ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AnalyticAt.analyticWithinAt
lemma AnalyticAt.analyticWithinAt {f : E → F} {s : Set E} {x : E} (h : AnalyticAt 𝕜 f x) :
    AnalyticWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {x : E} (h : AnalyticAt 𝕜 f x)
Goal: AnalyticWithinAt 𝕜 f s x
-- AnalyticOn.analyticWithinOn
lemma AnalyticOn.analyticWithinOn {f : E → F} {s : Set E} (h : AnalyticOn 𝕜 f s) :
    AnalyticWithinOn 𝕜 f s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : AnalyticOn 𝕜 f s)
Goal: AnalyticWithinOn 𝕜 f s
-- AnalyticWithinOn.continuousOn
lemma AnalyticWithinOn.continuousOn {f : E → F} {s : Set E} (h : AnalyticWithinOn 𝕜 f s) :
    ContinuousOn f s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : AnalyticWithinOn 𝕜 f s)
Goal: ContinuousOn f s
-- analyticWithinOn_of_locally_analyticWithinOn
lemma analyticWithinOn_of_locally_analyticWithinOn {f : E → F} {s : Set E}
    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ AnalyticWithinOn 𝕜 f (s ∩ u)) :
    AnalyticWithinOn 𝕜 f s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ AnalyticWithinOn 𝕜 f (s ∩ u))
Goal: AnalyticWithinOn 𝕜 f s
-- IsOpen.analyticWithinOn_iff_analyticOn
@[simp] lemma IsOpen.analyticWithinOn_iff_analyticOn {f : E → F} {s : Set E} (hs : IsOpen s) :
    AnalyticWithinOn 𝕜 f s ↔ AnalyticOn 𝕜 f s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hs : IsOpen s)
Goal: AnalyticWithinOn 𝕜 f s ↔ AnalyticOn 𝕜 f s
-- hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall
lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpace F] {f : E → F}
    {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ℝ≥0∞} :
    HasFPowerSeriesWithinOnBall f p s x r ↔
      ContinuousWithinAt f s x ∧ ∃ g, EqOn f g (s ∩ EMetric.ball x r) ∧
        HasFPowerSeriesOnBall g p x r 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ℝ≥0∞}
Goal: HasFPowerSeriesWithinOnBall f p s x r ↔
  ContinuousWithinAt f s x ∧ ∃ g, Set.EqOn f g (s ∩ EMetric.ball x r) ∧ HasFPowerSeriesOnBall g p x r
-- hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt
lemma hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt [CompleteSpace F] {f : E → F}
    {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} :
    HasFPowerSeriesWithinAt f p s x ↔
      ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ HasFPowerSeriesAt g p x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E}
Goal: HasFPowerSeriesWithinAt f p s x ↔ ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ HasFPowerSeriesAt g p x
-- analyticWithinAt_iff_exists_analyticAt
lemma analyticWithinAt_iff_exists_analyticAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E} :
    AnalyticWithinAt 𝕜 f s x ↔
      ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
Goal: AnalyticWithinAt 𝕜 f s x ↔ ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x
-- AnalyticWithinAt.exists_analyticAt
lemma AnalyticWithinAt.exists_analyticAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
    (h : AnalyticWithinAt 𝕜 f s x) : ∃ g, f x = g x ∧ f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {s : Set E} {x : E} (h : AnalyticWithinAt 𝕜 f s x)
Goal: ∃ g, f x = g x ∧ f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x
-- AnalyticWithinAt.congr_of_eventuallyEq
lemma AnalyticWithinAt.congr_of_eventuallyEq [CompleteSpace F] {f g : E → F} {s : Set E} {x : E}
    (hf : AnalyticWithinAt 𝕜 f s x) (hs : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
    AnalyticWithinAt 𝕜 g s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f g : E → F} {s : Set E} {x : E} (hf : AnalyticWithinAt 𝕜 f s x) (hs : f =ᶠ[𝓝[s] x] g) (hx : f x = g x)
Goal: AnalyticWithinAt 𝕜 g s x
-- AnalyticWithinAt.congr
lemma AnalyticWithinAt.congr [CompleteSpace F] {f g : E → F} {s : Set E} {x : E}
    (hf : AnalyticWithinAt 𝕜 f s x) (hs : EqOn f g s) (hx : f x = g x) :
    AnalyticWithinAt 𝕜 g s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f g : E → F} {s : Set E} {x : E} (hf : AnalyticWithinAt 𝕜 f s x) (hs : Set.EqOn f g s) (hx : f x = g x)
Goal: AnalyticWithinAt 𝕜 g s x
-- AnalyticWithinOn.congr
lemma AnalyticWithinOn.congr [CompleteSpace F] {f g : E → F} {s : Set E}
    (hf : AnalyticWithinOn 𝕜 f s) (hs : EqOn f g s) :
    AnalyticWithinOn 𝕜 g s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f g : E → F} {s : Set E} (hf : AnalyticWithinOn 𝕜 f s) (hs : Set.EqOn f g s)
Goal: AnalyticWithinOn 𝕜 g s
-- HasFPowerSeriesWithinOnBall.mono
lemma HasFPowerSeriesWithinOnBall.mono {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {s t : Set E} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesWithinOnBall f p t x r)
    (hs : s ⊆ t) : HasFPowerSeriesWithinOnBall f p s x r 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s t : Set E} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesWithinOnBall f p t x r) (hs : s ⊆ t)
Goal: HasFPowerSeriesWithinOnBall f p s x r
-- HasFPowerSeriesWithinAt.mono
lemma HasFPowerSeriesWithinAt.mono {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {s t : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p t x)
    (hs : s ⊆ t) : HasFPowerSeriesWithinAt f p s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s t : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p t x) (hs : s ⊆ t)
Goal: HasFPowerSeriesWithinAt f p s x
-- AnalyticWithinAt.mono
lemma AnalyticWithinAt.mono {f : E → F} {s t : Set E} {x : E} (h : AnalyticWithinAt 𝕜 f t x)
    (hs : s ⊆ t) : AnalyticWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s t : Set E} {x : E} (h : AnalyticWithinAt 𝕜 f t x) (hs : s ⊆ t)
Goal: AnalyticWithinAt 𝕜 f s x
-- AnalyticWithinOn.mono
lemma AnalyticWithinOn.mono {f : E → F} {s t : Set E} (h : AnalyticWithinOn 𝕜 f t)
    (hs : s ⊆ t) : AnalyticWithinOn 𝕜 f s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s t : Set E} (h : AnalyticWithinOn 𝕜 f t) (hs : s ⊆ t)
Goal: AnalyticWithinOn 𝕜 f s
-- AnalyticWithinAt.comp
lemma AnalyticWithinAt.comp [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F}
    {t : Set E} {x : E} (hf : AnalyticWithinAt 𝕜 f s (g x)) (hg : AnalyticWithinAt 𝕜 g t x)
    (h : MapsTo g t s) : AnalyticWithinAt 𝕜 (f ∘ g) t x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F} {t : Set E} {x : E} (hf : AnalyticWithinAt 𝕜 f s (g x)) (hg : AnalyticWithinAt 𝕜 g t x) (h : Set.MapsTo g t s)
Goal: AnalyticWithinAt 𝕜 (f ∘ g) t x
-- AnalyticWithinOn.comp
lemma AnalyticWithinOn.comp [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F}
    {t : Set E} (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g t) (h : MapsTo g t s) :
    AnalyticWithinOn 𝕜 (f ∘ g) t 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F} {t : Set E} (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g t) (h : Set.MapsTo g t s)
Goal: AnalyticWithinOn 𝕜 (f ∘ g) t
-- AnalyticWithinAt.contDiffWithinAt
lemma AnalyticWithinAt.contDiffWithinAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
    (h : AnalyticWithinAt 𝕜 f s x) {n : ℕ∞} : ContDiffWithinAt 𝕜 n f s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {s : Set E} {x : E} (h : AnalyticWithinAt 𝕜 f s x) {n : ℕ∞}
Goal: ContDiffWithinAt 𝕜 n f s x
-- AnalyticWithinOn.contDiffOn
lemma AnalyticWithinOn.contDiffOn [CompleteSpace F] {f : E → F} {s : Set E}
    (h : AnalyticWithinOn 𝕜 f s) {n : ℕ∞} : ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {s : Set E} (h : AnalyticWithinOn 𝕜 f s) {n : ℕ∞}
Goal: ContDiffOn 𝕜 n f s
-- HasFPowerSeriesWithinOnBall.prod
lemma HasFPowerSeriesWithinOnBall.prod {e : E} {f : E → F} {g : E → G} {s : Set E} {r t : ℝ≥0∞}
    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
    (hf : HasFPowerSeriesWithinOnBall f p s e r) (hg : HasFPowerSeriesWithinOnBall g q s e t) :
    HasFPowerSeriesWithinOnBall (fun x ↦ (f x, g x)) (p.prod q) s e (min r t) 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {e : E} {f : E → F} {g : E → G} {s : Set E} {r t : ℝ≥0∞} {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G} (hf : HasFPowerSeriesWithinOnBall f p s e r) (hg : HasFPowerSeriesWithinOnBall g q s e t)
Goal: HasFPowerSeriesWithinOnBall (fun x => (f x, g x)) (p.prod q) s e (min r t)
-- HasFPowerSeriesWithinAt.prod
lemma HasFPowerSeriesWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
    (hf : HasFPowerSeriesWithinAt f p s e) (hg : HasFPowerSeriesWithinAt g q s e) :
    HasFPowerSeriesWithinAt (fun x ↦ (f x, g x)) (p.prod q) s e 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {e : E} {f : E → F} {g : E → G} {s : Set E} {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G} (hf : HasFPowerSeriesWithinAt f p s e) (hg : HasFPowerSeriesWithinAt g q s e)
Goal: HasFPowerSeriesWithinAt (fun x => (f x, g x)) (p.prod q) s e
-- AnalyticWithinAt.prod
lemma AnalyticWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
    (hf : AnalyticWithinAt 𝕜 f s e) (hg : AnalyticWithinAt 𝕜 g s e) :
    AnalyticWithinAt 𝕜 (fun x ↦ (f x, g x)) s e 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {e : E} {f : E → F} {g : E → G} {s : Set E} (hf : AnalyticWithinAt 𝕜 f s e) (hg : AnalyticWithinAt 𝕜 g s e)
Goal: AnalyticWithinAt 𝕜 (fun x => (f x, g x)) s e
-- AnalyticWithinOn.prod
lemma AnalyticWithinOn.prod {f : E → F} {g : E → G} {s : Set E}
    (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g s) :
    AnalyticWithinOn 𝕜 (fun x ↦ (f x, g x)) s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {g : E → G} {s : Set E} (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g s)
Goal: AnalyticWithinOn 𝕜 (fun x => (f x, g x)) s
-- Basis.parallelepiped_basisFun
theorem Basis.parallelepiped_basisFun (ι : Type*) [Fintype ι] :
    (Pi.basisFun ℝ ι).parallelepiped = TopologicalSpace.PositiveCompacts.piIcc01 ι 
Arguments: (ι : Type u_1) [Fintype ι]
Goal: (Pi.basisFun ℝ ι).parallelepiped = TopologicalSpace.PositiveCompacts.piIcc01 ι
-- Basis.parallelepiped_eq_map
theorem Basis.parallelepiped_eq_map  {ι E : Type*} [Fintype ι] [NormedAddCommGroup E]
    [NormedSpace ℝ E] (b : Basis ι ℝ E) :
    b.parallelepiped = (PositiveCompacts.piIcc01 ι).map b.equivFun.symm
      b.equivFunL.symm.continuous b.equivFunL.symm.isOpenMap 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E)
Goal: b.parallelepiped =
  TopologicalSpace.PositiveCompacts.map ⇑b.equivFun.symm ⋯ ⋯ (TopologicalSpace.PositiveCompacts.piIcc01 ι)
-- Basis.map_addHaar
theorem Basis.map_addHaar {ι E F : Type*} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F]
    [NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E]
    [BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F]
    (b : Basis ι ℝ E) (f : E ≃L[ℝ] F) :
    map f b.addHaar = (b.map f.toLinearEquiv).addHaar 
Arguments: {ι : Type u_1} {E : Type u_2} {F : Type u_3} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E] [BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F] (b : Basis ι ℝ E) (f : E ≃L[ℝ] F)
Goal: MeasureTheory.Measure.map (⇑f) b.addHaar = (b.map f.toLinearEquiv).addHaar
-- MeasureTheory.addHaarMeasure_eq_volume
theorem addHaarMeasure_eq_volume : addHaarMeasure Icc01 = volume 
Goal: MeasureTheory.Measure.addHaarMeasure TopologicalSpace.PositiveCompacts.Icc01 = MeasureTheory.volume
-- MeasureTheory.addHaarMeasure_eq_volume_pi
theorem addHaarMeasure_eq_volume_pi (ι : Type*) [Fintype ι] :
    addHaarMeasure (piIcc01 ι) = volume 
Arguments: (ι : Type u_1) [Fintype ι]
Goal: MeasureTheory.Measure.addHaarMeasure (TopologicalSpace.PositiveCompacts.piIcc01 ι) = MeasureTheory.volume
-- MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates_aux
theorem addHaar_eq_zero_of_disjoint_translates_aux {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E)
    [IsAddHaarMeasure μ] {s : Set E} (u : ℕ → E) (sb : IsBounded s) (hu : IsBounded (range u))
    (hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : MeasurableSet s) : μ s = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {s : Set E} (u : ℕ → E) (sb : Bornology.IsBounded s) (hu : Bornology.IsBounded (Set.range u)) (hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : MeasurableSet s)
Goal: μ s = 0
-- MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates
theorem addHaar_eq_zero_of_disjoint_translates {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E)
    [IsAddHaarMeasure μ] {s : Set E} (u : ℕ → E) (hu : IsBounded (range u))
    (hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : MeasurableSet s) : μ s = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {s : Set E} (u : ℕ → E) (hu : Bornology.IsBounded (Set.range u)) (hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : MeasurableSet s)
Goal: μ s = 0
-- MeasureTheory.Measure.addHaar_submodule
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Submodule ℝ E) (hs : s ≠ ⊤)
Goal: μ ↑s = 0
-- MeasureTheory.Measure.addHaar_affineSubspace
theorem addHaar_affineSubspace {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
    (s : AffineSubspace ℝ E) (hs : s ≠ ⊤) : μ s = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : AffineSubspace ℝ E) (hs : s ≠ ⊤)
Goal: μ ↑s = 0
-- MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar
theorem map_linearMap_addHaar_pi_eq_smul_addHaar {ι : Type*} [Finite ι] {f : (ι → ℝ) →ₗ[ℝ] ι → ℝ}
    (hf : LinearMap.det f ≠ 0) (μ : Measure (ι → ℝ)) [IsAddHaarMeasure μ] :
    Measure.map f μ = ENNReal.ofReal (abs (LinearMap.det f)⁻¹) • μ 
Arguments: {ι : Type u_1} [Finite ι] {f : (ι → ℝ) →ₗ[ℝ] ι → ℝ} (hf : LinearMap.det f ≠ 0) (μ : MeasureTheory.Measure (ι → ℝ)) [μ.IsAddHaarMeasure]
Goal: MeasureTheory.Measure.map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ
-- MeasureTheory.Measure.map_linearMap_addHaar_eq_smul_addHaar
theorem map_linearMap_addHaar_eq_smul_addHaar {f : E →ₗ[ℝ] E} (hf : LinearMap.det f ≠ 0) :
    Measure.map f μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {f : E →ₗ[ℝ] E} (hf : LinearMap.det f ≠ 0)
Goal: MeasureTheory.Measure.map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ
-- MeasureTheory.Measure.addHaar_preimage_linearMap
@[simp]
theorem addHaar_preimage_linearMap {f : E →ₗ[ℝ] E} (hf : LinearMap.det f ≠ 0) (s : Set E) :
    μ (f ⁻¹' s) = ENNReal.ofReal |(LinearMap.det f)⁻¹| * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {f : E →ₗ[ℝ] E} (hf : LinearMap.det f ≠ 0) (s : Set E)
Goal: μ (⇑f ⁻¹' s) = ENNReal.ofReal |(LinearMap.det f)⁻¹| * μ s
-- MeasureTheory.Measure.addHaar_preimage_continuousLinearMap
@[simp]
theorem addHaar_preimage_continuousLinearMap {f : E →L[ℝ] E}
    (hf : LinearMap.det (f : E →ₗ[ℝ] E) ≠ 0) (s : Set E) :
    μ (f ⁻¹' s) = ENNReal.ofReal (abs (LinearMap.det (f : E →ₗ[ℝ] E))⁻¹) * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {f : E →L[ℝ] E} (hf : LinearMap.det ↑f ≠ 0) (s : Set E)
Goal: μ (⇑f ⁻¹' s) = ENNReal.ofReal |(LinearMap.det ↑f)⁻¹| * μ s
-- MeasureTheory.Measure.addHaar_preimage_linearEquiv
@[simp]
theorem addHaar_preimage_linearEquiv (f : E ≃ₗ[ℝ] E) (s : Set E) :
    μ (f ⁻¹' s) = ENNReal.ofReal |LinearMap.det (f.symm : E →ₗ[ℝ] E)| * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E ≃ₗ[ℝ] E) (s : Set E)
Goal: μ (⇑f ⁻¹' s) = ENNReal.ofReal |LinearMap.det ↑f.symm| * μ s
-- MeasureTheory.Measure.addHaar_preimage_continuousLinearEquiv
@[simp]
theorem addHaar_preimage_continuousLinearEquiv (f : E ≃L[ℝ] E) (s : Set E) :
    μ (f ⁻¹' s) = ENNReal.ofReal |LinearMap.det (f.symm : E →ₗ[ℝ] E)| * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E ≃L[ℝ] E) (s : Set E)
Goal: μ (⇑f ⁻¹' s) = ENNReal.ofReal |LinearMap.det ↑↑f.symm| * μ s
-- MeasureTheory.Measure.addHaar_image_linearMap
@[simp]
theorem addHaar_image_linearMap (f : E →ₗ[ℝ] E) (s : Set E) :
    μ (f '' s) = ENNReal.ofReal |LinearMap.det f| * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E →ₗ[ℝ] E) (s : Set E)
Goal: μ (⇑f '' s) = ENNReal.ofReal |LinearMap.det f| * μ s
-- MeasureTheory.Measure.addHaar_image_continuousLinearMap
@[simp]
theorem addHaar_image_continuousLinearMap (f : E →L[ℝ] E) (s : Set E) :
    μ (f '' s) = ENNReal.ofReal |LinearMap.det (f : E →ₗ[ℝ] E)| * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E →L[ℝ] E) (s : Set E)
Goal: μ (⇑f '' s) = ENNReal.ofReal |LinearMap.det ↑f| * μ s
-- MeasureTheory.Measure.addHaar_image_continuousLinearEquiv
@[simp]
theorem addHaar_image_continuousLinearEquiv (f : E ≃L[ℝ] E) (s : Set E) :
    μ (f '' s) = ENNReal.ofReal |LinearMap.det (f : E →ₗ[ℝ] E)| * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E ≃L[ℝ] E) (s : Set E)
Goal: μ (⇑f '' s) = ENNReal.ofReal |LinearMap.det ↑↑f| * μ s
-- MeasureTheory.Measure.LinearMap.quasiMeasurePreserving
theorem LinearMap.quasiMeasurePreserving (f : E →ₗ[ℝ] E) (hf : LinearMap.det f ≠ 0) :
    QuasiMeasurePreserving f μ μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E →ₗ[ℝ] E) (hf : LinearMap.det f ≠ 0)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (⇑f) μ μ
-- MeasureTheory.Measure.ContinuousLinearMap.quasiMeasurePreserving
theorem ContinuousLinearMap.quasiMeasurePreserving (f : E →L[ℝ] E) (hf : f.det ≠ 0) :
    QuasiMeasurePreserving f μ μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E →L[ℝ] E) (hf : f.det ≠ 0)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (⇑f) μ μ
-- MeasureTheory.Measure.map_addHaar_smul
theorem map_addHaar_smul {r : ℝ} (hr : r ≠ 0) :
    Measure.map (r • ·) μ = ENNReal.ofReal (abs (r ^ finrank ℝ E)⁻¹) • μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {r : ℝ} (hr : r ≠ 0)
Goal: MeasureTheory.Measure.map (fun x => r • x) μ = ENNReal.ofReal |(r ^ FiniteDimensional.finrank ℝ E)⁻¹| • μ