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

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
h : HasIntegral I l f vol y
h' : HasIntegral I l g vol y'
⊢ HasIntegral I l (f + g) vol (y + y')

Tactic:
simpa using

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
h : HasIntegral I l f vol y
h' : HasIntegral I l g vol y'
⊢ Tendsto (integralSum (f + g) vol) (toFilteriUnion I ⊤) (𝓝 (y + y'))
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y : F
hf : HasIntegral I l f vol y
⊢ HasIntegral I l (-f) vol (-y)

Tactic:
simpa using

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y : F
hf : HasIntegral I l f vol y
⊢ Tendsto (integralSum (-f) vol) (toFilteriUnion I ⊤) (𝓝 (-y))
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 : 0 < e.toBasis.det ⇑f.toBasis
⊢ 0 < e.toBasis.det ⇑f

Tactic:
simpa using

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 : 0 < e.toBasis.det ⇑f.toBasis
⊢ 0 < e.toBasis.det ⇑f
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
h : HasIntegral I l f vol y
h' : HasIntegral I l g vol y'
⊢ HasIntegral I l (f - g) vol (y - y')

Tactic:
simpa using

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
h : HasIntegral I l f vol y
h' : HasIntegral I l g vol y'
⊢ HasIntegral I l (f + -g) vol (y + -y')
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
⊢ HasIntegral I l (fun x ↦ 0) vol 0

Tactic:
simpa using

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
⊢ HasIntegral I l (fun x ↦ 0) vol ((vol I) 0)
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
e : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι
inst✝ : Nonempty ι
⊢ ∀ (i : ι), (e.toBasis.adjustToOrientation x) i = e i ∨ (e.toBasis.adjustToOrientation x) i = -e i

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
e : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι
inst✝ : Nonempty ι
⊢ ∀ (i : ι), (e.toBasis.adjustToOrientation x) i = e i ∨ (e.toBasis.adjustToOrientation x) i = -e i
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
e : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι
inst✝ : Nonempty ι
i : ι
⊢ (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
e : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι
inst✝ : Nonempty ι
i : ι
⊢ (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y : F
hf : HasIntegral I l f vol y
c : ℝ
⊢ HasIntegral I l (c • f) vol (c • y)

Tactic:
simpa using

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y : F
hf : HasIntegral I l f vol y
c : ℝ
⊢ Tendsto (integralSum (c • f) vol) (toFilteriUnion I ⊤) (𝓝 (c • y))
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
c : ℝ
hf : Integrable I l (c • f) vol
hc : c ≠ 0
⊢ Integrable I l f vol

Tactic:
simpa using

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
c : ℝ
hf : Integrable I l (c • f) vol
hc : c ≠ 0
⊢ Integrable I l f vol
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
h : SameRay ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) (-AlternatingMap.constLinearEquivOfIsEmpty 1)
⊢ False

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
h : SameRay ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) (-AlternatingMap.constLinearEquivOfIsEmpty 1)
⊢ False
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
c₁ c₂ : ℝ≥0
ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
h : Integrable I l f vol
hpos₁ : 0 < ε₁
hpos₂ : 0 < ε₂
h₁ : l.MemBaseSet I c₁ (h.convergenceR ε₁ c₁) π₁
h₂ : l.MemBaseSet I c₂ (h.convergenceR ε₂ c₂) π₂
HU : π₁.iUnion = π₂.iUnion
π : Prepartition I
hπU✝ : π.iUnion = ↑I \ π₁.iUnion
hπU : π.iUnion = ↑I \ π₂.iUnion
hπc₁ : l.bDistortion = true → π.distortion ≤ c₁
hπc₂ : l.bDistortion = true → π.distortion ≤ c₂
r : (ι → ℝ) → ↑(Set.Ioi 0) := fun x ↦ min (h.convergenceR ε₁ c₁ x) (h.convergenceR ε₂ c₂ x)
πr : TaggedPrepartition I := π.toSubordinate r
H₁ : dist (integralSum f vol (π₁.unionComplToSubordinate π hπU✝ r)) (integral I l f vol) ≤ ε₁
H₂ : dist (integralSum f vol (π₂.unionComplToSubordinate π hπU r)) (integral I l f vol) ≤ ε₂
⊢ dist (integralSum f vol π₁) (integralSum f vol π₂) ≤ ε₁ + ε₂

Tactic:
simpa using

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
c₁ c₂ : ℝ≥0
ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
h : Integrable I l f vol
hpos₁ : 0 < ε₁
hpos₂ : 0 < ε₂
h₁ : l.MemBaseSet I c₁ (h.convergenceR ε₁ c₁) π₁
h₂ : l.MemBaseSet I c₂ (h.convergenceR ε₂ c₂) π₂
HU : π₁.iUnion = π₂.iUnion
π : Prepartition I
hπU✝ : π.iUnion = ↑I \ π₁.iUnion
hπU : π.iUnion = ↑I \ π₂.iUnion
hπc₁ : l.bDistortion = true → π.distortion ≤ c₁
hπc₂ : l.bDistortion = true → π.distortion ≤ c₂
r : (ι → ℝ) → ↑(Set.Ioi 0) := fun x ↦ min (h.convergenceR ε₁ c₁ x) (h.convergenceR ε₂ c₂ x)
πr : TaggedPrepartition I := π.toSubordinate r
H₁ : dist (integralSum f vol (π₁.unionComplToSubordinate π hπU✝ r)) (integral I l f vol) ≤ ε₁
H₂ : dist (integralSum f vol (π₂.unionComplToSubordinate π hπU r)) (integral I l f vol) ≤ ε₂
⊢ dist (integralSum f vol π₁) (integralSum f vol π₂) ≤ ε₁ + ε₂
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
this : FiniteDimensional ℝ E
⊢ finrank ℝ E = Fintype.card (Fin n.succ)

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
this : FiniteDimensional ℝ E
⊢ finrank ℝ E = n + 1
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
hv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
this : FiniteDimensional ℝ E
⊢ finrank ℝ E = Fintype.card (Fin n.succ)

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
hv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
this : FiniteDimensional ℝ E
⊢ finrank ℝ E = n + 1
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n)
o : Orientation ℝ E (Fin n)
v : OrthonormalBasis (Fin n) ℝ E
⊢ |o.volumeForm ⇑v| = 1

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n)
o : Orientation ℝ E (Fin n)
v : OrthonormalBasis (Fin n) ℝ E
⊢ |v.toBasis.det ⇑v| = 1
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) := {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)
x : ι → ℝ
hx : x ∈ Box.Icc I \ U
⊢ oscillationWithin f (Box.Icc I) x = 0

Tactic:
simpa using

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) := {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)
x : ι → ℝ
hx : x ∈ Box.Icc I \ U
⊢ ContinuousWithinAt f (Box.Icc I) x
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

Goal state after:
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 = 0
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
g : ℕ → ℕ := fun n ↦ m✝ * n + k
m n : ℕ
hmn : g m = g n
⊢ m = n

Tactic:
simpa using

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
g : ℕ → ℕ := fun n ↦ m✝ * n + k
m n : ℕ
hmn : g m = g n
⊢ m = n
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case refine_3
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
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
hfi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (⇑(f n)) (↑J) μ
hgi : ∀ J ∈ π, IntegrableOn g (↑J) μ
hfgi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (fun x ↦ ‖(f n) x - g x‖) (↑J) μ
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∑ i ∈ π.boxes, ∫ (x : ι → ℝ) in ↑i, g x ∂μ) ≤
    ∫ (x : ι → ℝ) in ⋃ J ∈ π.boxes, ↑J, ‖(f N₀) x - g x‖ ∂μ

Tactic:
integral_biUnion_finset π.boxes (fun J _ => J.measurableSet_coe) π.pairwiseDisjoint (hfgi _)

Goal state after:
case refine_3
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
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
hfi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (⇑(f n)) (↑J) μ
hgi : ∀ J ∈ π, IntegrableOn g (↑J) μ
hfgi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (fun x ↦ ‖(f n) x - g x‖) (↑J) μ
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∑ i ∈ π.boxes, ∫ (x : ι → ℝ) in ↑i, g x ∂μ) ≤
    ∑ i ∈ π.boxes, ∫ (x : ι → ℝ) in ↑i, ‖(f N₀) x - g x‖ ∂μ
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
p q : ℕ
hpq : p < q
⊢ (fun x ↦ x ^ p) =o[cobounded R] fun x ↦ x ^ (p + (q - p))

Tactic:
simpa using

Goal state after:
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
p q : ℕ
hpq : p < q
⊢ (fun x ↦ x ^ p) =o[cobounded R] fun x ↦ x ^ p * x ^ (q - p)
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, g i‖ = ∑ i ∈ Finset.range n, g i
ε : ℝ
εpos : 0 < ε
⊢ ∃ N, ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, g i‖ = ∑ i ∈ Finset.range n, g i
ε : ℝ
εpos : 0 < ε
⊢ ∃ N, ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b
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
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
⊢ ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k

Tactic:
by
  induction n with
  | zero => simp
  | succ n ihn =>
    suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k by
      rw [sum_range_succ, ← sum_Ico_consecutive]
      exacts [add_le_add ihn this,
        (add_le_add_left (hu n.zero_le) _ : u 0 + 1 ≤ u n + 1),
        add_le_add_left (hu n.le_succ) _]
    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)
    convert! sum_le_sum this
    simp

Goal state after:
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: u : ℕ → ℕ
f : ℕ → ℝ≥0∞
C : ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
h_nonneg : ∀ (n : ℕ), 0 ≤ f n
hu : Monotone u
h_succ_diff : SuccDiffBounded C u
n : ℕ
⊢ ∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤
    (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (u n + 1), f x

Tactic:
simpa using

Goal state after:
u : ℕ → ℕ
f : ℕ → ℝ≥0∞
C : ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
h_nonneg : ∀ (n : ℕ), 0 ≤ f n
hu : Monotone u
h_succ_diff : SuccDiffBounded C u
n : ℕ
⊢ ∑ a ∈ range (n + 1), (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤
    (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (u n + 1), f x
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: f : ℕ → ℝ≥0∞
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
n : ℕ
⊢ ∑ a ∈ range n.succ, 2 ^ a * f (2 ^ a) ≤ f 1 + 2 • ∑ x ∈ Ico 2 (2 ^ n + 1), f x

Tactic:
simpa using

Goal state after:
f : ℕ → ℝ≥0∞
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
n : ℕ
⊢ ∑ a ∈ range (n + 1), 2 ^ a * f (2 ^ a) ≤ f 1 + 2 * ∑ x ∈ Ico 2 (2 ^ n + 1), f x
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: case h
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
hc : ∀ (a : D), ‖f a‖ ≤ c
⊢ ∀ (x : D), ‖f x‖ ≤ c * ‖1 x‖

Tactic:
simpa using

Goal state after:
case h
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
hc : ∀ (a : D), ‖f a‖ ≤ c
⊢ ∀ (x : D), ‖f x‖ ≤ c
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case mp
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
hf : ∀ (m n : ℕ), 1 < m → m ≤ n → ↑(f n) ≤ ↑(f m)
h_nonneg : ∀ (n : ℕ), 0 ≤ ↑(f n)
hC : ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * ↑(f (u k)) ≤ (↑(u 1) - ↑(u 0)) * ↑(f (u 0)) + ↑C * ∑' (k : ℕ), ↑(f k)
⊢ ∑' (b : ℕ), ↑(f b) = ∞

Tactic:
simpa using

Goal state after:
case mp
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
hf : ∀ (m n : ℕ), 1 < m → m ≤ n → ↑(f n) ≤ ↑(f m)
h_nonneg : ∀ (n : ℕ), 0 ≤ ↑(f n)
hC : ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * ↑(f (u k)) ≤ (↑(u 1) - ↑(u 0)) * ↑(f (u 0)) + ↑C * ∑' (k : ℕ), ↑(f k)
⊢ ∑' (b : ℕ), ↑(f b) = ∞
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(f b) = ∞
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞

Tactic:
replace hf : ∀ m n, 0 < m → m ≤ n → (f n : ℝ≥0∞) ≤ f m := fun m n hm hmn =>
      ENNReal.coe_le_coe.2 (hf hm hmn)

Goal state after:
case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(f b) = ∞
hf : ∀ (m n : ℕ), 0 < m → m ≤ n → ↑(f n) ≤ ↑(f m)
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(f b) = ∞
hf : ∀ (m n : ℕ), 0 < m → m ≤ n → ↑(f n) ≤ ↑(f m)
this : ∑ k ∈ range (u 0), ↑(f k) ≠ ∞
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞

Tactic:
simpa using

Goal state after:
case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(f b) = ∞
hf : ∀ (m n : ℕ), 0 < m → m ≤ n → ↑(f n) ≤ ↑(f m)
this : ∑ k ∈ range (u 0), ↑(f k) ≠ ∞
⊢ ∑' (b : ℕ), (↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
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}
y : D
hy : y ∈ tᶜ
⊢ ‖f y‖ ≤ c

Tactic:
simpa using

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}
y : D
hy : y ∈ tᶜ
⊢ ‖f y‖ ≤ c
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inr
p : ℝ
hp : p < 0
h : Summable fun n ↦ (↑n ^ p)⁻¹
k : ℕ
hk₁ : (↑k ^ p)⁻¹ < 1
hk₀ : 0 < ↑k
⊢ k = 0

Tactic:
simpa using

Goal state after:
case inr
p : ℝ
hp : p < 0
h : Summable fun n ↦ (↑n ^ p)⁻¹
k : ℕ
hk₁ : (↑k ^ p)⁻¹ < 1
hk₀ : 0 < ↑k
⊢ k = 0
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: ⊢ ¬Summable fun n ↦ 1 / ↑n

Tactic:
simpa using

Goal state after:
⊢ ¬Summable fun n ↦ 1 / ↑n