mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Matrix.HermitianFunctionalCalculus

Goal state before:
pp: n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
f : ℝ → ℝ
hA' : IsSelfAdjoint A
this : cfcHom hA' = hA.cfcAux
⊢ cfc f A = hA.cfc f

Tactic:
rw [cfc_apply f A hA' (by rw [continuousOn_iff_continuous_domRestrict]; fun_prop), this]

Goal state after:
n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
f : ℝ → ℝ
hA' : IsSelfAdjoint A
this : cfcHom hA' = hA.cfcAux
⊢ hA.cfcAux { toFun := (spectrum ℝ A).domRestrict f, continuous_toFun := ⋯ } = hA.cfc f
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
hT : T.IsSymmetric
⊢ T.charpoly.Splits

Tactic:
refine Polynomial.splits_iff_card_roots.mpr ?_

Goal state after:
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
hT : T.IsSymmetric
⊢ T.charpoly.roots.card = T.charpoly.natDegree
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : α → E
c : E
hf : MemLp f 1 μ
⊢ MemLp (fun x ↦ ⟪c, f x⟫) 1 μ

Tactic:
exact hf.const_inner c

Goal state after:
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
P : ℕ → X →L[𝕜] X
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
i j : ℕ
x : X
⊢ (P (min i j + 1)) x - (P (min i (j + 1))) x - ((P (min (i + 1) j)) x - (P (min i j)) x) =
    if i = j then (P (j + 1)) x - (P j) x else 0

Tactic:
split_ifs with h

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
P : ℕ → X →L[𝕜] X
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
i j : ℕ
x : X
h : i = j
⊢ (P (min i j + 1)) x - (P (min i (j + 1))) x - ((P (min (i + 1) j)) x - (P (min i j)) x) = (P (j + 1)) x - (P j) x

case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
P : ℕ → X →L[𝕜] X
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
i j : ℕ
x : X
h : ¬i = j
⊢ (P (min i j + 1)) x - (P (min i (j + 1))) x - ((P (min (i + 1) j)) x - (P (min i j)) x) = 0
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
s : Set α
st : s ⊆ t
s_compact : IsCompact s
s_closed : IsClosed[inst✝⁸] s
μs : μ (t \ s) < ↑η
hsμ : μ s < ∞
I1 : eLpNorm ((s.indicator fun _y ↦ c) - t.indicator fun _y ↦ c) p μ ≤ δ
k : Set α
k_compact : IsCompact k
sk : s ⊆ interior k
f : α → E
f_cont : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
I2 : eLpNorm (fun x ↦ f x - s.indicator (fun _y ↦ c) x) p μ ≤ δ
_f_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖
f_support : Function.support f ⊆ interior k
f_mem : MemLp f p μ
I3 : eLpNorm (f - t.indicator fun _y ↦ c) p μ ≤ ε
⊢ ∃ 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:
refine ⟨f, I3, f_cont, f_mem, HasCompactSupport.intro k_compact fun x hx => ?_⟩

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
hsμ : μ s < ∞
I1 : eLpNorm ((s.indicator fun _y ↦ c) - t.indicator fun _y ↦ c) p μ ≤ δ
k : Set α
k_compact : IsCompact k
sk : s ⊆ interior k
f : α → E
f_cont : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
I2 : eLpNorm (fun x ↦ f x - s.indicator (fun _y ↦ c) x) p μ ≤ δ
_f_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖
f_support : Function.support f ⊆ interior k
f_mem : MemLp f p μ
I3 : eLpNorm (f - t.indicator fun _y ↦ c) p μ ≤ ε
x : α
hx : x ∉ k
⊢ f x = 0
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
H :
  ∃ g,
    eLpNorm (f - g) p μ ≤ ε ∧
      Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ IsBounded (range g)
⊢ ∃ g, eLpNorm (f - ⇑g) p μ ≤ ε ∧ MemLp (⇑g) p μ

Tactic:
rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩

Goal state after:
α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
g : α → E
hg : eLpNorm (f - g) p μ ≤ ε
g_cont : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_mem : MemLp g p μ
g_bd : IsBounded (range g)
⊢ ∃ g, eLpNorm (f - ⇑g) p μ ≤ ε ∧ MemLp (⇑g) p μ
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Goal state before:
pp: D : Type u_4
E : Type u_5
V : Type u_7
F : Type u_8
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : NormedSpace ℝ D
inst✝³ : MeasurableSpace D
μ : Measure D
inst✝² : BorelSpace D
inst✝¹ : FiniteDimensional ℝ D
inst✝ : μ.IsAddHaarMeasure
f : 𝓢(D, E)
g : 𝓢(D, F)
L : E →L[ℝ] F →L[ℝ] V
v : D
⊢ ∫ (x : D), (L (f x)) ((∂_{v} g) x) ∂μ = -∫ (x : D), (L ((∂_{v} f) x)) (g x) ∂μ

Tactic:
apply integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable (v := v)
    (bilinLeftCLM L g.hasTemperateGrowth _).integrable
    (bilinLeftCLM L (∂_{v} g).hasTemperateGrowth _).integrable
    (bilinLeftCLM L g.hasTemperateGrowth _).integrable
  all_goals exact fun x _ ↦ (hasFDerivAt _ x).hasLineDerivAt v

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : DecidableEq ι
i : ι
a : G i
f : ↥(lp G 2)
j : ι
⊢ ⟪Pi.single i a j, ↑f j⟫ = if j = i then ⟪a, ↑f i⟫ else 0

Tactic:
split_ifs with h

Goal state after:
case pos
ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : DecidableEq ι
i : ι
a : G i
f : ↥(lp G 2)
j : ι
h : j = i
⊢ ⟪Pi.single i a j, ↑f j⟫ = ⟪a, ↑f i⟫

case neg
ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : DecidableEq ι
i : ι
a : G i
f : ↥(lp G 2)
j : ι
h : ¬j = i
⊢ ⟪Pi.single i a j, ↑f j⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: case neg
ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : DecidableEq ι
i : ι
a : G i
f : ↥(lp G 2)
j : ι
h : ¬j = i
⊢ ⟪Pi.single i a j, ↑f j⟫ = 0

Tactic:
simp [Pi.single_eq_of_ne h]

Goal state after:
Module: Mathlib.Analysis.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
⊢ ‖{ toFun := fun f ↦ ∑' (i : ι), (V i) (↑f i), map_add' := ⋯, map_smul' := ⋯ } f‖ = ‖f‖

Tactic:
suffices ‖∑' i : ι, V i (f i)‖ ^ (2 : ℝ≥0∞).toReal = ‖f‖ ^ (2 : ℝ≥0∞).toReal by
      exact Real.rpow_left_injOn H.ne' (norm_nonneg _) (norm_nonneg _) this

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
⊢ ‖∑' (i : ι), (V i) (↑f i)‖ ^ ENNReal.toReal 2 = ‖f‖ ^ ENNReal.toReal 2
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: case e_f
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
i : ι
x : G i
j : ι
⊢ (V j) (Pi.single i x j) = if j = i then (V i) x else 0

Tactic:
split_ifs with h

Goal state after:
case pos
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
i : ι
x : G i
j : ι
h : j = i
⊢ (V j) (Pi.single i x j) = (V i) x

case neg
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
i : ι
x : G i
j : ι
h : ¬j = i
⊢ (V j) (Pi.single i x j) = 0
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
m n : ℤ
x : AddCircle T
⊢ ↑((m + n) • x).toCircle = ↑(m • x).toCircle * ↑(n • x).toCircle

Tactic:
add_zsmul,

Goal state after:
T : ℝ
m n : ℤ
x : AddCircle T
⊢ ↑(m • x + n • x).toCircle = ↑(m • x).toCircle * ↑(n • x).toCircle
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
β : Type u_3
L : SummationFilter β
b : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
i j : ℕ
x : X
hx : (succSub D.P j) x = D.e j
⊢ (if i = j then D.e j else 0) = (if i = j then 1 else 0) • D.e i

Tactic:
split_ifs with h

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
i j : ℕ
x : X
hx : (succSub D.P j) x = D.e j
h : i = j
⊢ D.e j = 1 • D.e i

case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
i j : ℕ
x : X
hx : (succSub D.P j) x = D.e j
h : ¬i = j
⊢ 0 = 0 • D.e i
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
i j : ℤ
⊢ ∫ (x : AddCircle T), (fourier (j + -i)) x ∂haarAddCircle = if i = j then 1 else 0

Tactic:
split_ifs with h

Goal state after:
case pos
T : ℝ
hT : Fact (0 < T)
i j : ℤ
h : i = j
⊢ ∫ (x : AddCircle T), (fourier (j + -i)) x ∂haarAddCircle = 1

case neg
T : ℝ
hT : Fact (0 < T)
i j : ℤ
h : ¬i = j
⊢ ∫ (x : AddCircle T), (fourier (j + -i)) x ∂haarAddCircle = 0
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: case hcg
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
h't : μ t ≠ ∞
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝⁴] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
h : α → E := ⋯
⊢ Tendsto h (𝓝[s] x₀) (𝓝 0)

Tactic:
rw [← sub_self a]
      apply Tendsto.sub hcg
      apply tendsto_const_nhds.congr'
      filter_upwards [h'ts] with x hx using by simp [hx]

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✝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 : ℕ
f : 𝓢(E, F)
⊢ ∀ (k n : ℕ), ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • ⇑f) x‖ ≤ C

Tactic:
simpa [← Nat.cast_smul_eq_nsmul ℝ] using! ((c : ℝ) • f).decay'

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✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace ℝ D
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedSpace ℝ V
inst✝³ : NormedSpace 𝕜 V
inst✝² : MeasurableSpace D
μ : Measure D
hμ : μ.HasTemperateGrowth
inst✝¹ : BorelSpace D
inst✝ : SecondCountableTopology D
n : ℕ
h : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑n)) μ
m : ℕ × ℕ := (n, 0)
f : 𝓢(D, V)
x : D
⊢ ‖f x‖ ≤ 2 ^ n * ((Finset.Iic m).sup fun m' ↦ SchwartzMap.seminorm 𝕜 m'.1 m'.2) f / (1 + ‖x‖) ^ ↑n

Tactic:
le_div_iff₀' (by positivity),

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✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace ℝ D
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedSpace ℝ V
inst✝³ : NormedSpace 𝕜 V
inst✝² : MeasurableSpace D
μ : Measure D
hμ : μ.HasTemperateGrowth
inst✝¹ : BorelSpace D
inst✝ : SecondCountableTopology D
n : ℕ
h : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑n)) μ
m : ℕ × ℕ := (n, 0)
f : 𝓢(D, V)
x : D
⊢ (1 + ‖x‖) ^ ↑n * ‖f x‖ ≤ 2 ^ n * ((Finset.Iic m).sup fun m' ↦ SchwartzMap.seminorm 𝕜 m'.1 m'.2) f
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: case h₂
𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
k : ℕ
hk : eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) p μ < ∞
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
x : E
⊢ ‖(1 + ‖x‖) ^ k • f x‖₊ ≤ ↑2 ^ k * (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f).toNNReal

Tactic:
← norm_toNNReal,

Goal state after:
case h₂
𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
k : ℕ
hk : eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) p μ < ∞
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
x : E
⊢ ‖(1 + ‖x‖) ^ k • f x‖.toNNReal ≤ ↑2 ^ k * (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f).toNNReal
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: case hmg
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
⊢ Integrable g μ

Tactic:
exact hg

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
c : ℝ
f : ℝ → E
⊢ ∫ (x : ℝ) in Iic (-c), f (- -x) = ∫ (x : ℝ) in Iic (- - -c), f x

Tactic:
simp only [neg_neg]

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

Goal state before:
pp: c : ℝ
⊢ 𝓝 (rexp c) = 𝓝 (rexp c - 0)

Tactic:
rw [sub_zero]

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
⊢ Tendsto (fun t ↦ (t + m) ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1)))

Tactic:
← neg_neg (a + 1)

Goal state after:
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
⊢ Tendsto (fun t ↦ (t + m) ^ (- -(a + 1)) / - -(a + 1)) atTop (𝓝 (0 / - -(a + 1)))
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
hf' : Integrable (fun x ↦ x • f x) volume
w : ℝ
hf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume
L : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := (ContinuousLinearMap.mul ℝ ℝ).flip
this : Integrable (fun v ↦ (L v).smulRight (f v)) volume
⊢ Integrable (fun v ↦ fourierSMulRight L f v) volume

Tactic:
by
      simpa only [fourierSMulRight, neg_smul, neg_mul, Pi.smul_apply] using! this.smul (-2 * π * I)

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

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

Tactic:
rw [← integrableOn_univ, ← @Iio_union_Ici _ _ (0 : ℝ), integrableOn_union,
    integrableOn_Ici_iff_integrableOn_Ioi]
  refine ⟨?_, integrableOn_rpow_mul_exp_neg_mul_sq hb hs⟩
  rw [← (Measure.measurePreserving_neg (volume : Measure ℝ)).integrableOn_comp_preimage
      (Homeomorph.neg ℝ).measurableEmbedding]
  simp only [Function.comp_def, neg_sq, neg_preimage, neg_Iio, neg_zero]
  apply Integrable.mono' (integrableOn_rpow_mul_exp_neg_mul_sq hb hs)
  · apply Measurable.aestronglyMeasurable
    exact (measurable_id'.neg.pow measurable_const).mul
      ((measurable_id'.pow measurable_const).const_mul (-b)).exp
  · have : MeasurableSet (Ioi (0 : ℝ)) := measurableSet_Ioi
    filter_upwards [ae_restrict_mem this] with x hx
    have h'x : 0 ≤ x := le_of_lt hx
    rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (exp_pos _).le]
    apply mul_le_mul_of_nonneg_right _ (exp_pos _).le
    simpa [abs_of_nonneg h'x] using abs_rpow_le_abs_rpow (-x) s

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

Goal state before:
pp: case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ y ^ 2 ≤ c ^ 2

Tactic:
sq_le_sq

Goal state after:
case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ |y| ≤ |c|
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c T : ℝ
hT : 0 ≤ T
vert_norm_bound :
  ∀ {T : ℝ},
    0 ≤ T →
      ∀ {c y : ℝ},
        |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))
⊢ ?m.420 * |c - 0| ≤ 2 * |c| * rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))

Tactic:
rw [sub_zero]

Goal state after:
b : ℂ
hb : 0 < b.re
c T : ℝ
hT : 0 ≤ T
vert_norm_bound :
  ∀ {T : ℝ},
    0 ≤ T →
      ∀ {c y : ℝ},
        |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))
⊢ ?m.420 * |c| ≤ 2 * |c| * rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

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

Tactic:
refine ⟨by fun_prop, ?_⟩
  rw [← hasFiniteIntegral_norm_iff]
  simp_rw [norm_cexp_neg_mul_sq_add_mul_I' hb.ne', neg_sub _ (c ^ 2 * _),
    sub_eq_add_neg _ (b.re * _), Real.exp_add]
  suffices Integrable fun x : ℝ => exp (-(b.re * x ^ 2)) by
    exact (Integrable.comp_sub_right this (b.im * c / b.re)).hasFiniteIntegral.const_mul _
  simp_rw [← neg_mul]
  apply integrable_exp_neg_mul_sq hb

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

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
m : V
⊢ ∂_{m} (𝓕 f) = 𝓕 (-(2 * ↑π * Complex.I) • (smulLeftCLM E fun x ↦ inner ℝ x m) f)

Tactic:
change SchwartzMap.evalCLM ℝ V E m (fderivCLM ℝ V E (𝓕 f)) = _
  rw [fderivCLM_fourier_eq, ← fourier_evalCLM_eq]
  congr
  ext
  have : (inner ℝ · m).HasTemperateGrowth := ((innerSL ℝ).flip m).hasTemperateGrowth
  simp [this, innerSL_apply_apply ℝ]

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℂ F
inst✝² : CompleteSpace F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : FiniteDimensional ℝ E
f : ↥(Lp F 2 volume)
⊢ 𝓕⁻ (toTemperedDistribution (𝓕 (𝓕⁻ f))) = 𝓕⁻ (𝓕 (toTemperedDistribution (𝓕⁻ f)))

Tactic:
fourier_toTemperedDistribution_eq

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℂ F
inst✝² : CompleteSpace F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : FiniteDimensional ℝ E
f : ↥(Lp F 2 volume)
⊢ 𝓕⁻ (toTemperedDistribution (𝓕 (𝓕⁻ f))) = 𝓕⁻ (toTemperedDistribution (𝓕 (𝓕⁻ f)))