mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
t_bound : t ⊆ closedBall 0 1
r : ℝ
rpos : 0 < r
⊢ μ (s ∩ ({x} +ᵥ r • closedBall 0 1)) / μ (closedBall x r) ≤ μ (s ∩ closedBall x r) / μ (closedBall x r)

Tactic:
singleton_vadd,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
t_bound : t ⊆ closedBall 0 1
r : ℝ
rpos : 0 < r
⊢ μ (s ∩ (x +ᵥ r • closedBall 0 1)) / μ (closedBall x r) ≤ μ (s ∩ closedBall x r) / μ (closedBall x r)
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: case pos
ι : Type u_1
I : Box ι
s : Set ι
y : ι → ℝ
i : ι
hs : i ∈ s
⊢ y i ∈ Ioc ((I.lower i + I.upper i) / 2) (I.upper i) ↔
    y i ∈ Ioc (I.lower i) (I.upper i) ∧ ((I.lower i + I.upper i) / 2 < y i ↔ i ∈ s)

Tactic:
simp only [hs, iff_true, iff_false, not_lt]

Goal state after:
case pos
ι : Type u_1
I : Box ι
s : Set ι
y : ι → ℝ
i : ι
hs : i ∈ s
⊢ y i ∈ Ioc ((I.lower i + I.upper i) / 2) (I.upper i) ↔
    y i ∈ Ioc (I.lower i) (I.upper i) ∧ (I.lower i + I.upper i) / 2 < y i
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: case neg
ι : Type u_1
I : Box ι
s : Set ι
y : ι → ℝ
i : ι
hs : i ∉ s
⊢ y i ∈ Ioc (I.lower i) ((I.lower i + I.upper i) / 2) ↔
    y i ∈ Ioc (I.lower i) (I.upper i) ∧ ((I.lower i + I.upper i) / 2 < y i ↔ i ∈ s)

Tactic:
simp only [hs, iff_true, iff_false, not_lt]

Goal state after:
case neg
ι : Type u_1
I : Box ι
s : Set ι
y : ι → ℝ
i : ι
hs : i ∉ s
⊢ y i ∈ Ioc (I.lower i) ((I.lower i + I.upper i) / 2) ↔
    y i ∈ Ioc (I.lower i) (I.upper i) ∧ y i ≤ (I.lower i + I.upper i) / 2
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
inst✝ : Fintype ι
x : ι → ℝ
⊢ #({J ∈ π.boxes | x ∈ Box.Icc J}) ≤ 2 ^ Fintype.card ι

Tactic:
rw [← Fintype.card_set]

Goal state after:
ι : Type u_1
I : Box ι
π : Prepartition I
inst✝ : Fintype ι
x : ι → ℝ
⊢ #({J ∈ π.boxes | x ∈ Box.Icc J}) ≤ Fintype.card (Set ι)
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: case mpr
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
πi' : Box ι → (J : Box ι) → Prepartition J
J : Box ι
⊢ (∃ J', (∃ J'_1 ∈ π, J' ∈ πi J'_1) ∧ J ∈ πi' (π.biUnionIndex πi J') J') → ∃ J' ∈ π, ∃ J'_1 ∈ πi J', J ∈ πi' J' J'_1

Tactic:
· rintro ⟨J₁, ⟨J₂, hJ₂, hJ₁⟩, hJ⟩
    refine ⟨J₂, hJ₂, J₁, hJ₁, ?_⟩
    rwa [π.biUnionIndex_of_mem hJ₂ hJ₁] at hJ

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ∞
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
n : ℕ
npos : 0 < n
hn : μ (t \ closedBall 0 ↑n) < ε / 2 * μ t
L : Tendsto (fun r ↦ μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
r : ℝ
hr : μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t) < ε / 2
rpos : 0 < r
⊢ μ (s ∩ ({x} + r • t)) = μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n ∪ t \ closedBall 0 ↑n)))

Tactic:
inter_union_sdiff

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ∞
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
n : ℕ
npos : 0 < n
hn : μ (t \ closedBall 0 ↑n) < ε / 2 * μ t
L : Tendsto (fun r ↦ μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
r : ℝ
hr : μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t) < ε / 2
rpos : 0 < r
⊢ μ (s ∩ ({x} + r • t)) = μ (s ∩ ({x} + r • t))
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: case inr
ι : Type u_1
I : Box ι
i : ι
x : ℝ
h✝ : I.lower i < x
⊢ I.splitLower i x ≠ I.splitUpper i x

Tactic:
refine (disjoint_splitLower_splitUpper I i x).ne ?_
    rwa [Ne, splitLower_eq_bot, not_le]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
⊢ finrank ℤ ↥L = Fintype.card ι

Tactic:
ZLattice.rank ℝ,

Goal state after:
ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
⊢ finrank ℝ (ι → ℝ) = Fintype.card ι
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B x✝ : Box ι
hI₂ : x✝ ∈ ↑(Finset.image (fun ν ↦ box n ν) (admissibleIndex n B))
w✝ : ι → ℤ
left✝ : w✝ ∈ admissibleIndex n B
hI₁ : box n w✝ ∈ ↑(Finset.image (fun ν ↦ box n ν) (admissibleIndex n B))
h : box n w✝ ≠ x✝
⊢ Function.onFun Disjoint Box.toSet (box n w✝) x✝

Tactic:
obtain ⟨_, _, rfl⟩ := Finset.mem_image.mp hI₂

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B : Box ι
w✝¹ : ι → ℤ
left✝¹ : w✝¹ ∈ admissibleIndex n B
hI₁ : box n w✝¹ ∈ ↑(Finset.image (fun ν ↦ box n ν) (admissibleIndex n B))
w✝ : ι → ℤ
left✝ : w✝ ∈ admissibleIndex n B
hI₂ : box n w✝ ∈ ↑(Finset.image (fun ν ↦ box n ν) (admissibleIndex n B))
h : box n w✝¹ ≠ box n w✝
⊢ Function.onFun Disjoint Box.toSet (box n w✝¹) (box n w✝)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Finite ι
v : ι → ℝ
this : Fintype ι
⊢ v ∈ (↑n)⁻¹ • span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)) ↔ ∀ (i : ι), ↑n * v i ∈ Set.range ⇑(algebraMap ℤ ℝ)

Tactic:
ZSpan.smul _ (inv_ne_zero (NeZero.ne _)),

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Finite ι
v : ι → ℝ
this : Fintype ι
⊢ v ∈ span ℤ (Set.range ⇑((Pi.basisFun ℝ ι).isUnitSMul ⋯)) ↔ ∀ (i : ι), ↑n * v i ∈ Set.range ⇑(algebraMap ℤ ℝ)
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case neg
R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : LinearOrder R
inst✝⁵ : IsStrictOrderedRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
h : ¬e.orientation = x
⊢ (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))) i = e i ∨
    (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))) i = -e i

Tactic:
by_cases hi : i = Classical.arbitrary ι <;> simp [unitsSMul_apply, hi]

Goal state after:
Module: Mathlib.Analysis.SumOverResidueClass

Goal state before:
pp: R : Type u_1
inst✝⁵ : AddCommGroup R
inst✝⁴ : UniformSpace R
inst✝³ : IsUniformAddGroup R
inst✝² : CompleteSpace R
inst✝¹ : T0Space R
f : ℕ → R
hf : Summable f
N : ℕ
inst✝ : NeZero N
⊢ ∑ b, ∑' (c : ℕ), f (N.residueClassesEquiv.symm (b, c)) = ∑ j, ∑' (m : ℕ), f (j.val + N * m)

pp: R : Type u_1
inst✝⁵ : AddCommGroup R
inst✝⁴ : UniformSpace R
inst✝³ : IsUniformAddGroup R
inst✝² : CompleteSpace R
inst✝¹ : T0Space R
f : ℕ → R
hf : Summable f
N : ℕ
inst✝ : NeZero N
⊢ Summable fun c ↦ f (N.residueClassesEquiv.symm c)

Tactic:
residueClassesEquiv,

Goal state after:
R : Type u_1
inst✝⁵ : AddCommGroup R
inst✝⁴ : UniformSpace R
inst✝³ : IsUniformAddGroup R
inst✝² : CompleteSpace R
inst✝¹ : T0Space R
f : ℕ → R
hf : Summable f
N : ℕ
inst✝ : NeZero N
⊢ ∑ b,
      ∑' (c : ℕ),
        f
          ({ toFun := fun n ↦ (↑n, n / N), invFun := fun p ↦ p.1.val + N * p.2, left_inv := ⋯, right_inv := ⋯ }.symm
            (b, c)) =
    ∑ j, ∑' (m : ℕ), f (j.val + N * m)

R : Type u_1
inst✝⁵ : AddCommGroup R
inst✝⁴ : UniformSpace R
inst✝³ : IsUniformAddGroup R
inst✝² : CompleteSpace R
inst✝¹ : T0Space R
f : ℕ → R
hf : Summable f
N : ℕ
inst✝ : NeZero N
⊢ Summable fun c ↦ f (N.residueClassesEquiv.symm c)
Module: Mathlib.Analysis.SumOverResidueClass

Goal state before:
pp: R : Type u_1
inst✝⁵ : AddCommGroup R
inst✝⁴ : UniformSpace R
inst✝³ : IsUniformAddGroup R
inst✝² : CompleteSpace R
inst✝¹ : T0Space R
f : ℕ → R
hf : Summable f
N : ℕ
inst✝ : NeZero N
⊢ ∑' (n : ℕ), f n = ∑ j, ∑' (m : ℕ), f (j.val + N * m)

Tactic:
rw [← (residueClassesEquiv N).symm.tsum_eq f, Summable.tsum_prod, tsum_fintype,
    residueClassesEquiv, Equiv.coe_fn_symm_mk]
  exact hf.comp_injective (residueClassesEquiv N).symm.injective

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁
b₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂
⊢ covolume L₁ volume / covolume L₂ volume = ↑(L₁.toAddSubgroup.relIndex L₂.toAddSubgroup)

Tactic:
rw [AddSubgroup.relIndex_eq_natAbs_det L₁.toAddSubgroup L₂.toAddSubgroup h b₁ b₂,
    Nat.cast_natAbs, Int.cast_abs]

Goal state after:
ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁
b₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂
⊢ covolume L₁ volume / covolume L₂ volume = |↑(b₂.det fun i ↦ ⟨↑(b₁ i), ⋯⟩)|
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case hx
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := ⋯
μ' : Measure (ι → ℝ) := ⋯
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := ⋯
t₂ : Box ι → ι → ℝ := ⋯
B : Finset (Box ι) := ⋯
B' : Finset (Box ι) := ⋯
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
hJ : J ∈ B ∧ (J ∈ B → ¬↑J ⊆ U)
x : ι → ℝ
xJ : x ∈ J
xnU : x ∉ U
hx : x ∈ Box.Icc I \ U
⊢ f (t₁ J) ∈ f '' (Metric.ball x r ∩ Box.Icc I)

Tactic:
refine Set.mem_image_of_mem f ⟨?_, tag_mem_Icc _ J⟩

Goal state after:
case hx
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := fun J ↦ (π₁.infPrepartition π₂.toPrepartition).tag J
t₂ : Box ι → ι → ℝ := fun J ↦ (π₂.infPrepartition π₁.toPrepartition).tag J
B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes
B' : Finset (Box ι) := {J ∈ B | ↑J ⊆ U}
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
hJ : J ∈ B ∧ (J ∈ B → ¬↑J ⊆ U)
x : ι → ℝ
xJ : x ∈ J
xnU : x ∉ U
hx : x ∈ Box.Icc I \ U
⊢ t₁ J ∈ Metric.ball x r
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case hy
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := ⋯
μ' : Measure (ι → ℝ) := ⋯
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := ⋯
t₂ : Box ι → ι → ℝ := ⋯
B : Finset (Box ι) := ⋯
B' : Finset (Box ι) := ⋯
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
hJ : J ∈ B ∧ (J ∈ B → ¬↑J ⊆ U)
x : ι → ℝ
xJ : x ∈ J
xnU : x ∉ U
hx : x ∈ Box.Icc I \ U
⊢ f (t₂ J) ∈ f '' (Metric.ball x r ∩ Box.Icc I)

Tactic:
refine Set.mem_image_of_mem f ⟨?_, tag_mem_Icc _ J⟩

Goal state after:
case hy
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := fun J ↦ (π₁.infPrepartition π₂.toPrepartition).tag J
t₂ : Box ι → ι → ℝ := fun J ↦ (π₂.infPrepartition π₁.toPrepartition).tag J
B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes
B' : Finset (Box ι) := {J ∈ B | ↑J ⊆ U}
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
hJ : J ∈ B ∧ (J ∈ B → ¬↑J ⊆ U)
x : ι → ℝ
xJ : x ∈ J
xnU : x ∉ U
hx : x ∈ Box.Icc I \ U
⊢ t₂ J ∈ Metric.ball x r
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: case mpr.ht
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : ℝ
t : Set D
hcompact : IsCompact t
h : tᶜ ⊆ {x | ‖f x‖ ≤ c}
⊢ Bornology.IsBounded (f '' tᶜ)

Tactic:
rw [isBounded_iff_forall_norm_le]

Goal state after:
case mpr.ht
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : ℝ
t : Set D
hcompact : IsCompact t
h : tᶜ ⊆ {x | ‖f x‖ ≤ c}
⊢ ∃ C, ∀ x ∈ f '' tᶜ, ‖x‖ ≤ C
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
⊢ Bornology.IsBounded (range f) ↔ f =O[cocompact D] 1

Tactic:
by
  constructor <;> intro h
  · rw [isBounded_iff_forall_norm_le] at h
    obtain ⟨c, hc⟩ := h
    simp only [Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff] at hc
    rw [isBigO_iff]
    use c
    apply Eventually.of_forall
    simpa using hc
  · simp_rw [isBigO_iff, Filter.Eventually, Filter.mem_cocompact] at h
    simp only [Pi.one_apply, norm_one, mul_one] at h
    obtain ⟨c, t, hcompact, h⟩ := h
    rw [← Set.image_union_image_compl_eq_range (s := t)]
    apply IsBounded.union
    · apply (IsCompact.image hcompact hf).isBounded
    · rw [isBounded_iff_forall_norm_le]
      refine ⟨c, fun x hx ↦ ?_⟩
      rw [Set.mem_image] at hx
      obtain ⟨y, hy, rfl⟩ := hx
      simpa using mem_of_mem_of_subset hy h

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ Tendsto (fun n ↦ ↑(Nat.card ↑(s ∩ (↑n)⁻¹ • ↑L)) / ↑n ^ card ι) atTop (𝓝 (volume.real s / covolume L volume))

Tactic:
convert! tendsto_card_div_pow'' b hs₁ hs₂ ?_

Goal state after:
ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ volume.real s / covolume L volume = volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
inst✝ : Nonempty ι
⊢ finrank ℤ ↥L = card ι

Tactic:
ZLattice.rank ℝ,

Goal state after:
ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
inst✝ : Nonempty ι
⊢ finrank ℝ (ι → ℝ) = card ι
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ Tendsto (fun n ↦ ↑(Nat.card ↑(s ∩ (↑n)⁻¹ • ↑L)) / ↑n ^ finrank ℝ E) atTop (𝓝 (volume.real s / covolume L volume))

Tactic:
convert! tendsto_card_div_pow'' b hs₁ hs₂ ?_

Goal state after:
case e'_3
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
x✝ : ℕ
⊢ finrank ℝ E = Fintype.card (Free.ChooseBasisIndex ℤ ↥L)

case e'_5
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume.real s / covolume L volume = volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case inr
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
r : ℝ
hr : r < -↑(finrank ℤ ↥L)
x : E
h✝ : Nontrivial ↥L
H : IsClosed ↑L
t : ↥L
⊢ t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x_1}ᶜ →
    t ∈ ⇑(Function.Embedding.subtype fun x ↦ x ∈ L) ⁻¹' (Metric.closedBall 0 (2 * ‖x‖) ∩ ↑L)

Tactic:
ht

Goal state after:
case inr
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
r : ℝ
hr : r < -↑(finrank ℤ ↥L)
x : E
h✝ : Nontrivial ↥L
H : IsClosed ↑L
t : ↥L
ht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x_1}ᶜ
⊢ t ∈ ⇑(Function.Embedding.subtype fun x ↦ x ∈ L) ⁻¹' (Metric.closedBall 0 (2 * ‖x‖) ∩ ↑L)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mpr.inr
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ q
⊢ X i ∣ p * q

Tactic:
exact dvd_mul_of_dvd_right h p

Goal state after:
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
⊢ t ∈ s → (f + fun i ↦ if i = a then p.1 else 0) t = f t

Tactic:
ht

Goal state after:
case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ (f + fun i ↦ if i = a then p.1 else 0) t = f t