Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume.real {x | x ∈ X ∧ F x ≤ 1} / covolume L volume =
    volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})

pp: case convert_1
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := ⋯
⊢ Nonempty (Free.ChooseBasisIndex ℤ ↥L)

pp: case convert_2
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ Fintype.card (Free.ChooseBasisIndex ℤ ↥L) * F x

pp: case convert_3
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0

Tactic:
· simp only [measureReal_def]
    rw [volume_image_eq_volume_div_covolume' L b h₃.nullMeasurableSet, ENNReal.toReal_div,
      ENNReal.toReal_ofReal (covolume_pos L volume).le]

Goal state after:
case convert_1
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := ⋯
⊢ Nonempty (Free.ChooseBasisIndex ℤ ↥L)

case convert_2
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ Fintype.card (Free.ChooseBasisIndex ℤ ↥L) * F x

case convert_3
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0