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

Goal state before:
pp: case refine_2
ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I : Box ι
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π₁ π₂ : TaggedPrepartition I
h : Disjoint π₁.iUnion π₂.iUnion
J : Box ι
hJ : J ∈ π₂.boxes
⊢ (vol J) (f ((π₁.disjUnion π₂ h).tag J)) = (vol J) (f (π₂.tag J))

Tactic:
·

Goal state after:
case refine_2
ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I : Box ι
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π₁ π₂ : TaggedPrepartition I
h : Disjoint π₁.iUnion π₂.iUnion
J : Box ι
hJ : J ∈ π₂.boxes
⊢ (vol J) (f ((π₁.disjUnion π₂ h).tag J)) = (vol J) (f (π₂.tag J))
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case 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 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))

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:
·

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

Goal state before:
pp: case refine_1.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))

pp: case refine_1.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)

Tactic:
·

Goal state after:
case refine_1.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))
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_1.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)

Tactic:
·

Goal state after:
case refine_1.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.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:
·

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

Goal state before:
pp: 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))

pp: 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)

Tactic:
·

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))
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: 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)

Tactic:
·

Goal state after:
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✝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
h : ¬Integrable I l f vol
⊢ (if h : Integrable I l (-f) vol then Exists.choose h else 0) = -0

Tactic:
dif_neg (mt Integrable.of_neg h),

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
h : ¬Integrable I l f vol
⊢ 0 = -0
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case mp
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
⊢ e.toBasis.det = f.toBasis.det → e.toBasis.orientation = f.toBasis.orientation

pp: case mpr
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
⊢ e.toBasis.orientation = f.toBasis.orientation → e.toBasis.det = f.toBasis.det

Tactic:
·

Goal state after:
case mp
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
⊢ e.toBasis.det = f.toBasis.det → e.toBasis.orientation = f.toBasis.orientation
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case mpr
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
⊢ e.toBasis.orientation = f.toBasis.orientation → e.toBasis.det = f.toBasis.det

Tactic:
·

Goal state after:
case mpr
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
⊢ e.toBasis.orientation = f.toBasis.orientation → e.toBasis.det = f.toBasis.det
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case pos
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
h : e.orientation = x
⊢ e.orientation = x

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

Tactic:
·

Goal state after:
case pos
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
h : e.orientation = x
⊢ e.orientation = x
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case inl
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
⊢ integral I l (fun x ↦ 0 • f x) vol = 0 • integral I l f vol

pp: case inr
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
hc : c ≠ 0
⊢ integral I l (fun x ↦ c • f x) vol = c • integral I l f vol

Tactic:
·

Goal state after:
case inl
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
⊢ integral I l (fun x ↦ 0 • f x) vol = 0 • integral I l f vol
Module: Mathlib.LinearAlgebra.Orientation

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

Tactic:
·

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

Goal state before:
pp: case pos
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
hc : c ≠ 0
hf : Integrable I l f vol
⊢ integral I l (fun x ↦ c • f x) vol = c • integral I l f vol

pp: case neg
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
hc : c ≠ 0
hf : ¬Integrable I l f vol
⊢ integral I l (fun x ↦ c • f x) vol = c • integral I l f vol

Tactic:
·

Goal state after:
case pos
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
hc : c ≠ 0
hf : Integrable I l f vol
⊢ integral I l (fun x ↦ c • f x) vol = c • integral I l f vol
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case neg
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
hc : c ≠ 0
hf : ¬Integrable I l f vol
⊢ integral I l (fun x ↦ c • f x) vol = c • integral I l f vol

Tactic:
·

Goal state after:
case neg
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
hc : c ≠ 0
hf : ¬Integrable I l f vol
⊢ integral I l (fun x ↦ c • f x) vol = c • integral I l f vol
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det

pp: case succ
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))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det

Tactic:
·

Goal state after:
case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case pos
ι : Type u
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
g : (ι → ℝ) → ℝ
hg : ∀ x ∈ Box.Icc I, 0 ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hgi : Integrable I l g μ.toBoxAdditive.toSMul
⊢ 0 ≤ integral I l g μ.toBoxAdditive.toSMul

pp: case neg
ι : Type u
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
g : (ι → ℝ) → ℝ
hg : ∀ x ∈ Box.Icc I, 0 ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hgi : ¬Integrable I l g μ.toBoxAdditive.toSMul
⊢ 0 ≤ integral I l g μ.toBoxAdditive.toSMul

Tactic:
·

Goal state after:
case pos
ι : Type u
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
g : (ι → ℝ) → ℝ
hg : ∀ x ∈ Box.Icc I, 0 ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hgi : Integrable I l g μ.toBoxAdditive.toSMul
⊢ 0 ≤ integral I l g μ.toBoxAdditive.toSMul
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case neg
ι : Type u
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
g : (ι → ℝ) → ℝ
hg : ∀ x ∈ Box.Icc I, 0 ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hgi : ¬Integrable I l g μ.toBoxAdditive.toSMul
⊢ 0 ≤ integral I l g μ.toBoxAdditive.toSMul

Tactic:
·

Goal state after:
case neg
ι : Type u
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
g : (ι → ℝ) → ℝ
hg : ∀ x ∈ Box.Icc I, 0 ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hgi : ¬Integrable I l g μ.toBoxAdditive.toSMul
⊢ 0 ≤ integral I l g μ.toBoxAdditive.toSMul
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ
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))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det

Tactic:
·

Goal state after:
case succ
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))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det
Module: Mathlib.LinearAlgebra.Orientation

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

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

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : LinearOrder R
inst✝⁵ : IsStrictOrderedRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
h : e.orientation = x
⊢ e i = e i ∨ e i = -e i
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

pp: case neg
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:
·

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
⊢ ‖integral I l f μ.toBoxAdditive.toSMul‖ ≤ integral I l g μ.toBoxAdditive.toSMul
Module: Mathlib.LinearAlgebra.Orientation

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

Tactic:
·

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

Goal state before:
pp: case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ o.volumeForm = -b.toBasis.det

pp: case succ
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))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ o.volumeForm = -b.toBasis.det

Tactic:
·

Goal state after:
case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ o.volumeForm = -b.toBasis.det
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 => ?_
    refine norm_sum_le_of_le _ fun J _ => ?_
    simp only [BoxAdditiveMap.toSMul_apply, norm_smul, smul_eq_mul, Real.norm_eq_abs,
      μ.toBoxAdditive_apply, abs_of_nonneg measureReal_nonneg]
    gcongr
    exact hle _ <| π.tag_mem_Icc _

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

Goal state before:
pp: case neg
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:
·

Goal state after:
case neg
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
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case pos
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
h✝ : e.orientation = x
⊢ e.det = e.det ∨ e.det = -e.det

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

Tactic:
·

Goal state after:
case pos
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
h✝ : e.orientation = x
⊢ e.det = e.det ∨ e.det = -e.det
Module: Mathlib.LinearAlgebra.Orientation

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

Tactic:
·

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