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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SecondCountableTopologyEither α E
inst✝⁵ : IsFiniteMeasure μ
𝕜 : Type u_3
inst✝⁴ : Fact (1 ≤ p)
inst✝³ : NormedRing 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 E
f g : α →ᵇ E
inst✝ : μ.IsOpenPosMeasure
⊢ f = g → (toLp p μ 𝕜) f = (toLp p μ 𝕜) g

Tactic:
tauto

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : SMulCommClass ℝ 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : SMulCommClass ℝ 𝕜 F'
n : ℕ∞
K : Compacts E
inst✝ : LinearMap.CompatibleSMul F F' ℝ 𝕜
i : ℕ
T : F →L[𝕜] F'
f : 𝓓^{n}_{K}(E, F)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
hi : ↑i ≤ n
x : E
hx : x ∈ K
⊢ ‖iteratedFDeriv ℝ i (⇑T' ∘ ⇑f) x‖ ≤ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f

Tactic:
calc
      ‖iteratedFDeriv ℝ i (T' ∘ f) x‖
  _ = ‖T'.compContinuousMultilinearMap (iteratedFDeriv ℝ i f x)‖ := by
        rw [T'.iteratedFDeriv_comp_left f.contDiff.contDiffAt (mod_cast hi)]
  _ ≤ ‖T'‖ * ‖iteratedFDeriv ℝ i f x‖ := T'.norm_compContinuousMultilinearMap_le _
  _ ≤ ‖T'‖ * N[ℝ]_{K, n, i} f := by grw [norm_iteratedFDeriv_apply_le_seminorm ℝ hi]

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

Goal state before:
pp: case negSucc
A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℕ
⊢ abs ↑(n + 1) = ↑|Int.negSucc n|

Tactic:
abs_natCast,

Goal state after:
case negSucc
A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℕ
⊢ ↑(n + 1) = ↑|Int.negSucc n|
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
⊢ Continuous ⇑(monoLM 𝕜)

Tactic:
refine continuous_of_isBounded (ContDiffMapSupportedIn.withSeminorms _ _ _ _ _)
      (ContDiffMapSupportedIn.withSeminorms _ _ _ _ _) _ (fun i ↦ ⟨{i}, 1, fun f ↦ ?_⟩)

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
i : ℕ
f : 𝓓^{n₁}_{K₁}(E, F)
⊢ ((ContDiffMapSupportedIn.seminorm 𝕜 E F n₂ K₂ i).comp (monoLM 𝕜)) f ≤
    (1 • {i}.sup (ContDiffMapSupportedIn.seminorm 𝕜 E F n₁ K₁)) f
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
b : SchauderBasis 𝕜 X
n i : ℕ
⊢ (b.proj n) (↑b i) = if i < n then ↑b i else 0

Tactic:
by
  rw [proj, GeneralSchauderBasis.proj_apply_basis_mem]
  simp

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : AddCircle T → ℂ
p : ℝ≥0∞
hT' : 0 < T
h : MemLp f p (ENNReal.ofReal T • AddCircle.haarAddCircle)
this : MemLp f p (1 • AddCircle.haarAddCircle)
⊢ MemLp f p AddCircle.haarAddCircle

Tactic:
one_smul

Goal state after:
T : ℝ
hT : Fact (0 < T)
f : AddCircle T → ℂ
p : ℝ≥0∞
hT' : 0 < T
h : MemLp f p (ENNReal.ofReal T • AddCircle.haarAddCircle)
this : MemLp f p AddCircle.haarAddCircle
⊢ MemLp f p AddCircle.haarAddCircle
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✝ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV✝ hV : OrthogonalFamily 𝕜 G V
f : ↥(lp G 2)
H : 0 < ENNReal.toReal 2
⊢ ‖∑' (i : ι), (V i) (↑f i)‖ ^ ENNReal.toReal 2 = ‖f‖ ^ ENNReal.toReal 2

Tactic:
refine tendsto_nhds_unique ?_ (lp.hasSum_norm H f)

Goal state after:
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✝ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV✝ hV : OrthogonalFamily 𝕜 G V
f : ↥(lp G 2)
H : 0 < ENNReal.toReal 2
⊢ Tendsto (fun s ↦ ∑ b ∈ s, (fun i ↦ ‖↑f i‖ ^ ENNReal.toReal 2) b) (SummationFilter.unconditional ι).filter
    (𝓝 (‖∑' (i : ι), (V i) (↑f i)‖ ^ ENNReal.toReal 2))
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: case H
T : ℝ
n : ℤ
z✝ : ℝ
⊢ ↑(Circle.exp (2 * π / T * -(n • z✝))) = ↑(Circle.exp (-(2 * π / T * n • z✝)))

Tactic:
mul_neg

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

Goal state before:
pp: case false
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
⊢ CompleteSpace ↥(bif false then K else Kᗮ)

Tactic:
first | exact instOrthogonalCompleteSpace K | assumption

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

Goal state before:
pp: case true
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
⊢ CompleteSpace ↥(bif true then K else Kᗮ)

Tactic:
first | exact instOrthogonalCompleteSpace K | assumption

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
this : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x
⊢ fourierCoeff f n = (1 / T) • ∫ (x : ℝ) in a..a + T, (fun z ↦ (fourier (-n)) z • f z) ↑x

Tactic:
fourierCoeff,

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
this : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x
⊢ ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle =
    (1 / T) • ∫ (x : ℝ) in a..a + T, (fun z ↦ (fourier (-n)) z • f z) ↑x
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
this : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x
⊢ ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle =
    1 • ∫ (x : AddCircle T), (fourier (-n)) x • f x ∂haarAddCircle

Tactic:
one_smul

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
n : ℤ
a : ℝ
this : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x
⊢ ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle =
    ∫ (x : AddCircle T), (fourier (-n)) x • f x ∂haarAddCircle
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
c : ℂ
n : ℤ
⊢ fourierCoeff (c • f) n = c • fourierCoeff f n

Tactic:
fourierCoeff,

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : AddCircle T → E
c : ℂ
n : ℤ
⊢ ∫ (t : AddCircle T), (fourier (-n)) t • (c • f) t ∂haarAddCircle =
    c • ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
hab : a < b
f : ℝ → E
n : ℤ
⊢ 0 < b - a

Tactic:
linarith

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
n : ℤ
hab : a < b
⊢ 0 < b - a

Tactic:
linarith

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
c : ℂ
n : ℤ
hab : a < b
⊢ 0 < b - a

Tactic:
linarith

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
inst✝ : DecidableEq ι
h : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure
i : ι
⊢ 1 • v i = v i

Tactic:
one_smul

Goal state after:
ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
inst✝ : DecidableEq ι
h : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure
i : ι
⊢ v i = v i
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: a b : ℝ
f : ℝ → ℂ
hab : a < b
hL2 : MemLp f 2 (volume.restrict (Ioc a b))
⊢ 0 < b - a

Tactic:
linarith

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
n : ℤ
⊢ ∀ (x : AddCircle T), (fourier (-n)) x = (fourier (-n)) x

Tactic:
tauto

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

Goal state before:
pp: a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
⊢ 0 < b - a

Tactic:
linarith

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
hφpos : ∀ x ∈ s, 0 ≤ φ i x
h''i : IntegrableOn (fun x ↦ φ i x • g x) s μ
h : ∫ (a : α) in t, φ i a ∂μ = 0
h'i : 1 < δ
⊢ False

Tactic:
linarith

Goal state after:
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
v : V
n : ℕ
m : Fin n → W
⊢ (2 * π) ^ n * ((∏ x, |(L v) (m x)|) * ‖f v‖) ≤ (2 * π) ^ n * ((∏ x, ‖L‖ * ‖v‖ * ‖m x‖) * ‖f v‖)

Tactic:
gcongr with i _hi
      exact L.le_opNorm₂ v (m i)

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

Goal state before:
pp: 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 : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra ℝ 𝕜
inst✝ : NormedSpace 𝕜 F
c : 𝕜
f : 𝓢(E, F)
x : E
this : Function.HasTemperateGrowth fun x ↦ c
⊢ ((smulLeftCLM F fun x ↦ c) f) x = ((c • ContinuousLinearMap.id 𝕜 𝓢(E, F)) f) x

Tactic:
simp [this]

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 : V → E
w : W
aux : ∀ {g : V → E}, Integrable g μ → ∀ (x : W), Integrable (fun v ↦ e (-(L v) x) • g v) μ
hf : Integrable (fun v ↦ e (-(L v) w) • f v) μ
this : Integrable (fun v ↦ 1 • f v) μ
⊢ Integrable f μ

Tactic:
one_smul

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 : V → E
w : W
aux : ∀ {g : V → E}, Integrable g μ → ∀ (x : W), Integrable (fun v ↦ e (-(L v) x) • g v) μ
hf : Integrable (fun v ↦ e (-(L v) w) • f v) μ
this : Integrable (fun v ↦ f v) μ
⊢ Integrable f μ
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
K : ℝ≥0
g : D → E
hg : Function.HasTemperateGrowth g
h'g : AntilipschitzWith K g
x : D
⊢ ‖x‖ ≤ ↑K * ‖g x - g 0‖

Tactic:
rw [← dist_zero_right, ← dist_eq_norm]

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
K : ℝ≥0
g : D → E
hg : Function.HasTemperateGrowth g
h'g : AntilipschitzWith K g
x : D
⊢ dist x 0 ≤ ↑K * dist (g x) (g 0)
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
x₀ : F
hg : Integrable g μ
h'g : ContinuousAt g x₀
f : F → E := fun x ↦ g (x₀ - x)
If : Integrable f μ
this : Tendsto (fun c ↦ ∫ (x : F), (c ^ finrank ℝ F * φ (c • x)) • g (x₀ - x) ∂μ) atTop (𝓝 (g x₀))
c : ℝ
| ∫ (x : F), (c ^ finrank ℝ F * φ (c • (x₀ + x))) • g (x₀ - (x₀ + x)) ∂μ

Tactic:
← integral_neg_eq_self

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
x₀ : F
hg : Integrable g μ
h'g : ContinuousAt g x₀
f : F → E := fun x ↦ g (x₀ - x)
If : Integrable f μ
this : Tendsto (fun c ↦ ∫ (x : F), (c ^ finrank ℝ F * φ (c • x)) • g (x₀ - x) ∂μ) atTop (𝓝 (g x₀))
c : ℝ
| ∫ (x : F), (c ^ finrank ℝ F * φ (c • (x₀ + -x))) • g (x₀ - (x₀ + -x)) ∂μ
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a c m : ℝ
ha : a < -1
hc : -m < c
x : ℝ
hx : x ∈ Ici c
⊢ a + 1 ≠ 0

Tactic:
linarith

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

Goal state before:
pp: a c m : ℝ
ha : a < -1
hc : -m < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ (t + m) ^ (a + 1) / (a + 1)) ((x + m) ^ a) x
⊢ 0 < -(a + 1)

Tactic:
linarith

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

Goal state before:
pp: case inr
s : ℝ
h : IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume
hs : -1 < s
⊢ False

Tactic:
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.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
x : ℝ
hx : x ∈ Ici c
⊢ a + 1 < 0

Tactic:
linarith

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

Goal state before:
pp: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ t ^ (a + 1) / (a + 1)) (x ^ a) x
⊢ 0 < -(a + 1)

Tactic:
linarith

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

Goal state before:
pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
⊢ 0 < -(a.re + 1)

Tactic:
linarith

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

Goal state before:
pp: case hf
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
f g : ↥(Lp E 1 volume)
x : V
⊢ Integrable (fun v ↦ 𝐞 (-⟪v, x⟫) • ↑↑f v) volume

Tactic:
rw [Real.fourierIntegral_convergent_iff]
          exact L1.integrable_coeFn f

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
h : 0 < s
⊢ -1 < s - 1

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ rexp (-x) * x ^ (s.re - 1)) =ᵐ[volume.restrict (Ioi 0)] fun a ↦ ‖↑(rexp (-a)) * ↑a ^ (s - 1)‖

Tactic:
apply (ae_restrict_iff' measurableSet_Ioi).mpr

Goal state after:
case right
s : ℂ
hs : 0 < s.re
⊢ ∀ᵐ (x : ℝ), x ∈ Ioi 0 → (fun x ↦ rexp (-x) * x ^ (s.re - 1)) x = (fun a ↦ ‖↑(rexp (-a)) * ↑a ^ (s - 1)‖) x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
x : ℝ
hx : x ∈ Ioi 0
⊢ ↑(rexp (-x)) * ↑x ^ (↑s - 1) = ↑(rexp (-x)) * ↑(x ^ (s - 1))

Tactic:
ofReal_cpow (mem_Ioi.mp hx).le

Goal state after:
s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
x : ℝ
hx : x ∈ Ioi 0
⊢ ↑(rexp (-x)) * ↑x ^ (↑s - 1) = ↑(rexp (-x)) * ↑x ^ ↑(s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p b : ℝ
hb : 0 < b
hp : 1 < p
⊢ 0 < p - 1

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
⊢ 0 < s.re + 1

Tactic:
linarith

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

Goal state before:
pp: p s : ℝ
hs : -1 < s
hp : 0 < p
⊢ 0 < s + 1

Tactic:
linarith

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, |p.1| • f (↑polarCoord.symm p) =
    ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p)

Tactic:
apply setIntegral_congr_fun polarCoord.open_target.measurableSet fun x hx => ?_

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
x : ℝ × ℝ
hx : x ∈ polarCoord.target
⊢ |x.1| • f (↑polarCoord.symm x) = x.1 • f (↑polarCoord.symm x)
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 : 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) =
    (∫ (x : E 0), f 0 x ∂μ 0) * ∏ i, ∫ (x : E i.succ), f i.succ x ∂μ i.succ

Tactic:
rw [← n_ih, ← integral_prod_mul]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
this : Tendsto (s + 1).partialGamma atTop (𝓝 (s * s.GammaIntegral))
⊢ 0 < s.re + 1

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: n : ℕ
hn : ∀ (s : ℂ), -s.re < ↑n → GammaAux n s = GammaAux n (s + 1) / s
s : ℂ
h1 : -s.re < ↑n + 1
⊢ -(s.re + 1) < ↑n

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
n : ℕ
h1 : -s.re < ↑n + 1
⊢ -(s.re + 1) < ↑n

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : 0 ≤ 1 - s.re
⊢ 1 - s.re < ↑n + 1

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 1 - s.re ≤ 0

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
h2 : s ≠ 0
n : ℕ := ⌊1 - s.re⌋₊
t1 : -s.re < ↑n
⊢ -(s.re + 1) < ↑n

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
⊢ -s.re < 0

Tactic:
linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

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

Tactic:
linarith

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

Goal state before:
pp: 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
h :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
⊢ 𝓕 (𝓕⁻ f) v = f v

Tactic:
exact hf.fourier_fourierInv_eq h'f h.continuousAt

Goal state after: