-- Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
theorem MeasureTheory.measure_lt_one_eq_integral_div_gamma {p : ℝ} (hp : 0 < p) :
    μ {x : E | g x < 1} =
      .ofReal ((∫ (x : E), Real.exp (- (g x) ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1))

Proof state:
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁵ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝⁴ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝³ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝² : MeasurableSpace F := borel F
this✝¹ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this✝ : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
this : MeasurePreserving (⇑ψ) μ ν
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
    ENNReal.ofReal ((∫ (x : E), Real.exp (-‖ψ x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ F) / p + 1))

Tactic:
rfl

Result:
no goals

Proof state:
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this : NormedSpace ℝ F := NormedSpace.mk ⋯
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Tactic:
letI : TopologicalSpace F := UniformSpace.toTopologicalSpace

Result:
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝¹ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝ : NormedSpace ℝ F := NormedSpace.mk ⋯
this : TopologicalSpace F := UniformSpace.toTopologicalSpace
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Proof state:
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
x y z : F
⊢ dist x z = g (x - y + (y - z))

Tactic:
simp [F]

Result:
no goals

Proof state:
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Tactic:
letI : NormedSpace ℝ F :=
{ norm_smul_le := fun _ _ ↦ h5 _ _ }

Result:
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this : NormedSpace ℝ F := NormedSpace.mk ⋯
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Proof state:
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
    ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))

Tactic:
let ψ := @Homeomorph.toMeasurableEquiv E F tE mE _ _ _ _
  (@ContinuousLinearEquiv.toHomeomorph ℝ ℝ _ _ _ _ _ _ E tE _ F _ _ _ _ φ)

Result:
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
    ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))