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
u : ℕ → E
sb : Bornology.IsBounded s
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n ↦ {u n} + s)
h's : MeasurableSet s
⊢ μ s = 0

Tactic:
by
  by_contra h
  apply lt_irrefl ∞
  calc
    ∞ = ∑' _ : ℕ, μ s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm
    _ = ∑' n : ℕ, μ ({u n} + s) := by
      congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]
    _ = μ (⋃ n, {u n} + s) := Eq.symm <| measure_iUnion hs fun n => by
      simpa only [image_add_left, singleton_add] using! measurable_id.const_add _ h's
    _ = μ (range u + s) := by rw [← iUnion_add, iUnion_singleton_eq_range]
    _ < ∞ := (hu.add sb).measure_lt_top

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
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n ↦ {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), μ (s ∩ closedBall 0 R) = 0
⊢ ∑' (n : ℕ), μ (s ∩ closedBall 0 ↑n) = 0

Tactic:
by simp only [H, tsum_zero]

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

Goal state before:
pp: E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
⊢ μ.real (closedBall x r) = μ.real (closedBall 0 r)

Tactic:
by
  simp [measureReal_def, addHaar_closedBall_center]

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

Goal state before:
pp: case neg
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
r : ℝ
h : 0 ≤ r
⊢ μ (closedBall x r) = μ (ball x r)

Tactic:
addHaar_closedBall μ x h,

Goal state after:
case neg
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
r : ℝ
h : 0 ≤ r
⊢ ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) = μ (ball x r)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case pos
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
M : ∀ᵐ (x : ℝ) ∂μ, x ∉ s \ u
M' : ∀ᵐ (x : ℝ) ∂μ, ∀ i ∈ A, x ∈ s ∩ T i → p x
x : ℝ
hx : x ∉ s \ u
h'x : ∀ i ∈ A, x ∈ s ∩ T i → p x
xs : x ∈ s
Hx : x ∈ ⋃ i ∈ A, T i
⊢ p x

Tactic:
obtain ⟨p, pA, xp⟩ : ∃ p : ↥s × ↥s, p ∈ A ∧ x ∈ T p := by
      simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx

Goal state after:
case pos
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p✝ : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p✝ x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
M : ∀ᵐ (x : ℝ) ∂μ, x ∉ s \ u
M' : ∀ᵐ (x : ℝ) ∂μ, ∀ i ∈ A, x ∈ s ∩ T i → p✝ x
x : ℝ
hx : x ∉ s \ u
h'x : ∀ i ∈ A, x ∈ s ∩ T i → p✝ x
xs : x ∈ s
Hx : x ∈ ⋃ i ∈ A, T i
p : ↑s × ↑s
pA : p ∈ A
xp : x ∈ T p
⊢ p✝ x
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
r : ℝ
hr : r ∈ Ioi 0
x : E
⊢ ↑(2 ^ finrank ℝ E).toNNReal * μ (closedBall 0 r) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall 0 r)

Tactic:
Real.toNNReal_pow zero_le_two

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
r : ℝ
hr : r ∈ Ioi 0
x : E
⊢ ↑(Real.toNNReal 2 ^ finrank ℝ E) * μ (closedBall 0 r) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall 0 r)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case h'u
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
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := ⋯
ht' : t' = R⁻¹ • t
u' : Set E := ⋯
hu' : u' = R⁻¹ • u
⊢ μ u' ≠ 0

pp: case t_bound
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
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := ⋯
ht' : t' = R⁻¹ • t
u' : Set E := ⋯
hu' : u' = R⁻¹ • u
⊢ t' ⊆ closedBall 0 1

Tactic:
· simp only [u', h'u, (pow_pos Rpos _).ne', abs_nonpos_iff, addHaar_smul, not_false_iff,
        ENNReal.ofReal_eq_zero, inv_eq_zero, inv_pow, Ne, or_self_iff, mul_eq_zero]

Goal state after:
case t_bound
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
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := ⋯
ht' : t' = R⁻¹ • t
u' : Set E := ⋯
hu' : u' = R⁻¹ • u
⊢ t' ⊆ closedBall 0 1
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
p : Box ι → Prop
⊢ (π.filter fun J ↦ ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion

Tactic:
simp only [Prepartition.iUnion]

Goal state after:
ι : Type u_1
I : Box ι
π : Prepartition I
p : Box ι → Prop
⊢ ⋃ J ∈ π.filter fun J ↦ ¬p J, ↑J = (⋃ J ∈ π, ↑J) \ ⋃ J ∈ π.filter p, ↑J
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : OrthonormalBasis ι 𝕜 E
i : ι
⊢ b.repr (b i) = EuclideanSpace.single i 1

Tactic:
rw [← b.repr_symm_single i, LinearIsometryEquiv.apply_symm_apply]

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

Goal state before:
pp: case refine_1
K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
this✝¹ : Module.Finite ℤ ↥L
this✝ : Free ℤ ↥L
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := (Free.chooseBasis ℤ ↥L).indexEquiv b
this : Fintype ι
⊢ ⊤ ≤ span K ↑L

Tactic:
top_le_iff,

Goal state after:
case refine_1
K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
this✝¹ : Module.Finite ℤ ↥L
this✝ : Free ℤ ↥L
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := (Free.chooseBasis ℤ ↥L).indexEquiv b
this : Fintype ι
⊢ span K ↑L = ⊤
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
⊢ ∀ᶠ (t : Finset (ι × ℝ)) in atTop, π ⊓ splitMany I t = (splitMany I t).filter fun J ↦ ↑J ⊆ π.iUnion

Tactic:
refine (eventually_not_disjoint_imp_le_of_mem_splitMany π.boxes).mono fun t ht => ?_

Goal state after:
ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
t : Finset (ι × ℝ)
ht : ∀ (I_1 J : Box ι), J ∈ π.boxes → ∀ J' ∈ splitMany I_1 t, ¬Disjoint ↑J ↑J' → J' ≤ J
⊢ π ⊓ splitMany I t = (splitMany I t).filter fun J ↦ ↑J ⊆ π.iUnion
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁶ : RCLike 𝕜
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : Fintype ι
E' : Type u_7
inst✝² : Fintype ι'
inst✝¹ : NormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
b : OrthonormalBasis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 E'
e : ι ≃ ι'
i : ι
⊢ (b.equiv b' e) (b i) = b' (e i)

Tactic:
by
  classical
  simp only [OrthonormalBasis.equiv, LinearIsometryEquiv.trans_apply, OrthonormalBasis.repr_self]
  refine DFunLike.congr rfl ?_
  ext j
  simp

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := π.filter fun J ↦ ¬p J
this : Disjoint π₁.iUnion π₂.iUnion
⊢ (π₁.disjUnion π₂.toPrepartition this).distortion ≤ c

Tactic:
· have : (π.filter fun J => ¬p J).distortion ≤ c := (distortion_filter_le _ _).trans (hπ.3 hD)
    simpa [hc]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
⊢ Tendsto (⇑(embedBox I J h)) (l.toFilterDistortioniUnion I c ⊤)
    (⨆ c, l.toFilterDistortioniUnion J c (Prepartition.single J I h))

Tactic:
set π₀ := Prepartition.single J I h

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
⊢ Tendsto (⇑(embedBox I J h)) (l.toFilterDistortioniUnion I c ⊤) (⨆ c, l.toFilterDistortioniUnion J c π₀)
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝⁶ : RCLike 𝕜
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f : E ≃ₗᵢ[𝕜] G
b : OrthonormalBasis ι 𝕜 E
b' : OrthonormalBasis ι 𝕜 G
⊢ (LinearMap.toMatrix b.toBasis b'.toBasis) ↑f.toLinearEquiv ∈ Matrix.unitaryGroup ι 𝕜

Tactic:
simp [LinearMap.toMatrix_eq_basisToMatrix, ← coe_map, toMatrix_orthonormalBasis_mem_unitary]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: 𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
⊢ ∃ u b, v ⊆ ↑u ∧ ⇑b = Subtype.val

Tactic:
obtain ⟨u₀, hu₀s, hu₀, hu₀_max⟩ := exists_maximal_orthonormal hv
  rw [maximal_orthonormal_iff_orthogonalComplement_eq_bot hu₀] at hu₀_max
  have hu₀_finite : u₀.Finite := hu₀.linearIndependent.setFinite
  let u : Finset E := hu₀_finite.toFinset
  let fu : ↥u ≃ ↥u₀ := hu₀_finite.subtypeEquivToFinset.symm
  have hu : Orthonormal 𝕜 ((↑) : u → E) := by simpa using! hu₀.comp _ fu.injective
  refine ⟨u, OrthonormalBasis.mkOfOrthogonalEqBot hu ?_, ?_, ?_⟩
  · simpa [u] using! hu₀_max
  · simpa [u] using! hu₀s
  · simp

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
⊢ ∑ i ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset, F i / ↑n ^ card ι =
    ∑ J ∈ (prepartition n B).boxes, (volume.toBoxAdditive.toSMul J) (s.indicator F ((prepartition n B).tag J))

Tactic:
simp_rw [Set.indicator_apply, apply_ite, BoxAdditiveMap.toSMul_apply, Measure.toBoxAdditive_apply,
    smul_eq_mul, mul_zero, Finset.sum_ite, Finset.sum_const_zero, add_zero]

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
⊢ ∑ i ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset, F i / ↑n ^ card ι =
    ∑ x ∈ (prepartition n B).boxes with (prepartition n B).tag x ∈ s, volume.real ↑x * F ((prepartition n B).tag x)
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
⊢ i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j)

Tactic:
hij

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j)
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: case right
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
i j : ι
⊢ i ≠ j → (fun i j ↦ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0) i j

Tactic:
hij

Goal state after:
case right
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
i j : ι
hij : i ≠ j
⊢ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_4
⊢ reindex R M (Equiv.refl ι) = Equiv.refl (Orientation R M ι)

Tactic:
by
  rw [Orientation.reindex, AlternatingMap.domDomCongrₗ_refl, Module.Ray.map_refl]

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case h
R : Type u_1
inst✝⁵ : CommSemiring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : M [⋀^ι]→ₗ[R] R
hg : g ≠ 0
i : ι → M
⊢ (g.compLinearMap ↑f.symm) i = g i

Tactic:
rw [AlternatingMap.compLinearMap_apply]

Goal state after:
case h
R : Type u_1
inst✝⁵ : CommSemiring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : M [⋀^ι]→ₗ[R] R
hg : g ≠ 0
i : ι → M
⊢ (g fun i_1 ↦ ↑f.symm (i i_1)) = g i
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (μ ↑I < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), (μ.restrict ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c ((fun x ↦ r) c) π
x✝ : π.IsPartition
n : ℕ
m : ℝ≥0∞ := μ (π.filter fun J ↦ N (π.tag J) = n).iUnion
⊢ m < ↑(δ n) / ↑n

Tactic:
simp only [Measure.restrict_apply (hUo _).measurableSet] at hμU

Goal state after:
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (μ ↑I < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c ((fun x ↦ r) c) π
x✝ : π.IsPartition
n : ℕ
m : ℝ≥0∞ := μ (π.filter fun J ↦ N (π.tag J) = n).iUnion
hμU : ∀ (n : ℕ), μ (U n ∩ ↑I) < ↑(δ n) / ↑n
⊢ m < ↑(δ n) / ↑n
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case h
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
⊢ SameRay R x (AlternatingMap.constLinearEquivOfIsEmpty 1) ∨ SameRay R x (-AlternatingMap.constLinearEquivOfIsEmpty 1)

Tactic:
sameRay_or_sameRay_neg_iff_not_linearIndependent

Goal state after:
case h
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
⊢ ¬LinearIndependent R ![x, AlternatingMap.constLinearEquivOfIsEmpty 1]
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_2
ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Fintype ι
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
⊢ Tendsto (fun b ↦ ↑(Nat.card ↑(s ∩ (↑⌈b⌉₊)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) / b ^ card ι) atTop
    (𝓝 (volume.real s * 1 ^ card ι))

Tactic:
refine Tendsto.congr' (tendsto_card_div_pow₆ s) (Tendsto.mul ?_ (Tendsto.pow ?_ _))

Goal state after:
case refine_2.refine_1
ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Fintype ι
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
⊢ Tendsto (fun x ↦ ↑(Nat.card ↑(s ∩ (↑⌈x⌉₊)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) / ↑⌈x⌉₊ ^ card ι) atTop
    (𝓝 (volume.real s))

case refine_2.refine_2
ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Fintype ι
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
⊢ Tendsto (fun x ↦ ↑⌈x⌉₊ / x) atTop (𝓝 1)
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst✝ : CompleteSpace F
h : Integrable I l f vol
π₀ : Prepartition I
ε : ℝ
ε0 : 0 < ε
⊢ ∀ (x : TaggedPrepartition I),
    (∃ c, l.MemBaseSet I c (h.convergenceR ε c) x ∧ x.iUnion = π₀.iUnion) →
      integralSum f vol x ∈ closedBall (∑ J ∈ π₀.boxes, integral J l f vol) ε

Tactic:
rintro π ⟨c, hc, hU⟩

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst✝ : CompleteSpace F
h : Integrable I l f vol
π₀ : Prepartition I
ε : ℝ
ε0 : 0 < ε
π : TaggedPrepartition I
c : ℝ≥0
hc : l.MemBaseSet I c (h.convergenceR ε c) π
hU : π.iUnion = π₀.iUnion
⊢ integralSum f vol π ∈ closedBall (∑ J ∈ π₀.boxes, integral J l f vol) ε
Module: Mathlib.Analysis.SumOverResidueClass

Goal state before:
pp: R : Type u_1
inst✝² : AddCommGroup R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalAddGroup R
m : ℕ
hm : NeZero m
k : ℕ
f : ℕ → R
⊢ Summable ({n | ↑n = ↑k}.indicator f) ↔ Summable fun n ↦ f (m * n + k)

Tactic:
trans Summable ({n : ℕ | (n : ZMod m) = k ∧ k ≤ n}.indicator f)

Goal state after:
R : Type u_1
inst✝² : AddCommGroup R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalAddGroup R
m : ℕ
hm : NeZero m
k : ℕ
f : ℕ → R
⊢ Summable ({n | ↑n = ↑k}.indicator f) ↔ Summable ({n | ↑n = ↑k ∧ k ≤ n}.indicator f)

R : Type u_1
inst✝² : AddCommGroup R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalAddGroup R
m : ℕ
hm : NeZero m
k : ℕ
f : ℕ → R
⊢ Summable ({n | ↑n = ↑k ∧ k ≤ n}.indicator f) ↔ Summable fun n ↦ f (m * n + k)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
n : ℕ
⊢ ∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k) ≤ f 1 + 2 • ∑ k ∈ Ico 2 (2 ^ n + 1), f k

Tactic:
← sum_condensed_le' hf n

Goal state after:
M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
n : ℕ
⊢ ∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k) ≤ f 1 + 2 • ∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))
Module: Mathlib.Algebra.MonoidAlgebra.ToDirectSum

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝² : DecidableEq ι
inst✝¹ : Semiring M
inst✝ : (m : M) → Decidable (m ≠ 0)
i : ι
m : M
⊢ ((of (fun x ↦ M) i) m).toAddMonoidAlgebra.coeff = (AddMonoidAlgebra.single i m).coeff

Tactic:
exact DFinsupp.toFinsupp_single i m

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

Goal state before:
pp: a b c : ℕ
⊢ multinomial univ ![a, b, c] = (a + b + c)! / (a ! * b ! * c !)

Tactic:
rw [multinomial, Fin.sum_univ_three, Fin.prod_univ_three]
  rfl

Goal state after:
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp.g
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : X i * q = (p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1)) * r
this : X i ∣ p ∨ X i ∣ r
hip : p.modMonomial (Finsupp.single i 1) = 0
⊢ p.divMonomial (Finsupp.single i 1) ∣ q

Tactic:
hip,

Goal state after:
case mp.g
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : X i * q = (0 + X i * p.divMonomial (Finsupp.single i 1)) * r
this : X i ∣ p ∨ X i ∣ r
hip : p.modMonomial (Finsupp.single i 1) = 0
⊢ p.divMonomial (Finsupp.single i 1) ∣ q
Module: Mathlib.Data.DFinsupp.WellFounded

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = (x₁.piecewise x₂ p) j
hs : s i (x i) ((x₁.piecewise x₂ p) i)
⊢ ∃ a',
    InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧
      (fun x ↦ x.2.1.piecewise x.2.2 x.1) a' = x

Tactic:
piecewise_apply

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hs : s i (x i) (if i ∈ p then x₁ i else x₂ i)
⊢ ∃ a',
    InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧
      (fun x ↦ x.2.1.piecewise x.2.2 x.1) a' = x