ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsometryEquiv.neg_apply
@[to_additive (attr := simps! apply toEquiv) "Negation `x ↦ -x` as an `IsometryEquiv`."]
def inv : G ≃ᵢ G 
Arguments: (G : Type v) [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G]
Goal: ∀ (a : G), (IsometryEquiv.neg G) a = -a
-- IsometryEquiv.inv_apply
@[to_additive (attr := simps! apply toEquiv) "Negation `x ↦ -x` as an `IsometryEquiv`."]
def inv : G ≃ᵢ G 
Arguments: (G : Type v) [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
Goal: ∀ (a : G), (IsometryEquiv.inv G) a = a⁻¹
-- IsometryEquiv.inv_symm
@[to_additive (attr := simp)] theorem inv_symm : (inv G).symm = inv G 
Arguments: (G : Type v) [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
Goal: (IsometryEquiv.inv G).symm = IsometryEquiv.inv G
-- IsometryEquiv.neg_symm
@[to_additive (attr := simp)] theorem inv_symm : (inv G).symm = inv G 
Arguments: (G : Type v) [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G]
Goal: (IsometryEquiv.neg G).symm = IsometryEquiv.neg G
-- EMetric.vadd_ball
@[to_additive (attr := simp)]
theorem smul_ball (c : G) (x : X) (r : ℝ≥0∞) : c • ball x r = ball (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: c +ᵥ EMetric.ball x r = EMetric.ball (c +ᵥ x) r
-- EMetric.smul_ball
@[to_additive (attr := simp)]
theorem smul_ball (c : G) (x : X) (r : ℝ≥0∞) : c • ball x r = ball (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: c • EMetric.ball x r = EMetric.ball (c • x) r
-- EMetric.preimage_vadd_ball
@[to_additive (attr := simp)]
theorem preimage_smul_ball (c : G) (x : X) (r : ℝ≥0∞) :
    (c • ·) ⁻¹' ball x r = ball (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: (fun x => c +ᵥ x) ⁻¹' EMetric.ball x r = EMetric.ball (-c +ᵥ x) r
-- EMetric.preimage_smul_ball
@[to_additive (attr := simp)]
theorem preimage_smul_ball (c : G) (x : X) (r : ℝ≥0∞) :
    (c • ·) ⁻¹' ball x r = ball (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: (fun x => c • x) ⁻¹' EMetric.ball x r = EMetric.ball (c⁻¹ • x) r
-- EMetric.smul_closedBall
@[to_additive (attr := simp)]
theorem smul_closedBall (c : G) (x : X) (r : ℝ≥0∞) : c • closedBall x r = closedBall (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: c • EMetric.closedBall x r = EMetric.closedBall (c • x) r
-- EMetric.vadd_closedBall
@[to_additive (attr := simp)]
theorem smul_closedBall (c : G) (x : X) (r : ℝ≥0∞) : c • closedBall x r = closedBall (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: c +ᵥ EMetric.closedBall x r = EMetric.closedBall (c +ᵥ x) r
-- EMetric.preimage_smul_closedBall
@[to_additive (attr := simp)]
theorem preimage_smul_closedBall (c : G) (x : X) (r : ℝ≥0∞) :
    (c • ·) ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: (fun x => c • x) ⁻¹' EMetric.closedBall x r = EMetric.closedBall (c⁻¹ • x) r
-- EMetric.preimage_vadd_closedBall
@[to_additive (attr := simp)]
theorem preimage_smul_closedBall (c : G) (x : X) (r : ℝ≥0∞) :
    (c • ·) ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ≥0∞)
Goal: (fun x => c +ᵥ x) ⁻¹' EMetric.closedBall x r = EMetric.closedBall (-c +ᵥ x) r
-- EMetric.preimage_add_left_ball
@[to_additive (attr := simp)]
theorem preimage_mul_left_ball [IsometricSMul G G] (a b : G) (r : ℝ≥0∞) :
    (a * ·) ⁻¹' ball b r = ball (a⁻¹ * b) r 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => a + x) ⁻¹' EMetric.ball b r = EMetric.ball (-a + b) r
-- EMetric.preimage_mul_left_ball
@[to_additive (attr := simp)]
theorem preimage_mul_left_ball [IsometricSMul G G] (a b : G) (r : ℝ≥0∞) :
    (a * ·) ⁻¹' ball b r = ball (a⁻¹ * b) r 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => a * x) ⁻¹' EMetric.ball b r = EMetric.ball (a⁻¹ * b) r
-- EMetric.preimage_mul_right_ball
@[to_additive (attr := simp)]
theorem preimage_mul_right_ball [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
    (fun x => x * a) ⁻¹' ball b r = ball (b / a) r 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => x * a) ⁻¹' EMetric.ball b r = EMetric.ball (b / a) r
-- EMetric.preimage_add_right_ball
@[to_additive (attr := simp)]
theorem preimage_mul_right_ball [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
    (fun x => x * a) ⁻¹' ball b r = ball (b / a) r 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => x + a) ⁻¹' EMetric.ball b r = EMetric.ball (b - a) r
-- EMetric.preimage_mul_left_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_left_closedBall [IsometricSMul G G] (a b : G) (r : ℝ≥0∞) :
    (a * ·) ⁻¹' closedBall b r = closedBall (a⁻¹ * b) r 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => a * x) ⁻¹' EMetric.closedBall b r = EMetric.closedBall (a⁻¹ * b) r
-- EMetric.preimage_add_left_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_left_closedBall [IsometricSMul G G] (a b : G) (r : ℝ≥0∞) :
    (a * ·) ⁻¹' closedBall b r = closedBall (a⁻¹ * b) r 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => a + x) ⁻¹' EMetric.closedBall b r = EMetric.closedBall (-a + b) r
-- EMetric.preimage_add_right_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_right_closedBall [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
    (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => x + a) ⁻¹' EMetric.closedBall b r = EMetric.closedBall (b - a) r
-- EMetric.preimage_mul_right_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_right_closedBall [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
    (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞)
Goal: (fun x => x * a) ⁻¹' EMetric.closedBall b r = EMetric.closedBall (b / a) r
-- dist_smul
@[to_additive (attr := simp)]
theorem dist_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :
    dist (c • x) (c • y) = dist x y 
Arguments: {M : Type u} {X : Type w} [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X)
Goal: dist (c • x) (c • y) = dist x y
-- dist_vadd
@[to_additive (attr := simp)]
theorem dist_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :
    dist (c • x) (c • y) = dist x y 
Arguments: {M : Type u} {X : Type w} [PseudoMetricSpace X] [VAdd M X] [IsometricVAdd M X] (c : M) (x y : X)
Goal: dist (c +ᵥ x) (c +ᵥ y) = dist x y
-- nndist_smul
@[to_additive (attr := simp)]
theorem nndist_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :
    nndist (c • x) (c • y) = nndist x y 
Arguments: {M : Type u} {X : Type w} [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X)
Goal: nndist (c • x) (c • y) = nndist x y
-- nndist_vadd
@[to_additive (attr := simp)]
theorem nndist_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :
    nndist (c • x) (c • y) = nndist x y 
Arguments: {M : Type u} {X : Type w} [PseudoMetricSpace X] [VAdd M X] [IsometricVAdd M X] (c : M) (x y : X)
Goal: nndist (c +ᵥ x) (c +ᵥ y) = nndist x y
-- diam_vadd
@[to_additive (attr := simp)]
theorem diam_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (s : Set X) :
    Metric.diam (c • s) = Metric.diam s 
Arguments: {M : Type u} {X : Type w} [PseudoMetricSpace X] [VAdd M X] [IsometricVAdd M X] (c : M) (s : Set X)
Goal: Metric.diam (c +ᵥ s) = Metric.diam s
-- diam_smul
@[to_additive (attr := simp)]
theorem diam_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (s : Set X) :
    Metric.diam (c • s) = Metric.diam s 
Arguments: {M : Type u} {X : Type w} [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (s : Set X)
Goal: Metric.diam (c • s) = Metric.diam s
-- dist_add_left
@[to_additive (attr := simp)]
theorem dist_mul_left [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M) :
    dist (a * b) (a * c) = dist b c 
Arguments: {M : Type u} [PseudoMetricSpace M] [Add M] [IsometricVAdd M M] (a b c : M)
Goal: dist (a + b) (a + c) = dist b c
-- dist_mul_left
@[to_additive (attr := simp)]
theorem dist_mul_left [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M) :
    dist (a * b) (a * c) = dist b c 
Arguments: {M : Type u} [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M)
Goal: dist (a * b) (a * c) = dist b c
-- nndist_add_left
@[to_additive (attr := simp)]
theorem nndist_mul_left [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M) :
    nndist (a * b) (a * c) = nndist b c 
Arguments: {M : Type u} [PseudoMetricSpace M] [Add M] [IsometricVAdd M M] (a b c : M)
Goal: nndist (a + b) (a + c) = nndist b c
-- nndist_mul_left
@[to_additive (attr := simp)]
theorem nndist_mul_left [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M) :
    nndist (a * b) (a * c) = nndist b c 
Arguments: {M : Type u} [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M)
Goal: nndist (a * b) (a * c) = nndist b c
-- dist_add_right
@[to_additive (attr := simp)]
theorem dist_mul_right [Mul M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M) :
    dist (a * c) (b * c) = dist a b 
Arguments: {M : Type u} [Add M] [PseudoMetricSpace M] [IsometricVAdd Mᵃᵒᵖ M] (a b c : M)
Goal: dist (a + c) (b + c) = dist a b
-- dist_mul_right
@[to_additive (attr := simp)]
theorem dist_mul_right [Mul M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M) :
    dist (a * c) (b * c) = dist a b 
Arguments: {M : Type u} [Mul M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M)
Goal: dist (a * c) (b * c) = dist a b
-- nndist_add_right
@[to_additive (attr := simp)]
theorem nndist_mul_right [PseudoMetricSpace M] [Mul M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M) :
    nndist (a * c) (b * c) = nndist a b 
Arguments: {M : Type u} [PseudoMetricSpace M] [Add M] [IsometricVAdd Mᵃᵒᵖ M] (a b c : M)
Goal: nndist (a + c) (b + c) = nndist a b
-- nndist_mul_right
@[to_additive (attr := simp)]
theorem nndist_mul_right [PseudoMetricSpace M] [Mul M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M) :
    nndist (a * c) (b * c) = nndist a b 
Arguments: {M : Type u} [PseudoMetricSpace M] [Mul M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M)
Goal: nndist (a * c) (b * c) = nndist a b
-- dist_sub_right
@[to_additive (attr := simp)]
theorem dist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
    (a b c : M) : dist (a / c) (b / c) = dist a b 
Arguments: {M : Type u} [SubNegMonoid M] [PseudoMetricSpace M] [IsometricVAdd Mᵃᵒᵖ M] (a b c : M)
Goal: dist (a - c) (b - c) = dist a b
-- dist_div_right
@[to_additive (attr := simp)]
theorem dist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
    (a b c : M) : dist (a / c) (b / c) = dist a b 
Arguments: {M : Type u} [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M)
Goal: dist (a / c) (b / c) = dist a b
-- nndist_sub_right
@[to_additive (attr := simp)]
theorem nndist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
    (a b c : M) : nndist (a / c) (b / c) = nndist a b 
Arguments: {M : Type u} [SubNegMonoid M] [PseudoMetricSpace M] [IsometricVAdd Mᵃᵒᵖ M] (a b c : M)
Goal: nndist (a - c) (b - c) = nndist a b
-- nndist_div_right
@[to_additive (attr := simp)]
theorem nndist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
    (a b c : M) : nndist (a / c) (b / c) = nndist a b 
Arguments: {M : Type u} [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M)
Goal: nndist (a / c) (b / c) = nndist a b
-- dist_inv_inv
@[to_additive (attr := simp)]
theorem dist_inv_inv [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b : G) : dist a⁻¹ b⁻¹ = dist a b 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (a b : G)
Goal: dist a⁻¹ b⁻¹ = dist a b
-- dist_neg_neg
@[to_additive (attr := simp)]
theorem dist_inv_inv [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b : G) : dist a⁻¹ b⁻¹ = dist a b 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (a b : G)
Goal: dist (-a) (-b) = dist a b
-- nndist_inv_inv
@[to_additive (attr := simp)]
theorem nndist_inv_inv [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b : G) : nndist a⁻¹ b⁻¹ = nndist a b 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (a b : G)
Goal: nndist a⁻¹ b⁻¹ = nndist a b
-- nndist_neg_neg
@[to_additive (attr := simp)]
theorem nndist_inv_inv [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b : G) : nndist a⁻¹ b⁻¹ = nndist a b 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (a b : G)
Goal: nndist (-a) (-b) = nndist a b
-- dist_div_left
@[to_additive (attr := simp)]
theorem dist_div_left [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b c : G) : dist (a / b) (a / c) = dist b c 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (a b c : G)
Goal: dist (a / b) (a / c) = dist b c
-- dist_sub_left
@[to_additive (attr := simp)]
theorem dist_div_left [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b c : G) : dist (a / b) (a / c) = dist b c 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (a b c : G)
Goal: dist (a - b) (a - c) = dist b c
-- nndist_sub_left
@[to_additive (attr := simp)]
theorem nndist_div_left [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b c : G) : nndist (a / b) (a / c) = nndist b c 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (a b c : G)
Goal: nndist (a - b) (a - c) = nndist b c
-- nndist_div_left
@[to_additive (attr := simp)]
theorem nndist_div_left [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
    [IsometricSMul Gᵐᵒᵖ G] (a b c : G) : nndist (a / b) (a / c) = nndist b c 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (a b c : G)
Goal: nndist (a / b) (a / c) = nndist b c
-- Bornology.IsBounded.vadd
@[to_additive "Given an additive isometric action of `G` on `X`, the image of a bounded set in `X`
under translation by `c : G` is bounded"]
theorem Bornology.IsBounded.smul [PseudoMetricSpace X] [SMul G X] [IsometricSMul G X] {s : Set X}
    (hs : IsBounded s) (c : G) : IsBounded (c • s) 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [VAdd G X] [IsometricVAdd G X] {s : Set X} (hs : Bornology.IsBounded s) (c : G)
Goal: Bornology.IsBounded (c +ᵥ s)
-- Bornology.IsBounded.smul
@[to_additive "Given an additive isometric action of `G` on `X`, the image of a bounded set in `X`
under translation by `c : G` is bounded"]
theorem Bornology.IsBounded.smul [PseudoMetricSpace X] [SMul G X] [IsometricSMul G X] {s : Set X}
    (hs : IsBounded s) (c : G) : IsBounded (c • s) 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [SMul G X] [IsometricSMul G X] {s : Set X} (hs : Bornology.IsBounded s) (c : G)
Goal: Bornology.IsBounded (c • s)
-- Metric.smul_ball
@[to_additive (attr := simp)]
theorem smul_ball (c : G) (x : X) (r : ℝ) : c • ball x r = ball (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ)
Goal: c • Metric.ball x r = Metric.ball (c • x) r
-- Metric.vadd_ball
@[to_additive (attr := simp)]
theorem smul_ball (c : G) (x : X) (r : ℝ) : c • ball x r = ball (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ)
Goal: c +ᵥ Metric.ball x r = Metric.ball (c +ᵥ x) r
-- Metric.preimage_smul_ball
@[to_additive (attr := simp)]
theorem preimage_smul_ball (c : G) (x : X) (r : ℝ) : (c • ·) ⁻¹' ball x r = ball (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ)
Goal: (fun x => c • x) ⁻¹' Metric.ball x r = Metric.ball (c⁻¹ • x) r
-- Metric.preimage_vadd_ball
@[to_additive (attr := simp)]
theorem preimage_smul_ball (c : G) (x : X) (r : ℝ) : (c • ·) ⁻¹' ball x r = ball (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ)
Goal: (fun x => c +ᵥ x) ⁻¹' Metric.ball x r = Metric.ball (-c +ᵥ x) r
-- Metric.vadd_closedBall
@[to_additive (attr := simp)]
theorem smul_closedBall (c : G) (x : X) (r : ℝ) : c • closedBall x r = closedBall (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ)
Goal: c +ᵥ Metric.closedBall x r = Metric.closedBall (c +ᵥ x) r
-- Metric.smul_closedBall
@[to_additive (attr := simp)]
theorem smul_closedBall (c : G) (x : X) (r : ℝ) : c • closedBall x r = closedBall (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ)
Goal: c • Metric.closedBall x r = Metric.closedBall (c • x) r
-- Metric.preimage_smul_closedBall
@[to_additive (attr := simp)]
theorem preimage_smul_closedBall (c : G) (x : X) (r : ℝ) :
    (c • ·) ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ)
Goal: (fun x => c • x) ⁻¹' Metric.closedBall x r = Metric.closedBall (c⁻¹ • x) r
-- Metric.preimage_vadd_closedBall
@[to_additive (attr := simp)]
theorem preimage_smul_closedBall (c : G) (x : X) (r : ℝ) :
    (c • ·) ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ)
Goal: (fun x => c +ᵥ x) ⁻¹' Metric.closedBall x r = Metric.closedBall (-c +ᵥ x) r
-- Metric.smul_sphere
@[to_additive (attr := simp)]
theorem smul_sphere (c : G) (x : X) (r : ℝ) : c • sphere x r = sphere (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ)
Goal: c • Metric.sphere x r = Metric.sphere (c • x) r
-- Metric.vadd_sphere
@[to_additive (attr := simp)]
theorem smul_sphere (c : G) (x : X) (r : ℝ) : c • sphere x r = sphere (c • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ)
Goal: c +ᵥ Metric.sphere x r = Metric.sphere (c +ᵥ x) r
-- Metric.preimage_smul_sphere
@[to_additive (attr := simp)]
theorem preimage_smul_sphere (c : G) (x : X) (r : ℝ) :
    (c • ·) ⁻¹' sphere x r = sphere (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X) (r : ℝ)
Goal: (fun x => c • x) ⁻¹' Metric.sphere x r = Metric.sphere (c⁻¹ • x) r
-- Metric.preimage_vadd_sphere
@[to_additive (attr := simp)]
theorem preimage_smul_sphere (c : G) (x : X) (r : ℝ) :
    (c • ·) ⁻¹' sphere x r = sphere (c⁻¹ • x) r 
Arguments: {G : Type v} {X : Type w} [PseudoMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X) (r : ℝ)
Goal: (fun x => c +ᵥ x) ⁻¹' Metric.sphere x r = Metric.sphere (-c +ᵥ x) r
-- Metric.preimage_mul_left_ball
@[to_additive (attr := simp)]
theorem preimage_mul_left_ball [IsometricSMul G G] (a b : G) (r : ℝ) :
    (a * ·) ⁻¹' ball b r = ball (a⁻¹ * b) r 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul G G] (a b : G) (r : ℝ)
Goal: (fun x => a * x) ⁻¹' Metric.ball b r = Metric.ball (a⁻¹ * b) r
-- Metric.preimage_add_left_ball
@[to_additive (attr := simp)]
theorem preimage_mul_left_ball [IsometricSMul G G] (a b : G) (r : ℝ) :
    (a * ·) ⁻¹' ball b r = ball (a⁻¹ * b) r 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd G G] (a b : G) (r : ℝ)
Goal: (fun x => a + x) ⁻¹' Metric.ball b r = Metric.ball (-a + b) r
-- Metric.preimage_add_right_ball
@[to_additive (attr := simp)]
theorem preimage_mul_right_ball [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
    (fun x => x * a) ⁻¹' ball b r = ball (b / a) r 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (a b : G) (r : ℝ)
Goal: (fun x => x + a) ⁻¹' Metric.ball b r = Metric.ball (b - a) r
-- Metric.preimage_mul_right_ball
@[to_additive (attr := simp)]
theorem preimage_mul_right_ball [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
    (fun x => x * a) ⁻¹' ball b r = ball (b / a) r 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ)
Goal: (fun x => x * a) ⁻¹' Metric.ball b r = Metric.ball (b / a) r
-- Metric.preimage_add_left_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_left_closedBall [IsometricSMul G G] (a b : G) (r : ℝ) :
    (a * ·) ⁻¹' closedBall b r = closedBall (a⁻¹ * b) r 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd G G] (a b : G) (r : ℝ)
Goal: (fun x => a + x) ⁻¹' Metric.closedBall b r = Metric.closedBall (-a + b) r
-- Metric.preimage_mul_left_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_left_closedBall [IsometricSMul G G] (a b : G) (r : ℝ) :
    (a * ·) ⁻¹' closedBall b r = closedBall (a⁻¹ * b) r 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul G G] (a b : G) (r : ℝ)
Goal: (fun x => a * x) ⁻¹' Metric.closedBall b r = Metric.closedBall (a⁻¹ * b) r
-- Metric.preimage_add_right_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_right_closedBall [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
    (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r 
Arguments: {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (a b : G) (r : ℝ)
Goal: (fun x => x + a) ⁻¹' Metric.closedBall b r = Metric.closedBall (b - a) r
-- Metric.preimage_mul_right_closedBall
@[to_additive (attr := simp)]
theorem preimage_mul_right_closedBall [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
    (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r 
Arguments: {G : Type v} [Group G] [PseudoMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ)
Goal: (fun x => x * a) ⁻¹' Metric.closedBall b r = Metric.closedBall (b / a) r
-- Set.le_einfsep_iff
theorem le_einfsep_iff {d} :
    d ≤ s.einfsep ↔ ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ edist x y 
Arguments: {α : Type u_1} [EDist α] {s : Set α} {d : ENNReal}
Goal: d ≤ s.einfsep ↔ ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ edist x y
-- Set.einfsep_zero
theorem einfsep_zero : s.einfsep = 0 ↔ ∀ C > 0, ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < C 
Arguments: {α : Type u_1} [EDist α] {s : Set α}
Goal: s.einfsep = 0 ↔ ∀ C > 0, ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < C
-- Set.einfsep_pos
theorem einfsep_pos : 0 < s.einfsep ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y 
Arguments: {α : Type u_1} [EDist α] {s : Set α}
Goal: 0 < s.einfsep ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y
-- Set.einfsep_top
theorem einfsep_top :
    s.einfsep = ∞ ↔ ∀ x ∈ s, ∀ y ∈ s, x ≠ y → edist x y = ∞ 
Arguments: {α : Type u_1} [EDist α] {s : Set α}
Goal: s.einfsep = ⊤ ↔ ∀ x ∈ s, ∀ y ∈ s, x ≠ y → edist x y = ⊤