ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- smoothOn_iff_target
theorem smoothOn_iff_target :
    SmoothOn I I' f s ↔
      ContinuousOn f s ∧
        ∀ y : M', SmoothOn I 𝓘(𝕜, E') (extChartAt I' y ∘ f) (s ∩ f ⁻¹' (extChartAt I' y).source) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M}
Goal: SmoothOn I I' f s ↔
  ContinuousOn f s ∧ ∀ (y : M'), SmoothOn I 𝓘(𝕜, E') (↑(extChartAt I' y) ∘ f) (s ∩ f ⁻¹' (extChartAt I' y).source)
-- contMDiff_iff
theorem contMDiff_iff :
    ContMDiff I I' n f ↔
      Continuous f ∧
        ∀ (x : M) (y : M'),
          ContDiffOn 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
            ((extChartAt I x).target ∩
              (extChartAt I x).symm ⁻¹' (f ⁻¹' (extChartAt I' y).source)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {n : ℕ∞}
Goal: ContMDiff I I' n f ↔
  Continuous f ∧
    ∀ (x : M) (y : M'),
      ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (f ⁻¹' (extChartAt I' y).source))
-- contMDiff_iff_target
theorem contMDiff_iff_target :
    ContMDiff I I' n f ↔
      Continuous f ∧ ∀ y : M',
        ContMDiffOn I 𝓘(𝕜, E') n (extChartAt I' y ∘ f) (f ⁻¹' (extChartAt I' y).source) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {n : ℕ∞}
Goal: ContMDiff I I' n f ↔
  Continuous f ∧ ∀ (y : M'), ContMDiffOn I 𝓘(𝕜, E') n (↑(extChartAt I' y) ∘ f) (f ⁻¹' (extChartAt I' y).source)
-- smooth_iff
theorem smooth_iff :
    Smooth I I' f ↔
      Continuous f ∧
        ∀ (x : M) (y : M'),
          ContDiffOn 𝕜 ⊤ (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
            ((extChartAt I x).target ∩
              (extChartAt I x).symm ⁻¹' (f ⁻¹' (extChartAt I' y).source)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {f : M → M'}
Goal: Smooth I I' f ↔
  Continuous f ∧
    ∀ (x : M) (y : M'),
      ContDiffOn 𝕜 ⊤ (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (f ⁻¹' (extChartAt I' y).source))
-- smooth_iff_target
theorem smooth_iff_target :
    Smooth I I' f ↔
      Continuous f ∧
        ∀ y : M', SmoothOn I 𝓘(𝕜, E') (extChartAt I' y ∘ f) (f ⁻¹' (extChartAt I' y).source) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {f : M → M'}
Goal: Smooth I I' f ↔ Continuous f ∧ ∀ (y : M'), SmoothOn I 𝓘(𝕜, E') (↑(extChartAt I' y) ∘ f) (f ⁻¹' (extChartAt I' y).source)
-- ContMDiffWithinAt.of_succ
theorem ContMDiffWithinAt.of_succ {n : ℕ} (h : ContMDiffWithinAt I I' n.succ f s x) :
    ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ} (h : ContMDiffWithinAt I I' (↑n.succ) f s x)
Goal: ContMDiffWithinAt I I' (↑n) f s x
-- ContMDiffAt.of_succ
theorem ContMDiffAt.of_succ {n : ℕ} (h : ContMDiffAt I I' n.succ f x) : ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {n : ℕ} (h : ContMDiffAt I I' (↑n.succ) f x)
Goal: ContMDiffAt I I' (↑n) f x
-- ContMDiffOn.of_succ
theorem ContMDiffOn.of_succ {n : ℕ} (h : ContMDiffOn I I' n.succ f s) : ContMDiffOn I I' n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {n : ℕ} (h : ContMDiffOn I I' (↑n.succ) f s)
Goal: ContMDiffOn I I' (↑n) f s
-- ContMDiff.of_succ
theorem ContMDiff.of_succ {n : ℕ} (h : ContMDiff I I' n.succ f) : ContMDiff I I' n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {n : ℕ} (h : ContMDiff I I' (↑n.succ) f)
Goal: ContMDiff I I' (↑n) f
-- ContMDiffWithinAt.continuousWithinAt
theorem ContMDiffWithinAt.continuousWithinAt (hf : ContMDiffWithinAt I I' n f s x) :
    ContinuousWithinAt f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} (hf : ContMDiffWithinAt I I' n f s x)
Goal: ContinuousWithinAt f s x
-- ContMDiffAt.continuousAt
theorem ContMDiffAt.continuousAt (hf : ContMDiffAt I I' n f x) : ContinuousAt f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {n : ℕ∞} (hf : ContMDiffAt I I' n f x)
Goal: ContinuousAt f x
-- ContMDiffOn.continuousOn
theorem ContMDiffOn.continuousOn (hf : ContMDiffOn I I' n f s) : ContinuousOn f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {n : ℕ∞} (hf : ContMDiffOn I I' n f s)
Goal: ContinuousOn f s
-- ContMDiff.continuous
theorem ContMDiff.continuous (hf : ContMDiff I I' n f) : Continuous f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {n : ℕ∞} (hf : ContMDiff I I' n f)
Goal: Continuous f
-- contMDiffWithinAt_top
theorem contMDiffWithinAt_top :
    SmoothWithinAt I I' f s x ↔ ∀ n : ℕ, ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M}
Goal: SmoothWithinAt I I' f s x ↔ ∀ (n : ℕ), ContMDiffWithinAt I I' (↑n) f s x
-- contMDiffAt_top
theorem contMDiffAt_top : SmoothAt I I' f x ↔ ∀ n : ℕ, ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M}
Goal: SmoothAt I I' f x ↔ ∀ (n : ℕ), ContMDiffAt I I' (↑n) f x
-- contMDiffOn_top
theorem contMDiffOn_top : SmoothOn I I' f s ↔ ∀ n : ℕ, ContMDiffOn I I' n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M}
Goal: SmoothOn I I' f s ↔ ∀ (n : ℕ), ContMDiffOn I I' (↑n) f s
-- contMDiff_top
theorem contMDiff_top : Smooth I I' f ↔ ∀ n : ℕ, ContMDiff I I' n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'}
Goal: Smooth I I' f ↔ ∀ (n : ℕ), ContMDiff I I' (↑n) f
-- contMDiffWithinAt_iff_nat
theorem contMDiffWithinAt_iff_nat :
    ContMDiffWithinAt I I' n f s x ↔ ∀ m : ℕ, (m : ℕ∞) ≤ n → ContMDiffWithinAt I I' m f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞}
Goal: ContMDiffWithinAt I I' n f s x ↔ ∀ (m : ℕ), ↑m ≤ n → ContMDiffWithinAt I I' (↑m) f s x
-- ContMDiffWithinAt.mono_of_mem
theorem ContMDiffWithinAt.mono_of_mem (hf : ContMDiffWithinAt I I' n f s x) (hts : s ∈ 𝓝[t] x) :
    ContMDiffWithinAt I I' n f t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s t : Set M} {x : M} {n : ℕ∞} (hf : ContMDiffWithinAt I I' n f s x) (hts : s ∈ 𝓝[t] x)
Goal: ContMDiffWithinAt I I' n f t x
-- ContMDiffWithinAt.mono
theorem ContMDiffWithinAt.mono (hf : ContMDiffWithinAt I I' n f s x) (hts : t ⊆ s) :
    ContMDiffWithinAt I I' n f t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s t : Set M} {x : M} {n : ℕ∞} (hf : ContMDiffWithinAt I I' n f s x) (hts : t ⊆ s)
Goal: ContMDiffWithinAt I I' n f t x
-- contMDiffWithinAt_congr_nhds
theorem contMDiffWithinAt_congr_nhds (hst : 𝓝[s] x = 𝓝[t] x) :
    ContMDiffWithinAt I I' n f s x ↔ ContMDiffWithinAt I I' n f t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s t : Set M} {x : M} {n : ℕ∞} (hst : 𝓝[s] x = 𝓝[t] x)
Goal: ContMDiffWithinAt I I' n f s x ↔ ContMDiffWithinAt I I' n f t x
-- contMDiffWithinAt_insert_self
theorem contMDiffWithinAt_insert_self :
    ContMDiffWithinAt I I' n f (insert x s) x ↔ ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞}
Goal: ContMDiffWithinAt I I' n f (insert x s) x ↔ ContMDiffWithinAt I I' n f s x
-- ContMDiffWithinAt.insert
theorem ContMDiffWithinAt.insert (h : ContMDiffWithinAt I I' n f s x) :
    ContMDiffWithinAt I I' n f (insert x s) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} (h : ContMDiffWithinAt I I' n f s x)
Goal: ContMDiffWithinAt I I' n f (insert x s) x
-- ContMDiffAt.contMDiffWithinAt
theorem ContMDiffAt.contMDiffWithinAt (hf : ContMDiffAt I I' n f x) :
    ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} (hf : ContMDiffAt I I' n f x)
Goal: ContMDiffWithinAt I I' n f s x
-- SmoothAt.smoothWithinAt
theorem SmoothAt.smoothWithinAt (hf : SmoothAt I I' f x) : SmoothWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} (hf : SmoothAt I I' f x)
Goal: SmoothWithinAt I I' f s x
-- ContMDiffOn.mono
theorem ContMDiffOn.mono (hf : ContMDiffOn I I' n f s) (hts : t ⊆ s) : ContMDiffOn I I' n f t 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s t : Set M} {n : ℕ∞} (hf : ContMDiffOn I I' n f s) (hts : t ⊆ s)
Goal: ContMDiffOn I I' n f t
-- ContMDiff.contMDiffOn
theorem ContMDiff.contMDiffOn (hf : ContMDiff I I' n f) : ContMDiffOn I I' n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {n : ℕ∞} (hf : ContMDiff I I' n f)
Goal: ContMDiffOn I I' n f s
-- Smooth.smoothOn
theorem Smooth.smoothOn (hf : Smooth I I' f) : SmoothOn I I' f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} (hf : Smooth I I' f)
Goal: SmoothOn I I' f s
-- contMDiffWithinAt_inter'
theorem contMDiffWithinAt_inter' (ht : t ∈ 𝓝[s] x) :
    ContMDiffWithinAt I I' n f (s ∩ t) x ↔ ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s t : Set M} {x : M} {n : ℕ∞} (ht : t ∈ 𝓝[s] x)
Goal: ContMDiffWithinAt I I' n f (s ∩ t) x ↔ ContMDiffWithinAt I I' n f s x
-- contMDiffWithinAt_inter
theorem contMDiffWithinAt_inter (ht : t ∈ 𝓝 x) :
    ContMDiffWithinAt I I' n f (s ∩ t) x ↔ ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s t : Set M} {x : M} {n : ℕ∞} (ht : t ∈ 𝓝 x)
Goal: ContMDiffWithinAt I I' n f (s ∩ t) x ↔ ContMDiffWithinAt I I' n f s x
-- ContMDiffWithinAt.contMDiffAt
theorem ContMDiffWithinAt.contMDiffAt (h : ContMDiffWithinAt I I' n f s x) (ht : s ∈ 𝓝 x) :
    ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} (h : ContMDiffWithinAt I I' n f s x) (ht : s ∈ 𝓝 x)
Goal: ContMDiffAt I I' n f x
-- SmoothWithinAt.smoothAt
theorem SmoothWithinAt.smoothAt (h : SmoothWithinAt I I' f s x) (ht : s ∈ 𝓝 x) :
    SmoothAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} (h : SmoothWithinAt I I' f s x) (ht : s ∈ 𝓝 x)
Goal: SmoothAt I I' f x
-- ContMDiffOn.contMDiffAt
theorem ContMDiffOn.contMDiffAt (h : ContMDiffOn I I' n f s) (hx : s ∈ 𝓝 x) :
    ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} (h : ContMDiffOn I I' n f s) (hx : s ∈ 𝓝 x)
Goal: ContMDiffAt I I' n f x
-- SmoothOn.smoothAt
theorem SmoothOn.smoothAt (h : SmoothOn I I' f s) (hx : s ∈ 𝓝 x) : SmoothAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M} (h : SmoothOn I I' f s) (hx : s ∈ 𝓝 x)
Goal: SmoothAt I I' f x
-- contMDiffOn_iff_source_of_mem_maximalAtlas
theorem contMDiffOn_iff_source_of_mem_maximalAtlas (he : e ∈ maximalAtlas I M) (hs : s ⊆ e.source) :
    ContMDiffOn I I' n f s ↔
      ContMDiffOn 𝓘(𝕜, E) I' n (f ∘ (e.extend I).symm) (e.extend I '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {e : PartialHomeomorph M H} {f : M → M'} {s : Set M} {n : ℕ∞} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hs : s ⊆ e.source)
Goal: ContMDiffOn I I' n f s ↔ ContMDiffOn 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s)