ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContMDiffOn.div_const
@[to_additive]
theorem ContMDiffOn.div_const (hf : ContMDiffOn I' I n f s) :
    ContMDiffOn I' I n (fun x ↦ f x / c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {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 → G} {s : Set M} {n : ℕ∞} (c : G) (hf : ContMDiffOn I' I n f s)
Goal: ContMDiffOn I' I n (fun x => f x / c) s
-- ContMDiff.sub_const
@[to_additive]
theorem ContMDiff.div_const (hf : ContMDiff I' I n f) :
    ContMDiff I' I n (fun x ↦ f x / c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {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 → G} {n : ℕ∞} (c : G) (hf : ContMDiff I' I n f)
Goal: ContMDiff I' I n fun x => f x - c
-- ContMDiff.div_const
@[to_additive]
theorem ContMDiff.div_const (hf : ContMDiff I' I n f) :
    ContMDiff I' I n (fun x ↦ f x / c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {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 → G} {n : ℕ∞} (c : G) (hf : ContMDiff I' I n f)
Goal: ContMDiff I' I n fun x => f x / c
-- SmoothWithinAt.div_const
@[to_additive]
nonrec theorem SmoothWithinAt.div_const (hf : SmoothWithinAt I' I f s x) :
  SmoothWithinAt I' I (fun x ↦ f x / c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {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 → G} {s : Set M} {x : M} (c : G) (hf : SmoothWithinAt I' I f s x)
Goal: SmoothWithinAt I' I (fun x => f x / c) s x
-- SmoothWithinAt.sub_const
@[to_additive]
nonrec theorem SmoothWithinAt.div_const (hf : SmoothWithinAt I' I f s x) :
  SmoothWithinAt I' I (fun x ↦ f x / c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {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 → G} {s : Set M} {x : M} (c : G) (hf : SmoothWithinAt I' I f s x)
Goal: SmoothWithinAt I' I (fun x => f x - c) s x
-- SmoothAt.div_const
@[to_additive]
nonrec theorem SmoothAt.div_const (hf : SmoothAt I' I f x) :
    SmoothAt I' I (fun x ↦ f x / c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {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 → G} {x : M} (c : G) (hf : SmoothAt I' I f x)
Goal: SmoothAt I' I (fun x => f x / c) x
-- SmoothAt.sub_const
@[to_additive]
nonrec theorem SmoothAt.div_const (hf : SmoothAt I' I f x) :
    SmoothAt I' I (fun x ↦ f x / c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {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 → G} {x : M} (c : G) (hf : SmoothAt I' I f x)
Goal: SmoothAt I' I (fun x => f x - c) x
-- SmoothOn.div_const
@[to_additive]
nonrec theorem SmoothOn.div_const (hf : SmoothOn I' I f s) :
    SmoothOn I' I (fun x ↦ f x / c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {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 → G} {s : Set M} (c : G) (hf : SmoothOn I' I f s)
Goal: SmoothOn I' I (fun x => f x / c) s
-- SmoothOn.sub_const
@[to_additive]
nonrec theorem SmoothOn.div_const (hf : SmoothOn I' I f s) :
    SmoothOn I' I (fun x ↦ f x / c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {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 → G} {s : Set M} (c : G) (hf : SmoothOn I' I f s)
Goal: SmoothOn I' I (fun x => f x - c) s
-- Smooth.sub_const
@[to_additive]
nonrec theorem Smooth.div_const (hf : Smooth I' I f) : Smooth I' I (fun x ↦ f x / c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {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 → G} (c : G) (hf : Smooth I' I f)
Goal: Smooth I' I fun x => f x - c
-- Smooth.div_const
@[to_additive]
nonrec theorem Smooth.div_const (hf : Smooth I' I f) : Smooth I' I (fun x ↦ f x / c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {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 → G} (c : G) (hf : Smooth I' I f)
Goal: Smooth I' I fun x => f x / c
-- SmoothBumpFunction.closedBall_subset
structure SmoothBumpFunction (c : M) extends ContDiffBump (extChartAt I c c) where
  closedBall_subset : closedBall (extChartAt I c c) rOut ∩ range I ⊆ (extChartAt I c).target
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (self : SmoothBumpFunction I c)
Goal: Metric.closedBall (↑(extChartAt I c) c) self.rOut ∩ Set.range ↑I ⊆ (extChartAt I c).target
-- SmoothBumpFunction.coe_def
theorem coe_def : ⇑f = indicator (chartAt H c).source (f.toContDiffBump ∘ extChartAt I c) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: ↑f = (chartAt H c).source.indicator (↑f.toContDiffBump ∘ ↑(extChartAt I c))
-- SmoothBumpFunction.rOut_pos
theorem rOut_pos : 0 < f.rOut 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: 0 < f.rOut
-- SmoothBumpFunction.ball_subset
theorem ball_subset : ball (extChartAt I c c) f.rOut ∩ range I ⊆ (extChartAt I c).target 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: Metric.ball (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I ⊆ (extChartAt I c).target
-- SmoothBumpFunction.ball_inter_range_eq_ball_inter_target
theorem ball_inter_range_eq_ball_inter_target :
    ball (extChartAt I c c) f.rOut ∩ range I =
      ball (extChartAt I c c) f.rOut ∩ (extChartAt I c).target 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: Metric.ball (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I =
  Metric.ball (↑(extChartAt I c) c) f.rOut ∩ (extChartAt I c).target
-- SmoothBumpFunction.eqOn_source
theorem eqOn_source : EqOn f (f.toContDiffBump ∘ extChartAt I c) (chartAt H c).source 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: Set.EqOn (↑f) (↑f.toContDiffBump ∘ ↑(extChartAt I c)) (chartAt H c).source
-- SmoothBumpFunction.eventuallyEq_of_mem_source
theorem eventuallyEq_of_mem_source (hx : x ∈ (chartAt H c).source) :
    f =ᶠ[𝓝 x] f.toContDiffBump ∘ extChartAt I c 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {x : M} (hx : x ∈ (chartAt H c).source)
Goal: ↑f =ᶠ[𝓝 x] ↑f.toContDiffBump ∘ ↑(extChartAt I c)
-- SmoothBumpFunction.one_of_dist_le
theorem one_of_dist_le (hs : x ∈ (chartAt H c).source)
    (hd : dist (extChartAt I c x) (extChartAt I c c) ≤ f.rIn) : f x = 1 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {x : M} (hs : x ∈ (chartAt H c).source) (hd : dist (↑(extChartAt I c) x) (↑(extChartAt I c) c) ≤ f.rIn)
Goal: ↑f x = 1
-- SmoothBumpFunction.support_eq_inter_preimage
theorem support_eq_inter_preimage :
    support f = (chartAt H c).source ∩ extChartAt I c ⁻¹' ball (extChartAt I c c) f.rOut 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: Function.support ↑f = (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' Metric.ball (↑(extChartAt I c) c) f.rOut
-- SmoothBumpFunction.isOpen_support
theorem isOpen_support : IsOpen (support f) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: IsOpen (Function.support ↑f)
-- SmoothBumpFunction.support_eq_symm_image
theorem support_eq_symm_image :
    support f = (extChartAt I c).symm '' (ball (extChartAt I c c) f.rOut ∩ range I) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: Function.support ↑f = ↑(extChartAt I c).symm '' (Metric.ball (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I)
-- SmoothBumpFunction.support_subset_source
theorem support_subset_source : support f ⊆ (chartAt H c).source 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: Function.support ↑f ⊆ (chartAt H c).source
-- SmoothBumpFunction.image_eq_inter_preimage_of_subset_support
theorem image_eq_inter_preimage_of_subset_support {s : Set M} (hs : s ⊆ support f) :
    extChartAt I c '' s =
      closedBall (extChartAt I c c) f.rOut ∩ range I ∩ (extChartAt I c).symm ⁻¹' s 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {s : Set M} (hs : s ⊆ Function.support ↑f)
Goal: ↑(extChartAt I c) '' s = Metric.closedBall (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s
-- SmoothBumpFunction.mem_Icc
theorem mem_Icc : f x ∈ Icc (0 : ℝ) 1 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {x : M}
Goal: ↑f x ∈ Set.Icc 0 1
-- SmoothBumpFunction.nonneg
theorem nonneg : 0 ≤ f x 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {x : M}
Goal: 0 ≤ ↑f x
-- SmoothBumpFunction.le_one
theorem le_one : f x ≤ 1 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {x : M}
Goal: ↑f x ≤ 1
-- SmoothBumpFunction.eventuallyEq_one_of_dist_lt
theorem eventuallyEq_one_of_dist_lt (hs : x ∈ (chartAt H c).source)
    (hd : dist (extChartAt I c x) (extChartAt I c c) < f.rIn) : f =ᶠ[𝓝 x] 1 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {x : M} (hs : x ∈ (chartAt H c).source) (hd : dist (↑(extChartAt I c) x) (↑(extChartAt I c) c) < f.rIn)
Goal: ↑f =ᶠ[𝓝 x] 1
-- SmoothBumpFunction.eventuallyEq_one
theorem eventuallyEq_one : f =ᶠ[𝓝 c] 1 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: ↑f =ᶠ[𝓝 c] 1
-- SmoothBumpFunction.eq_one
@[simp]
theorem eq_one : f c = 1 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: ↑f c = 1
-- SmoothBumpFunction.support_mem_nhds
theorem support_mem_nhds : support f ∈ 𝓝 c 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: Function.support ↑f ∈ 𝓝 c
-- SmoothBumpFunction.tsupport_mem_nhds
theorem tsupport_mem_nhds : tsupport f ∈ 𝓝 c 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: tsupport ↑f ∈ 𝓝 c
-- SmoothBumpFunction.c_mem_support
theorem c_mem_support : c ∈ support f 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: c ∈ Function.support ↑f
-- SmoothBumpFunction.nonempty_support
theorem nonempty_support : (support f).Nonempty 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: (Function.support ↑f).Nonempty
-- SmoothBumpFunction.isCompact_symm_image_closedBall
theorem isCompact_symm_image_closedBall :
    IsCompact ((extChartAt I c).symm '' (closedBall (extChartAt I c c) f.rOut ∩ range I)) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c)
Goal: IsCompact (↑(extChartAt I c).symm '' (Metric.closedBall (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I))
-- SmoothBumpFunction.nhdsWithin_range_basis
theorem nhdsWithin_range_basis :
    (𝓝[range I] extChartAt I c c).HasBasis (fun _ : SmoothBumpFunction I c => True) fun f =>
      closedBall (extChartAt I c c) f.rOut ∩ range I 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M}
Goal: (𝓝[Set.range ↑I] ↑(extChartAt I c) c).HasBasis (fun x => True) fun f =>
  Metric.closedBall (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I
-- SmoothBumpFunction.isClosed_image_of_isClosed
theorem isClosed_image_of_isClosed {s : Set M} (hsc : IsClosed s) (hs : s ⊆ support f) :
    IsClosed (extChartAt I c '' s) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {s : Set M} (hsc : IsClosed s) (hs : s ⊆ Function.support ↑f)
Goal: IsClosed (↑(extChartAt I c) '' s)
-- SmoothBumpFunction.exists_r_pos_lt_subset_ball
theorem exists_r_pos_lt_subset_ball {s : Set M} (hsc : IsClosed s) (hs : s ⊆ support f) :
    ∃ r ∈ Ioo 0 f.rOut,
      s ⊆ (chartAt H c).source ∩ extChartAt I c ⁻¹' ball (extChartAt I c c) r 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {s : Set M} (hsc : IsClosed s) (hs : s ⊆ Function.support ↑f)
Goal: ∃ r ∈ Set.Ioo 0 f.rOut, s ⊆ (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' Metric.ball (↑(extChartAt I c) c) r
-- SmoothBumpFunction.updateRIn_rOut
@[simps rOut rIn]
def updateRIn (r : ℝ) (hr : r ∈ Ioo 0 f.rOut) : SmoothBumpFunction I c 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) (r : ℝ) (hr : r ∈ Set.Ioo 0 f.rOut)
Goal: (f.updateRIn r hr).rOut = f.rOut
-- SmoothBumpFunction.updateRIn_rIn
@[simps rOut rIn]
def updateRIn (r : ℝ) (hr : r ∈ Ioo 0 f.rOut) : SmoothBumpFunction I c 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) (r : ℝ) (hr : r ∈ Set.Ioo 0 f.rOut)
Goal: (f.updateRIn r hr).rIn = r
-- SmoothBumpFunction.support_updateRIn
@[simp]
theorem support_updateRIn {r : ℝ} (hr : r ∈ Ioo 0 f.rOut) :
    support (f.updateRIn r hr) = support f 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) {r : ℝ} (hr : r ∈ Set.Ioo 0 f.rOut)
Goal: Function.support ↑(f.updateRIn r hr) = Function.support ↑f
-- SmoothBumpFunction.isClosed_symm_image_closedBall
theorem isClosed_symm_image_closedBall :
    IsClosed ((extChartAt I c).symm '' (closedBall (extChartAt I c c) f.rOut ∩ range I)) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M]
Goal: IsClosed (↑(extChartAt I c).symm '' (Metric.closedBall (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I))
-- SmoothBumpFunction.tsupport_subset_symm_image_closedBall
theorem tsupport_subset_symm_image_closedBall :
    tsupport f ⊆ (extChartAt I c).symm '' (closedBall (extChartAt I c c) f.rOut ∩ range I) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M]
Goal: tsupport ↑f ⊆ ↑(extChartAt I c).symm '' (Metric.closedBall (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑I)
-- SmoothBumpFunction.tsupport_subset_extChartAt_source
theorem tsupport_subset_extChartAt_source : tsupport f ⊆ (extChartAt I c).source 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M]
Goal: tsupport ↑f ⊆ (extChartAt I c).source
-- SmoothBumpFunction.tsupport_subset_chartAt_source
theorem tsupport_subset_chartAt_source : tsupport f ⊆ (chartAt H c).source 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M]
Goal: tsupport ↑f ⊆ (chartAt H c).source
-- SmoothBumpFunction.hasCompactSupport
protected theorem hasCompactSupport : HasCompactSupport f 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M]
Goal: HasCompactSupport ↑f
-- SmoothBumpFunction.nhds_basis_tsupport
theorem nhds_basis_tsupport :
    (𝓝 c).HasBasis (fun _ : SmoothBumpFunction I c => True) fun f => tsupport f 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] (c : M) [T2Space M]
Goal: (𝓝 c).HasBasis (fun x => True) fun f => tsupport ↑f
-- SmoothBumpFunction.nhds_basis_support
theorem nhds_basis_support {s : Set M} (hs : s ∈ 𝓝 c) :
    (𝓝 c).HasBasis (fun f : SmoothBumpFunction I c => tsupport f ⊆ s) fun f => support f 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} [T2Space M] {s : Set M} (hs : s ∈ 𝓝 c)
Goal: (𝓝 c).HasBasis (fun f => tsupport ↑f ⊆ s) fun f => Function.support ↑f