l3lab/ntp-mathlib huggingface dataset:
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
  obtain hE | hE := subsingleton_or_nontrivial E
  · rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
      integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
      Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zero, Real.exp_zero,
      smul_eq_mul, mul_one, ofReal_toReal (measure_ne_top μ {0})]
  · have : (0:ℝ) < finrank ℝ E := Nat.cast_pos.mpr finrank_pos
    have : ((∫ y in Set.Ioi (0:ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_nat_cast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      

Proof state:
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ (∫ (y : ℝ) in Set.Ioi 0, y ^ (↑(finrank ℝ E) - 1) * Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1

Next tactic:
rw [integral_rpow_mul_exp_neg_rpow hp (by linarith), sub_add_cancel,
        Real.Gamma_add_one (ne_of_gt (by positivity))]
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
  obtain hE | hE := subsingleton_or_nontrivial E
  · rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
      integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
      Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zero, Real.exp_zero,
      smul_eq_mul, mul_one, ofReal_toReal (measure_ne_top μ {0})]
  · have : (0:ℝ) < finrank ℝ E := Nat.cast_pos.mpr finrank_pos
    have : ((∫ y in Set.Ioi (0:ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_nat_cast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      rw [integral_rpow_mul_exp_neg_rpow hp (by 

Proof state:
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ -1 < ↑(finrank ℝ E) - 1

Next tactic:
linarith
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
  obtain hE | hE := subsingleton_or_nontrivial E
  · rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
      integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
      Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zero, Real.exp_zero,
      smul_eq_mul, mul_one, ofReal_toReal (measure_ne_top μ {0})]
  · have : (0:ℝ) < finrank ℝ E := Nat.cast_pos.mpr finrank_pos
    have : ((∫ y in Set.Ioi (0:ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_nat_cast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      rw [integral_rpow_mul_exp_neg_rpow hp (by linarith), sub_add_cancel,
        Real.Gamma_add_one (ne_of_gt (by 

Proof state:
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ 0 < ↑(finrank ℝ E) / p

Next tactic:
positivity
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
  obtain hE | hE := subsingleton_or_nontrivial E
  · rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
      integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
      Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zero, Real.exp_zero,
      smul_eq_mul, mul_one, ofReal_toReal (measure_ne_top μ {0})]
  · have : (0:ℝ) < finrank ℝ E := Nat.cast_pos.mpr finrank_pos
    have : ((∫ y in Set.Ioi (0:ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_nat_cast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      rw [integral_rpow_mul_exp_neg_rpow hp (by linarith), sub_add_cancel,
        Real.Gamma_add_one (ne_of_gt (by positivity))]
      

Proof state:
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ 1 / p * Real.Gamma (↑(finrank ℝ E) / p) / (↑(finrank ℝ E) / p * Real.Gamma (↑(finrank ℝ E) / p)) * ↑(finrank ℝ E) = 1

Next tactic:
field_simp
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
  obtain hE | hE := subsingleton_or_nontrivial E
  · rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
      integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
      Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zero, Real.exp_zero,
      smul_eq_mul, mul_one, ofReal_toReal (measure_ne_top μ {0})]
  · have : (0:ℝ) < finrank ℝ E := Nat.cast_pos.mpr finrank_pos
    have : ((∫ y in Set.Ioi (0:ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_nat_cast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      rw [integral_rpow_mul_exp_neg_rpow hp (by linarith), sub_add_cancel,
        Real.Gamma_add_one (ne_of_gt (by positivity))]
      field_simp; 

Proof state:
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ Real.Gamma (↑(finrank ℝ E) / p) * ↑(finrank ℝ E) = ↑(finrank ℝ E) * Real.Gamma (↑(finrank ℝ E) / p)

Next tactic:
ring
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
  obtain hE | hE := subsingleton_or_nontrivial E
  · rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
      integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
      Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zero, Real.exp_zero,
      smul_eq_mul, mul_one, ofReal_toReal (measure_ne_top μ {0})]
  · have : (0:ℝ) < finrank ℝ E := Nat.cast_pos.mpr finrank_pos
    have : ((∫ y in Set.Ioi (0:ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_nat_cast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      rw [integral_rpow_mul_exp_neg_rpow hp (by linarith), sub_add_cancel,
        Real.Gamma_add_one (ne_of_gt (by positivity))]
      field_simp; ring
    

Proof state:
case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ ↑↑μ (Metric.ball 0 1) = ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Next tactic:
rw [integral_fun_norm_addHaar μ (fun x => Real.exp (- x ^ p)), nsmul_eq_mul, smul_eq_mul,
      mul_div_assoc, mul_div_assoc, mul_comm, mul_assoc, this, mul_one, ofReal_toReal]
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
  obtain hE | hE := subsingleton_or_nontrivial E
  · rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
      integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
      Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zero, Real.exp_zero,
      smul_eq_mul, mul_one, ofReal_toReal (measure_ne_top μ {0})]
  · have : (0:ℝ) < finrank ℝ E := Nat.cast_pos.mpr finrank_pos
    have : ((∫ y in Set.Ioi (0:ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_nat_cast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      rw [integral_rpow_mul_exp_neg_rpow hp (by linarith), sub_add_cancel,
        Real.Gamma_add_one (ne_of_gt (by positivity))]
      field_simp; ring
    rw [integral_fun_norm_addHaar μ (fun x => Real.exp (- x ^ p)), nsmul_eq_mul, smul_eq_mul,
      mul_div_assoc, mul_div_assoc, mul_comm, mul_assoc, this, mul_one, ofReal_toReal]
    

Proof state:
case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ ↑↑μ (Metric.ball 0 1) ≠ ⊤

Next tactic:
exact ne_of_lt measure_ball_lt_top
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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
⊢ ↑↑μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Next tactic:
letI F : Type _ := E
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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
⊢ ↑↑μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Next tactic:
letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
    dist_triangle := fun x y z => by convert h3 (x - y) (y - z) using 1; abel_nf
    edist := fun x y => .ofReal (g (x - y))
    edist_dist := fun _ _ => rfl
    eq_of_dist_eq_zero := by convert fun _ _ h => eq_of_sub_eq_zero (h4 h) }
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by 

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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 : F), dist x x = 0

Next tactic:
simp only [_root_.sub_self, h1, forall_const]
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by 

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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✝¹ x✝ : F
⊢ dist x✝¹ x✝ = dist x✝ x✝¹

Next tactic:
dsimp [dist]
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by dsimp [dist]; 

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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✝¹ x✝ : F
⊢ g (x✝¹ - x✝) = g (x✝ - x✝¹)

Next tactic:
rw [← h2, neg_sub]
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
    dist_triangle := fun x y z => by 

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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 ≤ dist x y + dist y z

Next tactic:
convert h3 (x - y) (y - z) using 1
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
    dist_triangle := fun x y z => by convert h3 (x - y) (y - z) using 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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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))

Next tactic:
abel_nf
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
    dist_triangle := fun x y z => by convert h3 (x - y) (y - z) using 1; abel_nf
    edist := fun x y => .ofReal (g (x - y))
    edist_dist := fun _ _ => rfl
    eq_of_dist_eq_zero := by 

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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 : F}, dist x y = 0 → x = y

Next tactic:
convert fun _ _ h => eq_of_sub_eq_zero (h4 h)
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
    dist_triangle := fun x y z => by convert h3 (x - y) (y - z) using 1; abel_nf
    edist := fun x y => .ofReal (g (x - y))
    edist_dist := fun _ _ => rfl
    eq_of_dist_eq_zero := by convert fun _ _ h => eq_of_sub_eq_zero (h4 h) }
  

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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))

Next tactic:
letI : NormedSpace ℝ F :=
  { norm_smul_le := fun _ _ =>  h5 _ _ }
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)) := by
  -- We copy `E` to a new type `F` on which we will put the norm defined by `g`
  letI F : Type _ := E
  letI : NormedAddCommGroup F :=
  { norm := g
    dist := fun x y => g (x - y)
    dist_self := by simp only [_root_.sub_self, h1, forall_const]
    dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
    dist_triangle := fun x y z => by convert h3 (x - y) (y - z) using 1; abel_nf
    edist := fun x y => .ofReal (g (x - y))
    edist_dist := fun _ _ => rfl
    eq_of_dist_eq_zero := by convert fun _ _ h => eq_of_sub_eq_zero (h4 h) }
  letI : NormedSpace ℝ F :=
  { norm_smul_le := fun _ _ =>  h5 _ _ }
  -- We put the new topology on F
  

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 : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
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 (_ : ∀ (x : ℝ) (x_1 : F), g (x • x_1) ≤ |x| * g x_1)
⊢ ↑↑μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Next tactic:
letI : TopologicalSpace F := UniformSpace.toTopologicalSpace