mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
hxy : fract b a✝ = fract b b✝
⊢ ⟦a✝⟧ = ⟦b✝⟧

Tactic:
]

Goal state after:
case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
hxy : fract b a✝ = fract b b✝
⊢ ⟦a✝⟧ = ⟦b✝⟧
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 : 0 ≤ r
s : Set E
⊢ ENNReal.ofReal (r ^ finrank ℝ E) * μ s = ENNReal.ofReal (r ^ finrank ℝ E) * μ s

Tactic:
]

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

Goal state before:
pp: case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
hxy : fract b a✝ = fract b b✝
⊢ fract b a✝ = fract b b✝

Tactic:
]

Goal state after:
case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
hxy : fract b a✝ = fract b b✝
⊢ fract b a✝ = fract b b✝
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
x : ↑(fundamentalDomain b)
⊢ (quotientEquiv b).symm x = Submodule.Quotient.mk ↑x

Tactic:
Equiv.symm_apply_eq,

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
x : ↑(fundamentalDomain b)
⊢ x = (quotientEquiv b) (Submodule.Quotient.mk ↑x)
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
x : ↑(fundamentalDomain b)
⊢ ↑x = fract b ↑x

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
x : ↑(fundamentalDomain b)
⊢ ↑x = fract b ↑x
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case inr.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
r : ℝ
hs' : s.Nonempty
hr : r ≠ 0
t : Set E
ht : MeasurableSet t
hst : μ (symmDiff s t) = 0
⊢ μ (symmDiff (r • s) (r • t)) = 0

Tactic:
]

Goal state after:
case inr.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
r : ℝ
hs' : s.Nonempty
hr : r ≠ 0
t : Set E
ht : MeasurableSet t
hst : μ (symmDiff s t) = 0
⊢ μ (symmDiff (r • s) (r • t)) = 0
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
val✝ : Fintype ι
x : ↥(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ (∀ (i : ι), ‖↑x i‖ < 1) ↔ x = 0

Tactic:
]

Goal state after:
case intro
ι : Type u_2
inst✝ : Finite ι
val✝ : Fintype ι
x : ↥(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ (∀ (i : ι), ‖↑x i‖ < 1) ↔ x = 0
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case inr.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
r : ℝ
hs' : s.Nonempty
hr : r ≠ 0
t : Set E
ht : MeasurableSet t
hst : μ (symmDiff s t) = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
val✝ : Fintype ι
x : ↥(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
i : ι
y : ℤ
hy : ↑y = ↑x i
⊢ y = 0 ↔ y = 0

Tactic:
]

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

Goal state before:
pp: E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
this : ball 0 r = (fun x_1 ↦ x + x_1) ⁻¹' ball x r
⊢ μ (ball x r) = μ (ball x r)

Tactic:
]

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

Goal state before:
pp: E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
this : closedBall 0 r = (fun x_1 ↦ x + x_1) ⁻¹' closedBall x r
⊢ μ (closedBall x r) = μ (closedBall x r)

Tactic:
]

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝ : Fintype ι
⊢ ∀ (a : E), (∀ (i : ι), (b.repr a) i ∈ Set.Ico 0 1) → ∀ (i : ι), (b.repr a) i ∈ Set.Icc 0 1

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝ : Fintype ι
⊢ ∀ (a : E), (∀ (i : ι), (b.repr a) i ∈ Set.Ico 0 1) → ∀ (i : ι), (b.repr a) i ∈ Set.Icc 0 1
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝ : Finite ι
x✝ : E
hx : ∀ (i : ι), (b.repr x✝) i ∈ Set.range ⇑(algebraMap ℤ ℝ)
⊢ ∀ (i : ι), ((Pi.basisFun ℝ ι).repr (b.equivFun x✝)) i ∈ Set.range ⇑(algebraMap ℤ ℝ)

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝ : Finite ι
x✝ : E
hx : ∀ (i : ι), (b.repr x✝) i ∈ Set.range ⇑(algebraMap ℤ ℝ)
⊢ ∀ (i : ι), ((Pi.basisFun ℝ ι).repr (b.equivFun x✝)) i ∈ Set.range ⇑(algebraMap ℤ ℝ)
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
x : E
r : ℝ
hr : 0 < r
⊢ μ (ball x r) = μ (ball x r)

Tactic:
]

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

Goal state before:
pp: case right.inr
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
this : s ⊆ s \ u ∪ ⋃ p ∈ A, s ∩ T p
a : ℝ
as : a ∈ s
b : ℝ
bs : b ∈ s
hab : a < b
⊢ ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x

Tactic:
· exact h a b as bs hab

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

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

Tactic:
]

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝¹ : ProperSpace E
inst✝ : Finite ι
s : Set E
hs : IsBounded s
this : DiscreteTopology ↥(span ℤ (Set.range ⇑b))
⊢ IsClosed ↑(span ℤ (Set.range ⇑b)).toAddSubgroup

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝¹ : ProperSpace E
inst✝ : Finite ι
s : Set E
hs : IsBounded s
this : DiscreteTopology ↥(span ℤ (Set.range ⇑b))
⊢ IsClosed ↑(span ℤ (Set.range ⇑b)).toAddSubgroup
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
M : ∀ᵐ (x : ℝ) ∂μ, x ∉ s \ u
⊢ ∀ i ∈ A, ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ T i → p x

Tactic:
]

Goal state after:
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
M : ∀ᵐ (x : ℝ) ∂μ, x ∉ s \ u
⊢ ∀ i ∈ A, ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ T i → p x
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case intro
E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
⊢ MeasurableSet (⇑↑b.equivFun ⁻¹' D)

pp: E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
⊢ fundamentalDomain b = ⇑↑b.equivFun ⁻¹' D

Tactic:
]

Goal state after:
case intro
E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
⊢ MeasurableSet (⇑↑b.equivFun ⁻¹' D)

E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
⊢ fundamentalDomain b = ⇑↑b.equivFun ⁻¹' D
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
x : E
r : ℝ
hr : 0 ≤ r
⊢ μ (closedBall x r) = μ (closedBall x r)

Tactic:
]

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

Goal state before:
pp: case e'_2
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ fundamentalDomain b₀ = fundamentalDomain b₀

Tactic:
]

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

Goal state before:
pp: case inr
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
M : ∀ᵐ (x : ℝ) ∂μ, x ∉ s \ u
a : ℝ
as : a ∈ s
b : ℝ
bs : b ∈ s
hab : a < b
⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x

Tactic:
· exact h a b as bs hab

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : Tendsto (fun r ↦ ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1)) (𝓝[<] 1) (𝓝 (μ (closedBall 0 1)))
r : ℝ
hr : r ∈ Ioo 0 1
⊢ μ (closedBall 0 r) ≤ μ (ball 0 1)

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : Tendsto (fun r ↦ ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1)) (𝓝[<] 1) (𝓝 (μ (closedBall 0 1)))
r : ℝ
hr : r ∈ Ioo 0 1
⊢ μ (closedBall 0 r) ≤ μ (ball 0 1)
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
x : E
r : ℝ
hr : 0 ≤ r
⊢ ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1)

Tactic:
]

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

Goal state before:
pp: case pos
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
r : ℝ
h : r < 0
⊢ μ ∅ = μ ∅

Tactic:
]

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

Goal state before:
pp: ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℝ (ι → ℝ)
⊢ ENNReal.ofReal |(Pi.basisFun ℝ ι).det ⇑b| = ENNReal.ofReal |(Matrix.of ⇑b).transpose.det|

Tactic:
]

Goal state after:
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℝ (ι → ℝ)
⊢ ENNReal.ofReal |(Pi.basisFun ℝ ι).det ⇑b| = ENNReal.ofReal |(Matrix.of ⇑b).transpose.det|
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

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

Tactic:
]

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this : FiniteDimensional ℝ E
⊢ μ (parallelepiped ⇑b \ fundamentalDomain b) = 0

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this : FiniteDimensional ℝ E
⊢ μ (parallelepiped ⇑b \ fundamentalDomain b) = 0
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: 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
x : E
r : ℝ
hr : r ≠ 0
h : 0 < r
⊢ 0 = 0

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: case inl
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hr : r ≠ 0
x y : E
s t : Set E
⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) * μ s * ((ENNReal.ofReal (|r| ^ finrank ℝ E))⁻¹ * (μ t)⁻¹) =
    ENNReal.ofReal (|r| ^ finrank ℝ E) * (ENNReal.ofReal (|r| ^ finrank ℝ E))⁻¹ * (μ s * (μ t)⁻¹)

pp: case ha
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) ≠ 0 ∨ μ t ≠ ∞

pp: case hb
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) ≠ ∞ ∨ μ t ≠ 0

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hr : r ≠ 0
x y : E
s t : Set E
⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) * μ s * ((ENNReal.ofReal (|r| ^ finrank ℝ E))⁻¹ * (μ t)⁻¹) =
    ENNReal.ofReal (|r| ^ finrank ℝ E) * (ENNReal.ofReal (|r| ^ finrank ℝ E))⁻¹ * (μ s * (μ t)⁻¹)

case ha
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) ≠ 0 ∨ μ t ≠ ∞

case hb
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) ≠ ∞ ∨ μ t ≠ 0
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
⊢ μ s * (μ t)⁻¹ = μ s * (μ t)⁻¹

pp: case h0
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) ≠ 0

pp: case ht
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) ≠ ∞

Tactic:
]

Goal state after:
case h0
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) ≠ 0

case ht
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) ≠ ∞
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
l u : ι → ℝ
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) = ⊥ ↔ ∃ i, u i ≤ l i

Tactic:
]

Goal state after:
ι : Type u_1
l u : ι → ℝ
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) = ⊥ ↔ ∃ i, u i ≤ l i
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
      ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
    l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
      u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper

Tactic:
]

Goal state after:
ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
      ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
    l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
      u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
l u : ι → ℝ
⊢ ↑(if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
    univ.pi fun i ↦ Ioc (l i) (u i)

Tactic:
]

Goal state after:
ι : Type u_1
l u : ι → ℝ
⊢ ↑(if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
    univ.pi fun i ↦ Ioc (l i) (u i)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
r : ℝ
hr : r ∈ Ioi 0
x : E
⊢ ↑(Real.toNNReal 2 ^ finrank ℝ E) * μ (closedBall 0 r) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall 0 r)

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
r : ℝ
hr : r ∈ Ioi 0
x : E
⊢ ↑(Real.toNNReal 2 ^ finrank ℝ E) * μ (closedBall 0 r) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall 0 r)
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
x : E
hx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ ∃ x_1, 0 ≤ (b.repr x) x_1 → 1 ≤ (b.repr x) x_1
i : ι
hi : 0 ≤ (b.repr x) i → 1 ≤ (b.repr x) i
this : (b.repr x) i = 1
⊢ ∑ x_1 ∈ Finset.univ.erase i, (b.repr x) x_1 • b x_1 +ᵥ b i ∈
    ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
x : E
hx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ ∃ x_1, 0 ≤ (b.repr x) x_1 → 1 ≤ (b.repr x) x_1
i : ι
hi : 0 ≤ (b.repr x) i → 1 ≤ (b.repr x) i
this : (b.repr x) i = 1
⊢ ∑ x_1 ∈ Finset.univ.erase i, (b.repr x) x_1 • b x_1 +ᵥ b i ∈
    ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))