ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Bornology.IsBounded.add
@[to_additive]
theorem Bornology.IsBounded.mul (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s * t) 
Arguments: {E : Type u_1} [SeminormedAddGroup E] {s t : Set E} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s + t)
-- Bornology.IsBounded.of_add
@[to_additive]
theorem Bornology.IsBounded.of_mul (hst : IsBounded (s * t)) : IsBounded s ∨ IsBounded t 
Arguments: {E : Type u_1} [SeminormedAddGroup E] {s t : Set E} (hst : Bornology.IsBounded (s + t))
Goal: Bornology.IsBounded s ∨ Bornology.IsBounded t
-- Bornology.IsBounded.of_mul
@[to_additive]
theorem Bornology.IsBounded.of_mul (hst : IsBounded (s * t)) : IsBounded s ∨ IsBounded t 
Arguments: {E : Type u_1} [SeminormedGroup E] {s t : Set E} (hst : Bornology.IsBounded (s * t))
Goal: Bornology.IsBounded s ∨ Bornology.IsBounded t
-- Bornology.IsBounded.neg
@[to_additive]
theorem Bornology.IsBounded.inv : IsBounded s → IsBounded s⁻¹ 
Arguments: {E : Type u_1} [SeminormedAddGroup E] {s : Set E}
Goal: Bornology.IsBounded s → Bornology.IsBounded (-s)
-- Bornology.IsBounded.inv
@[to_additive]
theorem Bornology.IsBounded.inv : IsBounded s → IsBounded s⁻¹ 
Arguments: {E : Type u_1} [SeminormedGroup E] {s : Set E}
Goal: Bornology.IsBounded s → Bornology.IsBounded s⁻¹
-- Bornology.IsBounded.div
@[to_additive]
theorem Bornology.IsBounded.div (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s / t) 
Arguments: {E : Type u_1} [SeminormedGroup E] {s t : Set E} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s / t)
-- Bornology.IsBounded.sub
@[to_additive]
theorem Bornology.IsBounded.div (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s / t) 
Arguments: {E : Type u_1} [SeminormedAddGroup E] {s t : Set E} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s - t)
-- infEdist_neg_neg
@[to_additive (attr := simp)]
theorem infEdist_inv_inv (x : E) (s : Set E) : infEdist x⁻¹ s⁻¹ = infEdist x s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (x : E) (s : Set E)
Goal: EMetric.infEdist (-x) (-s) = EMetric.infEdist x s
-- infEdist_inv_inv
@[to_additive (attr := simp)]
theorem infEdist_inv_inv (x : E) (s : Set E) : infEdist x⁻¹ s⁻¹ = infEdist x s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (x : E) (s : Set E)
Goal: EMetric.infEdist x⁻¹ s⁻¹ = EMetric.infEdist x s
-- infEdist_neg
@[to_additive]
theorem infEdist_inv (x : E) (s : Set E) : infEdist x⁻¹ s = infEdist x s⁻¹ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (x : E) (s : Set E)
Goal: EMetric.infEdist (-x) s = EMetric.infEdist x (-s)
-- infEdist_inv
@[to_additive]
theorem infEdist_inv (x : E) (s : Set E) : infEdist x⁻¹ s = infEdist x s⁻¹ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (x : E) (s : Set E)
Goal: EMetric.infEdist x⁻¹ s = EMetric.infEdist x s⁻¹
-- ediam_mul_le
@[to_additive]
theorem ediam_mul_le (x y : Set E) : EMetric.diam (x * y) ≤ EMetric.diam x + EMetric.diam y 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (x y : Set E)
Goal: EMetric.diam (x * y) ≤ EMetric.diam x + EMetric.diam y
-- ediam_add_le
@[to_additive]
theorem ediam_mul_le (x y : Set E) : EMetric.diam (x * y) ≤ EMetric.diam x + EMetric.diam y 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (x y : Set E)
Goal: EMetric.diam (x + y) ≤ EMetric.diam x + EMetric.diam y
-- neg_thickening
@[to_additive (attr := simp)]
theorem inv_thickening : (thickening δ s)⁻¹ = thickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E)
Goal: -Metric.thickening δ s = Metric.thickening δ (-s)
-- inv_thickening
@[to_additive (attr := simp)]
theorem inv_thickening : (thickening δ s)⁻¹ = thickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E)
Goal: (Metric.thickening δ s)⁻¹ = Metric.thickening δ s⁻¹
-- neg_cthickening
@[to_additive (attr := simp)]
theorem inv_cthickening : (cthickening δ s)⁻¹ = cthickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E)
Goal: -Metric.cthickening δ s = Metric.cthickening δ (-s)
-- inv_cthickening
@[to_additive (attr := simp)]
theorem inv_cthickening : (cthickening δ s)⁻¹ = cthickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E)
Goal: (Metric.cthickening δ s)⁻¹ = Metric.cthickening δ s⁻¹
-- inv_ball
@[to_additive (attr := simp)]
theorem inv_ball : (ball x δ)⁻¹ = ball x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: (Metric.ball x δ)⁻¹ = Metric.ball x⁻¹ δ
-- neg_ball
@[to_additive (attr := simp)]
theorem inv_ball : (ball x δ)⁻¹ = ball x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: -Metric.ball x δ = Metric.ball (-x) δ
-- neg_closedBall
@[to_additive (attr := simp)]
theorem inv_closedBall : (closedBall x δ)⁻¹ = closedBall x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: -Metric.closedBall x δ = Metric.closedBall (-x) δ
-- inv_closedBall
@[to_additive (attr := simp)]
theorem inv_closedBall : (closedBall x δ)⁻¹ = closedBall x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: (Metric.closedBall x δ)⁻¹ = Metric.closedBall x⁻¹ δ
-- singleton_add_ball
@[to_additive]
theorem singleton_mul_ball : {x} * ball y δ = ball (x * y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} + Metric.ball y δ = Metric.ball (x + y) δ
-- singleton_mul_ball
@[to_additive]
theorem singleton_mul_ball : {x} * ball y δ = ball (x * y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} * Metric.ball y δ = Metric.ball (x * y) δ
-- singleton_sub_ball
@[to_additive]
theorem singleton_div_ball : {x} / ball y δ = ball (x / y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} - Metric.ball y δ = Metric.ball (x - y) δ
-- singleton_div_ball
@[to_additive]
theorem singleton_div_ball : {x} / ball y δ = ball (x / y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} / Metric.ball y δ = Metric.ball (x / y) δ
-- ball_add_singleton
@[to_additive]
theorem ball_mul_singleton : ball x δ * {y} = ball (x * y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.ball x δ + {y} = Metric.ball (x + y) δ
-- ball_mul_singleton
@[to_additive]
theorem ball_mul_singleton : ball x δ * {y} = ball (x * y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.ball x δ * {y} = Metric.ball (x * y) δ
-- ball_div_singleton
@[to_additive]
theorem ball_div_singleton : ball x δ / {y} = ball (x / y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.ball x δ / {y} = Metric.ball (x / y) δ
-- ball_sub_singleton
@[to_additive]
theorem ball_div_singleton : ball x δ / {y} = ball (x / y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.ball x δ - {y} = Metric.ball (x - y) δ
-- singleton_add_ball_zero
@[to_additive]
theorem singleton_mul_ball_one : {x} * ball 1 δ = ball x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: {x} + Metric.ball 0 δ = Metric.ball x δ
-- singleton_mul_ball_one
@[to_additive]
theorem singleton_mul_ball_one : {x} * ball 1 δ = ball x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: {x} * Metric.ball 1 δ = Metric.ball x δ
-- singleton_div_ball_one
@[to_additive]
theorem singleton_div_ball_one : {x} / ball 1 δ = ball x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: {x} / Metric.ball 1 δ = Metric.ball x δ
-- singleton_sub_ball_zero
@[to_additive]
theorem singleton_div_ball_one : {x} / ball 1 δ = ball x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: {x} - Metric.ball 0 δ = Metric.ball x δ
-- ball_one_mul_singleton
@[to_additive]
theorem ball_one_mul_singleton : ball 1 δ * {x} = ball x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.ball 1 δ * {x} = Metric.ball x δ
-- ball_zero_add_singleton
@[to_additive]
theorem ball_one_mul_singleton : ball 1 δ * {x} = ball x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.ball 0 δ + {x} = Metric.ball x δ
-- ball_zero_sub_singleton
@[to_additive]
theorem ball_one_div_singleton : ball 1 δ / {x} = ball x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.ball 0 δ - {x} = Metric.ball (-x) δ
-- ball_one_div_singleton
@[to_additive]
theorem ball_one_div_singleton : ball 1 δ / {x} = ball x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.ball 1 δ / {x} = Metric.ball x⁻¹ δ
-- vadd_ball_zero
@[to_additive]
theorem smul_ball_one : x • ball (1 : E) δ = ball x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: x +ᵥ Metric.ball 0 δ = Metric.ball x δ
-- smul_ball_one
@[to_additive]
theorem smul_ball_one : x • ball (1 : E) δ = ball x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: x • Metric.ball 1 δ = Metric.ball x δ
-- singleton_mul_closedBall
@[to_additive (attr := simp 1100)]
theorem singleton_mul_closedBall : {x} * closedBall y δ = closedBall (x * y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} * Metric.closedBall y δ = Metric.closedBall (x * y) δ
-- singleton_add_closedBall
@[to_additive (attr := simp 1100)]
theorem singleton_mul_closedBall : {x} * closedBall y δ = closedBall (x * y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} + Metric.closedBall y δ = Metric.closedBall (x + y) δ
-- singleton_sub_closedBall
@[to_additive (attr := simp 1100)]
theorem singleton_div_closedBall : {x} / closedBall y δ = closedBall (x / y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} - Metric.closedBall y δ = Metric.closedBall (x - y) δ
-- singleton_div_closedBall
@[to_additive (attr := simp 1100)]
theorem singleton_div_closedBall : {x} / closedBall y δ = closedBall (x / y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: {x} / Metric.closedBall y δ = Metric.closedBall (x / y) δ
-- closedBall_mul_singleton
@[to_additive (attr := simp 1100)]
theorem closedBall_mul_singleton : closedBall x δ * {y} = closedBall (x * y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.closedBall x δ * {y} = Metric.closedBall (x * y) δ
-- closedBall_add_singleton
@[to_additive (attr := simp 1100)]
theorem closedBall_mul_singleton : closedBall x δ * {y} = closedBall (x * y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.closedBall x δ + {y} = Metric.closedBall (x + y) δ
-- closedBall_div_singleton
@[to_additive (attr := simp 1100)]
theorem closedBall_div_singleton : closedBall x δ / {y} = closedBall (x / y) δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.closedBall x δ / {y} = Metric.closedBall (x / y) δ
-- closedBall_sub_singleton
@[to_additive (attr := simp 1100)]
theorem closedBall_div_singleton : closedBall x δ / {y} = closedBall (x / y) δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x y : E)
Goal: Metric.closedBall x δ - {y} = Metric.closedBall (x - y) δ
-- singleton_add_closedBall_zero
@[to_additive]
theorem singleton_mul_closedBall_one : {x} * closedBall 1 δ = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: {x} + Metric.closedBall 0 δ = Metric.closedBall x δ
-- singleton_mul_closedBall_one
@[to_additive]
theorem singleton_mul_closedBall_one : {x} * closedBall 1 δ = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: {x} * Metric.closedBall 1 δ = Metric.closedBall x δ
-- singleton_sub_closedBall_zero
@[to_additive]
theorem singleton_div_closedBall_one : {x} / closedBall 1 δ = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: {x} - Metric.closedBall 0 δ = Metric.closedBall x δ
-- singleton_div_closedBall_one
@[to_additive]
theorem singleton_div_closedBall_one : {x} / closedBall 1 δ = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: {x} / Metric.closedBall 1 δ = Metric.closedBall x δ
-- closedBall_zero_add_singleton
@[to_additive]
theorem closedBall_one_mul_singleton : closedBall 1 δ * {x} = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.closedBall 0 δ + {x} = Metric.closedBall x δ
-- closedBall_one_mul_singleton
@[to_additive]
theorem closedBall_one_mul_singleton : closedBall 1 δ * {x} = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.closedBall 1 δ * {x} = Metric.closedBall x δ
-- closedBall_zero_sub_singleton
@[to_additive]
theorem closedBall_one_div_singleton : closedBall 1 δ / {x} = closedBall x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.closedBall 0 δ - {x} = Metric.closedBall (-x) δ
-- closedBall_one_div_singleton
@[to_additive]
theorem closedBall_one_div_singleton : closedBall 1 δ / {x} = closedBall x⁻¹ δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: Metric.closedBall 1 δ / {x} = Metric.closedBall x⁻¹ δ
-- vadd_closedBall_zero
@[to_additive (attr := simp 1100)]
theorem smul_closedBall_one : x • closedBall (1 : E) δ = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (x : E)
Goal: x +ᵥ Metric.closedBall 0 δ = Metric.closedBall x δ
-- smul_closedBall_one
@[to_additive (attr := simp 1100)]
theorem smul_closedBall_one : x • closedBall (1 : E) δ = closedBall x δ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (x : E)
Goal: x • Metric.closedBall 1 δ = Metric.closedBall x δ
-- mul_ball_one
@[to_additive]
theorem mul_ball_one : s * ball 1 δ = thickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E)
Goal: s * Metric.ball 1 δ = Metric.thickening δ s
-- add_ball_zero
@[to_additive]
theorem mul_ball_one : s * ball 1 δ = thickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E)
Goal: s + Metric.ball 0 δ = Metric.thickening δ s
-- div_ball_one
@[to_additive]
theorem div_ball_one : s / ball 1 δ = thickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E)
Goal: s / Metric.ball 1 δ = Metric.thickening δ s
-- sub_ball_zero
@[to_additive]
theorem div_ball_one : s / ball 1 δ = thickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E)
Goal: s - Metric.ball 0 δ = Metric.thickening δ s
-- ball_mul_one
@[to_additive]
theorem ball_mul_one : ball 1 δ * s = thickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E)
Goal: Metric.ball 1 δ * s = Metric.thickening δ s
-- ball_add_zero
@[to_additive]
theorem ball_mul_one : ball 1 δ * s = thickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E)
Goal: Metric.ball 0 δ + s = Metric.thickening δ s
-- ball_sub_zero
@[to_additive]
theorem ball_div_one : ball 1 δ / s = thickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E)
Goal: Metric.ball 0 δ - s = Metric.thickening δ (-s)
-- ball_div_one
@[to_additive]
theorem ball_div_one : ball 1 δ / s = thickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E)
Goal: Metric.ball 1 δ / s = Metric.thickening δ s⁻¹
-- mul_ball
@[to_additive (attr := simp)]
theorem mul_ball : s * ball x δ = x • thickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: s * Metric.ball x δ = x • Metric.thickening δ s
-- add_ball
@[to_additive (attr := simp)]
theorem mul_ball : s * ball x δ = x • thickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: s + Metric.ball x δ = x +ᵥ Metric.thickening δ s
-- div_ball
@[to_additive (attr := simp)]
theorem div_ball : s / ball x δ = x⁻¹ • thickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: s / Metric.ball x δ = x⁻¹ • Metric.thickening δ s
-- sub_ball
@[to_additive (attr := simp)]
theorem div_ball : s / ball x δ = x⁻¹ • thickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: s - Metric.ball x δ = -x +ᵥ Metric.thickening δ s
-- ball_add
@[to_additive (attr := simp)]
theorem ball_mul : ball x δ * s = x • thickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: Metric.ball x δ + s = x +ᵥ Metric.thickening δ s
-- ball_mul
@[to_additive (attr := simp)]
theorem ball_mul : ball x δ * s = x • thickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: Metric.ball x δ * s = x • Metric.thickening δ s
-- ball_div
@[to_additive (attr := simp)]
theorem ball_div : ball x δ / s = x • thickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: Metric.ball x δ / s = x • Metric.thickening δ s⁻¹
-- ball_sub
@[to_additive (attr := simp)]
theorem ball_div : ball x δ / s = x • thickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] (δ : ℝ) (s : Set E) (x : E)
Goal: Metric.ball x δ - s = x +ᵥ Metric.thickening δ (-s)
-- IsCompact.mul_closedBall_one
@[to_additive]
theorem IsCompact.mul_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s * closedBall (1 : E) δ = cthickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: s * Metric.closedBall 1 δ = Metric.cthickening δ s
-- IsCompact.add_closedBall_zero
@[to_additive]
theorem IsCompact.mul_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s * closedBall (1 : E) δ = cthickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: s + Metric.closedBall 0 δ = Metric.cthickening δ s
-- IsCompact.div_closedBall_one
@[to_additive]
theorem IsCompact.div_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s / closedBall 1 δ = cthickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: s / Metric.closedBall 1 δ = Metric.cthickening δ s
-- IsCompact.sub_closedBall_zero
@[to_additive]
theorem IsCompact.div_closedBall_one (hs : IsCompact s) (hδ : 0 ≤ δ) :
    s / closedBall 1 δ = cthickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: s - Metric.closedBall 0 δ = Metric.cthickening δ s
-- IsCompact.closedBall_one_mul
@[to_additive]
theorem IsCompact.closedBall_one_mul (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ * s = cthickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: Metric.closedBall 1 δ * s = Metric.cthickening δ s
-- IsCompact.closedBall_zero_add
@[to_additive]
theorem IsCompact.closedBall_one_mul (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ * s = cthickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: Metric.closedBall 0 δ + s = Metric.cthickening δ s
-- IsCompact.closedBall_one_div
@[to_additive]
theorem IsCompact.closedBall_one_div (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ / s = cthickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: Metric.closedBall 1 δ / s = Metric.cthickening δ s⁻¹
-- IsCompact.closedBall_zero_sub
@[to_additive]
theorem IsCompact.closedBall_one_div (hs : IsCompact s) (hδ : 0 ≤ δ) :
    closedBall 1 δ / s = cthickening δ s⁻¹ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ)
Goal: Metric.closedBall 0 δ - s = Metric.cthickening δ (-s)
-- IsCompact.mul_closedBall
@[to_additive]
theorem IsCompact.mul_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s * closedBall x δ = x • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: s * Metric.closedBall x δ = x • Metric.cthickening δ s
-- IsCompact.add_closedBall
@[to_additive]
theorem IsCompact.mul_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s * closedBall x δ = x • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: s + Metric.closedBall x δ = x +ᵥ Metric.cthickening δ s
-- IsCompact.sub_closedBall
@[to_additive]
theorem IsCompact.div_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s / closedBall x δ = x⁻¹ • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: s - Metric.closedBall x δ = -x +ᵥ Metric.cthickening δ s
-- IsCompact.div_closedBall
@[to_additive]
theorem IsCompact.div_closedBall (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    s / closedBall x δ = x⁻¹ • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: s / Metric.closedBall x δ = x⁻¹ • Metric.cthickening δ s
-- IsCompact.closedBall_add
@[to_additive]
theorem IsCompact.closedBall_mul (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: Metric.closedBall x δ + s = x +ᵥ Metric.cthickening δ s
-- IsCompact.closedBall_mul
@[to_additive]
theorem IsCompact.closedBall_mul (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: Metric.closedBall x δ * s = x • Metric.cthickening δ s
-- IsCompact.closedBall_sub
@[to_additive]
theorem IsCompact.closedBall_div (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: Metric.closedBall x δ + s = x +ᵥ Metric.cthickening δ s
-- IsCompact.closedBall_div
@[to_additive]
theorem IsCompact.closedBall_div (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
    closedBall x δ * s = x • cthickening δ s 
Arguments: {E : Type u_1} [SeminormedCommGroup E] {δ : ℝ} {s : Set E} (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E)
Goal: Metric.closedBall x δ * s = x • Metric.cthickening δ s
-- NormedAddGroupHom.bound'
structure NormedAddGroupHom (V W : Type*) [SeminormedAddCommGroup V]
  [SeminormedAddCommGroup W] where
    toFun : V → W
    map_add' : ∀ v₁ v₂, toFun (v₁ + v₂) = toFun v₁ + toFun v₂
    bound' : ∃ C, ∀ v, ‖toFun v‖ ≤ C * ‖v‖
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] (self : NormedAddGroupHom V W)
Goal: ∃ C, ∀ (v : V), ‖self.toFun v‖ ≤ C * ‖v‖
-- NormedAddGroupHom.map_add'
structure NormedAddGroupHom (V W : Type*) [SeminormedAddCommGroup V]
  [SeminormedAddCommGroup W] where
    toFun : V → W
    map_add' : ∀ v₁ v₂, toFun (v₁ + v₂) = toFun v₁ + toFun v₂
    bound' : ∃ C, ∀ v, ‖toFun v‖ ≤ C * ‖v‖
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] (self : NormedAddGroupHom V W) (v₁ v₂ : V)
Goal: self.toFun (v₁ + v₂) = self.toFun v₁ + self.toFun v₂
-- exists_pos_bound_of_bound
theorem exists_pos_bound_of_bound {V W : Type*} [SeminormedAddCommGroup V]
    [SeminormedAddCommGroup W] {f : V → W} (M : ℝ) (h : ∀ x, ‖f x‖ ≤ M * ‖x‖) :
    ∃ N, 0 < N ∧ ∀ x, ‖f x‖ ≤ N * ‖x‖ 
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] {f : V → W} (M : ℝ) (h : ∀ (x : V), ‖f x‖ ≤ M * ‖x‖)
Goal: ∃ N, 0 < N ∧ ∀ (x : V), ‖f x‖ ≤ N * ‖x‖
-- NormedAddGroupHom.coe_inj
theorem coe_inj (H : (f : V₁ → V₂) = g) : f = g 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f g : NormedAddGroupHom V₁ V₂} (H : ⇑f = ⇑g)
Goal: f = g
-- NormedAddGroupHom.coe_injective
theorem coe_injective : @Function.Injective (NormedAddGroupHom V₁ V₂) (V₁ → V₂) toFun 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂]
Goal: Function.Injective NormedAddGroupHom.toFun