ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Metric.dense_iff
theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: Dense s ↔ ∀ (x : α), ∀ r > 0, (Metric.ball x r ∩ s).Nonempty
-- Metric.denseRange_iff
theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α}
Goal: DenseRange f ↔ ∀ (x : α), ∀ r > 0, ∃ y, dist x (f y) < r
-- Inducing.isSeparable_preimage
protected theorem _root_.Inducing.isSeparable_preimage {f : β → α} [TopologicalSpace β]
    (hf : Inducing f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} [TopologicalSpace β] (hf : Inducing f) {s : Set α} (hs : TopologicalSpace.IsSeparable s)
Goal: TopologicalSpace.IsSeparable (f ⁻¹' s)
-- Embedding.isSeparable_preimage
protected theorem _root_.Embedding.isSeparable_preimage {f : β → α} [TopologicalSpace β]
    (hf : Embedding f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} [TopologicalSpace β] (hf : Embedding f) {s : Set α} (hs : TopologicalSpace.IsSeparable s)
Goal: TopologicalSpace.IsSeparable (f ⁻¹' s)
-- ContinuousOn.isSeparable_image
theorem _root_.ContinuousOn.isSeparable_image [TopologicalSpace β] {f : α → β} {s : Set α}
    (hf : ContinuousOn f s) (hs : IsSeparable s) : IsSeparable (f '' s) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hf : ContinuousOn f s) (hs : TopologicalSpace.IsSeparable s)
Goal: TopologicalSpace.IsSeparable (f '' s)
-- IsCompact.isSeparable
theorem IsCompact.isSeparable {s : Set α} (hs : IsCompact s) : IsSeparable s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsCompact s)
Goal: TopologicalSpace.IsSeparable s
-- finite_cover_balls_of_compact
theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s : Set α}
    (hs : IsCompact s) {e : ℝ} (he : 0 < e) :
    ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ x ∈ t, ball x e 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsCompact s) {e : ℝ} (he : 0 < e)
Goal: ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ x ∈ t, Metric.ball x e
-- Metric.secondCountable_of_almost_dense_set
theorem secondCountable_of_almost_dense_set
    (H : ∀ ε > (0 : ℝ), ∃ s : Set α, s.Countable ∧ ∀ x, ∃ y ∈ s, dist x y ≤ ε) :
    SecondCountableTopology α 
Arguments: {α : Type u} [PseudoMetricSpace α] (H : ∀ ε > 0, ∃ s, s.Countable ∧ ∀ (x : α), ∃ y ∈ s, dist x y ≤ ε)
Goal: SecondCountableTopology α
-- lebesgue_number_lemma_of_metric
theorem lebesgue_number_lemma_of_metric {s : Set α} {ι : Sort*} {c : ι → Set α} (hs : IsCompact s)
    (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {ι : Sort u_3} {c : ι → Set α} (hs : IsCompact s) (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i)
Goal: ∃ δ > 0, ∀ x ∈ s, ∃ i, Metric.ball x δ ⊆ c i
-- lebesgue_number_lemma_of_metric_sUnion
theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
    (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {c : Set (Set α)} (hs : IsCompact s) (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c)
Goal: ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, Metric.ball x δ ⊆ t
-- TopCat.isSheaf_of_isLimit
theorem isSheaf_of_isLimit [HasLimits C] {X : TopCat} (F : J ⥤ Presheaf.{v} C X)
    (H : ∀ j, (F.obj j).IsSheaf) {c : Cone F} (hc : IsLimit c) : c.pt.IsSheaf 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.Limits.HasLimits C] {X : TopCat} (F : J ⥤ TopCat.Presheaf C X) (H : ∀ (j : J), (F.obj j).IsSheaf) {c : CategoryTheory.Limits.Cone F} (hc : CategoryTheory.Limits.IsLimit c)
Goal: c.pt.IsSheaf
-- TopCat.limit_isSheaf
theorem limit_isSheaf [HasLimits C] {X : TopCat} (F : J ⥤ Presheaf.{v} C X)
    (H : ∀ j, (F.obj j).IsSheaf) : (limit F).IsSheaf 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.Limits.HasLimits C] {X : TopCat} (F : J ⥤ TopCat.Presheaf C X) (H : ∀ (j : J), (F.obj j).IsSheaf)
Goal: (CategoryTheory.Limits.limit F).IsSheaf
-- TopCat.Presheaf.isSheaf_iff_isSheaf_comp'
theorem isSheaf_iff_isSheaf_comp' {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
    (G : C ⥤ D) [G.ReflectsIsomorphisms] [HasLimitsOfSize.{v, v} C] [PreservesLimitsOfSize.{v, v} G]
    {X : TopCat.{v}} (F : Presheaf C X) : Presheaf.IsSheaf F ↔ Presheaf.IsSheaf (F ⋙ G) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : C ⥤ D) [G.ReflectsIsomorphisms] [CategoryTheory.Limits.HasLimitsOfSize.{v, v, v₁, u₁} C] [CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, v₁, v₂, u₁, u₂} G] {X : TopCat} (F : TopCat.Presheaf C X)
Goal: F.IsSheaf ↔ TopCat.Presheaf.IsSheaf (F ⋙ G)
-- TopCat.Presheaf.isSheaf_iff_isSheaf_comp
theorem isSheaf_iff_isSheaf_comp {C : Type u₁} [Category.{v} C] {D : Type u₂} [Category.{v} D]
    (G : C ⥤ D) [G.ReflectsIsomorphisms] [HasLimits C] [PreservesLimits G]
    {X : TopCat.{v}} (F : Presheaf C X) : Presheaf.IsSheaf F ↔ Presheaf.IsSheaf (F ⋙ G) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v, u₂} D] (G : C ⥤ D) [G.ReflectsIsomorphisms] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.Limits.PreservesLimits G] {X : TopCat} (F : TopCat.Presheaf C X)
Goal: F.IsSheaf ↔ TopCat.Presheaf.IsSheaf (F ⋙ G)
-- IsometricVAdd.isometry_vadd
class IsometricVAdd [PseudoEMetricSpace X] [VAdd M X] : Prop where
  protected isometry_vadd : ∀ c : M, Isometry ((c +ᵥ ·) : X → X)
Arguments: {M : Type u} {X : Type w} [PseudoEMetricSpace X] [VAdd M X] [self : IsometricVAdd M X] (c : M)
Goal: Isometry fun x => c +ᵥ x
-- IsometricSMul.isometry_smul
@[to_additive]
class IsometricSMul [PseudoEMetricSpace X] [SMul M X] : Prop where
  protected isometry_smul : ∀ c : M, Isometry ((c • ·) : X → X)
Arguments: {M : Type u} {X : Type w} [PseudoEMetricSpace X] [SMul M X] [self : IsometricSMul M X] (c : M)
Goal: Isometry fun x => c • x
-- isometry_vadd
@[to_additive]
theorem isometry_smul {M : Type u} (X : Type w) [PseudoEMetricSpace X] [SMul M X]
    [IsometricSMul M X] (c : M) : Isometry (c • · : X → X) 
Arguments: {M : Type u} (X : Type w) [PseudoEMetricSpace X] [VAdd M X] [IsometricVAdd M X] (c : M)
Goal: Isometry fun x => c +ᵥ x
-- isometry_smul
@[to_additive]
theorem isometry_smul {M : Type u} (X : Type w) [PseudoEMetricSpace X] [SMul M X]
    [IsometricSMul M X] (c : M) : Isometry (c • · : X → X) 
Arguments: {M : Type u} (X : Type w) [PseudoEMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M)
Goal: Isometry fun x => c • x
-- edist_smul_left
@[to_additive (attr := simp)]
theorem edist_smul_left [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :
    edist (c • x) (c • y) = edist x y 
Arguments: {M : Type u} {X : Type w} [PseudoEMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X)
Goal: edist (c • x) (c • y) = edist x y
-- edist_vadd_left
@[to_additive (attr := simp)]
theorem edist_smul_left [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :
    edist (c • x) (c • y) = edist x y 
Arguments: {M : Type u} {X : Type w} [PseudoEMetricSpace X] [VAdd M X] [IsometricVAdd M X] (c : M) (x y : X)
Goal: edist (c +ᵥ x) (c +ᵥ y) = edist x y
-- ediam_vadd
@[to_additive (attr := simp)]
theorem ediam_smul [SMul M X] [IsometricSMul M X] (c : M) (s : Set X) :
    EMetric.diam (c • s) = EMetric.diam s 
Arguments: {M : Type u} {X : Type w} [PseudoEMetricSpace X] [VAdd M X] [IsometricVAdd M X] (c : M) (s : Set X)
Goal: EMetric.diam (c +ᵥ s) = EMetric.diam s
-- ediam_smul
@[to_additive (attr := simp)]
theorem ediam_smul [SMul M X] [IsometricSMul M X] (c : M) (s : Set X) :
    EMetric.diam (c • s) = EMetric.diam s 
Arguments: {M : Type u} {X : Type w} [PseudoEMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (s : Set X)
Goal: EMetric.diam (c • s) = EMetric.diam s
-- isometry_add_left
@[to_additive]
theorem isometry_mul_left [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a : M) :
    Isometry (a * ·) 
Arguments: {M : Type u} [Add M] [PseudoEMetricSpace M] [IsometricVAdd M M] (a : M)
Goal: Isometry fun x => a + x
-- isometry_mul_left
@[to_additive]
theorem isometry_mul_left [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a : M) :
    Isometry (a * ·) 
Arguments: {M : Type u} [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a : M)
Goal: Isometry fun x => a * x
-- edist_mul_left
@[to_additive (attr := simp)]
theorem edist_mul_left [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a b c : M) :
    edist (a * b) (a * c) = edist b c 
Arguments: {M : Type u} [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a b c : M)
Goal: edist (a * b) (a * c) = edist b c
-- edist_add_left
@[to_additive (attr := simp)]
theorem edist_mul_left [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a b c : M) :
    edist (a * b) (a * c) = edist b c 
Arguments: {M : Type u} [Add M] [PseudoEMetricSpace M] [IsometricVAdd M M] (a b c : M)
Goal: edist (a + b) (a + c) = edist b c
-- isometry_mul_right
@[to_additive]
theorem isometry_mul_right [Mul M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a : M) :
    Isometry fun x => x * a 
Arguments: {M : Type u} [Mul M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a : M)
Goal: Isometry fun x => x * a
-- isometry_add_right
@[to_additive]
theorem isometry_mul_right [Mul M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a : M) :
    Isometry fun x => x * a 
Arguments: {M : Type u} [Add M] [PseudoEMetricSpace M] [IsometricVAdd Mᵃᵒᵖ M] (a : M)
Goal: Isometry fun x => x + a
-- edist_add_right
@[to_additive (attr := simp)]
theorem edist_mul_right [Mul M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M) :
    edist (a * c) (b * c) = edist a b 
Arguments: {M : Type u} [Add M] [PseudoEMetricSpace M] [IsometricVAdd Mᵃᵒᵖ M] (a b c : M)
Goal: edist (a + c) (b + c) = edist a b
-- edist_mul_right
@[to_additive (attr := simp)]
theorem edist_mul_right [Mul M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M) :
    edist (a * c) (b * c) = edist a b 
Arguments: {M : Type u} [Mul M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M)
Goal: edist (a * c) (b * c) = edist a b
-- edist_sub_right
@[to_additive (attr := simp)]
theorem edist_div_right [DivInvMonoid M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
    (a b c : M) : edist (a / c) (b / c) = edist a b 
Arguments: {M : Type u} [SubNegMonoid M] [PseudoEMetricSpace M] [IsometricVAdd Mᵃᵒᵖ M] (a b c : M)
Goal: edist (a - c) (b - c) = edist a b
-- edist_div_right
@[to_additive (attr := simp)]
theorem edist_div_right [DivInvMonoid M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
    (a b c : M) : edist (a / c) (b / c) = edist a b 
Arguments: {M : Type u} [DivInvMonoid M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M] (a b c : M)
Goal: edist (a / c) (b / c) = edist a b
-- edist_inv_inv
@[to_additive (attr := simp)]
theorem edist_inv_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
    (a b : G) : edist a⁻¹ b⁻¹ = edist a b 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (a b : G)
Goal: edist a⁻¹ b⁻¹ = edist a b
-- edist_neg_neg
@[to_additive (attr := simp)]
theorem edist_inv_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
    (a b : G) : edist a⁻¹ b⁻¹ = edist a b 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (a b : G)
Goal: edist (-a) (-b) = edist a b
-- isometry_neg
@[to_additive]
theorem isometry_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] :
    Isometry (Inv.inv : G → G) 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G]
Goal: Isometry Neg.neg
-- isometry_inv
@[to_additive]
theorem isometry_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] :
    Isometry (Inv.inv : G → G) 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
Goal: Isometry Inv.inv
-- edist_inv
@[to_additive]
theorem edist_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
    (x y : G) : edist x⁻¹ y = edist x y⁻¹ 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (x y : G)
Goal: edist x⁻¹ y = edist x y⁻¹
-- edist_neg
@[to_additive]
theorem edist_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
    (x y : G) : edist x⁻¹ y = edist x y⁻¹ 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (x y : G)
Goal: edist (-x) y = edist x (-y)
-- edist_sub_left
@[to_additive (attr := simp)]
theorem edist_div_left [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
    (a b c : G) : edist (a / b) (a / c) = edist b c 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (a b c : G)
Goal: edist (a - b) (a - c) = edist b c
-- edist_div_left
@[to_additive (attr := simp)]
theorem edist_div_left [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
    (a b c : G) : edist (a / b) (a / c) = edist b c 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (a b c : G)
Goal: edist (a / b) (a / c) = edist b c
-- IsometryEquiv.constVAdd_apply
@[to_additive (attr := simps! toEquiv apply) "If an additive group `G` acts on `X` by isometries,
then `IsometryEquiv.constVAdd` is the isometry of `X` given by addition of a constant element of the
group."]
def constSMul (c : G) : X ≃ᵢ X 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G) (x : X)
Goal: (IsometryEquiv.constVAdd c) x = c +ᵥ x
-- IsometryEquiv.constSMul_toEquiv
@[to_additive (attr := simps! toEquiv apply) "If an additive group `G` acts on `X` by isometries,
then `IsometryEquiv.constVAdd` is the isometry of `X` given by addition of a constant element of the
group."]
def constSMul (c : G) : X ≃ᵢ X 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G)
Goal: (IsometryEquiv.constSMul c).toEquiv = MulAction.toPerm c
-- IsometryEquiv.constVAdd_toEquiv
@[to_additive (attr := simps! toEquiv apply) "If an additive group `G` acts on `X` by isometries,
then `IsometryEquiv.constVAdd` is the isometry of `X` given by addition of a constant element of the
group."]
def constSMul (c : G) : X ≃ᵢ X 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G)
Goal: (IsometryEquiv.constVAdd c).toEquiv = AddAction.toPerm c
-- IsometryEquiv.constSMul_apply
@[to_additive (attr := simps! toEquiv apply) "If an additive group `G` acts on `X` by isometries,
then `IsometryEquiv.constVAdd` is the isometry of `X` given by addition of a constant element of the
group."]
def constSMul (c : G) : X ≃ᵢ X 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G) (x : X)
Goal: (IsometryEquiv.constSMul c) x = c • x
-- IsometryEquiv.constVAdd_symm
@[to_additive (attr := simp)]
theorem constSMul_symm (c : G) : (constSMul c : X ≃ᵢ X).symm = constSMul c⁻¹ 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [AddGroup G] [AddAction G X] [IsometricVAdd G X] (c : G)
Goal: (IsometryEquiv.constVAdd c).symm = IsometryEquiv.constVAdd (-c)
-- IsometryEquiv.constSMul_symm
@[to_additive (attr := simp)]
theorem constSMul_symm (c : G) : (constSMul c : X ≃ᵢ X).symm = constSMul c⁻¹ 
Arguments: {G : Type v} {X : Type w} [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X] (c : G)
Goal: (IsometryEquiv.constSMul c).symm = IsometryEquiv.constSMul c⁻¹
-- IsometryEquiv.addLeft_toEquiv
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ x + y` as an `IsometryEquiv`."]
def mulLeft [IsometricSMul G G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] (c : G)
Goal: (IsometryEquiv.addLeft c).toEquiv = Equiv.addLeft c
-- IsometryEquiv.mulLeft_apply
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ x + y` as an `IsometryEquiv`."]
def mulLeft [IsometricSMul G G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] (c x : G)
Goal: (IsometryEquiv.mulLeft c) x = c * x
-- IsometryEquiv.addLeft_apply
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ x + y` as an `IsometryEquiv`."]
def mulLeft [IsometricSMul G G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] (c x : G)
Goal: (IsometryEquiv.addLeft c) x = c + x
-- IsometryEquiv.mulLeft_toEquiv
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ x + y` as an `IsometryEquiv`."]
def mulLeft [IsometricSMul G G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] (c : G)
Goal: (IsometryEquiv.mulLeft c).toEquiv = Equiv.mulLeft c
-- IsometryEquiv.addLeft_symm
@[to_additive (attr := simp)]
theorem mulLeft_symm [IsometricSMul G G] (x : G) :
    (mulLeft x).symm = IsometryEquiv.mulLeft x⁻¹ 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] (x : G)
Goal: (IsometryEquiv.addLeft x).symm = IsometryEquiv.addLeft (-x)
-- IsometryEquiv.mulLeft_symm
@[to_additive (attr := simp)]
theorem mulLeft_symm [IsometricSMul G G] (x : G) :
    (mulLeft x).symm = IsometryEquiv.mulLeft x⁻¹ 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] (x : G)
Goal: (IsometryEquiv.mulLeft x).symm = IsometryEquiv.mulLeft x⁻¹
-- IsometryEquiv.addRight_toEquiv
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ y + x` as an `IsometryEquiv`."]
def mulRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (c : G)
Goal: (IsometryEquiv.addRight c).toEquiv = Equiv.addRight c
-- IsometryEquiv.mulRight_apply
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ y + x` as an `IsometryEquiv`."]
def mulRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (c x : G)
Goal: (IsometryEquiv.mulRight c) x = x * c
-- IsometryEquiv.mulRight_toEquiv
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ y + x` as an `IsometryEquiv`."]
def mulRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (c : G)
Goal: (IsometryEquiv.mulRight c).toEquiv = Equiv.mulRight c
-- IsometryEquiv.addRight_apply
@[to_additive (attr := simps! apply toEquiv) "Addition `y ↦ y + x` as an `IsometryEquiv`."]
def mulRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (c x : G)
Goal: (IsometryEquiv.addRight c) x = x + c
-- IsometryEquiv.mulRight_symm
@[to_additive (attr := simp)]
theorem mulRight_symm [IsometricSMul Gᵐᵒᵖ G] (x : G) : (mulRight x).symm = mulRight x⁻¹ 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (x : G)
Goal: (IsometryEquiv.mulRight x).symm = IsometryEquiv.mulRight x⁻¹
-- IsometryEquiv.addRight_symm
@[to_additive (attr := simp)]
theorem mulRight_symm [IsometricSMul Gᵐᵒᵖ G] (x : G) : (mulRight x).symm = mulRight x⁻¹ 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (x : G)
Goal: (IsometryEquiv.addRight x).symm = IsometryEquiv.addRight (-x)
-- IsometryEquiv.divRight_apply
@[to_additive (attr := simps! apply toEquiv) "Subtraction `y ↦ y - x` as an `IsometryEquiv`."]
def divRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (c b : G)
Goal: (IsometryEquiv.divRight c) b = b / c
-- IsometryEquiv.subRight_toEquiv
@[to_additive (attr := simps! apply toEquiv) "Subtraction `y ↦ y - x` as an `IsometryEquiv`."]
def divRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (c : G)
Goal: (IsometryEquiv.subRight c).toEquiv = Equiv.subRight c
-- IsometryEquiv.subRight_apply
@[to_additive (attr := simps! apply toEquiv) "Subtraction `y ↦ y - x` as an `IsometryEquiv`."]
def divRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (c b : G)
Goal: (IsometryEquiv.subRight c) b = b - c
-- IsometryEquiv.divRight_toEquiv
@[to_additive (attr := simps! apply toEquiv) "Subtraction `y ↦ y - x` as an `IsometryEquiv`."]
def divRight [IsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (c : G)
Goal: (IsometryEquiv.divRight c).toEquiv = Equiv.divRight c
-- IsometryEquiv.divRight_symm
@[to_additive (attr := simp)]
theorem divRight_symm [IsometricSMul Gᵐᵒᵖ G] (c : G) : (divRight c).symm = mulRight c 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul Gᵐᵒᵖ G] (c : G)
Goal: (IsometryEquiv.divRight c).symm = IsometryEquiv.mulRight c
-- IsometryEquiv.subRight_symm
@[to_additive (attr := simp)]
theorem divRight_symm [IsometricSMul Gᵐᵒᵖ G] (c : G) : (divRight c).symm = mulRight c 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd Gᵃᵒᵖ G] (c : G)
Goal: (IsometryEquiv.subRight c).symm = IsometryEquiv.addRight c
-- IsometryEquiv.subLeft_toEquiv
@[to_additive (attr := simps! apply symm_apply toEquiv)
  "Subtraction `y ↦ x - y` as an `IsometryEquiv`."]
def divLeft (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (c : G)
Goal: (IsometryEquiv.subLeft c).toEquiv = Equiv.subLeft c
-- IsometryEquiv.divLeft_toEquiv
@[to_additive (attr := simps! apply symm_apply toEquiv)
  "Subtraction `y ↦ x - y` as an `IsometryEquiv`."]
def divLeft (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (c : G)
Goal: (IsometryEquiv.divLeft c).toEquiv = Equiv.divLeft c
-- IsometryEquiv.subLeft_symm_apply
@[to_additive (attr := simps! apply symm_apply toEquiv)
  "Subtraction `y ↦ x - y` as an `IsometryEquiv`."]
def divLeft (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (c b : G)
Goal: (IsometryEquiv.subLeft c).symm b = -b + c
-- IsometryEquiv.divLeft_apply
@[to_additive (attr := simps! apply symm_apply toEquiv)
  "Subtraction `y ↦ x - y` as an `IsometryEquiv`."]
def divLeft (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (c b : G)
Goal: (IsometryEquiv.divLeft c) b = c / b
-- IsometryEquiv.subLeft_apply
@[to_additive (attr := simps! apply symm_apply toEquiv)
  "Subtraction `y ↦ x - y` as an `IsometryEquiv`."]
def divLeft (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsometricVAdd G G] [IsometricVAdd Gᵃᵒᵖ G] (c b : G)
Goal: (IsometryEquiv.subLeft c) b = c - b
-- IsometryEquiv.divLeft_symm_apply
@[to_additive (attr := simps! apply symm_apply toEquiv)
  "Subtraction `y ↦ x - y` as an `IsometryEquiv`."]
def divLeft (c : G) : G ≃ᵢ G 
Arguments: {G : Type v} [Group G] [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G] (c b : G)
Goal: (IsometryEquiv.divLeft c).symm b = b⁻¹ * c
-- IsometryEquiv.neg_toEquiv
@[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: (IsometryEquiv.neg G).toEquiv = Equiv.neg G
-- IsometryEquiv.inv_toEquiv
@[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: (IsometryEquiv.inv G).toEquiv = Equiv.inv G