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
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ

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
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case e'_3
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
inst✝ : SFinite μ
hf : AEMeasurable f (μ.restrict s)
hg : AEMeasurable g (μ.restrict s)
hs : MeasurableSet s
h₁ : (fun y ↦ ofReal ((g - f) y)) =ᵐ[μ.restrict s] fun y ↦ ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)
h₂ :
  ((μ.restrict s).prod volume) (regionBetween f g s) =
    ((μ.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)
⊢ (μ.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =
    ((μ.prod volume).restrict (s ×ˢ univ)) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)

Tactic:
]

Goal state after:
case e'_3
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
inst✝ : SFinite μ
hf : AEMeasurable f (μ.restrict s)
hg : AEMeasurable g (μ.restrict s)
hs : MeasurableSet s
h₁ : (fun y ↦ ofReal ((g - f) y)) =ᵐ[μ.restrict s] fun y ↦ ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)
h₂ :
  ((μ.restrict s).prod volume) (regionBetween f g s) =
    ((μ.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)
⊢ (μ.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =
    ((μ.prod volume).restrict (s ×ˢ univ)) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) 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
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
fg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e
Ce : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
Cg : Continuous ⇑g
Cesymm : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑e.symm
⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ

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
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
fg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e
Ce : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
Cg : Continuous ⇑g
Cesymm : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑e.symm
⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝² : NormedField K
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace K E
b : Basis ι K E
c : K
hc : c ≠ 0
⊢ span ℤ (Set.range fun i ↦ c • b i) = span ℤ (Set.range ⇑(b.isUnitSMul ⋯))

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝² : NormedField K
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace K E
b : Basis ι K E
c : K
hc : c ≠ 0
⊢ span ℤ (Set.range fun i ↦ c • b i) = span ℤ (Set.range ⇑(b.isUnitSMul ⋯))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
f_mble : AEMeasurable f μ
g_mble : AEMeasurable g μ
s : Set α
s_mble : NullMeasurableSet s μ
⊢ NullMeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Ioo (f p.1) (g p.1)} (μ.prod volume)

Tactic:
refine NullMeasurableSet.inter
          (s_mble.preimage quasiMeasurePreserving_fst) (NullMeasurableSet.inter ?_ ?_)
  · exact nullMeasurableSet_lt (by fun_prop) measurable_snd.aemeasurable
  · exact nullMeasurableSet_lt measurable_snd.aemeasurable (by fun_prop)

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝² : NormedField K
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace K E
b : Basis ι K E
c : K
hc : c ≠ 0
x✝ : ι
⊢ c • b x✝ = c • b x✝

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
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝¹ : FiniteDimensional ℝ (ι → ℝ)
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
fg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e
Ce : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
Cg : Continuous ⇑g
Cesymm : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑e.symm
this : (map (⇑e) μ).IsAddHaarMeasure
ecomp : ⇑e.symm ∘ ⇑e = id
⊢ map (⇑e.symm) (map (⇑g) (map (⇑e) μ)) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ

Tactic:
map_linearMap_addHaar_pi_eq_smul_addHaar hf (map e μ),

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
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝¹ : FiniteDimensional ℝ (ι → ℝ)
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
fg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e
Ce : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
Cg : Continuous ⇑g
Cesymm : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑e.symm
this : (map (⇑e) μ).IsAddHaarMeasure
ecomp : ⇑e.symm ∘ ⇑e = id
⊢ map (⇑e.symm) (ENNReal.ofReal |(LinearMap.det g)⁻¹| • map (⇑e) μ) = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
b : Basis ι K E
inst✝² : LinearOrder K
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
f : E ≃ₗ[K] F
x : F
⊢ x ∈ ⇑f '' fundamentalDomain b ↔ x ∈ ⇑f.toEquiv '' fundamentalDomain b

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
b : Basis ι K E
inst✝² : LinearOrder K
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
f : E ≃ₗ[K] F
x : F
⊢ x ∈ ⇑f '' fundamentalDomain b ↔ x ∈ ⇑f.toEquiv '' fundamentalDomain b
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
f_mble : AEMeasurable f μ
g_mble : AEMeasurable g μ
s : Set α
s_mble : NullMeasurableSet s μ
⊢ NullMeasurableSet {p | g p.1 < p.2}ᶜ (μ.prod volume)

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
f_mble : AEMeasurable f μ
g_mble : AEMeasurable g μ
s : Set α
s_mble : NullMeasurableSet s μ
⊢ NullMeasurableSet {p | g p.1 < p.2}ᶜ (μ.prod volume)
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
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌊(algebraMap ℤ K) z⌋ = (algebraMap ℤ K) z

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
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌊(algebraMap ℤ K) z⌋ = (algebraMap ℤ K) z
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
f_mble : AEMeasurable f μ
g_mble : AEMeasurable g μ
s : Set α
s_mble : NullMeasurableSet s μ
⊢ NullMeasurableSet {p | p.2 < f p.1}ᶜ (μ.prod volume)

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
f_mble : AEMeasurable f μ
g_mble : AEMeasurable g μ
s : Set α
s_mble : NullMeasurableSet s μ
⊢ NullMeasurableSet {p | p.2 < f p.1}ᶜ (μ.prod volume)
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
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝¹ : FiniteDimensional ℝ (ι → ℝ)
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
fg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e
Ce : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
Cg : Continuous ⇑g
Cesymm : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑e.symm
this : (map (⇑e) μ).IsAddHaarMeasure
ecomp : ⇑e.symm ∘ ⇑e = id
⊢ ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ

Tactic:
]

Goal state after:
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
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌈(algebraMap ℤ K) z⌉ = (algebraMap ℤ K) z

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
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌈(algebraMap ℤ K) z⌉ = (algebraMap ℤ K) z
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
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
s : Set E
⊢ (ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ) s = ENNReal.ofReal |(LinearMap.det f)⁻¹| * μ s

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
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
s : Set E
⊢ (ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ) s = ENNReal.ofReal |(LinearMap.det f)⁻¹| * μ s
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
m : E
i : ι
⊢ (b.repr m) i - ↑⌊(b.repr m) i⌋ = (b.repr m) i - ↑⌊(b.repr m) i⌋

Tactic:
]

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

Goal state before:
pp: case 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
m v : E
h : v ∈ span ℤ (Set.range ⇑b)
i : ι
⊢ (b.repr v) i = (b.repr v) i

Tactic:
]

Goal state after:
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
m v : E
h : v ∈ span ℤ (Set.range ⇑b)
⊢ fract b m = fract b m

Tactic:
]

Goal state after:
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
m n : E
⊢ (∀ (i : ι), (b.repr (fract b n)) i = (b.repr (fract b m)) i) ↔ -m + n ∈ span ℤ (Set.range ⇑b)

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
m n : E
⊢ (∀ (i : ι), (b.repr (fract b n)) i = (b.repr (fract b m)) i) ↔ -m + n ∈ span ℤ (Set.range ⇑b)
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
f : E →ₗ[ℝ] E
s : Set E
hf : LinearMap.det f ≠ 0
g : E ≃L[ℝ] E := (f.equivOfDetNeZero hf).toContinuousLinearEquiv
⊢ ENNReal.ofReal |LinearMap.det ↑↑g.symm.symm| * μ s = ENNReal.ofReal |LinearMap.det f| * μ s

Tactic:
]

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
f : E →ₗ[ℝ] E
s : Set E
hf : LinearMap.det f ≠ 0
g : E ≃L[ℝ] E := (f.equivOfDetNeZero hf).toContinuousLinearEquiv
⊢ ENNReal.ofReal |LinearMap.det ↑↑g.symm.symm| * μ s = ENNReal.ofReal |LinearMap.det f| * μ s
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
⊢ ‖fract b m‖ = ‖fract b m‖

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
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
⊢ ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ ≪ μ

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
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
⊢ ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ ≪ μ
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
i : ι
x✝ : i ∈ Finset.univ
⊢ ‖Int.fract ((b.repr m) i)‖ ≤ ‖1‖

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
i : ι
x✝ : i ∈ Finset.univ
⊢ ‖Int.fract ((b.repr m) i)‖ ≤ ‖1‖
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
i : ι
x✝ : i ∈ Finset.univ
⊢ Int.fract ((b.repr m) i) ≤ 1

Tactic:
]

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
i : ι
x✝ : i ∈ Finset.univ
⊢ Int.fract ((b.repr m) i) ≤ 1
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : Unique ι
k : K
x✝ : ι
⊢ ↑⌊k⌋ = ↑⌊k⌋

Tactic:
]

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

Goal state before:
pp: ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : Unique ι
k : K
x✝ : ι
⊢ Int.fract k = Int.fract k

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
⊢ ENNReal.ofReal |(r ^ finrank ℝ E)⁻¹| • μ ≪ μ

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
⊢ ENNReal.ofReal |(r ^ finrank ℝ E)⁻¹| • μ ≪ μ
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case intro
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Finite ι
inst✝ : HasSolidNorm K
val✝ : Fintype ι
x : E
hx : x ∈ fundamentalDomain b
⊢ ‖fract b x‖ ≤ ∑ j, ‖b j‖

Tactic:
]

Goal state after:
case intro
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Finite ι
inst✝ : HasSolidNorm K
val✝ : Fintype ι
x : E
hx : x ∈ fundamentalDomain b
⊢ ‖fract b x‖ ≤ ∑ j, ‖b j‖
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.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
x : ℝ
hx : x ∈ s
h'x : x ∈ ⋃ i ∈ A, T i
⊢ x ∈ s \ u ∪ ⋃ p ∈ A, s ∩ T p

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.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
x : ℝ
hx : x ∈ s
h'x : x ∈ ⋃ i ∈ A, T i
⊢ x ∈ s \ u ∪ ⋃ p ∈ A, s ∩ T p
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
s : Set E
⊢ ENNReal.ofReal |(r ^ finrank ℝ E)⁻¹| * μ s = ENNReal.ofReal |(r ^ finrank ℝ E)⁻¹| * μ s

Tactic:
]

Goal state after:
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
y : ↥(span ℤ (Set.range ⇑b))
x : E
⊢ fract b (y +ᵥ x) = y +ᵥ x ↔ fract b (y +ᵥ x) = y +ᵥ x

Tactic:
]

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
r : ℝ
s : Set E
h : r ≠ 0
⊢ ENNReal.ofReal |r ^ finrank ℝ E| * μ s = ENNReal.ofReal |r ^ finrank ℝ E| * μ s

Tactic:
]

Goal state after:
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
hs : s.Nonempty
⊢ μ {0} = ENNReal.ofReal |0 ^ finrank ℝ E| * μ s

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
hs : s.Nonempty
⊢ μ {0} = ENNReal.ofReal |0 ^ finrank ℝ E| * μ s
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_1
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
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ -x✝¹ + x✝ ∈ span ℤ (Set.range ⇑b)

Tactic:
]

Goal state after:
case refine_1
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
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ -x✝¹ + x✝ ∈ span ℤ (Set.range ⇑b)
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.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
⊢ (∀ᵐ (x : ℝ) ∂μ.restrict (s \ u), p x) ∧ ∀ i ∈ A, ∀ᵐ (x : ℝ) ∂μ.restrict (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.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
⊢ (∀ᵐ (x : ℝ) ∂μ.restrict (s \ u), p x) ∧ ∀ i ∈ A, ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ T i), p x