 1
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x : F), 0 ≤ φ x
h'φ : ∫ (x : F), φ x ∂μ = 1
h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)
g : F → E
hg : Integrable g μ
h'g : ContinuousAt g 0
I : Integrable φ μ
this : Tendsto (fun c ↦ ∫ (x : F) in closedBall 0 c, φ x ∂μ) atTop (𝓝 1)
c : ℝ
hc : 0 < c
⊢ ∫ (x : F) in closedBall 0 c, φ x ∂μ = ∫ (x : F) in closedBall 0 1, c ^ finrank ℝ F * φ (c • x) ∂μ

Tactic:
integral_const_mul,

Goal state after:
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x : F), 0 ≤ φ x
h'φ : ∫ (x : F), φ x ∂μ = 1
h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)
g : F → E
hg : Integrable g μ
h'g : ContinuousAt g 0
I : Integrable φ μ
this : Tendsto (fun c ↦ ∫ (x : F) in closedBall 0 c, φ x ∂μ) atTop (𝓝 1)
c : ℝ
hc : 0 < c
⊢ ∫ (x : F) in closedBall 0 c, φ x ∂μ = c ^ finrank ℝ F * ∫ (a : F) in closedBall 0 1, φ (c • a) ∂μ
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
h : IntegrableOn (fun x ↦ ‖↑x ^ s‖) (Ioi t) volume
a : ℝ
ha : a ∈ Ioi t
⊢ ‖↑a ^ s‖ = a ^ s.re

Tactic:
rw [Complex.norm_cpow_eq_rpow_re_of_pos (ht.trans ha)]

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

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs : s.re ≤ 0
x : ℝ
hx : x ∈ Ioi t
⊢ deriv (fun x ↦ ‖↑x ^ s‖) x = ?m.36 x

Tactic:
by
    rw [deriv_norm_ofReal_cpow _ (ht.trans hx)]

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

Goal state before:
pp: E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_4
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
inst✝² : MeasurableSpace V
W : Type u_5
inst✝¹ : AddCommGroup W
inst✝ : Module ℝ W
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
μ : Measure V
f : V → E
w : W
⊢ ∫ (v : V), Complex.exp (↑(-(2 * π * (L v) w)) * Complex.I) • f v ∂μ =
    ∫ (v : V), Complex.exp (↑(-2 * π * (L v) w) * Complex.I) • f v ∂μ

Tactic:
neg_mul

Goal state after:
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
w : V
⊢ ∫ (v : V), Complex.exp (↑(-(2 * π * ⟪v, w⟫)) * Complex.I) • f v =
    ∫ (v : V), Complex.exp (↑(-2 * π * ⟪v, w⟫) * Complex.I) • f v

Tactic:
neg_mul

Goal state after:
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 : ↥(Lp E 1 volume)
x : V
⊢ (fourierTransformInv f) x = 𝓕⁻ (↑↑f) x

Tactic:
by
  simp [Lp.fourierTransformInv, fourierInv_eq_fourier_neg]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p s : ℝ
hs : -1 < s
hp : 0 < p
ht : 0 < (s + 1) / p
x : ℝ
hx : x ∈ Ioi 0
⊢ x ^ (p - 1 + (s + 1 - p)) = x ^ s

Tactic:
ring_nf

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
c T : ℝ
⊢ ‖cexp (-b * (↑T + ↑c * I) ^ 2)‖ = rexp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2))

Tactic:
Complex.norm_exp,

Goal state after:
b : ℂ
c T : ℝ
⊢ rexp (-b * (↑T + ↑c * I) ^ 2).re = rexp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
X : ℝ
hX : X ∈ Ici 0
⊢ (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) X = s * s.partialGamma X - ↑(rexp (-X)) * ↑X ^ s

Tactic:
ring_nf

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
c T : ℝ
⊢ rexp
      (-(b.re * ((T + (c * 0 - 0 * 1)) * (T + (c * 0 - 0 * 1)) - (0 + (c * 1 + 0 * 0)) * (0 + (c * 1 + 0 * 0))) -
          b.im * ((T + (c * 0 - 0 * 1)) * (0 + (c * 1 + 0 * 0)) + (0 + (c * 1 + 0 * 0)) * (T + (c * 0 - 0 * 1))))) =
    rexp (-(b.re * (T * T) - 2 * b.im * c * T - b.re * (c * c)))

Tactic:
ring_nf

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
this : Integrable (fun x ↦ rexp (-(b.re * x ^ 2))) volume
⊢ HasFiniteIntegral (fun a ↦ rexp (c ^ 2 * (b.im ^ 2 / b.re + b.re)) * rexp (-(b.re * (a - b.im * c / b.re) ^ 2)))
    volume

Tactic:
exact (Integrable.comp_sub_right this (b.im * c / b.re)).hasFiniteIntegral.const_mul _

Goal state after:
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
⊢ ∀ x ∈ Ioi 0, 0 x ≤ (fun x ↦ rexp (-x) * x ^ (s - 1)) x

Tactic:
exact fun x hx => (mul_pos (exp_pos _) (rpow_pos_of_pos hx _)).le

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

Goal state before:
pp: case h'φ
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 := ⋯
⊢ ∫ (x : V), φ x = 1

Tactic:
integral_const_mul,

Goal state after:
case h'φ
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)
⊢ π ^ (↑(finrank ℝ V) / 2) * ∫ (a : V), rexp (-π ^ 2 * ‖a‖ ^ 2) = 1
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: case h'φ
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)
⊢ π ^ (↑(finrank ℝ V) / 2 + (↑1 - ↑2) * (↑(finrank ℝ V) / 2)) = 1

Tactic:
ring_nf

Goal state after:
case h'φ
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)
⊢ π ^ 0 = 1
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
a b : ℂ
⊢ (a - b * I) * (a - b * I) = (-a + b * I) ^ 2

Tactic:
ring_nf

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case e_a
b : ℂ
hb : 0 < b.re
c : ℂ
this✝ : b ≠ 0
h : (-↑π * b).re < 0
t : ℝ
this :
  ∀ (x : ℝ),
    ↑(-2 * π * x * t) * I + -↑π * b * ↑x ^ 2 + 2 * ↑π * c * ↑x =
      -↑π * b * ↑x ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * c) * ↑x + 0
⊢ 0 * 4 * b - -(2 ^ 2 * ↑π * (-(I * ↑t) + c) ^ 2) = -(↑π * 4 * (↑t + I * c) ^ 2)

Tactic:
ring_nf

Goal state after:
case e_a
b : ℂ
hb : 0 < b.re
c : ℂ
this✝ : b ≠ 0
h : (-↑π * b).re < 0
t : ℝ
this :
  ∀ (x : ℝ),
    ↑(-2 * π * x * t) * I + -↑π * b * ↑x ^ 2 + 2 * ↑π * c * ↑x =
      -↑π * b * ↑x ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * c) * ↑x + 0
⊢ -(↑π * I * ↑t * c * 8) + ↑π * I ^ 2 * ↑t ^ 2 * 4 + ↑π * c ^ 2 * 4 =
    -(↑π * I * ↑t * c * 8) - ↑π * I ^ 2 * c ^ 2 * 4 - ↑π * ↑t ^ 2 * 4