ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- torusIntegral_succAbove
theorem torusIntegral_succAbove {f : ℂⁿ⁺¹ → E} {c : ℂⁿ⁺¹} {R : ℝⁿ⁺¹} (hf : TorusIntegrable f c R)
    (i : Fin (n + 1)) :
    (∯ x in T(c, R), f x) =
      ∮ x in C(c i, R i), ∯ y in T(c ∘ i.succAbove, R ∘ i.succAbove), f (i.insertNth x y) 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : (Fin (n + 1) → ℂ) → E} {c : Fin (n + 1) → ℂ} {R : Fin (n + 1) → ℝ} (hf : TorusIntegrable f c R) (i : Fin (n + 1))
Goal: (∯ (x : Fin (n + 1) → ℂ) in T(c, R), f x) =
  ∮ (x : ℂ) in C(c i, R i), ∯ (y : Fin n → ℂ) in T(c ∘ i.succAbove, R ∘ i.succAbove), f (i.insertNth x y)
-- torusIntegral_succ
theorem torusIntegral_succ {f : ℂⁿ⁺¹ → E} {c : ℂⁿ⁺¹} {R : ℝⁿ⁺¹} (hf : TorusIntegrable f c R) :
    (∯ x in T(c, R), f x) =
      ∮ x in C(c 0, R 0), ∯ y in T(c ∘ Fin.succ, R ∘ Fin.succ), f (Fin.cons x y) 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : (Fin (n + 1) → ℂ) → E} {c : Fin (n + 1) → ℂ} {R : Fin (n + 1) → ℝ} (hf : TorusIntegrable f c R)
Goal: (∯ (x : Fin (n + 1) → ℂ) in T(c, R), f x) =
  ∮ (x : ℂ) in C(c 0, R 0), ∯ (y : Fin n → ℂ) in T(c ∘ Fin.succ, R ∘ Fin.succ), f (Fin.cons x y)
-- measurable_oneLePart
@[to_additive (attr := measurability)]
theorem measurable_oneLePart : Measurable (oneLePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [Group α] [MeasurableSpace α] [MeasurableSup α]
Goal: Measurable oneLePart
-- measurable_posPart
@[to_additive (attr := measurability)]
theorem measurable_oneLePart : Measurable (oneLePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [MeasurableSpace α] [MeasurableSup α]
Goal: Measurable posPart
-- Measurable.posPart
@[to_additive (attr := measurability)]
protected theorem Measurable.oneLePart : Measurable fun x ↦ oneLePart (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [AddGroup α] [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) [MeasurableSup α]
Goal: Measurable fun x => (f x)⁺
-- Measurable.oneLePart
@[to_additive (attr := measurability)]
protected theorem Measurable.oneLePart : Measurable fun x ↦ oneLePart (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Group α] [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) [MeasurableSup α]
Goal: Measurable fun x => (f x)⁺ᵐ
-- measurable_negPart
@[to_additive (attr := measurability)]
theorem measurable_leOnePart : Measurable (leOnePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [MeasurableSpace α] [MeasurableSup α] [MeasurableNeg α]
Goal: Measurable negPart
-- measurable_leOnePart
@[to_additive (attr := measurability)]
theorem measurable_leOnePart : Measurable (leOnePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [Group α] [MeasurableSpace α] [MeasurableSup α] [MeasurableInv α]
Goal: Measurable leOnePart
-- Measurable.leOnePart
@[to_additive (attr := measurability)]
protected theorem Measurable.leOnePart : Measurable fun x ↦ leOnePart (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Group α] [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) [MeasurableSup α] [MeasurableInv α]
Goal: Measurable fun x => (f x)⁻ᵐ
-- Measurable.negPart
@[to_additive (attr := measurability)]
protected theorem Measurable.leOnePart : Measurable fun x ↦ leOnePart (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [AddGroup α] [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) [MeasurableSup α] [MeasurableNeg α]
Goal: Measurable fun x => (f x)⁻
-- measurable_abs
@[to_additive (attr := measurability)]
theorem measurable_mabs : Measurable (mabs : α → α) 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [MeasurableSpace α] [MeasurableNeg α] [MeasurableSup₂ α]
Goal: Measurable abs
-- measurable_mabs
@[to_additive (attr := measurability)]
theorem measurable_mabs : Measurable (mabs : α → α) 
Arguments: {α : Type u_1} [Lattice α] [Group α] [MeasurableSpace α] [MeasurableInv α] [MeasurableSup₂ α]
Goal: Measurable mabs
-- Measurable.mabs
@[to_additive (attr := measurability)]
protected theorem Measurable.mabs : Measurable fun x ↦ mabs (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Group α] [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) [MeasurableInv α] [MeasurableSup₂ α]
Goal: Measurable fun x => mabs (f x)
-- Measurable.abs
@[to_additive (attr := measurability)]
protected theorem Measurable.mabs : Measurable fun x ↦ mabs (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [AddGroup α] [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) [MeasurableNeg α] [MeasurableSup₂ α]
Goal: Measurable fun x => |f x|
-- MeasurableSup.measurable_const_sup
class MeasurableSup (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_const_sup : ∀ c : M, Measurable (c ⊔ ·)
  measurable_sup_const : ∀ c : M, Measurable (· ⊔ c)
Arguments: {M : Type u_1} [MeasurableSpace M] [Sup M] [self : MeasurableSup M] (c : M)
Goal: Measurable fun x => c ⊔ x
-- MeasurableSup.measurable_sup_const
class MeasurableSup (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_const_sup : ∀ c : M, Measurable (c ⊔ ·)
  measurable_sup_const : ∀ c : M, Measurable (· ⊔ c)
Arguments: {M : Type u_1} [MeasurableSpace M] [Sup M] [self : MeasurableSup M] (c : M)
Goal: Measurable fun x => x ⊔ c
-- MeasurableSup₂.measurable_sup
class MeasurableSup₂ (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_sup : Measurable fun p : M × M => p.1 ⊔ p.2
Arguments: {M : Type u_1} [MeasurableSpace M] [Sup M] [self : MeasurableSup₂ M]
Goal: Measurable fun p => p.1 ⊔ p.2
-- MeasurableInf.measurable_const_inf
class MeasurableInf (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_const_inf : ∀ c : M, Measurable (c ⊓ ·)
  measurable_inf_const : ∀ c : M, Measurable (· ⊓ c)
Arguments: {M : Type u_1} [MeasurableSpace M] [Inf M] [self : MeasurableInf M] (c : M)
Goal: Measurable fun x => c ⊓ x
-- MeasurableInf.measurable_inf_const
class MeasurableInf (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_const_inf : ∀ c : M, Measurable (c ⊓ ·)
  measurable_inf_const : ∀ c : M, Measurable (· ⊓ c)
Arguments: {M : Type u_1} [MeasurableSpace M] [Inf M] [self : MeasurableInf M] (c : M)
Goal: Measurable fun x => x ⊓ c
-- MeasurableInf₂.measurable_inf
class MeasurableInf₂ (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_inf : Measurable fun p : M × M => p.1 ⊓ p.2
Arguments: {M : Type u_1} [MeasurableSpace M] [Inf M] [self : MeasurableInf₂ M]
Goal: Measurable fun p => p.1 ⊓ p.2
-- Measurable.const_sup
@[measurability]
theorem Measurable.const_sup (hf : Measurable f) (c : M) : Measurable fun x => c ⊔ f x 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f : α → M} [Sup M] [MeasurableSup M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => c ⊔ f x
-- AEMeasurable.const_sup
@[measurability]
theorem AEMeasurable.const_sup (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => c ⊔ f x) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → M} [Sup M] [MeasurableSup M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => c ⊔ f x) μ
-- Measurable.sup_const
@[measurability]
theorem Measurable.sup_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊔ c 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f : α → M} [Sup M] [MeasurableSup M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => f x ⊔ c
-- AEMeasurable.sup_const
@[measurability]
theorem AEMeasurable.sup_const (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => f x ⊔ c) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → M} [Sup M] [MeasurableSup M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => f x ⊔ c) μ
-- Measurable.sup'
@[measurability]
theorem Measurable.sup' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊔ g) 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f g : α → M} [Sup M] [MeasurableSup₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable (f ⊔ g)
-- Measurable.sup
@[measurability]
theorem Measurable.sup (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊔ g a 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f g : α → M} [Sup M] [MeasurableSup₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => f a ⊔ g a
-- AEMeasurable.sup'
@[measurability]
theorem AEMeasurable.sup' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f ⊔ g) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → M} [Sup M] [MeasurableSup₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (f ⊔ g) μ
-- AEMeasurable.sup
@[measurability]
theorem AEMeasurable.sup (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a ⊔ g a) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → M} [Sup M] [MeasurableSup₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => f a ⊔ g a) μ
-- Measurable.const_inf
@[measurability]
theorem Measurable.const_inf (hf : Measurable f) (c : M) : Measurable fun x => c ⊓ f x 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f : α → M} [Inf M] [MeasurableInf M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => c ⊓ f x
-- AEMeasurable.const_inf
@[measurability]
theorem AEMeasurable.const_inf (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => c ⊓ f x) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → M} [Inf M] [MeasurableInf M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => c ⊓ f x) μ
-- Measurable.inf_const
@[measurability]
theorem Measurable.inf_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊓ c 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f : α → M} [Inf M] [MeasurableInf M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => f x ⊓ c
-- AEMeasurable.inf_const
@[measurability]
theorem AEMeasurable.inf_const (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => f x ⊓ c) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → M} [Inf M] [MeasurableInf M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => f x ⊓ c) μ
-- Measurable.inf'
@[measurability]
theorem Measurable.inf' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊓ g) 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f g : α → M} [Inf M] [MeasurableInf₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable (f ⊓ g)
-- Measurable.inf
@[measurability]
theorem Measurable.inf (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊓ g a 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {f g : α → M} [Inf M] [MeasurableInf₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => f a ⊓ g a
-- AEMeasurable.inf'
@[measurability]
theorem AEMeasurable.inf' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f ⊓ g) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → M} [Inf M] [MeasurableInf₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (f ⊓ g) μ
-- AEMeasurable.inf
@[measurability]
theorem AEMeasurable.inf (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a ⊓ g a) μ 
Arguments: {M : Type u_1} [MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → M} [Inf M] [MeasurableInf₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => f a ⊓ g a) μ
-- Finset.measurable_sup'
@[measurability]
theorem Finset.measurable_sup' {ι : Type*} {s : Finset ι} (hs : s.Nonempty) {f : ι → δ → α}
    (hf : ∀ n ∈ s, Measurable (f n)) : Measurable (s.sup' hs f) 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {δ : Type u_3} [MeasurableSpace δ] [SemilatticeSup α] [MeasurableSup₂ α] {ι : Type u_4} {s : Finset ι} (hs : s.Nonempty) {f : ι → δ → α} (hf : ∀ n ∈ s, Measurable (f n))
Goal: Measurable (s.sup' hs f)
-- Finset.measurable_range_sup'
@[measurability]
theorem Finset.measurable_range_sup' {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k)) :
    Measurable ((range (n + 1)).sup' nonempty_range_succ f) 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {δ : Type u_3} [MeasurableSpace δ] [SemilatticeSup α] [MeasurableSup₂ α] {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k))
Goal: Measurable ((Finset.range (n + 1)).sup' ⋯ f)
-- Finset.measurable_range_sup''
@[measurability]
theorem Finset.measurable_range_sup'' {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k)) :
    Measurable fun x => (range (n + 1)).sup' nonempty_range_succ fun k => f k x 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {δ : Type u_3} [MeasurableSpace δ] [SemilatticeSup α] [MeasurableSup₂ α] {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k))
Goal: Measurable fun x => (Finset.range (n + 1)).sup' ⋯ fun k => f k x
-- MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite
theorem exists_nat_measurableEquiv_range_coe_fin_of_finite [Finite α] :
    ∃ n : ℕ, Nonempty (α ≃ᵐ range ((↑) : Fin n → ℝ)) 
Arguments: (α : Type u_1) [MeasurableSpace α] [StandardBorelSpace α] [Finite α]
Goal: ∃ n, Nonempty (α ≃ᵐ ↑(Set.range fun x => ↑↑x))
-- MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable
theorem measurableEquiv_range_coe_nat_of_infinite_of_countable [Infinite α] [Countable α] :
    Nonempty (α ≃ᵐ range ((↑) : ℕ → ℝ)) 
Arguments: (α : Type u_1) [MeasurableSpace α] [StandardBorelSpace α] [Infinite α] [Countable α]
Goal: Nonempty (α ≃ᵐ ↑(Set.range Nat.cast))
-- MeasureTheory.exists_subset_real_measurableEquiv
theorem exists_subset_real_measurableEquiv : ∃ s : Set ℝ, MeasurableSet s ∧ Nonempty (α ≃ᵐ s) 
Arguments: (α : Type u_1) [MeasurableSpace α] [StandardBorelSpace α]
Goal: ∃ s, MeasurableSet s ∧ Nonempty (α ≃ᵐ ↑s)
-- MeasureTheory.exists_measurableEmbedding_real
theorem exists_measurableEmbedding_real : ∃ f : α → ℝ, MeasurableEmbedding f 
Arguments: (α : Type u_1) [MeasurableSpace α] [StandardBorelSpace α]
Goal: ∃ f, MeasurableEmbedding f
-- MeasureTheory.measurableEmbedding_embeddingReal
lemma measurableEmbedding_embeddingReal (Ω : Type*) [MeasurableSpace Ω] [StandardBorelSpace Ω] :
    MeasurableEmbedding (embeddingReal Ω) 
Arguments: (Ω : Type u_2) [MeasurableSpace Ω] [StandardBorelSpace Ω]
Goal: MeasurableEmbedding (MeasureTheory.embeddingReal Ω)
-- MeasurableSet.eventuallyMeasurableSet
theorem MeasurableSet.eventuallyMeasurableSet (hs : MeasurableSet s) :
    EventuallyMeasurableSet m l s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l] {s : Set α} (hs : MeasurableSet s)
Goal: EventuallyMeasurableSet m l s
-- EventuallyMeasurableSpace.measurable_le
theorem EventuallyMeasurableSpace.measurable_le : m ≤ EventuallyMeasurableSpace m l 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l]
Goal: m ≤ EventuallyMeasurableSpace m l
-- eventuallyMeasurableSet_of_mem_filter
theorem eventuallyMeasurableSet_of_mem_filter (hs : s ∈ l) : EventuallyMeasurableSet m l s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l] {s : Set α} (hs : s ∈ l)
Goal: EventuallyMeasurableSet m l s
-- EventuallyMeasurableSet.congr
theorem EventuallyMeasurableSet.congr
    (ht : EventuallyMeasurableSet m l t) (hst : s =ᶠ[l] t) : EventuallyMeasurableSet m l s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l] {s t : Set α} (ht : EventuallyMeasurableSet m l t) (hst : s =ᶠ[l] t)
Goal: EventuallyMeasurableSet m l s
-- Measurable.eventuallyMeasurable
theorem Measurable.eventuallyMeasurable (hf : Measurable f) : EventuallyMeasurable m l f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l] {β : Type u_2} [MeasurableSpace β] {f : α → β} (hf : Measurable f)
Goal: EventuallyMeasurable m l f
-- Measurable.comp_eventuallyMeasurable
theorem Measurable.comp_eventuallyMeasurable (hh : Measurable h) (hf : EventuallyMeasurable m l f) :
    EventuallyMeasurable m l (h ∘ f) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l] {β : Type u_2} {γ : Type u_3} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {h : β → γ} (hh : Measurable h) (hf : EventuallyMeasurable m l f)
Goal: EventuallyMeasurable m l (h ∘ f)
-- EventuallyMeasurable.congr
theorem EventuallyMeasurable.congr
    (hf : EventuallyMeasurable m l f) (hgf : g =ᶠ[l] f) : EventuallyMeasurable m l g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l] {β : Type u_2} [MeasurableSpace β] {f g : α → β} (hf : EventuallyMeasurable m l f) (hgf : g =ᶠ[l] f)
Goal: EventuallyMeasurable m l g
-- Measurable.eventuallyMeasurable_of_eventuallyEq
theorem Measurable.eventuallyMeasurable_of_eventuallyEq
    (hf : Measurable f) (hgf : g =ᶠ[l] f) : EventuallyMeasurable m l g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {l : Filter α} [CountableInterFilter l] {β : Type u_2} [MeasurableSpace β] {f g : α → β} (hf : Measurable f) (hgf : g =ᶠ[l] f)
Goal: EventuallyMeasurable m l g
-- AddCircle.measurable_mk'
@[measurability]
protected theorem AddCircle.measurable_mk' {a : ℝ} :
    Measurable (β := AddCircle a) ((↑) : ℝ → AddCircle a) 
Arguments: {a : ℝ}
Goal: Measurable QuotientAddGroup.mk
-- isAddFundamentalDomain_Ioc
theorem isAddFundamentalDomain_Ioc {T : ℝ} (hT : 0 < T) (t : ℝ)
    (μ : Measure ℝ := by volume_tac) :
    IsAddFundamentalDomain (AddSubgroup.zmultiples T) (Ioc t (t + T)) μ 
Arguments: {T : ℝ} (hT : 0 < T) (t : ℝ) (μ : MeasureTheory.Measure ℝ := by volume_tac)
Goal: MeasureTheory.IsAddFundamentalDomain (↥(AddSubgroup.zmultiples T)) (Set.Ioc t (t + T)) μ
-- isAddFundamentalDomain_Ioc'
theorem isAddFundamentalDomain_Ioc' {T : ℝ} (hT : 0 < T) (t : ℝ) (μ : Measure ℝ := by volume_tac) :
    IsAddFundamentalDomain (AddSubgroup.op <| .zmultiples T) (Ioc t (t + T)) μ 
Arguments: {T : ℝ} (hT : 0 < T) (t : ℝ) (μ : MeasureTheory.Measure ℝ := by volume_tac)
Goal: MeasureTheory.IsAddFundamentalDomain (↥(AddSubgroup.zmultiples T).op) (Set.Ioc t (t + T)) μ
-- AddCircle.measure_univ
@[simp]
protected theorem measure_univ : volume (Set.univ : Set (AddCircle T)) = ENNReal.ofReal T 
Arguments: (T : ℝ) [hT : Fact (0 < T)]
Goal: MeasureTheory.volume Set.univ = ENNReal.ofReal T
-- AddCircle.measurePreserving_mk
protected theorem measurePreserving_mk (t : ℝ) :
    MeasurePreserving (β := AddCircle T) ((↑) : ℝ → AddCircle T)
      (volume.restrict (Ioc t (t + T))) 
Arguments: (T : ℝ) [hT : Fact (0 < T)] (t : ℝ)
Goal: MeasureTheory.MeasurePreserving QuotientAddGroup.mk (MeasureTheory.volume.restrict (Set.Ioc t (t + T)))
  MeasureTheory.volume
-- AddCircle.add_projection_respects_measure
lemma add_projection_respects_measure (t : ℝ) {U : Set (AddCircle T)} (meas_U : MeasurableSet U) :
    volume U = volume (QuotientAddGroup.mk ⁻¹' U ∩ (Ioc t (t + T))) 
Arguments: (T : ℝ) [hT : Fact (0 < T)] (t : ℝ) {U : Set (AddCircle T)} (meas_U : MeasurableSet U)
Goal: MeasureTheory.volume U = MeasureTheory.volume (QuotientAddGroup.mk ⁻¹' U ∩ Set.Ioc t (t + T))
-- AddCircle.volume_closedBall
theorem volume_closedBall {x : AddCircle T} (ε : ℝ) :
    volume (Metric.closedBall x ε) = ENNReal.ofReal (min T (2 * ε)) 
Arguments: (T : ℝ) [hT : Fact (0 < T)] {x : AddCircle T} (ε : ℝ)
Goal: MeasureTheory.volume (Metric.closedBall x ε) = ENNReal.ofReal (min T (2 * ε))
-- AddCircle.lintegral_preimage
protected theorem lintegral_preimage (t : ℝ) (f : AddCircle T → ℝ≥0∞) :
    (∫⁻ a in Ioc t (t + T), f a) = ∫⁻ b : AddCircle T, f b 
Arguments: (T : ℝ) [hT : Fact (0 < T)] (t : ℝ) (f : AddCircle T → ℝ≥0∞)
Goal: ∫⁻ (a : ℝ) in Set.Ioc t (t + T), f ↑a = ∫⁻ (b : AddCircle T), f b
-- AddCircle.integral_preimage
protected theorem integral_preimage (t : ℝ) (f : AddCircle T → E) :
    (∫ a in Ioc t (t + T), f a) = ∫ b : AddCircle T, f b 
Arguments: (T : ℝ) [hT : Fact (0 < T)] {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (t : ℝ) (f : AddCircle T → E)
Goal: ∫ (a : ℝ) in Set.Ioc t (t + T), f ↑a = ∫ (b : AddCircle T), f b
-- AddCircle.intervalIntegral_preimage
protected theorem intervalIntegral_preimage (t : ℝ) (f : AddCircle T → E) :
    ∫ a in t..t + T, f a = ∫ b : AddCircle T, f b 
Arguments: (T : ℝ) [hT : Fact (0 < T)] {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (t : ℝ) (f : AddCircle T → E)
Goal: ∫ (a : ℝ) in t..t + T, f ↑a = ∫ (b : AddCircle T), f b
-- UnitAddCircle.measure_univ
protected theorem measure_univ : volume (Set.univ : Set UnitAddCircle) = 1 
Goal: MeasureTheory.volume Set.univ = 1
-- UnitAddCircle.measurePreserving_mk
protected theorem measurePreserving_mk (t : ℝ) :
    MeasurePreserving (β := UnitAddCircle) ((↑) : ℝ → UnitAddCircle)
      (volume.restrict (Ioc t (t + 1))) 
Arguments: (t : ℝ)
Goal: MeasureTheory.MeasurePreserving QuotientAddGroup.mk (MeasureTheory.volume.restrict (Set.Ioc t (t + 1)))
  MeasureTheory.volume
-- UnitAddCircle.lintegral_preimage
protected theorem lintegral_preimage (t : ℝ) (f : UnitAddCircle → ℝ≥0∞) :
    (∫⁻ a in Ioc t (t + 1), f a) = ∫⁻ b : UnitAddCircle, f b 
Arguments: (t : ℝ) (f : UnitAddCircle → ℝ≥0∞)
Goal: ∫⁻ (a : ℝ) in Set.Ioc t (t + 1), f ↑a = ∫⁻ (b : UnitAddCircle), f b
-- UnitAddCircle.integral_preimage
protected theorem integral_preimage (t : ℝ) (f : UnitAddCircle → E) :
    (∫ a in Ioc t (t + 1), f a) = ∫ b : UnitAddCircle, f b 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (t : ℝ) (f : UnitAddCircle → E)
Goal: ∫ (a : ℝ) in Set.Ioc t (t + 1), f ↑a = ∫ (b : UnitAddCircle), f b
-- UnitAddCircle.intervalIntegral_preimage
protected theorem intervalIntegral_preimage (t : ℝ) (f : UnitAddCircle → E) :
    ∫ a in t..t + 1, f a = ∫ b : UnitAddCircle, f b 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (t : ℝ) (f : UnitAddCircle → E)
Goal: ∫ (a : ℝ) in t..t + 1, f ↑a = ∫ (b : UnitAddCircle), f b
-- Function.Periodic.intervalIntegral_add_eq_of_pos
theorem intervalIntegral_add_eq_of_pos (hf : Periodic f T) (hT : 0 < T) (t s : ℝ) :
    ∫ x in t..t + T, f x = ∫ x in s..s + T, f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {T : ℝ} (hf : Function.Periodic f T) (hT : 0 < T) (t s : ℝ)
Goal: ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in s..s + T, f x
-- Function.Periodic.intervalIntegral_add_eq
theorem intervalIntegral_add_eq (hf : Periodic f T) (t s : ℝ) :
    ∫ x in t..t + T, f x = ∫ x in s..s + T, f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {T : ℝ} (hf : Function.Periodic f T) (t s : ℝ)
Goal: ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in s..s + T, f x
-- Function.Periodic.intervalIntegral_add_eq_add
theorem intervalIntegral_add_eq_add (hf : Periodic f T) (t s : ℝ)
    (h_int : ∀ t₁ t₂, IntervalIntegrable f MeasureSpace.volume t₁ t₂) :
    ∫ x in t..s + T, f x = (∫ x in t..s, f x) + ∫ x in t..t + T, f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {T : ℝ} (hf : Function.Periodic f T) (t s : ℝ) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f MeasureTheory.volume t₁ t₂)
Goal: ∫ (x : ℝ) in t..s + T, f x = (∫ (x : ℝ) in t..s, f x) + ∫ (x : ℝ) in t..t + T, f x
-- Function.Periodic.intervalIntegral_add_zsmul_eq
theorem intervalIntegral_add_zsmul_eq (hf : Periodic f T) (n : ℤ) (t : ℝ)
    (h_int : ∀ t₁ t₂, IntervalIntegrable f MeasureSpace.volume t₁ t₂) :
    ∫ x in t..t + n • T, f x = n • ∫ x in t..t + T, f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {T : ℝ} (hf : Function.Periodic f T) (n : ℤ) (t : ℝ) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f MeasureTheory.volume t₁ t₂)
Goal: ∫ (x : ℝ) in t..t + n • T, f x = n • ∫ (x : ℝ) in t..t + T, f x