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

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
⊢ Bornology.IsBounded (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})

pp: case refine_3
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
⊢ MeasurableSet (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})

pp: case refine_4
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ x • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ⊆
    y • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1}

Tactic:
·

Goal state after:
case refine_2
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
⊢ Bornology.IsBounded (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})
Module: Mathlib.Algebra.MonoidAlgebra.Ideal

Goal state before:
pp: case mpr
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : k[G]
RHS : Ideal k[G] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (⇑(of k G) '' s)

Tactic:
·

Goal state after:
case mpr
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : k[G]
RHS : Ideal k[G] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (⇑(of k G) '' s)
Module: Mathlib.Algebra.MonoidAlgebra.Ideal

Goal state before:
pp: case mp
k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] := ⋯
⊢ x ∈ Ideal.span (of' k A '' s) → x ∈ RHS

pp: case mpr
k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (of' k A '' s)

Tactic:
·

Goal state after:
case mp
k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] := ⋯
⊢ x ∈ Ideal.span (of' k A '' s) → x ∈ RHS
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case refine_3
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
⊢ MeasurableSet (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})

pp: case refine_4
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ x • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ⊆
    y • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1}

Tactic:
·

Goal state after:
case refine_3
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
⊢ MeasurableSet (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})
Module: Mathlib.Algebra.MonoidAlgebra.Ideal

Goal state before:
pp: case mpr
k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (of' k A '' s)

Tactic:
·

Goal state after:
case mpr
k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (of' k A '' s)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case refine_4
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ x • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ⊆
    y • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1}

Tactic:
·

Goal state after:
case refine_4
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ x • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ⊆
    y • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1}
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case pos
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : d = 0
⊢ ∑ p ∈ Fintype.piFinset fun x ↦ Icc (-↑n) ↑n, ‖∑ i, p i • b i‖ ^ r ≤
    2 * ↑d * 3 ^ (d - 1) * normBound b ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)

pp: case neg
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : ¬d = 0
⊢ ∑ p ∈ Fintype.piFinset fun x ↦ Icc (-↑n) ↑n, ‖∑ i, p i • b i‖ ^ r ≤
    2 * ↑d * 3 ^ (d - 1) * normBound b ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)

Tactic:
·

Goal state after:
case pos
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : d = 0
⊢ ∑ p ∈ Fintype.piFinset fun x ↦ Icc (-↑n) ↑n, ‖∑ i, p i • b i‖ ^ r ≤
    2 * ↑d * 3 ^ (d - 1) * normBound b ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)
Module: Mathlib.Data.Finset.SMulAntidiagonal

Goal state before:
pp: case mp
G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt → a = hs.min hns ∧ b = ht.min hnt

pp: case mpr
G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a = hs.min hns ∧ b = ht.min hnt → a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt

Tactic:
·

Goal state after:
case mp
G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt → a = hs.min hns ∧ b = ht.min hnt
Module: Mathlib.Data.Finset.SMulAntidiagonal

Goal state before:
pp: case mpr
G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a = hs.min hns ∧ b = ht.min hnt → a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt

Tactic:
·

Goal state after:
case mpr
G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a = hs.min hns ∧ b = ht.min hnt → a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ volume.real s / covolume L volume = volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

pp: ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0

Tactic:
·

Goal state after:
ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ volume.real s / covolume L volume = volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0

Tactic:
·

Goal state after:
ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * 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
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ 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: ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * 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
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0

Tactic:
·

Goal state after:
ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * 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
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ 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})
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * 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
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0

Tactic:
·

Goal state after:
ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * 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
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e'_3
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
x✝ : ℕ
⊢ finrank ℝ E = Fintype.card (Free.ChooseBasisIndex ℤ ↥L)

pp: case e'_5
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume.real s / covolume L volume = volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0

Tactic:
·

Goal state after:
case e'_3
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
x✝ : ℕ
⊢ finrank ℝ E = Fintype.card (Free.ChooseBasisIndex ℤ ↥L)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e'_5
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume.real s / covolume L volume = volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0

Tactic:
·

Goal state after:
case e'_5
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume.real s / covolume L volume = volume.real (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0

Tactic:
·

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : 1 ≤ d
hs0 : s 0 = {0}
hs : ∀ {a b : ℕ}, a ≤ b → s a ⊆ s b
this : ∀ (k : ℕ), #(s (k + 1) \ s k) ≤ 2 * d * (2 * k + 3) ^ (d - 1)
ε : ℝ := normBound b
hε : 0 < ε
⊢ 2 * ↑d * 3 ^ (d - 1) * ε ^ r * ∑ k ∈ range n, (↑k + 1) ^ (d - 1) * (↑k + 1) ^ r =
    2 * ↑d * 3 ^ (d - 1) * ε ^ r * ∑ k ∈ range n, ↑(k + 1) ^ (↑d - 1 + r)

Tactic:
by
        congr with k
        rw [← Real.rpow_natCast, ← Real.rpow_add (by positivity), Nat.cast_sub hd]
        norm_cast

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

Goal state before:
pp: case inl
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Subsingleton ↥L
⊢ ∃ A > 0,
    ∀ r < -↑(finrank ℤ ↥L), ∀ (s : Finset ↥L), ∑ z ∈ s, ‖z‖ ^ r ≤ A ^ r * ∑' (k : ℕ), ↑k ^ (↑(finrank ℤ ↥L) - 1 + r)

pp: case inr
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Nontrivial ↥L
⊢ ∃ A > 0,
    ∀ r < -↑(finrank ℤ ↥L), ∀ (s : Finset ↥L), ∑ z ∈ s, ‖z‖ ^ r ≤ A ^ r * ∑' (k : ℕ), ↑k ^ (↑(finrank ℤ ↥L) - 1 + r)

Tactic:
·

Goal state after:
case inl
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Subsingleton ↥L
⊢ ∃ A > 0,
    ∀ r < -↑(finrank ℤ ↥L), ∀ (s : Finset ↥L), ∑ z ∈ s, ‖z‖ ^ r ≤ A ^ r * ∑' (k : ℕ), ↑k ^ (↑(finrank ℤ ↥L) - 1 + r)
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Nontrivial ↥L
I : Type u_1 := Free.ChooseBasisIndex ℤ ↥L
this : Fintype I
b : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L
d : ℕ := Fintype.card I
⊢ d ≠ 0

Tactic:
simp [d]

Goal state after: