l3lab/ntp-mathlib huggingface dataset:
theorem w_nonneg (D : ℝ) (x : E) : 0 ≤ w D x := by
  apply mul_nonneg _ (u_nonneg _)
  apply inv_nonneg.2
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ (∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E

Next tactic:
apply mul_nonneg (u_int_pos E).le
theorem w_nonneg (D : ℝ) (x : E) : 0 ≤ w D x := by
  apply mul_nonneg _ (u_nonneg _)
  apply inv_nonneg.2
  apply mul_nonneg (u_int_pos E).le
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ |D| ^ finrank ℝ E

Next tactic:
norm_cast
theorem w_nonneg (D : ℝ) (x : E) : 0 ≤ w D x := by
  apply mul_nonneg _ (u_nonneg _)
  apply inv_nonneg.2
  apply mul_nonneg (u_int_pos E).le
  norm_cast
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ |D| ^ finrank ℝ E

Next tactic:
apply pow_nonneg (abs_nonneg D)
theorem w_mul_φ_nonneg (D : ℝ) (x y : E) : 0 ≤ w D y * φ (x - y) :=
  mul_nonneg (w_nonneg D y) (indicator_nonneg (by 

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x y : E
⊢ ∀ a ∈ closedBall 0 1, 0 ≤ 1

Next tactic:
simp only [zero_le_one, imp_true_iff]
theorem w_integral {D : ℝ} (Dpos : 0 < D) : ∫ x : E, w D x ∂μ = 1 := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ ∫ (x : E), w D x ∂μ = 1

Next tactic:
simp_rw [w, integral_smul]
theorem w_integral {D : ℝ} (Dpos : 0 < D) : ∫ x : E, w D x ∂μ = 1 := by
  simp_rw [w, integral_smul]
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • ∫ (a : E), u (D⁻¹ • a) ∂μ = 1

Next tactic:
rw [integral_comp_inv_smul_of_nonneg μ (u : E → ℝ) Dpos.le, abs_of_nonneg Dpos.le, mul_comm]
theorem w_integral {D : ℝ} (Dpos : 0 < D) : ∫ x : E, w D x ∂μ = 1 := by
  simp_rw [w, integral_smul]
  rw [integral_comp_inv_smul_of_nonneg μ (u : E → ℝ) Dpos.le, abs_of_nonneg Dpos.le, mul_comm]
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ (D ^ finrank ℝ E * ∫ (x : E), u x ∂μ)⁻¹ • D ^ finrank ℝ E • ∫ (x : E), u x ∂μ = 1

Next tactic:
field_simp [(u_int_pos E).ne']
theorem w_support {D : ℝ} (Dpos : 0 < D) : support (w D : E → ℝ) = ball 0 D := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ support (w D) = ball 0 D

Next tactic:
have B : D • ball (0 : E) 1 = ball 0 D := by
    rw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]
theorem w_support {D : ℝ} (Dpos : 0 < D) : support (w D : E → ℝ) = ball 0 D := by
  have B : D • ball (0 : E) 1 = ball 0 D := by
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ D • ball 0 1 = ball 0 D

Next tactic:
rw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]
theorem w_support {D : ℝ} (Dpos : 0 < D) : support (w D : E → ℝ) = ball 0 D := by
  have B : D • ball (0 : E) 1 = ball 0 D := by
    rw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
B : D • ball 0 1 = ball 0 D
⊢ support (w D) = ball 0 D

Next tactic:
have C : D ^ finrank ℝ E ≠ 0 := by
    norm_cast
    exact pow_ne_zero _ Dpos.ne'
theorem w_support {D : ℝ} (Dpos : 0 < D) : support (w D : E → ℝ) = ball 0 D := by
  have B : D • ball (0 : E) 1 = ball 0 D := by
    rw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]
  have C : D ^ finrank ℝ E ≠ 0 := by
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
B : D • ball 0 1 = ball 0 D
⊢ D ^ finrank ℝ E ≠ 0

Next tactic:
norm_cast
theorem w_support {D : ℝ} (Dpos : 0 < D) : support (w D : E → ℝ) = ball 0 D := by
  have B : D • ball (0 : E) 1 = ball 0 D := by
    rw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]
  have C : D ^ finrank ℝ E ≠ 0 := by
    norm_cast
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
B : D • ball 0 1 = ball 0 D
⊢ ¬D ^ finrank ℝ E = 0

Next tactic:
exact pow_ne_zero _ Dpos.ne'
theorem w_support {D : ℝ} (Dpos : 0 < D) : support (w D : E → ℝ) = ball 0 D := by
  have B : D • ball (0 : E) 1 = ball 0 D := by
    rw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]
  have C : D ^ finrank ℝ E ≠ 0 := by
    norm_cast
    exact pow_ne_zero _ Dpos.ne'
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
B : D • ball 0 1 = ball 0 D
C : D ^ finrank ℝ E ≠ 0
⊢ support (w D) = ball 0 D

Next tactic:
simp only [w_def, Algebra.id.smul_eq_mul, support_mul, support_inv, univ_inter,
    support_comp_inv_smul₀ Dpos.ne', u_support, B, support_const (u_int_pos E).ne', support_const C,
    abs_of_nonneg Dpos.le]
theorem w_compact_support {D : ℝ} (Dpos : 0 < D) : HasCompactSupport (w D : E → ℝ) := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ HasCompactSupport (w D)

Next tactic:
rw [hasCompactSupport_def, w_support E Dpos, closure_ball (0 : E) Dpos.ne']
theorem w_compact_support {D : ℝ} (Dpos : 0 < D) : HasCompactSupport (w D : E → ℝ) := by
  rw [hasCompactSupport_def, w_support E Dpos, closure_ball (0 : E) Dpos.ne']
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ IsCompact (closedBall 0 D)

Next tactic:
exact isCompact_closedBall _ _
theorem y_neg (D : ℝ) (x : E) : y D (-x) = y D x := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ y D (-x) = y D x

Next tactic:
apply convolution_neg_of_neg_eq
theorem y_neg (D : ℝ) (x : E) : y D (-x) = y D x := by
  apply convolution_neg_of_neg_eq
  · 

Proof state:
case h1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ ∀ᵐ (x : E) ∂μ, w D (-x) = w D x

Next tactic:
apply eventually_of_forall fun x => _
theorem y_neg (D : ℝ) (x : E) : y D (-x) = y D x := by
  apply convolution_neg_of_neg_eq
  · apply eventually_of_forall fun x => _
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ ∀ (x : E), w D (-x) = w D x

Next tactic:
simp only [w_def, Real.rpow_nat_cast, mul_inv_rev, smul_neg, u_neg, smul_eq_mul, forall_const]
theorem y_neg (D : ℝ) (x : E) : y D (-x) = y D x := by
  apply convolution_neg_of_neg_eq
  · apply eventually_of_forall fun x => _
    simp only [w_def, Real.rpow_nat_cast, mul_inv_rev, smul_neg, u_neg, smul_eq_mul, forall_const]
  · 

Proof state:
case h2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ ∀ᵐ (x : E) ∂μ, φ (-x) = φ x

Next tactic:
apply eventually_of_forall fun x => _
theorem y_neg (D : ℝ) (x : E) : y D (-x) = y D x := by
  apply convolution_neg_of_neg_eq
  · apply eventually_of_forall fun x => _
    simp only [w_def, Real.rpow_nat_cast, mul_inv_rev, smul_neg, u_neg, smul_eq_mul, forall_const]
  · apply eventually_of_forall fun x => _
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ ∀ (x : E), φ (-x) = φ x

Next tactic:
simp only [φ, indicator, mem_closedBall, dist_zero_right, norm_neg, forall_const]
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ y D x = 1

Next tactic:
change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ w D ⋆[lsmul ℝ ℝ, x] φ = 1

Next tactic:
have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    linarith only [mem_ball.1 (C hy), h'y]
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ ∀ y ∈ ball x D, φ y = 1

Next tactic:
have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ ball x D ⊆ ball 0 1

Next tactic:
apply ball_subset_ball'
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      

Proof state:
case h
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ D + dist x 0 ≤ 1

Next tactic:
simp only [mem_closedBall] at hx
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      

Proof state:
case h
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : dist x 0 ≤ 1 - D
⊢ D + dist x 0 ≤ 1

Next tactic:
linarith only [hx]
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
C : ball x D ⊆ ball 0 1
⊢ ∀ y ∈ ball x D, φ y = 1

Next tactic:
intro y hy
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
C : ball x D ⊆ ball 0 1
y : E
hy : y ∈ ball x D
⊢ φ y = 1

Next tactic:
simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
C : ball x D ⊆ ball 0 1
y : E
hy : y ∈ ball x D
⊢ 1 < dist y 0 → False

Next tactic:
intro h'y
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
C : ball x D ⊆ ball 0 1
y : E
hy : y ∈ ball x D
h'y : 1 < dist y 0
⊢ False

Next tactic:
linarith only [mem_ball.1 (C hy), h'y]
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    linarith only [mem_ball.1 (C hy), h'y]
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
B : ∀ y ∈ ball x D, φ y = 1
⊢ w D ⋆[lsmul ℝ ℝ, x] φ = 1

Next tactic:
have Bx : φ x = 1 := B _ (mem_ball_self Dpos)
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    linarith only [mem_ball.1 (C hy), h'y]
  have Bx : φ x = 1 := B _ (mem_ball_self Dpos)
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
B : ∀ y ∈ ball x D, φ y = 1
Bx : φ x = 1
⊢ w D ⋆[lsmul ℝ ℝ, x] φ = 1

Next tactic:
have B' : ∀ y, y ∈ ball x D → φ y = φ x := by rw [Bx]; exact B
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    linarith only [mem_ball.1 (C hy), h'y]
  have Bx : φ x = 1 := B _ (mem_ball_self Dpos)
  have B' : ∀ y, y ∈ ball x D → φ y = φ x := by 

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
B : ∀ y ∈ ball x D, φ y = 1
Bx : φ x = 1
⊢ ∀ y ∈ ball x D, φ y = φ x

Next tactic:
rw [Bx]
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    linarith only [mem_ball.1 (C hy), h'y]
  have Bx : φ x = 1 := B _ (mem_ball_self Dpos)
  have B' : ∀ y, y ∈ ball x D → φ y = φ x := by rw [Bx]; 

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
B : ∀ y ∈ ball x D, φ y = 1
Bx : φ x = 1
⊢ ∀ y ∈ ball x D, φ y = 1

Next tactic:
exact B
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    linarith only [mem_ball.1 (C hy), h'y]
  have Bx : φ x = 1 := B _ (mem_ball_self Dpos)
  have B' : ∀ y, y ∈ ball x D → φ y = φ x := by rw [Bx]; exact B
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
B : ∀ y ∈ ball x D, φ y = 1
Bx : φ x = 1
B' : ∀ y ∈ ball x D, φ y = φ x
⊢ w D ⋆[lsmul ℝ ℝ, x] φ = 1

Next tactic:
rw [convolution_eq_right' _ (le_of_eq (w_support E Dpos)) B']
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
    (hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
  have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
    have C : ball x D ⊆ ball 0 1 := by
      apply ball_subset_ball'
      simp only [mem_closedBall] at hx
      linarith only [hx]
    intro y hy
    simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
    intro h'y
    linarith only [mem_ball.1 (C hy), h'y]
  have Bx : φ x = 1 := B _ (mem_ball_self Dpos)
  have B' : ∀ y, y ∈ ball x D → φ y = φ x := by rw [Bx]; exact B
  rw [convolution_eq_right' _ (le_of_eq (w_support E Dpos)) B']
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
B : ∀ y ∈ ball x D, φ y = 1
Bx : φ x = 1
B' : ∀ y ∈ ball x D, φ y = φ x
⊢ ∫ (t : E), ((lsmul ℝ ℝ) (w D t)) (φ x) ∂μ = 1

Next tactic:
simp only [lsmul_apply, Algebra.id.smul_eq_mul, integral_mul_right, w_integral E Dpos, Bx,
    one_mul]
theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
    y D x = 0 := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∉ ball 0 (1 + D)
⊢ y D x = 0

Next tactic:
change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 0
theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
    y D x = 0 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 0
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∉ ball 0 (1 + D)
⊢ w D ⋆[lsmul ℝ ℝ, x] φ = 0

Next tactic:
have B : ∀ y, y ∈ ball x D → φ y = 0 := by
    intro y hy
    simp only [φ, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero]
    intro h'y
    have C : ball y D ⊆ ball 0 (1 + D) := by
      apply ball_subset_ball'
      rw [← dist_zero_right] at h'y
      linarith only [h'y]
    exact hx (C (mem_ball_comm.1 hy))
theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
    y D x = 0 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 0
  have B : ∀ y, y ∈ ball x D → φ y = 0 := by
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∉ ball 0 (1 + D)
⊢ ∀ y ∈ ball x D, φ y = 0

Next tactic:
intro y hy
theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
    y D x = 0 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 0
  have B : ∀ y, y ∈ ball x D → φ y = 0 := by
    intro y hy
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∉ ball 0 (1 + D)
y : E
hy : y ∈ ball x D
⊢ φ y = 0

Next tactic:
simp only [φ, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero]
theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
    y D x = 0 := by
  change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 0
  have B : ∀ y, y ∈ ball x D → φ y = 0 := by
    intro y hy
    simp only [φ, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero]
    

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∉ ball 0 (1 + D)
y : E
hy : y ∈ ball x D
⊢ ‖y‖ ≤ 1 → False

Next tactic:
intro h'y