mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
a b : V
⊢ G.Adj a b ↔ ∃ e, ↑e = s(a, b)

Tactic:
by
  simp only [mem_edgeSet, exists_prop, SetCoe.exists, exists_eq_right]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
s t : Set (Sym2 V)
⊢ fromEdgeSet (s ∩ t) = fromEdgeSet s ⊓ fromEdgeSet t

Tactic:
ext v w
  simp only [fromEdgeSet_adj, Set.mem_inter_iff, Ne, inf_adj]
  tauto

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
a b : V
h : ¬G.Adj a b
hn : a ≠ b
⊢ ∀ x ∈ G.incidenceSet a, x ∉ G.incidenceSet b

Tactic:
intro u ha hb

Goal state after:
V : Type u
G : SimpleGraph V
a b : V
h : ¬G.Adj a b
hn : a ≠ b
u : Sym2 V
ha : u ∈ G.incidenceSet a
hb : u ∈ G.incidenceSet b
⊢ False
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
⊢ s(v, w) ∈ G.incidenceSet v ↔ G.Adj v w

Tactic:
simp [incidenceSet]

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder 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
v : V
hv : v ∈ s.direction
⊢ s.SSameSide x (v +ᵥ y) ↔ s.SSameSide x y

Tactic:
by
  rw [sSameSide_comm, sSameSide_vadd_left_iff hv, sSameSide_comm]

Goal state after:
Module: Mathlib.Analysis.Convex.StoneSeparation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q u v x y z : E
hz : z ∈ segment 𝕜 x y
hu : u ∈ segment 𝕜 x p
hv : v ∈ segment 𝕜 y q
⊢ ¬Disjoint (segment 𝕜 u v) ((convexHull 𝕜) {p, q, z})

Tactic:
rw [not_disjoint_iff]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q u v x y z : E
hz : z ∈ segment 𝕜 x y
hu : u ∈ segment 𝕜 x p
hv : v ∈ segment 𝕜 y q
⊢ ∃ x ∈ segment 𝕜 u v, x ∈ (convexHull 𝕜) {p, q, z}
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp.inl
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 p₁ : P
h : p₁ ∈ s
p₁' : P
hp₁' : p₁' ∈ s
p₂' : P
hp₂' : p₂' ∈ s
h0 : x -ᵥ p₁' = 0
⊢ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
rw [vsub_eq_zero_iff_eq] at h0
      rw [h0]
      exact Or.inl hp₁'

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp.inl
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 p₁ : P
h : p₁ ∈ s
p₁' : P
hp₁' : p₁' ∈ s
p₂' : P
hp₂' : p₂' ∈ s
h0 : x -ᵥ p₁' = 0
⊢ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

Tactic:
rw [vsub_eq_zero_iff_eq] at h0
      rw [h0]
      exact Or.inl hp₁'

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
hs : SignType.sign (w₁ i) = SignType.sign (w₂ i)
h0 : w₁ i ≠ 0
h0' : w₂ i ≠ 0
j : Fin (n + 1)
hj : j ≠ i
hj' : s.points j ∈ affineSpan R (Set.range (s.faceOpposite i).points)
w₃ : Fin (n + 1) → R := w₂ - w₂ i • (w₁ i)⁻¹ • (w₁ - Pi.single j 1)
hw₃1 : ∑ k, w₃ k = 1
hw₃i : w₃ i = 0
h : 0 < w₁ i
⊢ 0 ≤ w₂ i * (w₁ i)⁻¹

Tactic:
· rw [sign_pos h, eq_comm, sign_eq_one_iff] at hs
    positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case inr.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
h :
  (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)
h0 : w₁ i ≠ 0
h0' : w₂ i ≠ 0
hs : SignType.sign (w₁ i) = -SignType.sign (w₂ i)
h' : 0 < w₁ i
⊢ False

Tactic:
sign_pos h',

Goal state after:
case inr.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
h :
  (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)
h0 : w₁ i ≠ 0
h0' : w₂ i ≠ 0
hs : 1 = -SignType.sign (w₂ i)
h' : 0 < w₁ i
⊢ False
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case refine_2.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
x✝ : SignType.sign (w₁ i) = -SignType.sign (w₂ i) ∧ w₁ i ≠ 0
hs : SignType.sign (w₁ i) = -SignType.sign (w₂ i)
h0 : w₁ i ≠ 0
h' : 0 < w₁ i
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).SOppSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)

Tactic:
sign_pos h',

Goal state after:
case refine_2.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
x✝ : SignType.sign (w₁ i) = -SignType.sign (w₂ i) ∧ w₁ i ≠ 0
hs : 1 = -SignType.sign (w₂ i)
h0 : w₁ i ≠ 0
h' : 0 < w₁ i
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).SOppSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: case pos
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)
s : ℝ
a : α
h : s ∈ Ioc 0 (f a)
⊢ (Ioc 0 (f a)).indicator (fun t ↦ ENNReal.ofReal (g t)) s =
    ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * (Ici s).indicator (fun x ↦ 1) (f a)

pp: case neg
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)
s : ℝ
a : α
h : s ∉ Ioc 0 (f a)
⊢ (Ioc 0 (f a)).indicator (fun t ↦ ENNReal.ofReal (g t)) s =
    ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * (Ici s).indicator (fun x ↦ 1) (f a)

Tactic:
· simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,
          mul_one]

Goal state after:
case neg
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)
s : ℝ
a : α
h : s ∉ Ioc 0 (f a)
⊢ (Ioc 0 (f a)).indicator (fun t ↦ ENNReal.ofReal (g t)) s =
    ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * (Ici s).indicator (fun x ↦ 1) (f a)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case neg.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
h0 : ¬w₁ i = 0
h0' : ¬w₂ i = 0
h : SignType.sign (w₁ i) = -SignType.sign (w₂ i)
h' : 0 < w₁ i
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)

Tactic:
sign_pos h',

Goal state after:
case neg.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
h0 : ¬w₁ i = 0
h0' : ¬w₂ i = 0
h : 1 = -SignType.sign (w₂ i)
h' : 0 < w₁ i
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w : Fin (n + 1) → R
hw : ∑ j, w j = 1
i : Fin (n + 1)
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide (s.points i)
      ((Finset.affineCombination R Finset.univ s.points) w) ↔
    w i ≤ 0

Tactic:
by
  rw [← Finset.univ.affineCombination_piSingle R s.points (Finset.mem_univ i),
    s.wOppSide_affineSpan_faceOpposite_iff (Fintype.sum_pi_single' _ _) hw, eq_comm,
    neg_eq_iff_eq_neg]
  simp [sign_eq_neg_one_iff, le_iff_eq_or_lt, or_comm]

Goal state after:
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: case left
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
x : α
i : Fin n
hx : (x, i) ∈ graph f
y : α
j : Fin n
hy : (y, j) ∈ graph f
h✝ : x < y
⊢ graph.proj ⟨(x, i), hx⟩ ≤ graph.proj ⟨(y, j), hy⟩

pp: case right
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
x : α
i : Fin n
hx : (x, i) ∈ graph f
j : Fin n
hy : (x, j) ∈ graph f
h✝ : i ≤ j
⊢ graph.proj ⟨(x, i), hx⟩ ≤ graph.proj ⟨(x, j), hy⟩

Tactic:
· exact le_of_lt ‹_›

Goal state after:
case right
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
x : α
i : Fin n
hx : (x, i) ∈ graph f
j : Fin n
hy : (x, j) ∈ graph f
h✝ : i ≤ j
⊢ graph.proj ⟨(x, i), hx⟩ ≤ graph.proj ⟨(x, j), hy⟩
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
⊢ ContinuousOn f (spectrum ℝ A)

Tactic:
by rw [continuousOn_iff_continuous_domRestrict]; fun_prop

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
v : E
i : ι
⊢ ↑(b.repr v) i = ⟪b i, v⟫

Tactic:
classical
  rw [← b.repr.inner_map_map (b i) v, b.repr_self, lp.inner_single_left]
  simp

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
h : Summable (fourierCoeff ⇑f)
sum_L2 : HasSum (fun i ↦ fourierCoeff (⇑f) i • fourierLp 2 i) ((toLp 2 haarAddCircle ℂ) f)
⊢ HasSum (fun i ↦ fourierCoeff (⇑f) i • fourier i) f

Tactic:
refine ContinuousMap.hasSum_of_hasSum_Lp (.of_norm ?_) sum_L2

Goal state after:
T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
h : Summable (fourierCoeff ⇑f)
sum_L2 : HasSum (fun i ↦ fourierCoeff (⇑f) i • fourierLp 2 i) ((toLp 2 haarAddCircle ℂ) f)
⊢ Summable fun a ↦ ‖fourierCoeff (⇑f) a • fourier a‖
Module: Mathlib.Analysis.Fourier.AddCircle

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

pp: case inr
T : ℝ
hT : Fact (0 < T)
n m : ℤ
hmn : m ≠ n
⊢ inner ℂ (fourierLp 2 m) (fourierLp 2 n) = Pi.single n 1 m

Tactic:
· rw [inner_self_eq_norm_sq_to_K, (orthonormal_fourier (hT := hT)).1 m]; simp

Goal state after:
case inr
T : ℝ
hT : Fact (0 < T)
n m : ℤ
hmn : m ≠ n
⊢ inner ℂ (fourierLp 2 m) (fourierLp 2 n) = Pi.single n 1 m
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

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 : 𝓢(E, F)
k : ℕ
d : ℝ
left✝ : 0 < d
hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d
x : E
hx : x ≠ 0
⊢ ‖f x‖ ≤ d * ‖‖x‖ ^ (-↑k)‖

Tactic:
rw [Real.norm_of_nonneg (by positivity), zpow_neg, ← div_eq_mul_inv, le_div_iff₀' (by positivity)]

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 : 𝓢(E, F)
k : ℕ
d : ℝ
left✝ : 0 < d
hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d
x : E
hx : x ≠ 0
⊢ ‖x‖ ^ ↑k * ‖f x‖ ≤ d
Module: Mathlib.MeasureTheory.Integral.PeakFunction

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

Tactic:
by
  suffices Tendsto (fun i : ι ↦ ∫ x in univ, φ i x • g x ∂μ) l (𝓝 a) by simpa
  exact tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto MeasurableSet.univ ht (x₀ := x₀)
    (subset_univ _) (by simpa [nhdsWithin_univ]) h't (by simpa)
    (by simpa [← compl_eq_univ_sdiff] using hlφ) hiφ
    (by simpa) (by simpa) (by simpa [nhdsWithin_univ])

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

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : ℕ∞ω
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ
h'f : AEStronglyMeasurable f μ
n : ℕ
hn : ↑n < N
w : W
I₁ : Integrable (fun v ↦ fourierPowSMulRight L f v n) μ
v : V
⊢ ‖((fun a ↦ ‖a‖) * fun x ↦ ‖fourierPowSMulRight L f x n‖) v‖ ≤ (2 * π * ‖L‖) ^ n * (‖v‖ ^ (n + 1) * ‖f v‖)

Tactic:
simp only [Pi.mul_apply, norm_mul, norm_norm]

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : ℕ∞ω
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ
h'f : AEStronglyMeasurable f μ
n : ℕ
hn : ↑n < N
w : W
I₁ : Integrable (fun v ↦ fourierPowSMulRight L f v n) μ
v : V
⊢ ‖v‖ * ‖fourierPowSMulRight L f v n‖ ≤ (2 * π * ‖L‖) ^ n * (‖v‖ ^ (n + 1) * ‖f v‖)
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : ℕ∞
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ
h'f : AEStronglyMeasurable f μ
n : ℕ
hn : ↑n ≤ N
⊢ iteratedFDeriv ℝ n (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) =
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) fun v ↦ fourierPowSMulRight L f v n

Tactic:
by
  ext w : 1
  exact ((hasFTaylorSeriesUpTo_fourierIntegral' L hf h'f).eq_iteratedFDeriv
    (mod_cast hn) w).symm

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

Goal state before:
pp: E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
H :
  ((fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fderiv ℝ (iteratedFDeriv ℝ n f)) w) (m 0)) (Fin.tail m) =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ x, -(L (m x)) w) • ∫ (v : V), 𝐞 (-(L v) w) • f v ∂μ
⊢ (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ (n + 1) f) w) m =
    (fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w (n + 1)) m

Tactic:
rw [fourierIntegral_continuousMultilinearMap_apply' (h'f _ hn)]

Goal state after:
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
H :
  ((fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fderiv ℝ (iteratedFDeriv ℝ n f)) w) (m 0)) (Fin.tail m) =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ x, -(L (m x)) w) • ∫ (v : V), 𝐞 (-(L v) w) • f v ∂μ
⊢ fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun x ↦ (iteratedFDeriv ℝ (n + 1) f x) m) w =
    (fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w (n + 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✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 F
a x✝¹ x✝ : E
⊢ edist x✝¹ x✝ ≤ ↑1 * edist ((fun x ↦ x - a) x✝¹) ((fun x ↦ x - a) x✝)

Tactic:
by simp [edist_dist, dist_eq_norm]

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

Goal state before:
pp: case succ
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
h'n : ↑n < N
⊢ ((fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fderiv ℝ (iteratedFDeriv ℝ n f)) w) (m 0)) (Fin.tail m) =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ x, -(L (m x)) w) • ∫ (v : V), 𝐞 (-(L v) w) • f v ∂μ

Tactic:
rw [fourierIntegral_fderiv _ (h'f n h'n.le)
      (hf.differentiable_iteratedFDeriv (mod_cast h'n)) J]

Goal state after:
case succ
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
h'n : ↑n < N
⊢ ((fourierSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f)) w)
        (m 0))
      (Fin.tail m) =
    (-(2 * ↑π * I)) ^ (n + 1) • (∏ x, -(L (m x)) w) • ∫ (v : V), 𝐞 (-(L v) w) • f v ∂μ
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: case h
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
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‖) μ
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
w : W
I : ∀ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), Integrable (fun v ↦ ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ
v : V
i : ℕ
hi : i ≤ n
j : ℕ
hj : j ≤ k
⊢ (j, i) ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1)

Tactic:
simp only [Finset.mem_product, Finset.mem_range_succ_iff]

Goal state after:
case h
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
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‖) μ
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
w : W
I : ∀ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), Integrable (fun v ↦ ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ
v : V
i : ℕ
hi : i ≤ n
j : ℕ
hj : j ≤ k
⊢ j ≤ k ∧ i ≤ n