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
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
⊢ μ {a | u n < f a} < ∞

Tactic:
lt_top_iff_ne_top

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
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
⊢ μ {a | u n < f a} ≠ ∞
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹³ : NontriviallyNormedField 𝕜
E✝ : Type u_3
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
Ω✝ Ω₁ Ω₂ : Opens E✝
F✝ : Type u_4
inst✝¹⁰ : NormedAddCommGroup F✝
inst✝⁹ : NormedSpace ℝ F✝
inst✝⁸ : NormedSpace 𝕜 F✝
F' : Type u_5
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : NormedSpace 𝕜 F'
n✝ n₁ n₂ k : ℕ∞
B : Type u_6
E : Type u_7
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
Ω : Opens E
F : Type u_8
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
inst✝ : TestFunctionClass B Ω F n
f : B
C : ℝ
hC : ∀ (x : E), ‖f x‖ ≤ C
⊢ ∃ C, ∀ (x y : E), dist (f x) (f y) ≤ C

Tactic:
exact map_bounded (BoundedContinuousFunction.ofNormedAddCommGroup f (map_continuous f) C hC)

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

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 : E → F
hf : HasTemperateGrowth f
⊢ HasTemperateGrowth (-f)

Tactic:
refine ⟨hf.1.neg, fun n ↦ ?_⟩

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 : E → F
hf : HasTemperateGrowth f
n : ℕ
⊢ ∃ k C, ∀ (x : E), ‖iteratedFDeriv ℝ n (-f) x‖ ≤ C * (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Distribution.TemperateGrowth

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 : E →L[ℝ] F
x : E
⊢ ‖f x‖ ≤ ‖f‖ * (1 + ‖x‖) ^ 1

Tactic:
exact (f.le_opNorm x).trans (by simp [mul_add])

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

Goal state before:
pp: case mpr
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
r : ℝ
hr : 0 < r
a : ℝ
⊢ a ∈ T.rayleighQuotient '' sphere 0 r → a ∈ T.rayleighQuotient '' {0}ᶜ

Tactic:
· rintro ⟨x, hx, hxT⟩
    exact ⟨x, ne_zero_of_mem_sphere hr.ne' ⟨x, hx⟩, hxT⟩

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

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
M : ℝ := ⨆ x, |T.rayleighQuotient x|
nonneg : 0 ≤ M
hM : ∀ (x : E), |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ^ 2
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
key : re ⟪↑T x, y⟫_𝕜 + re ⟪y, ↑T x⟫_𝕜 = re (2 * ↑(re ⟪↑T x, y⟫_𝕜))
⊢ re ⟪T x, y⟫_𝕜 ≤ M * (‖x + y‖ ^ 2 + ‖x - y‖ ^ 2) / 4

Tactic:
← hT,

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
M : ℝ := ⨆ x, |T.rayleighQuotient x|
nonneg : 0 ≤ M
hM : ∀ (x : E), |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ^ 2
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
key : re ⟪↑T x, y⟫_𝕜 + re ⟪↑T y, x⟫_𝕜 = re (2 * ↑(re ⟪↑T x, y⟫_𝕜))
⊢ re ⟪T x, y⟫_𝕜 ≤ M * (‖x + y‖ ^ 2 + ‖x - y‖ ^ 2) / 4
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
⊢ σ₁₂ c⁻¹ • K ⊆ U

Tactic:
map_inv₀,

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
⊢ (σ₁₂ c)⁻¹ • K ⊆ U
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‖
this : x ≠ 0
η : 𝕜
hη : η ≠ 0
h₁ : ‖η • x‖ < 1
h₂ : 1 / ‖C‖ ≤ ‖η • x‖
h₃ : ‖η‖⁻¹ ≤ 1⁻¹ * ‖C‖ * ‖x‖
⊢ ∃ x, ‖x‖ ≤ 1 ∧ ‖C‖⁻¹ ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε

Tactic:
refine ⟨η • x, h₁.le, by simpa using h₂, ?_⟩

Goal state after:
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‖
this : x ≠ 0
η : 𝕜
hη : η ≠ 0
h₁ : ‖η • x‖ < 1
h₂ : 1 / ‖C‖ ≤ ‖η • x‖
h₃ : ‖η‖⁻¹ ≤ 1⁻¹ * ‖C‖ * ‖x‖
⊢ ‖(T - μ • 1) (η • x)‖ < ε
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Eigs

Goal state before:
pp: case inr
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
R : Type u_2
inst✝¹ : CommRing R
B : Matrix n n R
inst✝ : IsDomain R
hAps : B.charpoly.Splits
h✝ : Nonempty n
⊢ B.trace = B.charpoly.roots.sum

Tactic:
trace_eq_neg_charpoly_nextCoeff,

Goal state after:
case inr
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
R : Type u_2
inst✝¹ : CommRing R
B : Matrix n n R
inst✝ : IsDomain R
hAps : B.charpoly.Splits
h✝ : Nonempty n
⊢ -B.charpoly.nextCoeff = B.charpoly.roots.sum
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✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup E✝
inst✝¹³ : NormedSpace ℝ E✝
inst✝¹² : NormedAddCommGroup F✝
inst✝¹¹ : NormedSpace ℝ F✝
inst✝¹⁰ : NormedSpace 𝕜 F✝
inst✝⁹ : SMulCommClass ℝ 𝕜 F✝
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : SMulCommClass ℝ 𝕜 F'
n✝ n₁ n₂ k : ℕ∞
K✝ K₁ K₂ : Compacts E✝
B : Type u_5
E : Type u_6
F : Type u_7
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
inst✝ : ContDiffMapSupportedInClass B E F n K
f : B
this : HasCompactSupport ⇑f
C : ℝ
hC : ∀ (x : E), ‖f x‖ ≤ C
⊢ ∃ C, ∀ (x y : E), dist (f x) (f y) ≤ C

Tactic:
exact map_bounded (BoundedContinuousFunction.ofNormedAddCommGroup f (map_continuous f) C hC)

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

Goal state before:
pp: m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike 𝕜
A M : Matrix n n 𝕜
hM : M.PosSemidef
x✝² x✝¹ : n → 𝕜
x✝ : 𝕜
⊢ M *ᵥ x✝¹ ⬝ᵥ star (x✝ • x✝²) = M *ᵥ x✝¹ ⬝ᵥ star x✝ • star x✝²

Tactic:
← star_smul

Goal state after:
m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike 𝕜
A M : Matrix n n 𝕜
hM : M.PosSemidef
x✝² x✝¹ : n → 𝕜
x✝ : 𝕜
⊢ M *ᵥ x✝¹ ⬝ᵥ star (x✝ • x✝²) = M *ᵥ x✝¹ ⬝ᵥ star (x✝ • x✝²)
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
⊢ IsUniformAddGroup 𝓓^{n}_{K}(E, F)

Tactic:
rw [ContDiffMapSupportedIn.uniformSpace_eq_iInf]
  exact isUniformAddGroup_iInf fun _ ↦ IsUniformAddGroup.comap _

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 fun x ↦ ↑(NNReal.sqrt x.toNNReal)).det = ↑√(RCLike.re A.det)

Tactic:
simp only [IsHermitian.cfc, Real.coe_sqrt, Real.coe_toNNReal', det_map, det_diagonal,
    Function.comp_apply, hA.isHermitian.det_eq_prod_eigenvalues, ← RCLike.ofReal_prod,
    RCLike.ofReal_re, Real.sqrt_prod _ fun _ _ ↦ hA.eigenvalues_nonneg _]

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
⊢ ↑(∏ i, √(max (⋯.eigenvalues i) 0)) = ↑(∏ i, √(⋯.eigenvalues i))
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
X : Type u_2
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
β : Type u_3
b : UnconditionalSchauderBasis β 𝕜 X
inst✝ : CompleteSpace X
x : X
A₀ : Finset β
hA₀ : ∀ (t : Finset β), Disjoint t A₀ → ‖∑ i ∈ t, (b.coord i) x • ↑b i‖ < 1
A : Finset β
hdecomp :
  (GeneralSchauderBasis.proj b A) x =
    (GeneralSchauderBasis.proj b (A ∩ A₀)) x + (GeneralSchauderBasis.proj b (A \ A₀)) x
⊢ ‖(GeneralSchauderBasis.proj b (A \ A₀)) x‖ < 1

Tactic:
GeneralSchauderBasis.proj_apply

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
X : Type u_2
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
β : Type u_3
b : UnconditionalSchauderBasis β 𝕜 X
inst✝ : CompleteSpace X
x : X
A₀ : Finset β
hA₀ : ∀ (t : Finset β), Disjoint t A₀ → ‖∑ i ∈ t, (b.coord i) x • ↑b i‖ < 1
A : Finset β
hdecomp :
  (GeneralSchauderBasis.proj b A) x =
    (GeneralSchauderBasis.proj b (A ∩ A₀)) x + (GeneralSchauderBasis.proj b (A \ A₀)) x
⊢ ‖∑ i ∈ A \ A₀, (b.coord i) x • ↑b i‖ < 1
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε✝ : ε✝ ≠ 0
c : E
t : Set α
ht : MeasurableSet t
htμ : μ t < ∞
ε : ℝ≥0∞
hε : ε ≠ 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ :
  ∀ (f g : α → E),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ → eLpNorm f p μ ≤ δ → eLpNorm g p μ ≤ δ → eLpNorm (f + g) p μ < ε
η : ℝ≥0
ηpos : 0 < η
hη : ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun _x ↦ c) p μ ≤ δ
hη_pos' : 0 < ↑η
⊢ ∃ g,
    eLpNorm (g - t.indicator fun x ↦ c) p μ ≤ ε ∧
      Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ HasCompactSupport g

Tactic:
obtain ⟨s, st, s_compact, s_closed, μs⟩ :
      ∃ s, s ⊆ t ∧ IsCompact s ∧ IsClosed s ∧ μ (t \ s) < η :=
    ht.exists_isCompact_isClosed_sdiff_lt htμ.ne hη_pos'.ne'

Goal state after:
α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε✝ : ε✝ ≠ 0
c : E
t : Set α
ht : MeasurableSet t
htμ : μ t < ∞
ε : ℝ≥0∞
hε : ε ≠ 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ :
  ∀ (f g : α → E),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ → eLpNorm f p μ ≤ δ → eLpNorm g p μ ≤ δ → eLpNorm (f + g) p μ < ε
η : ℝ≥0
ηpos : 0 < η
hη : ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun _x ↦ c) p μ ≤ δ
hη_pos' : 0 < ↑η
s : Set α
st : s ⊆ t
s_compact : IsCompact s
s_closed : IsClosed[inst✝⁸] s
μs : μ (t \ s) < ↑η
⊢ ∃ g,
    eLpNorm (g - t.indicator fun x ↦ c) p μ ≤ ε ∧
      Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ HasCompactSupport g
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 : ℕ
x : X
⊢ (GeneralSchauderBasis.proj b (Finset.range n)) x = ∑ i ∈ Finset.range n, (b.coord i) x • ↑b i

Tactic:
GeneralSchauderBasis.proj_apply

Goal state after:
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 : ℕ
x : X
⊢ ∑ i ∈ Finset.range n, (b.coord i) x • ↑b i = ∑ i ∈ Finset.range n, (b.coord i) x • ↑b i
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Goal state before:
pp: case hf
E : Type u_5
F₁ : Type u_9
F₂ : Type u_10
F₃ : Type u_11
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : FiniteDimensional ℝ E
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace ℝ F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace ℝ F₃
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : 𝓢(E, F₁)
g : 𝓢(E, F₂)
L : F₁ →L[ℝ] F₂ →L[ℝ] F₃
⊢ ∀ i ∈ Finset.univ,
    Integrable (fun x ↦ (L ((∂_{(stdOrthonormalBasis ℝ E) i} (∂_{(stdOrthonormalBasis ℝ E) i} f)) x)) (g x)) μ

Tactic:
exact fun _ _ ↦ (pairing L (∂_{_} <| ∂_{_} f) g).integrable

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
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) / (-2 * ↑π * I * ↑n) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))

Tactic:
rw [(by ring : ((b - a : ℝ) : ℂ) / (-2 * π * I * n) = ((b - a : ℝ) : ℂ) * (1 / (-2 * π * I * n)))]

Goal state after:
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
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: case refine_1
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
⊢ IntegrableOn (fun x ↦ ‖φ i x • g x‖) (s \ u) μ

Tactic:
exact IntegrableOn.mono_set h''i.norm sdiff_subset

Goal state after:
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
⊢ c ∈ Ioi 0 → ∫ (x : F) in closedBall 0 c, φ x ∂μ = ∫ (x : F) in closedBall 0 1, c ^ finrank ℝ F * φ (c • x) ∂μ

Tactic:
(hc : 0 < c)

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 = ∫ (x : F) in closedBall 0 1, c ^ finrank ℝ F * φ (c • x) ∂μ
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
c : ℝ
⊢ c ∈ Ioi 0 → AEStronglyMeasurable (fun x ↦ c ^ finrank ℝ F * φ (c • x)) μ

Tactic:
(hc : 0 < c)

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
c : ℝ
hc : 0 < c
⊢ AEStronglyMeasurable (fun x ↦ c ^ finrank ℝ F * φ (c • x)) μ
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
m : MeasurableEmbedding fun x ↦ -x := Homeomorph.measurableEmbedding (Homeomorph.neg ℝ)
⊢ IntegrableOn (fun x ↦ f |x|) (Iic 0) (Measure.map Neg.neg volume)

Tactic:
rw [m.integrableOn_map_iff]

Goal state after:
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
m : MeasurableEmbedding fun x ↦ -x := Homeomorph.measurableEmbedding (Homeomorph.neg ℝ)
⊢ IntegrableOn ((fun x ↦ f |x|) ∘ fun x ↦ -x) ((fun x ↦ -x) ⁻¹' Iic 0) volume
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

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

Tactic:
· have : ¬ Integrable (fun x => f |x|) := by
      contrapose hf
      exact hf.integrableOn
    rw [← eq, integral_undef hf, integral_undef this, mul_zero]

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
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‖) volume
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
w : V
⊢ ‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖ ≤
    (2 * π) ^ k * (2 * ↑k + 2) ^ n *
      ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖

Tactic:
by
  have Z : ‖w‖ ^ n * (‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖) ≤
      ‖w‖ ^ n * ((2 * (π * ‖innerSL (E := V) ℝ‖)) ^ k * ((2 * k + 2) ^ n *
          ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1),
            ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂volume)) := by
    have := VectorFourier.pow_mul_norm_iteratedFDeriv_fourierIntegral_le (innerSL ℝ) hf h'f hk hn
      w w
    simp only [innerSL_apply_apply _ w w, real_inner_self_eq_norm_sq w, abs_pow, abs_norm,
      mul_assoc] at this
    rwa [pow_two, mul_pow, mul_assoc] at this
  rcases eq_or_ne n 0 with rfl | hn
  · simp only [pow_zero, one_mul, mul_one, zero_add, Finset.range_one, Finset.product_singleton,
      Finset.sum_map, Function.Embedding.coeFn_mk, norm_iteratedFDeriv_zero] at Z ⊢
    apply Z.trans
    conv_rhs => rw [← mul_one π]
    gcongr
    exact norm_innerSL_le _
  rcases eq_or_ne w 0 with rfl | hw
  · simp [hn]
    positivity
  rw [mul_le_mul_iff_right₀ (pow_pos (by simp [hw]) n)] at Z
  apply Z.trans
  conv_rhs => rw [← mul_one π]
  simp only [mul_assoc]
  gcongr
  exact norm_innerSL_le _

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

Goal state before:
pp: b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume

Tactic:
exact integrableOn_rpow_mul_exp_neg_mul_rpow hs two_pos hb

Goal state after: