ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- contMDiffOn_chart
theorem contMDiffOn_chart : ContMDiffOn I I n (chartAt H x) (chartAt H x).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] {x : M} {n : ℕ∞}
Goal: ContMDiffOn I I n (↑(chartAt H x)) (chartAt H x).source
-- contMDiffOn_chart_symm
theorem contMDiffOn_chart_symm : ContMDiffOn I I n (chartAt H x).symm (chartAt H x).target 
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] {x : M} {n : ℕ∞}
Goal: ContMDiffOn I I n (↑(chartAt H x).symm) (chartAt H x).target
-- contMDiffAt_extend
theorem contMDiffAt_extend {x : M} (he : e ∈ maximalAtlas I M) (hx : x ∈ e.source) :
    ContMDiffAt I 𝓘(𝕜, E) n (e.extend I) 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] [SmoothManifoldWithCorners I M] {e : PartialHomeomorph M H} {n : ℕ∞} {x : M} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hx : x ∈ e.source)
Goal: ContMDiffAt I 𝓘(𝕜, E) n (↑(e.extend I)) x
-- contMDiffAt_extChartAt'
theorem contMDiffAt_extChartAt' {x' : M} (h : x' ∈ (chartAt H x).source) :
    ContMDiffAt I 𝓘(𝕜, E) n (extChartAt I x) 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] [SmoothManifoldWithCorners I M] {x : M} {n : ℕ∞} {x' : M} (h : x' ∈ (chartAt H x).source)
Goal: ContMDiffAt I 𝓘(𝕜, E) n (↑(extChartAt I x)) x'
-- contMDiffAt_extChartAt
theorem contMDiffAt_extChartAt : ContMDiffAt I 𝓘(𝕜, E) n (extChartAt I x) 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] [SmoothManifoldWithCorners I M] {x : M} {n : ℕ∞}
Goal: ContMDiffAt I 𝓘(𝕜, E) n (↑(extChartAt I x)) x
-- contMDiffOn_extChartAt
theorem contMDiffOn_extChartAt : ContMDiffOn I 𝓘(𝕜, E) n (extChartAt I x) (chartAt H x).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] {x : M} {n : ℕ∞}
Goal: ContMDiffOn I 𝓘(𝕜, E) n (↑(extChartAt I x)) (chartAt H x).source
-- contMDiffOn_extend_symm
theorem contMDiffOn_extend_symm (he : e ∈ maximalAtlas I M) :
    ContMDiffOn 𝓘(𝕜, E) I n (e.extend I).symm (I '' e.target) 
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 : PartialHomeomorph M H} {n : ℕ∞} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M)
Goal: ContMDiffOn 𝓘(𝕜, E) I n (↑(e.extend I).symm) (↑I '' e.target)
-- contMDiffOn_extChartAt_symm
theorem contMDiffOn_extChartAt_symm (x : M) :
    ContMDiffOn 𝓘(𝕜, E) I n (extChartAt I x).symm (extChartAt I x).target 
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] {n : ℕ∞} (x : M)
Goal: ContMDiffOn 𝓘(𝕜, E) I n (↑(extChartAt I x).symm) (extChartAt I x).target
-- contMDiffOn_of_mem_contDiffGroupoid
theorem contMDiffOn_of_mem_contDiffGroupoid {e' : PartialHomeomorph H H}
    (h : e' ∈ contDiffGroupoid ⊤ I) : ContMDiffOn I I n e' e'.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} {n : ℕ∞} {e' : PartialHomeomorph H H} (h : e' ∈ contDiffGroupoid ⊤ I)
Goal: ContMDiffOn I I n (↑e') e'.source
-- isLocalStructomorphOn_contDiffGroupoid_iff_aux
theorem isLocalStructomorphOn_contDiffGroupoid_iff_aux {f : PartialHomeomorph M M'}
    (hf : LiftPropOn (contDiffGroupoid ⊤ I).IsLocalStructomorphWithinAt f f.source) :
    SmoothOn I I f f.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] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H M'] [IsM' : SmoothManifoldWithCorners I M'] {f : PartialHomeomorph M M'} (hf : ChartedSpace.LiftPropOn (contDiffGroupoid ⊤ I).IsLocalStructomorphWithinAt (↑f) f.source)
Goal: SmoothOn I I (↑f) f.source
-- isLocalStructomorphOn_contDiffGroupoid_iff
theorem isLocalStructomorphOn_contDiffGroupoid_iff (f : PartialHomeomorph M M') :
    LiftPropOn (contDiffGroupoid ⊤ I).IsLocalStructomorphWithinAt f f.source ↔
      SmoothOn I I f f.source ∧ SmoothOn I I f.symm f.target 
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] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H M'] [IsM' : SmoothManifoldWithCorners I M'] (f : PartialHomeomorph M M')
Goal: ChartedSpace.LiftPropOn (contDiffGroupoid ⊤ I).IsLocalStructomorphWithinAt (↑f) f.source ↔
  SmoothOn I I (↑f) f.source ∧ SmoothOn I I (↑f.symm) f.target
-- AlgebraicGeometry.PresheafedSpace.stalkMap_germ
@[elementwise, reassoc]
theorem stalkMap_germ {X Y : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (U : Opens Y)
    (x : (Opens.map α.base).obj U) :
    Y.presheaf.germ ⟨α.base x.1, x.2⟩ ≫ α.stalkMap ↑x = α.c.app (op U) ≫ X.presheaf.germ x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : AlgebraicGeometry.PresheafedSpace C} (α : X ⟶ Y) (U : TopologicalSpace.Opens ↑↑Y) (x : ↥((TopologicalSpace.Opens.map α.base).obj U))
Goal: Y.presheaf.germ ⟨α.base ↑x, ⋯⟩ ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap α ↑x =
  α.c.app (Opposite.op U) ≫ X.presheaf.germ x
-- AlgebraicGeometry.PresheafedSpace.stalkMap_germ'
@[simp, elementwise, reassoc]
theorem stalkMap_germ' {X Y : PresheafedSpace.{_, _, v} C}
    (α : X ⟶ Y) (U : Opens Y) (x : X) (hx : α.base x ∈ U) :
    Y.presheaf.germ ⟨α.base x, hx⟩ ≫ α.stalkMap x = α.c.app (op U) ≫
      X.presheaf.germ (U := (Opens.map α.base).obj U) ⟨x, hx⟩ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : AlgebraicGeometry.PresheafedSpace C} (α : X ⟶ Y) (U : TopologicalSpace.Opens ↑↑Y) (x : ↑↑X) (hx : α.base x ∈ U)
Goal: Y.presheaf.germ ⟨α.base x, hx⟩ ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap α x =
  α.c.app (Opposite.op U) ≫ X.presheaf.germ ⟨x, hx⟩
-- AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ
@[elementwise, reassoc]
theorem restrictStalkIso_hom_eq_germ {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (V : Opens U) (x : U) (hx : x ∈ V) :
    (X.restrict h).presheaf.germ ⟨x, hx⟩ ≫ (restrictStalkIso X h x).hom =
    X.presheaf.germ ⟨f x, show f x ∈ h.isOpenMap.functor.obj V from ⟨x, hx, rfl⟩⟩ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {U : TopCat} (X : AlgebraicGeometry.PresheafedSpace C) {f : U ⟶ ↑X} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑U) (x : ↑U) (hx : x ∈ V)
Goal: (X.restrict h).presheaf.germ ⟨x, hx⟩ ≫ (X.restrictStalkIso h x).hom = X.presheaf.germ ⟨f x, ⋯⟩
-- AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ
@[simp, elementwise, reassoc]
theorem restrictStalkIso_inv_eq_germ {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (V : Opens U) (x : U) (hx : x ∈ V) :
    X.presheaf.germ ⟨f x, show f x ∈ h.isOpenMap.functor.obj V from ⟨x, hx, rfl⟩⟩ ≫
        (restrictStalkIso X h x).inv =
      (X.restrict h).presheaf.germ ⟨x, hx⟩ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {U : TopCat} (X : AlgebraicGeometry.PresheafedSpace C) {f : U ⟶ ↑X} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑U) (x : ↑U) (hx : x ∈ V)
Goal: X.presheaf.germ ⟨f x, ⋯⟩ ≫ (X.restrictStalkIso h x).inv = (X.restrict h).presheaf.germ ⟨x, hx⟩
-- AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_ofRestrict
theorem restrictStalkIso_inv_eq_ofRestrict {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
    (X.restrictStalkIso h x).inv = (X.ofRestrict h).stalkMap x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {U : TopCat} (X : AlgebraicGeometry.PresheafedSpace C) {f : U ⟶ ↑X} (h : OpenEmbedding ⇑f) (x : ↑U)
Goal: (X.restrictStalkIso h x).inv = AlgebraicGeometry.PresheafedSpace.Hom.stalkMap (X.ofRestrict h) x
-- AlgebraicGeometry.PresheafedSpace.stalkMap.id
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    (𝟙 X : X ⟶ X).stalkMap x = 𝟙 (X.presheaf.stalk x) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] (X : AlgebraicGeometry.PresheafedSpace C) (x : ↑↑X)
Goal: AlgebraicGeometry.PresheafedSpace.Hom.stalkMap (𝟙 X) x = 𝟙 (X.presheaf.stalk x)
-- AlgebraicGeometry.PresheafedSpace.stalkMap.comp
@[simp]
theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
    (α ≫ β).stalkMap x =
      (β.stalkMap (α.base x) : Z.presheaf.stalk (β.base (α.base x)) ⟶ Y.presheaf.stalk (α.base x)) ≫
        (α.stalkMap x : Y.presheaf.stalk (α.base x) ⟶ X.presheaf.stalk x) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y Z : AlgebraicGeometry.PresheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : ↑↑X)
Goal: AlgebraicGeometry.PresheafedSpace.Hom.stalkMap (α ≫ β) x =
  AlgebraicGeometry.PresheafedSpace.Hom.stalkMap β (α.base x) ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap α x
-- AlgebraicGeometry.PresheafedSpace.stalkMap.congr
theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
    (h₁ : α = β) (x x' : X) (h₂ : x = x') :
    α.stalkMap x ≫ eqToHom (show X.presheaf.stalk x = X.presheaf.stalk x' by rw [h₂]) =
      eqToHom (show Y.presheaf.stalk (α.base x) = Y.presheaf.stalk (β.base x') by rw [h₁, h₂]) ≫
        β.stalkMap x' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : AlgebraicGeometry.PresheafedSpace C} (α β : X ⟶ Y) (h₁ : α = β) (x x' : ↑↑X) (h₂ : x = x')
Goal: AlgebraicGeometry.PresheafedSpace.Hom.stalkMap α x ≫ CategoryTheory.eqToHom ⋯ =
  CategoryTheory.eqToHom ⋯ ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap β x'
-- AlgebraicGeometry.PresheafedSpace.stalkMap.congr_hom
theorem congr_hom {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y) (h : α = β) (x : X) :
    α.stalkMap x =
      eqToHom (show Y.presheaf.stalk (α.base x) =
        Y.presheaf.stalk (β.base x) by rw [h]) ≫ β.stalkMap x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : AlgebraicGeometry.PresheafedSpace C} (α β : X ⟶ Y) (h : α = β) (x : ↑↑X)
Goal: AlgebraicGeometry.PresheafedSpace.Hom.stalkMap α x =
  CategoryTheory.eqToHom ⋯ ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap β x
-- AlgebraicGeometry.PresheafedSpace.stalkMap.congr_point
theorem congr_point {X Y : PresheafedSpace.{_, _, v} C}
    (α : X ⟶ Y) (x x' : X) (h : x = x') :
    α.stalkMap x ≫ eqToHom (show X.presheaf.stalk x = X.presheaf.stalk x' by rw [h]) =
      eqToHom (show Y.presheaf.stalk (α.base x) =
        Y.presheaf.stalk (α.base x') by rw [h]) ≫ α.stalkMap x' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : AlgebraicGeometry.PresheafedSpace C} (α : X ⟶ Y) (x x' : ↑↑X) (h : x = x')
Goal: AlgebraicGeometry.PresheafedSpace.Hom.stalkMap α x ≫ CategoryTheory.eqToHom ⋯ =
  CategoryTheory.eqToHom ⋯ ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap α x'
-- AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap
@[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkMap {X Y : PresheafedSpace.{_, _, v} C}
    (f : X ⟶ Y) {x y : X} (h : x ⤳ y) :
    Y.presheaf.stalkSpecializes (f.base.map_specializes h) ≫ f.stalkMap x =
      f.stalkMap y ≫ X.presheaf.stalkSpecializes h 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : AlgebraicGeometry.PresheafedSpace C} (f : X ⟶ Y) {x y : ↑↑X} (h : x ⤳ y)
Goal: Y.presheaf.stalkSpecializes ⋯ ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f x =
  AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f y ≫ X.presheaf.stalkSpecializes h
-- IsIntegralCurve.isIntegralCurveOn
lemma IsIntegralCurve.isIntegralCurveOn (h : IsIntegralCurve γ v) (s : Set ℝ) :
    IsIntegralCurveOn γ v s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} (h : IsIntegralCurve γ v) (s : Set ℝ)
Goal: IsIntegralCurveOn γ v s
-- isIntegralCurve_iff_isIntegralCurveOn
lemma isIntegralCurve_iff_isIntegralCurveOn : IsIntegralCurve γ v ↔ IsIntegralCurveOn γ v univ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x}
Goal: IsIntegralCurve γ v ↔ IsIntegralCurveOn γ v Set.univ
-- isIntegralCurveAt_iff
lemma isIntegralCurveAt_iff :
    IsIntegralCurveAt γ v t₀ ↔ ∃ s ∈ 𝓝 t₀, IsIntegralCurveOn γ v s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ}
Goal: IsIntegralCurveAt γ v t₀ ↔ ∃ s ∈ 𝓝 t₀, IsIntegralCurveOn γ v s
-- isIntegralCurveAt_iff'
lemma isIntegralCurveAt_iff' :
    IsIntegralCurveAt γ v t₀ ↔ ∃ ε > 0, IsIntegralCurveOn γ v (Metric.ball t₀ ε) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ}
Goal: IsIntegralCurveAt γ v t₀ ↔ ∃ ε > 0, IsIntegralCurveOn γ v (Metric.ball t₀ ε)
-- IsIntegralCurve.isIntegralCurveAt
lemma IsIntegralCurve.isIntegralCurveAt (h : IsIntegralCurve γ v) (t : ℝ) :
    IsIntegralCurveAt γ v t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} (h : IsIntegralCurve γ v) (t : ℝ)
Goal: IsIntegralCurveAt γ v t
-- isIntegralCurve_iff_isIntegralCurveAt
lemma isIntegralCurve_iff_isIntegralCurveAt :
    IsIntegralCurve γ v ↔ ∀ t : ℝ, IsIntegralCurveAt γ v t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x}
Goal: IsIntegralCurve γ v ↔ ∀ (t : ℝ), IsIntegralCurveAt γ v t
-- IsIntegralCurveOn.mono
lemma IsIntegralCurveOn.mono (h : IsIntegralCurveOn γ v s) (hs : s' ⊆ s) :
    IsIntegralCurveOn γ v s' 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s s' : Set ℝ} (h : IsIntegralCurveOn γ v s) (hs : s' ⊆ s)
Goal: IsIntegralCurveOn γ v s'
-- IsIntegralCurveOn.of_union
lemma IsIntegralCurveOn.of_union (h : IsIntegralCurveOn γ v s) (h' : IsIntegralCurveOn γ v s') :
    IsIntegralCurveOn γ v (s ∪ s') 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s s' : Set ℝ} (h : IsIntegralCurveOn γ v s) (h' : IsIntegralCurveOn γ v s')
Goal: IsIntegralCurveOn γ v (s ∪ s')
-- IsIntegralCurveAt.hasMFDerivAt
lemma IsIntegralCurveAt.hasMFDerivAt (h : IsIntegralCurveAt γ v t₀) :
    HasMFDerivAt 𝓘(ℝ, ℝ) I γ t₀ ((1 : ℝ →L[ℝ] ℝ).smulRight (v (γ t₀))) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} (h : IsIntegralCurveAt γ v t₀)
Goal: HasMFDerivAt 𝓘(ℝ, ℝ) I γ t₀ (ContinuousLinearMap.smulRight 1 (v (γ t₀)))
-- IsIntegralCurveOn.isIntegralCurveAt
lemma IsIntegralCurveOn.isIntegralCurveAt (h : IsIntegralCurveOn γ v s) (hs : s ∈ 𝓝 t₀) :
    IsIntegralCurveAt γ v t₀ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} {t₀ : ℝ} (h : IsIntegralCurveOn γ v s) (hs : s ∈ 𝓝 t₀)
Goal: IsIntegralCurveAt γ v t₀
-- IsIntegralCurveAt.isIntegralCurveOn
lemma IsIntegralCurveAt.isIntegralCurveOn (h : ∀ t ∈ s, IsIntegralCurveAt γ v t) :
    IsIntegralCurveOn γ v s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} (h : ∀ t ∈ s, IsIntegralCurveAt γ v t)
Goal: IsIntegralCurveOn γ v s
-- isIntegralCurveOn_iff_isIntegralCurveAt
lemma isIntegralCurveOn_iff_isIntegralCurveAt (hs : IsOpen s) :
    IsIntegralCurveOn γ v s ↔ ∀ t ∈ s, IsIntegralCurveAt γ v t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} (hs : IsOpen s)
Goal: IsIntegralCurveOn γ v s ↔ ∀ t ∈ s, IsIntegralCurveAt γ v t
-- IsIntegralCurveOn.continuousAt
lemma IsIntegralCurveOn.continuousAt (hγ : IsIntegralCurveOn γ v s) (ht : t₀ ∈ s) :
    ContinuousAt γ t₀ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} {t₀ : ℝ} (hγ : IsIntegralCurveOn γ v s) (ht : t₀ ∈ s)
Goal: ContinuousAt γ t₀
-- IsIntegralCurveOn.continuousOn
lemma IsIntegralCurveOn.continuousOn (hγ : IsIntegralCurveOn γ v s) :
    ContinuousOn γ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} (hγ : IsIntegralCurveOn γ v s)
Goal: ContinuousOn γ s
-- IsIntegralCurveAt.continuousAt
lemma IsIntegralCurveAt.continuousAt (hγ : IsIntegralCurveAt γ v t₀) :
    ContinuousAt γ t₀ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} (hγ : IsIntegralCurveAt γ v t₀)
Goal: ContinuousAt γ t₀
-- IsIntegralCurve.continuous
lemma IsIntegralCurve.continuous (hγ : IsIntegralCurve γ v) : Continuous γ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} (hγ : IsIntegralCurve γ v)
Goal: Continuous γ
-- IsIntegralCurveOn.hasDerivAt
lemma IsIntegralCurveOn.hasDerivAt (hγ : IsIntegralCurveOn γ v s) {t : ℝ} (ht : t ∈ s)
    (hsrc : γ t ∈ (extChartAt I (γ t₀)).source) :
    HasDerivAt ((extChartAt I (γ t₀)) ∘ γ)
      (tangentCoordChange I (γ t) (γ t₀) (γ t) (v (γ t))) t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} {t₀ : ℝ} (hγ : IsIntegralCurveOn γ v s) {t : ℝ} (ht : t ∈ s) (hsrc : γ t ∈ (extChartAt I (γ t₀)).source)
Goal: HasDerivAt (↑(extChartAt I (γ t₀)) ∘ γ) ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t))) t
-- IsIntegralCurveAt.eventually_hasDerivAt
lemma IsIntegralCurveAt.eventually_hasDerivAt (hγ : IsIntegralCurveAt γ v t₀) :
    ∀ᶠ t in 𝓝 t₀, HasDerivAt ((extChartAt I (γ t₀)) ∘ γ)
      (tangentCoordChange I (γ t) (γ t₀) (γ t) (v (γ t))) t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} (hγ : IsIntegralCurveAt γ v t₀)
Goal: ∀ᶠ (t : ℝ) in 𝓝 t₀, HasDerivAt (↑(extChartAt I (γ t₀)) ∘ γ) ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t))) t
-- IsIntegralCurveOn.comp_add
lemma IsIntegralCurveOn.comp_add (hγ : IsIntegralCurveOn γ v s) (dt : ℝ) :
    IsIntegralCurveOn (γ ∘ (· + dt)) v { t | t + dt ∈ s } 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} (hγ : IsIntegralCurveOn γ v s) (dt : ℝ)
Goal: IsIntegralCurveOn (γ ∘ fun x => x + dt) v {t | t + dt ∈ s}
-- isIntegralCurveOn_comp_add
lemma isIntegralCurveOn_comp_add {dt : ℝ} :
    IsIntegralCurveOn γ v s ↔ IsIntegralCurveOn (γ ∘ (· + dt)) v { t | t + dt ∈ s } 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {s : Set ℝ} {dt : ℝ}
Goal: IsIntegralCurveOn γ v s ↔ IsIntegralCurveOn (γ ∘ fun x => x + dt) v {t | t + dt ∈ s}
-- IsIntegralCurveAt.comp_add
lemma IsIntegralCurveAt.comp_add (hγ : IsIntegralCurveAt γ v t₀) (dt : ℝ) :
    IsIntegralCurveAt (γ ∘ (· + dt)) v (t₀ - dt) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {γ : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} (hγ : IsIntegralCurveAt γ v t₀) (dt : ℝ)
Goal: IsIntegralCurveAt (γ ∘ fun x => x + dt) v (t₀ - dt)