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
r : ℝ
s : Set E
⊢ μ (r • s) = ENNReal.ofReal |r ^ finrank ℝ E| * μ s

Tactic:
rcases ne_or_eq r 0 with (h | rfl)

Goal state after:
case inl
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
r : ℝ
s : Set E
h : r ≠ 0
⊢ μ (r • s) = ENNReal.ofReal |r ^ finrank ℝ E| * μ s

case inr
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
⊢ μ (0 • s) = ENNReal.ofReal |0 ^ finrank ℝ E| * μ s
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
r : ℝ
hr : r ≠ 0
x y : E
s t : Set E
⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) * (ENNReal.ofReal (|r| ^ finrank ℝ E))⁻¹ * (μ s * (μ t)⁻¹) = μ s / μ t

Tactic:
by
      rw [ENNReal.mul_inv_cancel, one_mul, div_eq_mul_inv]
      · simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne]
      · simp only [ENNReal.ofReal_ne_top, Ne, not_false_iff]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
I J : Finset ι'
hIJ : I ⊆ J
x y : EuclideanSpace 𝕜 ↥J
⊢ toLp 2 (Finset.restrict₂ hIJ (x + y).ofLp) =
    toLp 2 (Finset.restrict₂ hIJ x.ofLp) + toLp 2 (Finset.restrict₂ hIJ y.ofLp)

Tactic:
ext; simp

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
I J : Finset ι'
hIJ : I ⊆ J
m : 𝕜
x : EuclideanSpace 𝕜 ↥J
⊢ toLp 2 (Finset.restrict₂ hIJ (m • x).ofLp) = (RingHom.id 𝕜) m • toLp 2 (Finset.restrict₂ hIJ x.ofLp)

Tactic:
ext; simp

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

Goal state before:
pp: case inl
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
r : ℝ
⊢ μ (s ∩ ({x} + r • t)) = 0

Tactic:
rw [← nonpos_iff_eq_zero]

Goal state after:
case inl
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
r : ℝ
⊢ μ (s ∩ ({x} + r • t)) ≤ 0
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁸ : RCLike 𝕜
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
F' : Type u_6
inst✝³ : NormedAddCommGroup F'
inst✝² : InnerProductSpace ℝ F'
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
e₁ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] (i : ι) → ↥(V i) := ⋯
e₂ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] E := ⋯
v w : PiLp 2 fun i ↦ ↥(V i)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
· congr <;> simp

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

Goal state before:
pp: ι : Type u_7
𝕜 : Type u_8
inst✝¹ : Unique ι
inst✝ : RCLike 𝕜
⊢ ⇑(OrthonormalBasis.singleton ι 𝕜) = 1

Tactic:
ext; simp

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

Goal state before:
pp: 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
i : ι
⊢ (ofZLatticeBasis K L b) i = ↑(b i)

Tactic:
simp [Basis.ofZLatticeBasis]

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

Goal state before:
pp: 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
x : ↥L
i✝ i : ι
⊢ Finsupp.single i 1 = Finsupp.single i ((algebraMap ℤ K) 1)

Tactic:
map_one

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : NormedField K
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
L : Submodule ℤ E
hL : span K ↑L = ⊤
e : F →ₗ[K] E
he : ↑L ⊆ ↑e.range
⊢ span K ↑(ZLattice.comap K L e) = ⊤

Tactic:
rw [ZLattice.coe_comap, Submodule.span_preimage_eq (Submodule.nonempty L) he, hL, comap_top]

Goal state after:
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
a b : OrthonormalBasis ι 𝕜 E
this : ‖(a.toBasis.toMatrix ⇑b).det‖ ^ 2 = 1
⊢ ‖a.toBasis.det ⇑b‖ = 1

Tactic:
rwa [pow_eq_one_iff_of_nonneg (norm_nonneg _) two_ne_zero] at this

Goal state after:
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
D : Set E
x : E
hf : ContinuousWithinAt f D x
⊢ oscillationWithin f D x = 0

Tactic:
rw [← nonpos_iff_eq_zero]

Goal state after:
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
D : Set E
x : E
hf : ContinuousWithinAt f D x
⊢ oscillationWithin f D x ≤ 0
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: 𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
m : Type u_7
n : Type u_8
inst✝ : Fintype m
A B : Matrix m n 𝕜
i j : n
⊢ ⟪Aᵀ i, Bᵀ j⟫ₑ = (Aᴴ * B) i j

Tactic:
by
  simp [PiLp.inner_apply, dotProduct, mul_apply', mul_comm]

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

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Finite ι
x y : ι → ℝ
hx : x ∈ (↑n)⁻¹ • span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))
hy : y ∈ (↑n)⁻¹ • span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))
h : index n x = index n y
⊢ x = y

Tactic:
rw [← tag_index_eq_self_of_mem_smul_span n hx, ← tag_index_eq_self_of_mem_smul_span n hy, h]

Goal state after:
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
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ f n ∈ span 𝕜 (f '' Set.Iio n)

Tactic:
← span_gramSchmidt_Iio 𝕜 f n,

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
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ f n ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_4
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))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = 1 / ↑n ^ card ι * F x✝

Tactic:
mul_comm_div,

Goal state after:
case refine_4
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))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = 1 * (F x✝ / ↑n ^ card ι)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_4
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))))
x✝ : ι → ℝ
hx : x✝ ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset
⊢ F x✝ / ↑n ^ card ι = volume.real ↑(box n (index n x✝)) * F ((prepartition n B).tag (box n (index n x✝)))

Tactic:
rw [Set.mem_toFinset] at hx
    rw [measureReal_def, volume_box,
      prepartition_tag n (mem_admissibleIndex_of_mem_box n hB (hs₀ hx.1)),
      tag_index_eq_self_of_mem_smul_span n hx.2, ENNReal.toReal_div,
      ENNReal.toReal_one, ENNReal.toReal_pow, ENNReal.toReal_natCast, mul_comm_div, one_mul]

Goal state after:
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
⊢ (b.repr (gramSchmidt 𝕜 (⇑b) i)) j = 0

Tactic:
have : gramSchmidt 𝕜 b i ∈ span 𝕜 (gramSchmidt 𝕜 b '' Set.Iio j) :=
    subset_span ((Set.mem_image _ _ _).2 ⟨i, hij, rfl⟩)

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)
⊢ (b.repr (gramSchmidt 𝕜 (⇑b) i)) j = 0
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
J : Box ι
hJ : J ∈ π.filter fun J ↦ N (π.tag J) = n
⊢ ‖μ.real ↑J • f (π.tag J)‖ ≤ μ.real ↑J * ↑n

Tactic:
TaggedPrepartition.mem_filter

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)
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
J : Box ι
hJ : J ∈ π ∧ N (π.tag J) = n
⊢ ‖μ.real ↑J • f (π.tag J)‖ ≤ μ.real ↑J * ↑n
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e f : OrthonormalBasis ι ℝ E
h : e.toBasis.orientation = f.toBasis.orientation
⊢ e.toBasis.det ⇑f = 1

Tactic:
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e f : OrthonormalBasis ι ℝ E
h : e.toBasis.orientation = f.toBasis.orientation
⊢ ¬e.toBasis.det ⇑f = -1
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case add
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
f g : SimpleFunc (ι → ℝ) E
a✝ : Disjoint (Function.support ⇑f) (Function.support ⇑g)
hfi : HasIntegral I l (⇑f) μ.toBoxAdditive.toSMul (integral (μ.restrict ↑I) f)
hgi : HasIntegral I l (⇑g) μ.toBoxAdditive.toSMul (integral (μ.restrict ↑I) g)
⊢ HasIntegral I l (⇑(f + g)) μ.toBoxAdditive.toSMul (integral (μ.restrict ↑I) (f + g))

Tactic:
| @add f g _ hfi hgi =>

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

Goal state before:
pp: case pos
ι : Type u
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
g : (ι → ℝ) → ℝ
hle : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hg : Integrable I l g μ.toBoxAdditive.toSMul
hfi : Integrable I l f μ.toBoxAdditive.toSMul
⊢ ‖integral I l f μ.toBoxAdditive.toSMul‖ ≤ integral I l g μ.toBoxAdditive.toSMul

Tactic:
refine le_of_tendsto_of_tendsto' hfi.hasIntegral.norm hg.hasIntegral fun π => ?_

Goal state after:
case pos
ι : Type u
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
g : (ι → ℝ) → ℝ
hle : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hg : Integrable I l g μ.toBoxAdditive.toSMul
hfi : Integrable I l f μ.toBoxAdditive.toSMul
π : TaggedPrepartition I
⊢ ‖integralSum f μ.toBoxAdditive.toSMul π‖ ≤ integralSum g μ.toBoxAdditive.toSMul π
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π →
          π.IsPartition →
            dist (integralSum g μ.toBoxAdditive.toSMul π) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤ (μ.real ↑I + 1 + 1) * ↑ε

Tactic:
set r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ) := fun c x => (hfi' <| Nx x).convergenceR (δ <| Nx x) c x

Goal state after:
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π →
          π.IsPartition →
            dist (integralSum g μ.toBoxAdditive.toSMul π) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤ (μ.real ↑I + 1 + 1) * ↑ε
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ ↑↑K.orthogonalDecomposition =
    ↑(WithLp.prodContinuousLinearEquiv 2 𝕜 ↥K ↥Kᗮ).symm ∘SL K.orthogonalProjectionOnto.prod Kᗮ.orthogonalProjectionOnto

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ WithLp.fstL 2 𝕜 ↥K ↥Kᗮ ∘SL ↑↑K.orthogonalDecomposition = K.orthogonalProjectionOnto

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ WithLp.sndL 2 𝕜 ↥K ↥Kᗮ ∘SL ↑↑K.orthogonalDecomposition = Kᗮ.orthogonalProjectionOnto

Tactic:
ext; simp

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

Goal state before:
pp: case e'_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
x : Fin (n + 1) → E
this : FiniteDimensional ℝ E
⊢ x = ⇑φ.symm ∘ ⇑φ ∘ x

Tactic:
· ext
    simp

Goal state after:
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
hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
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
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),
        l.MemBaseSet I c₁ (r c₁) π₁ →
          π₁.IsPartition →
            l.MemBaseSet I c₂ (r c₂) π₂ →
              π₂.IsPartition →
                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε

Tactic:
rcases exists_pos_mul_lt ε0 (2 * μ.toBoxAdditive I) with ⟨ε₁, ε₁0, hε₁⟩

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
hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
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
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),
        l.MemBaseSet I c₁ (r c₁) π₁ →
          π₁.IsPartition →
            l.MemBaseSet I c₂ (r c₂) π₂ →
              π₂.IsPartition →
                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
⊢ Subtype.val ∘ ⇑b = ⇑(Basis.ofZLatticeBasis ℝ L b)

Tactic:
ext; simp

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

Goal state before:
pp: E : Type u_2
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
ι : Type u_3
inst✝ : Fintype ι
b : Basis ι ℤ ↥L
s : Set E
hs : NullMeasurableSet s volume
⊢ volume (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s) = volume s / ENNReal.ofReal (covolume L volume)

Tactic:
let e : Fin (finrank ℝ E) ≃ ι :=
    Fintype.equivOfCardEq (by rw [Fintype.card_fin, finrank_eq_card_basis (b.ofZLatticeBasis ℝ)])

Goal state after:
E : Type u_2
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
ι : Type u_3
inst✝ : Fintype ι
b : Basis ι ℤ ↥L
s : Set E
hs : NullMeasurableSet s volume
e : Fin (finrank ℝ E) ≃ ι := Fintype.equivOfCardEq ⋯
⊢ volume (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s) = volume s / ENNReal.ofReal (covolume L volume)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case succ
M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
⊢ (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k

Tactic:
have : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k := fun k hk =>
      hf (Nat.lt_of_le_of_lt (Nat.succ_le_of_lt (h_pos n)) <| (Nat.lt_succ_of_le le_rfl).trans_le
        (mem_Ico.mp hk).1) (Nat.le_of_lt_succ <| (mem_Ico.mp hk).2)

Goal state after:
case succ
M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k
this : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k
⊢ (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k