mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
hs : s.WSameSide x y
ho : s.WOppSide y x
⊢ x ∈ s ∨ y ∈ s

Tactic:
by_contra h

Goal state after:
case mp
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
hs : s.WSameSide x y
ho : s.WOppSide y x
h : ¬(x ∈ s ∨ y ∈ s)
⊢ False
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp.inr.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p) = r₂ • (y -ᵥ p₂)
⊢ ∃ a, 0 ≤ a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y

Tactic:
refine ⟨r₁ / r₂, (div_pos hr₁ hr₂).le, p₂, hp₂, ?_⟩

Goal state after:
case mp.inr.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p) = r₂ • (y -ᵥ p₂)
⊢ (r₁ / r₂) • (x -ᵥ p) +ᵥ p₂ = y
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case pos
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x p : P
hp : p ∈ ↑s
this : Nonempty ↥s
hx : x ∈ s
⊢ IsConnected {y | True}

Tactic:
have := AddTorsor.connectedSpace V P

Goal state after:
case pos
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x p : P
hp : p ∈ ↑s
this✝ : Nonempty ↥s
hx : x ∈ s
this : ConnectedSpace P
⊢ IsConnected {y | True}
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: r : ℝ
n : ℕ
hnr : ↑n < r
hr : 0 < r
x : ℝ
hx : x ∈ Ioc 0 1
⊢ -r⁻¹ ≤ 0

Tactic:
simpa using hr.le

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
t : ℝ
ht : 0 < t
hT' : (algebraMap ℝ 𝕜) t ∈ resolventSet 𝕜 T
⊢ ∃ c > 0, ∀ (x : E), T.rayleighQuotient x ≤ (t ^ 2 + ‖T‖ ^ 2) / (2 * t) - c

Tactic:
by_cases hT0 : T = 0
  · exact ⟨t ^ 2 / (2 * t), by positivity, by simp [hT0]⟩
  obtain ⟨c, hc0, hc⟩ := antilipschitzWith_iff_exists_mul_le_norm.mp
    (antilipschitz_of_isEmbedding _ (isHomeomorph_of_isUnit hT').isEmbedding)
  refine ⟨min (c ^ 2 / (2 * t)) ((t ^ 2 + ‖T‖ ^ 2) / (2 * t)), by positivity, fun x ↦ ?_⟩
  by_cases hx : x = 0
  · simp [hx]
  suffices T.rayleighQuotient x ≤ (t ^ 2 + ‖T‖ ^ 2) / (2 * t) - c ^ 2 / (2 * t) by
    grw [this, min_le_left]
  rw [rayleighQuotient, reApplyInnerSelf_apply]
  specialize hc x
  rw [← sq_le_sq₀ (by positivity) (by positivity), sub_apply, algebraMap_apply,
    norm_sub_sq (𝕜 := 𝕜), inner_re_symm] at hc
  grw [le_opNorm] at hc
  simp [inner_smul_right, norm_smul, abs_of_pos ht] at hc
  field_simp
  grind

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
H : IsLocalExtrOn T.reApplyInnerSelf {x | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • 2 • (innerSL ℝ) x₀ + b • 2 • (innerSL ℝ) (T x₀) = 0
⊢ (toDualMap ℝ F) (a • x₀ + b • T x₀) = (toDualMap ℝ F) 0

Tactic:
simp only [LinearIsometry.map_add, LinearIsometry.map_zero]

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
H : IsLocalExtrOn T.reApplyInnerSelf {x | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • 2 • (innerSL ℝ) x₀ + b • 2 • (innerSL ℝ) (T x₀) = 0
⊢ (toDualMap ℝ F) (a • x₀) + (toDualMap ℝ F) (b • T x₀) = 0
Module: Mathlib.Analysis.Normed.Operator.Compact.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
h : IsCompactOperator id
⊢ FiniteDimensional 𝕜 E

Tactic:
have := LocallyCompactSpace.of_isCompactOperator_id h
  exact FiniteDimensional.of_locallyCompactSpace 𝕜

Goal state after:
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
⊢ A.rank = (diagonal hA.eigenvalues).rank

Tactic:
conv_lhs => rw [hA.spectral_theorem, conjStarAlgAut_apply, ← coe_star]

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
⊢ (↑hA.eigenvectorUnitary * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * ↑(star hA.eigenvectorUnitary)).rank =
    (diagonal hA.eigenvalues).rank
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
⊢ ∃ C, ∀ (x y : E), dist (f x) (f y) ≤ C

Tactic:
have := HasCompactSupport.intro K.isCompact (map_zero_on_compl f)
    rcases (map_continuous f).bounded_above_of_compact_support this with ⟨C, hC⟩
    exact map_bounded (BoundedContinuousFunction.ofNormedAddCommGroup f (map_continuous f) C hC)

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✝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
f g : 𝓓^{n}_{K}(E, F)
⊢ EqOn (⇑f - ⇑g) 0 (↑K)ᶜ

Tactic:
rw [← sub_zero 0]

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
f g : 𝓓^{n}_{K}(E, F)
⊢ EqOn (⇑f - ⇑g) (0 - 0) (↑K)ᶜ
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
⊢ ((Multiset.map (⇑RCLike.re) T.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) = List.ofFn (hT.eigenvalues hn)

Tactic:
by
  simp_rw [hT.roots_charpoly_eq_eigenvalues, Fin.univ_val_map, Multiset.map_coe, List.map_ofFn,
    Function.comp_def, RCLike.ofReal_re, Multiset.coe_sort]
  have := hn.symm
  convert! List.mergeSort_of_pairwise ?_
  simp_rw [decide_eq_true_eq, ← List.sortedGE_iff_pairwise]
  convert! (hT.eigenvalues_antitone hn).sortedGE_ofFn

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

Goal state before:
pp: case pos
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 k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E (E →L[ℝ] F) k K i) ((fderivLM 𝕜 n k) f) ≤
    (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f

Tactic:
rw [ContDiffMapSupportedIn.seminorm_le_iff 𝕜 (apply_nonneg ..)]

Goal state after:
case pos
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 k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
⊢ ↑i ≤ k →
    ∀ x ∈ K, ‖iteratedFDeriv ℝ i (⇑((fderivLM 𝕜 n k) f)) x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
s : Set α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
f : ↥(Lp E 2 μ)
⊢ ⟪indicatorConstLp 2 hs hμs c, f⟫ = ⟪c, ∫ (x : α) in s, ↑↑f x ∂μ⟫

Tactic:
← integral_inner (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs),

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
s : Set α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
f : ↥(Lp E 2 μ)
⊢ ⟪indicatorConstLp 2 hs hμs c, f⟫ = ∫ (x : α) in s, ⟪c, ↑↑f x⟫ ∂μ
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
⊢ (Finset.range n).card = n

Tactic:
Finset.card_range

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
⊢ n = n
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
x : AddCircle T
⊢ (fourier 1) x = ↑x.toCircle

Tactic:
by rw [fourier_apply, one_zsmul]

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

Goal state before:
pp: T : ℝ
⊢ range fourier ⊆ ↑(star (adjoin ℂ (range fourier)))

Tactic:
rintro - ⟨n, rfl⟩

Goal state after:
T : ℝ
n : ℤ
⊢ fourier n ∈ ↑(star (adjoin ℂ (range fourier)))
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: case hf
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
⊢ 0 ≤ᵐ[μ.restrict t] fun x ↦ ‖φ i x‖ * δ

Tactic:
exact Eventually.of_forall fun x => mul_nonneg (norm_nonneg _) δpos.le

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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : 𝓢(E, F)
⊢ ∀ (k n : ℕ), ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f - ⇑g) x‖ ≤ C

Tactic:
by
      intro k n
      refine ⟨f.seminormAux k n + g.seminormAux k n, fun x => ?_⟩
      grw [← f.le_seminormAux k n x, ← g.le_seminormAux k n x]
      rw [sub_eq_add_neg]
      rw [← decay_neg_aux k n g x]
      exact decay_add_le_aux k n f (-g) 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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup D
inst✝¹ : NormedSpace ℝ D
inst✝ : NormedSpace 𝕜 F
g : D → E
hg : Function.HasTemperateGrowth g
k n l : ℕ
C : ℝ
hC : 0 ≤ C
hgrowth : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N g x‖ ≤ C * (1 + ‖x‖) ^ l
kg : ℕ
Cg : ℝ
hg_upper' : ∀ (x : D), ‖x‖ ≤ Cg * (1 + ‖g x‖) ^ kg
hCg : 1 ≤ 1 + Cg
k' : ℕ := kg * (k + l * n)
f : 𝓢(E, F)
x : D
seminorm_f : ℝ := ((Finset.Iic (k', n)).sup (schwartzSeminormFamily 𝕜 E F)) f
hg_upper'' : (1 + ‖x‖) ^ (k + l * n) ≤ (1 + Cg) ^ (k + l * n) * (1 + ‖g x‖) ^ k'
hbound : ∀ i ≤ n, ‖iteratedFDeriv ℝ i (⇑f) (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k'
N : ℕ
hN₁ : 1 ≤ N
hN₂ : N ≤ n
hN₁' : N ≠ 0
⊢ 1 ≤ C + 1

Tactic:
simp [hC]

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
⊢ ‖f.toLp p μ‖ = (eLpNorm (⇑f) p μ).toReal

Tactic:
by
  rw [Lp.norm_def, eLpNorm_congr_ae (coeFn_toLp f p μ)]

Goal state after:
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
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ ∫ (x : ℝ), f |x| = ∫ (x : ℝ) in univ, f |x|

Tactic:
restrict_univ

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
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ ∫ (x : ℝ), f |x| = ∫ (x : ℝ), f |x|
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: 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)
⊢ fourierTransform (f + g) = fourierTransform f + fourierTransform g

Tactic:
ext x
        simp only [Lp.fourierTransform_apply, BoundedContinuousFunction.coe_add, Pi.add_apply,
          Real.fourier_eq]
        rw [← integral_add]
        · apply integral_congr_ae
          filter_upwards [Lp.coeFn_add f g] with x h₁
          rw [h₁]
          simp
        · rw [Real.fourierIntegral_convergent_iff]
          exact L1.integrable_coeFn f
        · rw [Real.fourierIntegral_convergent_iff]
          exact L1.integrable_coeFn g

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

Goal state before:
pp: b : ℝ
h : IntegrableOn (fun x ↦ rexp (-b * x ^ 2)) (Ioi 0) volume
hb : b ≤ 0
⊢ False

Tactic:
have : ∫⁻ _ : ℝ in Ioi 0, 1 ≤ ∫⁻ x : ℝ in Ioi 0, ‖exp (-b * x ^ 2)‖₊ := by
    apply lintegral_mono (fun x ↦ _)
    simp only [neg_mul, ENNReal.one_le_coe_iff, ← toNNReal_one, toNNReal_le_iff_le_coe,
      Real.norm_of_nonneg (exp_pos _).le, coe_nnnorm, one_le_exp_iff, Right.nonneg_neg_iff]
    exact fun x ↦ mul_nonpos_of_nonpos_of_nonneg hb (sq_nonneg x)

Goal state after:
b : ℝ
h : IntegrableOn (fun x ↦ rexp (-b * x ^ 2)) (Ioi 0) volume
hb : b ≤ 0
this : ∫⁻ (x : ℝ) in Ioi 0, 1 ≤ ∫⁻ (x : ℝ) in Ioi 0, ↑‖rexp (-b * x ^ 2)‖₊
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero
s : ℂ
h1 : 0 < s.re
⊢ s.GammaIntegral = s.GammaIntegral * s / s

Tactic:
mul_div_cancel_right₀

Goal state after:
case zero
s : ℂ
h1 : 0 < s.re
⊢ s.GammaIntegral = s.GammaIntegral

case zero.hb
s : ℂ
h1 : 0 < s.re
⊢ s ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℝ × ℝ) → ℝ≥0∞
⊢ (∫⁻ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ polarCoord.target,
      (∏ i, ENNReal.ofReal (p i).1) * f fun i ↦ ↑polarCoord.symm (p i)) =
    ∫⁻ (p : ι → ℝ × ℝ), f p

Tactic:
by
  rw [← setLIntegral_univ f, ← setLIntegral_congr pi_polarCoord_symm_target_ae_eq_univ]
  convert!
    (lintegral_image_eq_lintegral_abs_det_fderiv_mul volume measurableSet_pi_polarCoord_target
        (fun p _ ↦ (hasFDerivAt_pi_polarCoord_symm p).hasFDerivWithinAt) injOn_pi_polarCoord_symm
        f).symm using 1
  refine setLIntegral_congr_fun measurableSet_pi_polarCoord_target (fun x hx ↦ ?_)
  simp_rw [det_fderivPiPolarCoordSymm, Finset.abs_prod, ENNReal.ofReal_prod_of_nonneg (fun _ _ ↦
    abs_nonneg _), abs_fst_of_mem_pi_polarCoord_target hx]

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ ∫ (x : ℝ), cexp (-b * (↑x + ↑c * I) ^ 2) = (↑π / b) ^ (1 / 2)

Tactic:
refine
    tendsto_nhds_unique
      (intervalIntegral_tendsto_integral (integrable_cexp_neg_mul_sq_add_real_mul_I hb c)
        tendsto_neg_atTop_atBot tendsto_id)
      ?_

Goal state after:
b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in -i..id i, cexp (-b * (↑x + ↑c * I) ^ 2)) atTop (𝓝 ((↑π / b) ^ (1 / 2)))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
t : ℂ
⊢ (↑π / b) ^ (1 / 2) * cexp (-((I * t) ^ 2 / -(4 * b))) = (↑π / b) ^ (1 / 2) * cexp (-t ^ 2 / (4 * b))

Tactic:
div_neg,

Goal state after:
b : ℂ
hb : 0 < b.re
t : ℂ
⊢ (↑π / b) ^ (1 / 2) * cexp (- -((I * t) ^ 2 / (4 * b))) = (↑π / b) ^ (1 / 2) * cexp (-t ^ 2 / (4 * b))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case e'_6
b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
v : ι → ℝ
⊢ 0 ≤ ∑ x, v x ^ 2

Tactic:
· exact Finset.sum_nonneg (fun i _hi ↦ by positivity)

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

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

Tactic:
integral_fintype_prod_volume_eq_prod (f := fun i (v : ℝ) ↦ cexp (-b i * v ^ 2 + c i * v))

Goal state after:
ι : Type u_2
inst✝ : Fintype ι
b : ι → ℂ
hb : ∀ (i : ι), 0 < (b i).re
c : ι → ℂ
⊢ ∏ i, ∫ (x : ℝ), cexp (-b i * ↑x ^ 2 + c i * ↑x) = ∏ i, (↑π / b i) ^ (1 / 2) * cexp (c i ^ 2 / (4 * b i))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

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

Tactic:
Finset.sum_div,

Goal state after:
b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ι → ℂ
⊢ (↑π / b) ^ (↑Finset.univ.card * 2⁻¹) * cexp (∑ x, c x ^ 2 / (4 * b)) =
    (↑π / b) ^ (↑Finset.univ.card / 2) * cexp (∑ i, c i ^ 2 / (4 * b))
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: case a
E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ {x} ⊆ dsupport ⇑(TemperedDistribution.delta x)

Tactic:
rintro x rfl

Goal state after:
case a
E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ x ∈ dsupport ⇑(TemperedDistribution.delta x)
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: case a
E : Type u_11
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
inst✝ : FiniteDimensional ℝ E
x : E
hx : x ∈ Ω
⊢ {x} ⊆ dsupport ⇑(delta x)

Tactic:
rintro x rfl

Goal state after:
case a
E : Type u_11
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
inst✝ : FiniteDimensional ℝ E
x : E
hx : x ∈ Ω
⊢ x ∈ dsupport ⇑(delta x)
Module: Mathlib.Analysis.Fourier.AddCircleMulti

Goal state before:
pp: d : Type u_1
inst✝¹ : Fintype d
inst✝ : DecidableEq d
z : d → AddCircle 1
i : d
⊢ (mFourier (Pi.single i 1)) z = (fourier 1) (z i)

Tactic:
by
  simp_rw [mFourier, ContinuousMap.coe_mk]
  have := Finset.prod_mul_prod_compl {i} (fun j ↦ fourier ((Pi.single i (1 : ℤ) : d → ℤ) j) (z j))
  rw [Finset.prod_singleton, Finset.prod_congr rfl (fun j hj ↦ ?_)] at this
  · rw [← this, Finset.prod_const_one, mul_one, Pi.single_eq_same]
  · rw [Finset.mem_compl, Finset.mem_singleton] at hj
    simp only [Pi.single_eq_of_ne hj, fourier_zero]

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

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℂ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul ℂ F
c : ℂ
f : 𝓢'(E, F)
⊢ (smulLeftCLM F fun x ↦ c) f = c • f

Tactic:
ext1; simp

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

Goal state before:
pp: d : Type u_1
inst✝¹ : Fintype d
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
⊢ (span ℂ (range (mFourierLp p))).topologicalClosure = ⊤

Tactic:
simpa only [map_span, ContinuousLinearMap.coe_coe, ← range_comp, Function.comp_def] using
    (ContinuousMap.toLp_denseRange ℂ volume ℂ hp).topologicalClosure_map_submodule
      span_mFourier_closure_eq_top

Goal state after:
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
a : 𝕜
f : ι → 𝕜
⊢ ⟪const ι a, f⟫ₙ_[𝕜] = 𝔼 i, f i * (starRingEnd 𝕜) a

Tactic:
simp [wInner_cWeight_eq_expect]

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
h : ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s)‖ₑ < ∞
⊢ ∫⁻ (a : E), ‖((1 + ‖a‖ ^ 2) ^ (-s / 2)) ^ 2‖ₑ < ∞

Tactic:
convert h

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
h : ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s)‖ₑ < ∞
x✝ : E
⊢ ((1 + ‖x✝‖ ^ 2) ^ (-s / 2)) ^ 2 = (1 + ‖x✝‖ ^ 2) ^ (-s)
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: n : ℕ
⊢ (μ * ↑ζ) n = 1 n

Tactic:
induction n using recOnPosPrimePosCoprime with

Goal state after: