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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
⊢ ∫⁻ (y : ℝ), ∫⁻ (x : α), (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) y ∂μ =
    ∫⁻ (a : ℝ), (Ioi 0).indicator (fun t ↦ μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a

Tactic:
apply congr_arg
    funext s
    have aux₁ :
      (fun x => (Ioc 0 (f x)).indicator (fun t : ℝ => ENNReal.ofReal (g t)) s) = fun x =>
        ENNReal.ofReal (g s) * (Ioi (0 : ℝ)).indicator (fun _ => 1) s *
          (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f x) := by
      funext a
      by_cases h : s ∈ Ioc (0 : ℝ) (f a)
      · simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
          mul_one]
      · have h_copy := h
        simp only [mem_Ioc, not_and, not_le] at h
        by_cases h' : 0 < s
        · simp only [h_copy, h h', indicator_of_notMem, not_false_iff, mem_Ici, not_le, mul_zero]
        · have : s ∉ Ioi (0 : ℝ) := h'
          simp only [this, h', indicator_of_notMem, not_false_iff, mul_zero,
            zero_mul, mem_Ioc, false_and]
    simp_rw [aux₁]
    rw [lintegral_const_mul']
    swap
    · apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top
      by_cases h : (0 : ℝ) < s <;> · simp [h]
    simp_rw [show
        (fun a => (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f a)) = fun a =>
          {a : α | s ≤ f a}.indicator (fun _ => 1) a
        by funext a; by_cases h : s ≤ f a <;> simp [h]]
    rw [lintegral_indicator₀]
    swap; · exact f_mble.nullMeasurable measurableSet_Ici
    rw [lintegral_one, Measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,
      mul_assoc,
      show
        (Ioi 0).indicator (fun _x : ℝ => (1 : ℝ≥0∞)) s * μ {a : α | s ≤ f a} =
          (Ioi 0).indicator (fun _x : ℝ => 1 * μ {a : α | s ≤ f a}) s
        by by_cases h : 0 < s <;> simp [h]]
    simp_rw [mul_comm _ (ENNReal.ofReal _), one_mul]
    rfl

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 g : E → F
hf : HasTemperateGrowth f
hg : HasTemperateGrowth g
⊢ HasTemperateGrowth (f + g)

Tactic:
rw [hasTemperateGrowth_iff_isBigO] at *
  refine ⟨hf.1.add hg.1, fun n ↦ ?_⟩
  obtain ⟨k₁, h₁⟩ := hf.2 n
  obtain ⟨k₂, h₂⟩ := hg.2 n
  use max k₁ k₂
  rw [iteratedFDeriv_add (hf.1.of_le <| mod_cast le_top) (hg.1.of_le <| mod_cast le_top)]
  have : 1 ≤ᶠ[⊤] fun (x : E) ↦ 1 + ‖x‖ := by
    filter_upwards with _ using (le_add_iff_nonneg_right _).mpr (by positivity)
  exact (h₁.trans (IsBigO.pow_of_le_right this (k₁.le_max_left k₂))).add
    (h₂.trans (IsBigO.pow_of_le_right this (k₁.le_max_right k₂)))

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 →L[ℝ] F
⊢ HasTemperateGrowth ⇑f

Tactic:
by
  apply Function.HasTemperateGrowth.of_fderiv ?_ f.differentiable (k := 1) (C := ‖f‖) (fun x ↦ ?_)
  · have : fderiv ℝ f = fun _ ↦ f := by ext1 v; simp only [ContinuousLinearMap.fderiv]
    simp [this]
  · exact (f.le_opNorm x).trans (by simp [mul_add])

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
x : E
⊢ (-T).rayleighQuotient x = -T.rayleighQuotient x

Tactic:
simp [rayleighQuotient, reApplyInnerSelf_apply, neg_div]

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

Goal state before:
pp: case h
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
⊢ ‖iteratedDeriv n (fun x ↦ x ^ r) x‖ ≤
    (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k

Tactic:
iteratedDeriv_eq_iterate,

Goal state after:
case h
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
⊢ ‖deriv^[n] (fun x ↦ x ^ r) x‖ ≤ (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k
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
⊢ Integrable (fun x ↦ (B (f x)) (φ x)) μ

Tactic:
by
    rwa [integrableOn_iff_integrable_of_support_subset] at this
    refine subset_trans ?_ (subset_tsupport f)
    exact fun x hx hfx ↦ hx (by simp [hfx])

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
U : Set M₂
hU : U ∈ 𝓝 0
V : Set M₂
hV : V ∈ 𝓝 0
hVU : ∀ v ∈ V, ∀ w ∈ V, v + w ∈ U
SV : Set M₁ × Set M₂ := (closedBall 0 1, -V)
v : M₁ →SL[σ₁₂] M₂
hv : v ∈ {f | IsCompactOperator ⇑f}
huv : v - u ∈ {f | ∀ x ∈ SV.1, f x ∈ SV.2}
T : Set M₂
hT : T.Finite
hTv✝ : closure[inst✝⁴.toTopologicalSpace] (⇑↑v '' closedBall 0 1) ⊆ ⋃ y ∈ T, y +ᵥ V
x : M₁
hTv : x ∈ ⋃ i ∈ T, ⇑v ⁻¹' (i +ᵥ V)
hx : x ∈ closedBall 0 1
⊢ x ∈ ⋃ i ∈ T, ⇑u ⁻¹' (i +ᵥ U)

Tactic:
rw [mem_iUnion₂] at hTv ⊢

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
U : Set M₂
hU : U ∈ 𝓝 0
V : Set M₂
hV : V ∈ 𝓝 0
hVU : ∀ v ∈ V, ∀ w ∈ V, v + w ∈ U
SV : Set M₁ × Set M₂ := (closedBall 0 1, -V)
v : M₁ →SL[σ₁₂] M₂
hv : v ∈ {f | IsCompactOperator ⇑f}
huv : v - u ∈ {f | ∀ x ∈ SV.1, f x ∈ SV.2}
T : Set M₂
hT : T.Finite
hTv✝ : closure[inst✝⁴.toTopologicalSpace] (⇑↑v '' closedBall 0 1) ⊆ ⋃ y ∈ T, y +ᵥ V
x : M₁
hTv : ∃ i, ∃ (_ : i ∈ T), x ∈ ⇑v ⁻¹' (i +ᵥ V)
hx : x ∈ closedBall 0 1
⊢ ∃ i, ∃ (_ : i ∈ T), x ∈ ⇑u ⁻¹' (i +ᵥ U)
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: case mp
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
⊢ HasEigenvalue (↑T) μ → μ ∈ spectrum 𝕜 T

pp: case mpr
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
⊢ μ ∈ spectrum 𝕜 T → HasEigenvalue (↑T) μ

Tactic:
· intro hμ'
    rw [ContinuousLinearMap.spectrum_eq]
    exact hμ'.mem_spectrum

Goal state after:
case mpr
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
⊢ μ ∈ spectrum 𝕜 T → HasEigenvalue (↑T) μ
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
h : hA.eigenvalues = 0
⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ 0)) = 0

Tactic:
Pi.comp_zero,

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
h : hA.eigenvalues = 0
⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (Function.const n ↑0)) = 0
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_2
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : A.IsHermitian
h_ne : hA.eigenvalues = 0
this : A = ((Unitary.conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ 0))
⊢ A = 0

Tactic:
Pi.comp_zero,

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_2
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : A.IsHermitian
h_ne : hA.eigenvalues = 0
this : A = ((Unitary.conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (Function.const n ↑0))
⊢ A = 0
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n : ℕ∞
K : Compacts E
i : ℕ
hin : n < ↑i
⊢ ContDiffMapSupportedIn.seminorm 𝕜 E F n K i = ⊥

Tactic:
have : ¬(i ≤ n) := by simpa using hin

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n : ℕ∞
K : Compacts E
i : ℕ
hin : n < ↑i
this : ¬↑i ≤ n
⊢ ContDiffMapSupportedIn.seminorm 𝕜 E F n K i = ⊥
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 : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ (∫⁻ (a : α), ‖↑↑f a‖ₑ ^ 2 ∂μ) ^ (2⁻¹ * ↑2) = ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ

Tactic:
simp [enorm_eq_nnnorm]

Goal state after:
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∞
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
hp : p ≠ ∞
hf : ¬f =ᵐ[μ] 0
hs₁ : μ (tsupport f) ≠ ∞
⊢ 0 < (μ (tsupport f)).toReal

Tactic:
rw [← Measure.measure_support_eq_zero_iff _] at hf

Goal state after:
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∞
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
hp : p ≠ ∞
hf : ¬μ (Function.support f) = 0
hs₁ : μ (tsupport f) ≠ ∞
⊢ 0 < (μ (tsupport f)).toReal
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
hε₂ : 0 < ε / 2
hε₂' : 0 < ENNReal.ofReal (ε / 2)
g : E → F
hg₁ : HasCompactSupport g
hg₂ : eLpNorm (f - g) p μ ≤ ENNReal.ofReal (ε / 2)
hg₃ : Continuous g
hg₄ : MemLp g p μ
g' : E → F
hg'₁ : HasCompactSupport g'
hg'₂ : ContDiff ℝ ∞ g'
hg'₃ : eLpNorm (g - g') ?m.157 μ ≤ ENNReal.ofReal (ε / 2)
this : f - g' = f - g - (g' - g)
⊢ eLpNorm (f - g - (g' - g)) p μ ≤ ENNReal.ofReal ε

Tactic:
eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂,

Goal state after:
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
hε₂ : 0 < ε / 2
hε₂' : 0 < ENNReal.ofReal (ε / 2)
g : E → F
hg₁ : HasCompactSupport g
hg₂ : eLpNorm (f - g) p μ ≤ ENNReal.ofReal (ε / 2)
hg₃ : Continuous g
hg₄ : MemLp g p μ
g' : E → F
hg'₁ : HasCompactSupport g'
hg'₂ : ContDiff ℝ ∞ g'
hg'₃ : eLpNorm (g - g') ?m.157 μ ≤ ENNReal.ofReal (ε / 2)
this : f - g' = f - g - (g' - g)
⊢ eLpNorm (f - g) p μ + eLpNorm (g' - g) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:

Tactic:
_ = ∑' i, ⟪c • f i, g i⟫ := ?_
        _ = ∑' i, conj c * ⟪f i, g i⟫ := by simp only [inner_smul_left]
        _ = conj c * ∑' i, ⟪f i, g i⟫ := tsum_mul_left
        _ = _ := ?_

Goal state after:
case calc_1
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
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
f g : ↥(lp G 2)
c : 𝕜
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

case calc_2
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
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
f g : ↥(lp G 2)
c : 𝕜
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: case insert
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
ι : Type u_2
f : ι → AddCircle T → E
a : ι
s : Finset ι
ha : a ∉ s
iha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)
hf : ∀ i ∈ insert a s, Integrable (f i) haarAddCircle
⊢ fourierCoeff (∑ i ∈ insert a s, f i) = ∑ i ∈ insert a s, fourierCoeff (f i)

Tactic:
| insert a s ha iha =>

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

Goal state before:
pp: case inl
T : ℝ
hT : Fact (0 < T)
m : ℤ
⊢ ↑‖fourierLp 2 m‖ ^ 2 = Pi.single m 1 m

Tactic:
(orthonormal_fourier (hT := hT)).1 m

Goal state after:
case inl
T : ℝ
hT : Fact (0 < T)
m : ℤ
⊢ ↑1 ^ 2 = Pi.single m 1 m
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✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SMulCommClass ℝ 𝕜 F
inst✝² : NormedField 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : SMulCommClass ℝ 𝕜' F
c : 𝕜
f : 𝓢(E, F)
k n : ℕ
x : E
⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • ⇑f) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ * ‖c‖

Tactic:
by
            rw [mul_comm _ ‖c‖, ← mul_assoc]
            exact decay_smul_aux k n f c x

Goal state after:
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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
⊢ ∀ (a : 𝕜) (f : 𝓢(E, F)) (x : E), (L x).smulRight ((a • f) x) = (RingHom.id 𝕜) a • (L x).smulRight (f x)

Tactic:
by intro c g x; ext v; simpa using smul_comm (L x v) c (g x)

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
u : Set F
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
hu : 0 ∈ u
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
h'u : ball 0 δ ⊆ u
M : ℝ
Mpos : M > 0
hM : ∀ ⦃x : F⦄, x ∈ (closedBall 0 M)ᶜ → ‖x‖ ^ finrank ℝ F * φ x < δ ^ finrank ℝ F * ε
c : ℝ
hc : M / δ < c
x : F
hx : x ∈ uᶜ
cpos : 0 < c
hδx : δ ≤ ‖x‖
⊢ ‖x‖ ^ finrank ℝ F * (c ^ finrank ℝ F * φ (c • x)) = ‖c • x‖ ^ finrank ℝ F * φ (c • x)

Tactic:
simp [norm_smul, abs_of_pos cpos, mul_pow]; ring

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

Goal state before:
pp: 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 : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
⊢ (eLpNorm (↑↑(f.toLp p μ)) p μ).toReal = (eLpNorm (⇑f) p μ).toReal

Tactic:
eLpNorm_congr_ae (coeFn_toLp f p μ)

Goal state after:
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 : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
⊢ (eLpNorm (⇑f) p μ).toReal = (eLpNorm (⇑f) p μ).toReal
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
⊢ ∫ (x : ℝ), f |x| = 2 * ∫ (x : ℝ) in Ioi 0, f x

Tactic:
have eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x := by
    refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
    rw [abs_eq_self.mpr (le_of_lt (by exact hx))]

Goal state after:
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
⊢ ∫ (x : ℝ), f |x| = 2 * ∫ (x : ℝ) in Ioi 0, f x
Module: Mathlib.MeasureTheory.Integral.Asymptotics

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : NormedAddCommGroup E
g : α → F
l : Filter α
ι : Type u_4
inst✝¹ : MeasurableSpace ι
f : ι × α → E
s : Set ι
μ : Measure ι
inst✝ : Norm F
hf : f =O[𝓟 s ×ˢ l] (g ∘ Prod.snd)
hfm : ∀ᶠ (x : α) in l, AEStronglyMeasurable (fun i ↦ f (i, x)) (μ.restrict s)
hs : MeasurableSet s
hμ : μ s < ⊤
C : ℝ
hC : ∀ᶠ (x : ι × α) in 𝓟 s ×ˢ l, ‖f x‖ ≤ C * ‖(g ∘ Prod.snd) x‖
t : Set (ι × α)
htl : t ∈ 𝓟 s ×ˢ l
ht : ∀ y ∈ t, ‖f y‖ ≤ C * ‖(g ∘ Prod.snd) y‖
u : Set ι
hu : u ∈ 𝓟 s
v : Set α
hv : v ∈ l
huv : u ×ˢ v ⊆ t
w : Set α
hwl : w ∈ l
hw : ∀ y ∈ w, AEStronglyMeasurable (fun i ↦ f (i, y)) (μ.restrict s)
x : α
hx : x ∈ w ∩ v
this : IsFiniteMeasure (μ.restrict s)
y : ι
hy : y ∈ s
⊢ ‖f (y, x)‖ ≤ C * ‖g x‖

Tactic:
exact ht (y, x) <| huv ⟨hu hy, hx.2⟩

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

Goal state before:
pp: case left
s : ℂ
hs : 0 < s.re
x : ℝ
hx : x ∈ Ioi 0
this : ContinuousAt (fun x ↦ x ^ (s - 1)) ↑x
⊢ ContinuousAt (fun x ↦ ↑x ^ (s - 1)) x

Tactic:
exact ContinuousAt.comp this continuous_ofReal.continuousAt

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

Goal state before:
pp: p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ (b ^ (-p⁻¹)) ^ s * (x✝ ^ s * rexp (-x✝ ^ p)) = (b ^ (-p⁻¹) * x✝) ^ s * rexp (-b * (b ^ (-p⁻¹) * x✝) ^ p)

Tactic:
rw [← mul_assoc, mul_rpow, mul_rpow, ← rpow_mul (z := p), neg_mul, neg_mul, inv_mul_cancel₀,
      rpow_neg_one, mul_inv_cancel_left₀]

Goal state after:
case h
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ b ≠ 0

p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ p ≠ 0

case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b

case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b ^ (-p⁻¹)

case hy
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ x✝

case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b ^ (-p⁻¹)

case hy
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ x✝
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: x : ℝ × ℝ
hx : x ∈ polarCoord.sourceᶜ
⊢ x ∈ ↑(LinearMap.snd ℝ ℝ ℝ).ker

Tactic:
simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_ofPred_eq, not_lt,
      Classical.not_not] at hx

Goal state after:
x : ℝ × ℝ
hx : x.1 ≤ 0 ∧ x.2 = 0
⊢ x ∈ ↑(LinearMap.snd ℝ ℝ ℝ).ker
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ Integrable (fun x ↦ cexp (-b * ↑x ^ 2)) volume

Tactic:
by
  refine ⟨by fun_prop, ?_⟩
  rw [← hasFiniteIntegral_norm_iff]
  simp_rw [norm_cexp_neg_mul_sq]
  exact (integrable_exp_neg_mul_sq hb).2

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
this : HasFiniteIntegral (fun a ↦ ‖a * rexp (-b.re * a ^ 2)‖) volume
x✝ : ℝ
⊢ ‖↑x✝‖ * rexp (-b.re * x✝ ^ 2) = ‖x✝‖ * ‖rexp (-b.re * x✝ ^ 2)‖

Tactic:
norm_real,

Goal state after:
b : ℂ
hb : 0 < b.re
this : HasFiniteIntegral (fun a ↦ ‖a * rexp (-b.re * a ^ 2)‖) volume
x✝ : ℝ
⊢ ‖x✝‖ * rexp (-b.re * x✝ ^ 2) = ‖x✝‖ * ‖rexp (-b.re * x✝ ^ 2)‖
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
⊢ ∫⁻ (x : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal |(fderivPolarCoordSymm x).det| * f (↑polarCoord.symm x) =
    ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal |p.1| • f (↑polarCoord.symm p)

Tactic:
simp_rw [det_fderivPolarCoordSymm]; rfl

Goal state after: