ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.IsAddRightInvariant.map_add_right_eq_self
class IsAddRightInvariant [Add G] (μ : Measure G) : Prop where
  map_add_right_eq_self : ∀ g : G, map (· + g) μ = μ
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [self : μ.IsAddRightInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => x + g) μ = μ
-- MeasureTheory.Measure.IsMulRightInvariant.map_mul_right_eq_self
@[to_additive existing]
class IsMulRightInvariant [Mul G] (μ : Measure G) : Prop where
  map_mul_right_eq_self : ∀ g : G, map (· * g) μ = μ
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [self : μ.IsMulRightInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => x * g) μ = μ
-- MeasureTheory.map_add_left_eq_self
@[to_additive]
theorem map_mul_left_eq_self (μ : Measure G) [IsMulLeftInvariant μ] (g : G) :
    map (g * ·) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => g + x) μ = μ
-- MeasureTheory.map_mul_left_eq_self
@[to_additive]
theorem map_mul_left_eq_self (μ : Measure G) [IsMulLeftInvariant μ] (g : G) :
    map (g * ·) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => g * x) μ = μ
-- MeasureTheory.map_add_right_eq_self
@[to_additive]
theorem map_mul_right_eq_self (μ : Measure G) [IsMulRightInvariant μ] (g : G) : map (· * g) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => x + g) μ = μ
-- MeasureTheory.map_mul_right_eq_self
@[to_additive]
theorem map_mul_right_eq_self (μ : Measure G) [IsMulRightInvariant μ] (g : G) : map (· * g) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => x * g) μ = μ
-- MeasureTheory.measurePreserving_mul_left
@[to_additive]
theorem measurePreserving_mul_left (μ : Measure G) [IsMulLeftInvariant μ] (g : G) :
    MeasurePreserving (g * ·) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun x => g * x) μ μ
-- MeasureTheory.measurePreserving_add_left
@[to_additive]
theorem measurePreserving_mul_left (μ : Measure G) [IsMulLeftInvariant μ] (g : G) :
    MeasurePreserving (g * ·) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun x => g + x) μ μ
-- MeasureTheory.MeasurePreserving.mul_left
@[to_additive]
theorem MeasurePreserving.mul_left (μ : Measure G) [IsMulLeftInvariant μ] (g : G) {X : Type*}
    [MeasurableSpace X] {μ' : Measure X} {f : X → G} (hf : MeasurePreserving f μ' μ) :
    MeasurePreserving (fun x => g * f x) μ' μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G) {X : Type u_4} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ)
Goal: MeasureTheory.MeasurePreserving (fun x => g * f x) μ' μ
-- MeasureTheory.MeasurePreserving.add_left
@[to_additive]
theorem MeasurePreserving.mul_left (μ : Measure G) [IsMulLeftInvariant μ] (g : G) {X : Type*}
    [MeasurableSpace X] {μ' : Measure X} {f : X → G} (hf : MeasurePreserving f μ' μ) :
    MeasurePreserving (fun x => g * f x) μ' μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G) {X : Type u_4} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ)
Goal: MeasureTheory.MeasurePreserving (fun x => g + f x) μ' μ
-- MeasureTheory.measurePreserving_add_right
@[to_additive]
theorem measurePreserving_mul_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) :
    MeasurePreserving (· * g) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun x => x + g) μ μ
-- MeasureTheory.measurePreserving_mul_right
@[to_additive]
theorem measurePreserving_mul_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) :
    MeasurePreserving (· * g) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun x => x * g) μ μ
-- MeasureTheory.MeasurePreserving.mul_right
@[to_additive]
theorem MeasurePreserving.mul_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) {X : Type*}
    [MeasurableSpace X] {μ' : Measure X} {f : X → G} (hf : MeasurePreserving f μ' μ) :
    MeasurePreserving (fun x => f x * g) μ' μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) {X : Type u_4} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ)
Goal: MeasureTheory.MeasurePreserving (fun x => f x * g) μ' μ
-- MeasureTheory.MeasurePreserving.add_right
@[to_additive]
theorem MeasurePreserving.mul_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) {X : Type*}
    [MeasurableSpace X] {μ' : Measure X} {f : X → G} (hf : MeasurePreserving f μ' μ) :
    MeasurePreserving (fun x => f x * g) μ' μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) {X : Type u_4} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ)
Goal: MeasureTheory.MeasurePreserving (fun x => f x + g) μ' μ
-- MeasureTheory.forall_measure_preimage_add_iff
@[to_additive " An alternative way to prove that `μ` is left invariant under addition. "]
theorem forall_measure_preimage_mul_iff (μ : Measure G) :
    (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => g * h) ⁻¹' A) = μ A) ↔
      IsMulLeftInvariant μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G)
Goal: (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => g + h) ⁻¹' A) = μ A) ↔ μ.IsAddLeftInvariant
-- MeasureTheory.forall_measure_preimage_mul_iff
@[to_additive " An alternative way to prove that `μ` is left invariant under addition. "]
theorem forall_measure_preimage_mul_iff (μ : Measure G) :
    (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => g * h) ⁻¹' A) = μ A) ↔
      IsMulLeftInvariant μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G)
Goal: (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => g * h) ⁻¹' A) = μ A) ↔ μ.IsMulLeftInvariant
-- MeasureTheory.forall_measure_preimage_add_right_iff
@[to_additive " An alternative way to prove that `μ` is right invariant under addition. "]
theorem forall_measure_preimage_mul_right_iff (μ : Measure G) :
    (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => h * g) ⁻¹' A) = μ A) ↔
      IsMulRightInvariant μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G)
Goal: (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => h + g) ⁻¹' A) = μ A) ↔ μ.IsAddRightInvariant
-- MeasureTheory.forall_measure_preimage_mul_right_iff
@[to_additive " An alternative way to prove that `μ` is right invariant under addition. "]
theorem forall_measure_preimage_mul_right_iff (μ : Measure G) :
    (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => h * g) ⁻¹' A) = μ A) ↔
      IsMulRightInvariant μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G)
Goal: (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => h * g) ⁻¹' A) = μ A) ↔ μ.IsMulRightInvariant
-- MeasureTheory.isAddLeftInvariant_map
@[to_additive]
theorem isMulLeftInvariant_map {H : Type*} [MeasurableSpace H] [Mul H] [MeasurableMul H]
    [IsMulLeftInvariant μ] (f : G →ₙ* H) (hf : Measurable f) (h_surj : Surjective f) :
    IsMulLeftInvariant (Measure.map f μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [MeasurableAdd G] {H : Type u_4} [MeasurableSpace H] [Add H] [MeasurableAdd H] [μ.IsAddLeftInvariant] (f : AddHom G H) (hf : Measurable ⇑f) (h_surj : Function.Surjective ⇑f)
Goal: (MeasureTheory.Measure.map (⇑f) μ).IsAddLeftInvariant
-- MeasureTheory.isMulLeftInvariant_map
@[to_additive]
theorem isMulLeftInvariant_map {H : Type*} [MeasurableSpace H] [Mul H] [MeasurableMul H]
    [IsMulLeftInvariant μ] (f : G →ₙ* H) (hf : Measurable f) (h_surj : Surjective f) :
    IsMulLeftInvariant (Measure.map f μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [MeasurableMul G] {H : Type u_4} [MeasurableSpace H] [Mul H] [MeasurableMul H] [μ.IsMulLeftInvariant] (f : G →ₙ* H) (hf : Measurable ⇑f) (h_surj : Function.Surjective ⇑f)
Goal: (MeasureTheory.Measure.map (⇑f) μ).IsMulLeftInvariant
-- MeasureTheory.isMulLeftInvariant_map_smul
@[to_additive "The image of a left invariant measure under a left additive action is left invariant,
assuming that the action preserves addition."]
theorem isMulLeftInvariant_map_smul
    {α} [SMul α G] [SMulCommClass α G G] [MeasurableSpace α] [MeasurableSMul α G]
    [IsMulLeftInvariant μ] (a : α) :
    IsMulLeftInvariant (map (a • · : G → G) μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Semigroup G] [MeasurableMul G] {μ : MeasureTheory.Measure G} {α : Type u_4} [SMul α G] [SMulCommClass α G G] [MeasurableSpace α] [MeasurableSMul α G] [μ.IsMulLeftInvariant] (a : α)
Goal: (MeasureTheory.Measure.map (fun x => a • x) μ).IsMulLeftInvariant
-- MeasureTheory.isAddLeftInvariant_map_vadd
@[to_additive "The image of a left invariant measure under a left additive action is left invariant,
assuming that the action preserves addition."]
theorem isMulLeftInvariant_map_smul
    {α} [SMul α G] [SMulCommClass α G G] [MeasurableSpace α] [MeasurableSMul α G]
    [IsMulLeftInvariant μ] (a : α) :
    IsMulLeftInvariant (map (a • · : G → G) μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddSemigroup G] [MeasurableAdd G] {μ : MeasureTheory.Measure G} {α : Type u_4} [VAdd α G] [VAddCommClass α G G] [MeasurableSpace α] [MeasurableVAdd α G] [μ.IsAddLeftInvariant] (a : α)
Goal: (MeasureTheory.Measure.map (fun x => a +ᵥ x) μ).IsAddLeftInvariant
-- MeasureTheory.isAddRightInvariant_map_vadd
@[to_additive "The image of a right invariant measure under a left additive action is right
 invariant, assuming that the action preserves addition."]
theorem isMulRightInvariant_map_smul
    {α} [SMul α G] [SMulCommClass α Gᵐᵒᵖ G] [MeasurableSpace α] [MeasurableSMul α G]
    [IsMulRightInvariant μ] (a : α) :
    IsMulRightInvariant (map (a • · : G → G) μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddSemigroup G] [MeasurableAdd G] {μ : MeasureTheory.Measure G} {α : Type u_4} [VAdd α G] [VAddCommClass α Gᵃᵒᵖ G] [MeasurableSpace α] [MeasurableVAdd α G] [μ.IsAddRightInvariant] (a : α)
Goal: (MeasureTheory.Measure.map (fun x => a +ᵥ x) μ).IsAddRightInvariant
-- MeasureTheory.isMulRightInvariant_map_smul
@[to_additive "The image of a right invariant measure under a left additive action is right
 invariant, assuming that the action preserves addition."]
theorem isMulRightInvariant_map_smul
    {α} [SMul α G] [SMulCommClass α Gᵐᵒᵖ G] [MeasurableSpace α] [MeasurableSMul α G]
    [IsMulRightInvariant μ] (a : α) :
    IsMulRightInvariant (map (a • · : G → G) μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Semigroup G] [MeasurableMul G] {μ : MeasureTheory.Measure G} {α : Type u_4} [SMul α G] [SMulCommClass α Gᵐᵒᵖ G] [MeasurableSpace α] [MeasurableSMul α G] [μ.IsMulRightInvariant] (a : α)
Goal: (MeasureTheory.Measure.map (fun x => a • x) μ).IsMulRightInvariant
-- MeasureTheory.map_div_right_eq_self
@[to_additive]
theorem map_div_right_eq_self (μ : Measure G) [IsMulRightInvariant μ] (g : G) :
    map (· / g) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [DivInvMonoid G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => x / g) μ = μ
-- MeasureTheory.map_sub_right_eq_self
@[to_additive]
theorem map_div_right_eq_self (μ : Measure G) [IsMulRightInvariant μ] (g : G) :
    map (· / g) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [SubNegMonoid G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => x - g) μ = μ
-- MeasureTheory.measurePreserving_sub_right
@[to_additive]
theorem measurePreserving_div_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) :
    MeasurePreserving (· / g) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun x => x - g) μ μ
-- MeasureTheory.measurePreserving_div_right
@[to_additive]
theorem measurePreserving_div_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) :
    MeasurePreserving (· / g) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun x => x / g) μ μ
-- MeasureTheory.measure_preimage_add
@[to_additive (attr := simp)
"We shorten this from `measure_preimage_add_left`, since left invariant is the preferred option for
measures in this formalization."]
theorem measure_preimage_mul (μ : Measure G) [IsMulLeftInvariant μ] (g : G) (A : Set G) :
    μ ((fun h => g * h) ⁻¹' A) = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G) (A : Set G)
Goal: μ ((fun h => g + h) ⁻¹' A) = μ A
-- MeasureTheory.measure_preimage_mul
@[to_additive (attr := simp)
"We shorten this from `measure_preimage_add_left`, since left invariant is the preferred option for
measures in this formalization."]
theorem measure_preimage_mul (μ : Measure G) [IsMulLeftInvariant μ] (g : G) (A : Set G) :
    μ ((fun h => g * h) ⁻¹' A) = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G) (A : Set G)
Goal: μ ((fun h => g * h) ⁻¹' A) = μ A
-- MeasureTheory.measure_preimage_mul_right
@[to_additive (attr := simp)]
theorem measure_preimage_mul_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) (A : Set G) :
    μ ((fun h => h * g) ⁻¹' A) = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) (A : Set G)
Goal: μ ((fun h => h * g) ⁻¹' A) = μ A
-- MeasureTheory.measure_preimage_add_right
@[to_additive (attr := simp)]
theorem measure_preimage_mul_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) (A : Set G) :
    μ ((fun h => h * g) ⁻¹' A) = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) (A : Set G)
Goal: μ ((fun h => h + g) ⁻¹' A) = μ A
-- MeasureTheory.map_mul_left_ae
@[to_additive]
theorem map_mul_left_ae (μ : Measure G) [IsMulLeftInvariant μ] (x : G) :
    Filter.map (fun h => x * h) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (x : G)
Goal: Filter.map (fun h => x * h) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.map_add_left_ae
@[to_additive]
theorem map_mul_left_ae (μ : Measure G) [IsMulLeftInvariant μ] (x : G) :
    Filter.map (fun h => x * h) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (x : G)
Goal: Filter.map (fun h => x + h) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.map_add_right_ae
@[to_additive]
theorem map_mul_right_ae (μ : Measure G) [IsMulRightInvariant μ] (x : G) :
    Filter.map (fun h => h * x) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (x : G)
Goal: Filter.map (fun h => h + x) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.map_mul_right_ae
@[to_additive]
theorem map_mul_right_ae (μ : Measure G) [IsMulRightInvariant μ] (x : G) :
    Filter.map (fun h => h * x) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (x : G)
Goal: Filter.map (fun h => h * x) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.map_div_right_ae
@[to_additive]
theorem map_div_right_ae (μ : Measure G) [IsMulRightInvariant μ] (x : G) :
    Filter.map (fun t => t / x) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (x : G)
Goal: Filter.map (fun t => t / x) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.map_sub_right_ae
@[to_additive]
theorem map_div_right_ae (μ : Measure G) [IsMulRightInvariant μ] (x : G) :
    Filter.map (fun t => t / x) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (x : G)
Goal: Filter.map (fun t => t - x) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.eventually_add_left_iff
@[to_additive]
theorem eventually_mul_left_iff (μ : Measure G) [IsMulLeftInvariant μ] (t : G) {p : G → Prop} :
    (∀ᵐ x ∂μ, p (t * x)) ↔ ∀ᵐ x ∂μ, p x 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (t : G) {p : G → Prop}
Goal: (∀ᵐ (x : G) ∂μ, p (t + x)) ↔ ∀ᵐ (x : G) ∂μ, p x
-- MeasureTheory.eventually_mul_left_iff
@[to_additive]
theorem eventually_mul_left_iff (μ : Measure G) [IsMulLeftInvariant μ] (t : G) {p : G → Prop} :
    (∀ᵐ x ∂μ, p (t * x)) ↔ ∀ᵐ x ∂μ, p x 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (t : G) {p : G → Prop}
Goal: (∀ᵐ (x : G) ∂μ, p (t * x)) ↔ ∀ᵐ (x : G) ∂μ, p x
-- MeasureTheory.eventually_add_right_iff
@[to_additive]
theorem eventually_mul_right_iff (μ : Measure G) [IsMulRightInvariant μ] (t : G) {p : G → Prop} :
    (∀ᵐ x ∂μ, p (x * t)) ↔ ∀ᵐ x ∂μ, p x 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (t : G) {p : G → Prop}
Goal: (∀ᵐ (x : G) ∂μ, p (x + t)) ↔ ∀ᵐ (x : G) ∂μ, p x
-- MeasureTheory.eventually_mul_right_iff
@[to_additive]
theorem eventually_mul_right_iff (μ : Measure G) [IsMulRightInvariant μ] (t : G) {p : G → Prop} :
    (∀ᵐ x ∂μ, p (x * t)) ↔ ∀ᵐ x ∂μ, p x 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (t : G) {p : G → Prop}
Goal: (∀ᵐ (x : G) ∂μ, p (x * t)) ↔ ∀ᵐ (x : G) ∂μ, p x
-- MeasureTheory.eventually_sub_right_iff
@[to_additive]
theorem eventually_div_right_iff (μ : Measure G) [IsMulRightInvariant μ] (t : G) {p : G → Prop} :
    (∀ᵐ x ∂μ, p (x / t)) ↔ ∀ᵐ x ∂μ, p x 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (t : G) {p : G → Prop}
Goal: (∀ᵐ (x : G) ∂μ, p (x - t)) ↔ ∀ᵐ (x : G) ∂μ, p x
-- MeasureTheory.eventually_div_right_iff
@[to_additive]
theorem eventually_div_right_iff (μ : Measure G) [IsMulRightInvariant μ] (t : G) {p : G → Prop} :
    (∀ᵐ x ∂μ, p (x / t)) ↔ ∀ᵐ x ∂μ, p x 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (t : G) {p : G → Prop}
Goal: (∀ᵐ (x : G) ∂μ, p (x / t)) ↔ ∀ᵐ (x : G) ∂μ, p x
-- MeasureTheory.Measure.IsNegInvariant.neg_eq_self
class IsNegInvariant [Neg G] (μ : Measure G) : Prop where
  neg_eq_self : μ.neg = μ
Arguments: {G : Type u_2} [MeasurableSpace G] [Neg G] {μ : MeasureTheory.Measure G} [self : μ.IsNegInvariant]
Goal: μ.neg = μ
-- MeasureTheory.Measure.IsInvInvariant.inv_eq_self
@[to_additive existing]
class IsInvInvariant [Inv G] (μ : Measure G) : Prop where
  inv_eq_self : μ.inv = μ
Arguments: {G : Type u_2} [MeasurableSpace G] [Inv G] {μ : MeasureTheory.Measure G} [self : μ.IsInvInvariant]
Goal: μ.inv = μ
-- MeasureTheory.Measure.neg_def
@[to_additive]
theorem inv_def (μ : Measure G) : μ.inv = Measure.map inv μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G)
Goal: μ.neg = MeasureTheory.Measure.map Neg.neg μ
-- MeasureTheory.Measure.inv_def
@[to_additive]
theorem inv_def (μ : Measure G) : μ.inv = Measure.map inv μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G)
Goal: μ.inv = MeasureTheory.Measure.map Inv.inv μ
-- MeasureTheory.Measure.inv_eq_self
@[to_additive (attr := simp)]
theorem inv_eq_self (μ : Measure G) [IsInvInvariant μ] : μ.inv = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant]
Goal: μ.inv = μ
-- MeasureTheory.Measure.neg_eq_self
@[to_additive (attr := simp)]
theorem inv_eq_self (μ : Measure G) [IsInvInvariant μ] : μ.inv = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant]
Goal: μ.neg = μ
-- MeasureTheory.Measure.map_inv_eq_self
@[to_additive (attr := simp)]
theorem map_inv_eq_self (μ : Measure G) [IsInvInvariant μ] : map Inv.inv μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant]
Goal: MeasureTheory.Measure.map Inv.inv μ = μ
-- MeasureTheory.Measure.map_neg_eq_self
@[to_additive (attr := simp)]
theorem map_inv_eq_self (μ : Measure G) [IsInvInvariant μ] : map Inv.inv μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant]
Goal: MeasureTheory.Measure.map Neg.neg μ = μ
-- MeasureTheory.Measure.measurePreserving_neg
@[to_additive]
theorem measurePreserving_inv (μ : Measure G) [IsInvInvariant μ] : MeasurePreserving Inv.inv μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Neg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant]
Goal: MeasureTheory.MeasurePreserving Neg.neg μ μ
-- MeasureTheory.Measure.measurePreserving_inv
@[to_additive]
theorem measurePreserving_inv (μ : Measure G) [IsInvInvariant μ] : MeasurePreserving Inv.inv μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Inv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant]
Goal: MeasureTheory.MeasurePreserving Inv.inv μ μ
-- MeasureTheory.Measure.neg_apply
@[to_additive (attr := simp)]
theorem inv_apply (μ : Measure G) (s : Set G) : μ.inv s = μ s⁻¹ 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) (s : Set G)
Goal: μ.neg s = μ (-s)
-- MeasureTheory.Measure.inv_apply
@[to_additive (attr := simp)]
theorem inv_apply (μ : Measure G) (s : Set G) : μ.inv s = μ s⁻¹ 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) (s : Set G)
Goal: μ.inv s = μ s⁻¹
-- MeasureTheory.Measure.neg_neg
@[to_additive (attr := simp)]
protected theorem inv_inv (μ : Measure G) : μ.inv.inv = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G)
Goal: μ.neg.neg = μ
-- MeasureTheory.Measure.inv_inv
@[to_additive (attr := simp)]
protected theorem inv_inv (μ : Measure G) : μ.inv.inv = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G)
Goal: μ.inv.inv = μ
-- MeasureTheory.Measure.measure_neg
@[to_additive (attr := simp)]
theorem measure_inv (μ : Measure G) [IsInvInvariant μ] (A : Set G) : μ A⁻¹ = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] (A : Set G)
Goal: μ (-A) = μ A