mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: n : ℕ
I : Box (Fin (n + 1))
i : Fin (n + 1)
x : ℝ
hx : x ∈ Ioc (I.lower i) (I.upper i)
y : Fin n → ℝ
hy : y ∈ ↑(I.face i)
⊢ i.insertNth x y ∈ ↑I

Tactic:
simp_rw [mem_coe, mem_def, i.forall_iff_succAbove, Fin.insertNth_apply_same,
    Fin.insertNth_apply_succAbove]

Goal state after:
n : ℕ
I : Box (Fin (n + 1))
i : Fin (n + 1)
x : ℝ
hx : x ∈ Ioc (I.lower i) (I.upper i)
y : Fin n → ℝ
hy : y ∈ ↑(I.face i)
⊢ x ∈ Ioc (I.lower i) (I.upper i) ∧ ∀ (i_1 : Fin n), y i_1 ∈ Ioc (I.lower (i.succAbove i_1)) (I.upper (i.succAbove i_1))
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
boxes : Finset (WithBot (Box ι))
le_of_mem : ∀ J ∈ boxes, J ≤ ↑I
pairwise_disjoint : (↑boxes).Pairwise Disjoint
H : ∀ J ∈ π, ∃ J' ∈ boxes, ↑J ≤ J'
⊢ π ≤ ofWithBot boxes le_of_mem pairwise_disjoint

Tactic:
intro J hJ
  rcases H J hJ with ⟨J', J'mem, hle⟩
  lift J' to Box ι using ne_bot_of_le_ne_bot WithBot.coe_ne_bot hle
  exact ⟨J', mem_ofWithBot.2 J'mem, WithBot.coe_le_coe.1 hle⟩

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
v : E
i : ι
⊢ (b.repr v).ofLp i = ⟪b i, v⟫

Tactic:
by
  classical
    rw [← b.repr.inner_map_map (b i) v, b.repr_self i, EuclideanSpace.inner_single_left]
    simp only [one_mul, map_one]

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

Goal state before:
pp: case refine_1
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
r : (ι → ℝ) → ↑(Ioi 0)
J : Box ι
x✝ : J ≤ I
πi : (J' : Box ι) → TaggedPrepartition J'
hP : ∀ J' ∈ splitCenter J, (πi J').IsPartition
hHen : ∀ J' ∈ splitCenter J, (πi J').IsHenstock
hr : ∀ J' ∈ splitCenter J, (πi J').IsSubordinate r
a✝ : ∀ J' ∈ splitCenter J, (πi J').distortion = J'.distortion
n : Box ι → Box ι → ℕ
hn : ∀ J' ∈ splitCenter J, ∀ J ∈ πi J', ∀ (i : ι), J.upper i - J.lower i = (J'.upper i - J'.lower i) / 2 ^ n J' J
⊢ ∃ π,
    π.IsPartition ∧
      π.IsHenstock ∧
        π.IsSubordinate r ∧
          (∀ J_1 ∈ π, ∃ m, ∀ (i : ι), J_1.upper i - J_1.lower i = (J.upper i - J.lower i) / 2 ^ m) ∧
            π.distortion = J.distortion

Tactic:
have hP : ((splitCenter J).biUnionTagged πi).IsPartition :=
      (isPartition_splitCenter _).biUnionTagged hP

Goal state after:
case refine_1
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
r : (ι → ℝ) → ↑(Ioi 0)
J : Box ι
x✝ : J ≤ I
πi : (J' : Box ι) → TaggedPrepartition J'
hP✝ : ∀ J' ∈ splitCenter J, (πi J').IsPartition
hHen : ∀ J' ∈ splitCenter J, (πi J').IsHenstock
hr : ∀ J' ∈ splitCenter J, (πi J').IsSubordinate r
a✝ : ∀ J' ∈ splitCenter J, (πi J').distortion = J'.distortion
n : Box ι → Box ι → ℕ
hn : ∀ J' ∈ splitCenter J, ∀ J ∈ πi J', ∀ (i : ι), J.upper i - J.lower i = (J'.upper i - J'.lower i) / 2 ^ n J' J
hP : ((splitCenter J).biUnionTagged πi).IsPartition
⊢ ∃ π,
    π.IsPartition ∧
      π.IsHenstock ∧
        π.IsSubordinate r ∧
          (∀ J_1 ∈ π, ∃ m, ∀ (i : ι), J_1.upper i - J_1.lower i = (J.upper i - J.lower i) / 2 ^ m) ∧
            π.distortion = J.distortion
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
I : Box ι
π : Prepartition I
h : π.IsPartition
s : Finset (ι × ℝ)
hs : π ⊓ splitMany I s = (splitMany I s).filter fun J ↦ ↑J ⊆ π.iUnion
⊢ splitMany I s ≤ π

Tactic:
h.iUnion_eq,

Goal state after:
ι : Type u_1
inst✝ : Finite ι
I : Box ι
π : Prepartition I
h : π.IsPartition
s : Finset (ι × ℝ)
hs : π ⊓ splitMany I s = (splitMany I s).filter fun J ↦ ↑J ⊆ ↑I
⊢ splitMany I s ≤ π
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
h : π.IsPartition
⊢ ↑I \ π.iUnion = ∅

Tactic:
h.iUnion_eq,

Goal state after:
ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
h : π.IsPartition
⊢ ↑I \ ↑I = ∅
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι'
a : OrthonormalBasis ι 𝕜 E
b : OrthonormalBasis ι' 𝕜 E
⊢ a.toBasis.toMatrix ⇑b * (a.toBasis.toMatrix ⇑b)ᴴ = 1

Tactic:
classical
  rw [Matrix.mul_eq_one_comm_of_equiv (a.toBasis.indexEquiv b.toBasis),
    a.toMatrix_orthonormalBasis_conjTranspose_mul_self b]

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

Goal state before:
pp: ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
I₀ : WithTop (Box ι)
I✝ : Box ι
i : ι
R : Type ?u.17
inst✝¹ : Monoid R
inst✝ : DistribMulAction R M
r : R
f : ι →ᵇᵃ[I₀] M
I : Box ι
hI : ↑I ≤ I₀
π : Prepartition I
hπ : π.IsPartition
⊢ ∑ Ji ∈ π.boxes, (r • ⇑f) Ji = (r • ⇑f) I

Tactic:
simp only [Pi.smul_apply, ← smul_sum, sum_partition_boxes _ hI hπ]

Goal state after:
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
x : E
hx : x ∈ K
n : Set F
hn₁ : f ⁻¹' n ∈ 𝓝[D] x
hn₂ : ediam n < ε
r : ℝ≥0∞
r0 : r > 0
hr : eball x r ∩ D ⊆ f ⁻¹' n
⊢ ∀ (r' : ℝ), ENNReal.ofReal r' ≤ r → ediam (f '' (eball x (ENNReal.ofReal r') ∩ D)) ≤ ε

Tactic:
intro r' hr'

Goal state after:
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
x : E
hx : x ∈ K
n : Set F
hn₁ : f ⁻¹' n ∈ 𝓝[D] x
hn₂ : ediam n < ε
r : ℝ≥0∞
r0 : r > 0
hr : eball x r ∩ D ⊆ f ⁻¹' n
r' : ℝ
hr' : ENNReal.ofReal r' ≤ r
⊢ ediam (f '' (eball x (ENNReal.ofReal r') ∩ D)) ≤ ε
Module: Mathlib.Analysis.BoxIntegral.Partition.Additive

Goal state before:
pp: ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I : Box ι
i✝ : ι
G : Type u
inst✝ : AddCommGroup G
I₀ : Box (Fin (n + 1))
i : Fin (n + 1)
f : ℝ → Box (Fin n) → G
fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → Fin n →ᵇᵃ[↑(I₀.face i)] G
hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J
J : Box (Fin (n + 1))
hJ : J ≤ I₀
j : Fin (n + 1)
x : ℝ
⊢ x ∈ Set.Ioo (J.lower j) (J.upper j) →
    Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitLower j x) +
        Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitUpper j x) =
      f (J.upper i) (J.face i) - f (J.lower i) (J.face i)

Tactic:
refine i.succAboveCases (fun hx => ?_) (fun j hx => ?_) j

Goal state after:
case refine_1
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I : Box ι
i✝ : ι
G : Type u
inst✝ : AddCommGroup G
I₀ : Box (Fin (n + 1))
i : Fin (n + 1)
f : ℝ → Box (Fin n) → G
fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → Fin n →ᵇᵃ[↑(I₀.face i)] G
hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J
J : Box (Fin (n + 1))
hJ : J ≤ I₀
j : Fin (n + 1)
x : ℝ
hx : x ∈ Set.Ioo (J.lower i) (J.upper i)
⊢ Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitLower i x) +
      Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitUpper i x) =
    f (J.upper i) (J.face i) - f (J.lower i) (J.face i)

case refine_2
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I : Box ι
i✝ : ι
G : Type u
inst✝ : AddCommGroup G
I₀ : Box (Fin (n + 1))
i : Fin (n + 1)
f : ℝ → Box (Fin n) → G
fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → Fin n →ᵇᵃ[↑(I₀.face i)] G
hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J
J : Box (Fin (n + 1))
hJ : J ≤ I₀
j✝ : Fin (n + 1)
x : ℝ
j : Fin n
hx : x ∈ Set.Ioo (J.lower (i.succAbove j)) (J.upper (i.succAbove j))
⊢ Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitLower (i.succAbove j) x) +
      Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitUpper (i.succAbove j) x) =
    f (J.upper i) (J.face i) - f (J.lower i) (J.face i)
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
i j : ι
hij : i ≤ j
⊢ f i ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j)

Tactic:
gramSchmidt_def' 𝕜 f i

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
i j : ι
hij : i ≤ j
⊢ gramSchmidt 𝕜 f i + ∑ i_1 < i, (𝕜 ∙ gramSchmidt 𝕜 f i_1).starProjection (f i) ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j)
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
i j : ι
hij : i < j
b : Basis ι 𝕜 E
this : gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (gramSchmidt 𝕜 ⇑b '' Set.Iio j)
⊢ gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (⇑b '' Set.Iio j)

Tactic:
← span_gramSchmidt_Iio 𝕜 b j

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
i j : ι
hij : i < j
b : Basis ι 𝕜 E
this : gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (gramSchmidt 𝕜 ⇑b '' Set.Iio j)
⊢ gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (gramSchmidt 𝕜 ⇑b '' Set.Iio j)
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] M
⊢ (LinearEquiv.det f)⁻¹ • e.orientation = e.orientation ↔ 0 < LinearMap.det ↑f

Tactic:
units_inv_smul,

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] M
⊢ LinearEquiv.det f • e.orientation = e.orientation ↔ 0 < LinearMap.det ↑f
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] M
⊢ (LinearEquiv.det f)⁻¹ • e.orientation = -e.orientation ↔ LinearMap.det ↑f < 0

Tactic:
units_inv_smul,

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] M
⊢ LinearEquiv.det f • e.orientation = -e.orientation ↔ LinearMap.det ↑f < 0
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: 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
⊢ (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i

Tactic:
rw [adjustToOrientation]
  split_ifs with h
  · simp
  · by_cases hi : i = Classical.arbitrary ι <;> simp [unitsSMul_apply, hi]

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case inr
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ (LinearEquiv.det f)⁻¹ • x = x ↔ 0 < LinearMap.det ↑f

Tactic:
units_inv_smul,

Goal state after:
case inr
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ LinearEquiv.det f • x = x ↔ 0 < LinearMap.det ↑f
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case inr
R : Type u_1
inst✝⁵ : Field 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✝ : Fintype ι
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
H : 0 < finrank R M
this : FiniteDimensional R M
⊢ (LinearEquiv.det f)⁻¹ • x = -x ↔ LinearMap.det ↑f < 0

Tactic:
units_inv_smul,

Goal state after:
case inr
R : Type u_1
inst✝⁵ : Field 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✝ : Fintype ι
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
H : 0 < finrank R M
this : FiniteDimensional R M
⊢ LinearEquiv.det f • x = -x ↔ LinearMap.det ↑f < 0
Module: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

Goal state before:
pp: E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : FiniteDimensional ℝ E
h : finrank ℝ E = 1
v : E
hv : v ≠ 0
⊢ ‖v‖⁻¹ ≠ 0

Tactic:
simpa using hv

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: α : Type u_1
u v : α → ℝ
l : Filter α
hv : 0 ≤ v
h : u ~[l] v
r : ℝ
φ : α → ℝ
hφ : Tendsto φ l (𝓝 1)
huφv : u =ᶠ[l] φ * v
hφr : Tendsto ((fun x ↦ x ^ r) ∘ φ) l (𝓝 1)
| u ^ r =ᶠ[l] (fun x ↦ x ^ r) ∘ φ * v ^ r

Tactic:
conv =>

Goal state after:
α : Type u_1
u v : α → ℝ
l : Filter α
hv : 0 ≤ v
h : u ~[l] v
r : ℝ
φ : α → ℝ
hφ : Tendsto φ l (𝓝 1)
huφv : u =ᶠ[l] φ * v
hφr : Tendsto ((fun x ↦ x ^ r) ∘ φ) l (𝓝 1)
| fun x ↦ φ x ^ r * v x ^ r
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
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0

Tactic:
frontier_equivFun,

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 (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' frontier 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
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0

Tactic:
frontier_equivFun,

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
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ volume (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
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
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0

Tactic:
frontier_equivFun,

Goal state after:
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 (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' frontier s) = 0
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case convert_3
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * 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
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0

Tactic:
frontier_equivFun,

Goal state after:
case convert_3
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * 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
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
Module: Mathlib.Data.Nat.Factorial.DoubleFactorial

Goal state before:
pp: k : ℕ
⊢ (k + 1).succ * ((k + 1)‼ * k‼) = (k + 2) * k‼ * (k + 1)‼

Tactic:
mul_comm _ k‼,

Goal state after:
k : ℕ
⊢ (k + 1).succ * (k‼ * (k + 1)‼) = (k + 2) * k‼ * (k + 1)‼
Module: Mathlib.Algebra.MvPolynomial.Comap

Goal state before:
pp: σ : Type u_1
τ : Type u_2
υ : Type u_3
R : Type u_4
inst✝ : CommSemiring R
f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R
g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R
x : υ → R
i : σ
⊢ g (f (X i)) = (aeval fun i ↦ g (X i)) (f (X i))

Tactic:
suffices g = aeval fun i => g (X i) by rw [← this]

Goal state after:
σ : Type u_1
τ : Type u_2
υ : Type u_3
R : Type u_4
inst✝ : CommSemiring R
f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R
g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R
x : υ → R
i : σ
⊢ g = aeval fun i ↦ g (X i)
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: 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
this : ∀ J ∈ B', ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * (2 * C)
⊢ μ' (⋃ J ∈ B', ↑J) ≤ μ' U

Tactic:
apply μ'.mono

Goal state after:
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
this : ∀ J ∈ B', ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * (2 * C)
⊢ ⋃ J ∈ B', ↑J ⊆ U
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case inr.succ
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
n : σ →₀ ℕ
hR : Nontrivial R
d : ℕ
hd :
  ∀ (n : σ →₀ ℕ),
    Finsupp.degree n = d →
      ∀ (p q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r
⊢ ∀ (n : σ →₀ ℕ),
    Finsupp.degree n = d + 1 →
      ∀ (p q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r

Tactic:
intro n hn p q

Goal state after:
case inr.succ
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p✝ q✝ : MvPolynomial σ R
inst✝ : IsCancelMulZero R
n✝ : σ →₀ ℕ
hR : Nontrivial R
d : ℕ
hd :
  ∀ (n : σ →₀ ℕ),
    Finsupp.degree n = d →
      ∀ (p q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r
n : σ →₀ ℕ
hn : Finsupp.degree n = d + 1
p q : MvPolynomial σ R
⊢ p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r