ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TangentBundle.continuousLinearMapAt_model_space
@[simp high, mfld_simps]
theorem continuousLinearMapAt_model_space (b b' : F) :
    (trivializationAt F (TangentSpace 𝓘(𝕜, F)) b).continuousLinearMapAt 𝕜 b' = (1 : F →L[𝕜] F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (b b' : F)
Goal: Trivialization.continuousLinearMapAt 𝕜 (trivializationAt F (TangentSpace 𝓘(𝕜, F)) b) b' = 1
-- tangentBundle_model_space_chartAt
@[simp, mfld_simps]
theorem tangentBundle_model_space_chartAt (p : TangentBundle I H) :
    (chartAt (ModelProd H E) p).toPartialEquiv = (TotalSpace.toProd H E).toPartialEquiv 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (p : TangentBundle I H)
Goal: (chartAt (ModelProd H E) p).toPartialEquiv = (Bundle.TotalSpace.toProd H E).toPartialEquiv
-- tangentBundle_model_space_coe_chartAt
@[simp, mfld_simps]
theorem tangentBundle_model_space_coe_chartAt (p : TangentBundle I H) :
    ⇑(chartAt (ModelProd H E) p) = TotalSpace.toProd H E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (p : TangentBundle I H)
Goal: ↑(chartAt (ModelProd H E) p) = ⇑(Bundle.TotalSpace.toProd H E)
-- tangentBundle_model_space_coe_chartAt_symm
@[simp, mfld_simps]
theorem tangentBundle_model_space_coe_chartAt_symm (p : TangentBundle I H) :
    ((chartAt (ModelProd H E) p).symm : ModelProd H E → TangentBundle I H) =
      (TotalSpace.toProd H E).symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (p : TangentBundle I H)
Goal: ↑(chartAt (ModelProd H E) p).symm = ⇑(Bundle.TotalSpace.toProd H E).symm
-- tangentBundleCore_coordChange_model_space
theorem tangentBundleCore_coordChange_model_space (x x' z : H) :
    (tangentBundleCore I H).coordChange (achart H x) (achart H x') z =
    ContinuousLinearMap.id 𝕜 E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (x x' z : H)
Goal: (tangentBundleCore I H).coordChange (achart H x) (achart H x') z = ContinuousLinearMap.id 𝕜 E
-- tangentBundleModelSpaceHomeomorph_coe
@[simp, mfld_simps]
theorem tangentBundleModelSpaceHomeomorph_coe :
    (tangentBundleModelSpaceHomeomorph H I : TangentBundle I H → ModelProd H E) =
      TotalSpace.toProd H E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (H : Type u_4) [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)
Goal: ⇑(tangentBundleModelSpaceHomeomorph H I) = ⇑(Bundle.TotalSpace.toProd H E)
-- tangentBundleModelSpaceHomeomorph_coe_symm
@[simp, mfld_simps]
theorem tangentBundleModelSpaceHomeomorph_coe_symm :
    ((tangentBundleModelSpaceHomeomorph H I).symm : ModelProd H E → TangentBundle I H) =
      (TotalSpace.toProd H E).symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (H : Type u_4) [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)
Goal: ⇑(tangentBundleModelSpaceHomeomorph H I).symm = ⇑(Bundle.TotalSpace.toProd H E).symm
-- inCoordinates_tangent_bundle_core_model_space
theorem inCoordinates_tangent_bundle_core_model_space (x₀ x : H) (y₀ y : H') (ϕ : E →L[𝕜] E') :
    inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x y₀ y ϕ = ϕ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {H' : Type u_5} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') (x₀ x : H) (y₀ y : H') (ϕ : E →L[𝕜] E')
Goal: ContinuousLinearMap.inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x y₀ y ϕ = ϕ
-- inTangentCoordinates_model_space
theorem inTangentCoordinates_model_space (f : N → H) (g : N → H') (ϕ : N → E →L[𝕜] E') (x₀ : N) :
    inTangentCoordinates I I' f g ϕ x₀ = ϕ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {H' : Type u_5} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {N : Type u_9} (f : N → H) (g : N → H') (ϕ : N → E →L[𝕜] E') (x₀ : N)
Goal: inTangentCoordinates I I' f g ϕ x₀ = ϕ
-- inTangentCoordinates_eq
theorem inTangentCoordinates_eq (f : N → M) (g : N → M') (ϕ : N → E →L[𝕜] E') {x₀ x : N}
    (hx : f x ∈ (chartAt H (f x₀)).source) (hy : g x ∈ (chartAt H' (g x₀)).source) :
    inTangentCoordinates I I' f g ϕ x₀ x =
      (tangentBundleCore I' M').coordChange (achart H' (g x)) (achart H' (g x₀)) (g x) ∘L
        ϕ x ∘L (tangentBundleCore I M).coordChange (achart H (f x₀)) (achart H (f x)) (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {H' : Type u_5} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {N : Type u_9} (f : N → M) (g : N → M') (ϕ : N → E →L[𝕜] E') {x₀ x : N} (hx : f x ∈ (chartAt H (f x₀)).source) (hy : g x ∈ (chartAt H' (g x₀)).source)
Goal: inTangentCoordinates I I' f g ϕ x₀ x =
  ((tangentBundleCore I' M').coordChange (achart H' (g x)) (achart H' (g x₀)) (g x)).comp
    ((ϕ x).comp ((tangentBundleCore I M).coordChange (achart H (f x₀)) (achart H (f x)) (f x)))
-- ContMDiffSection.contMDiff_toFun
structure ContMDiffSection where
    protected toFun : ∀ x, V x
    protected contMDiff_toFun : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦
    TotalSpace.mk' F x (toFun x)
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (self : Cₛ^n⟮I; F, V⟯)
Goal: ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => Bundle.TotalSpace.mk' F x (self.toFun x)
-- ContMDiffSection.coeFn_mk
@[simp]
theorem coeFn_mk (s : ∀ x, V x)
    (hs : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => TotalSpace.mk x (s x)) :
    (mk s hs : ∀ x, V x) = s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : (x : M) → V x) (hs : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => { proj := x, snd := s x })
Goal: ⇑{ toFun := s, contMDiff_toFun := hs } = s
-- ContMDiffSection.contMDiff
protected theorem contMDiff (s : Cₛ^n⟮I; F, V⟯) :
    ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => TotalSpace.mk' F x (s x : V x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : Cₛ^n⟮I; F, V⟯)
Goal: ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x => Bundle.TotalSpace.mk' F x (s x)
-- ContMDiffSection.smooth
protected theorem smooth (s : Cₛ^∞⟮I; F, V⟯) :
    Smooth I (I.prod 𝓘(𝕜, F)) fun x => TotalSpace.mk' F x (s x : V x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : Cₛ^⊤⟮I; F, V⟯)
Goal: Smooth I (I.prod 𝓘(𝕜, F)) fun x => Bundle.TotalSpace.mk' F x (s x)
-- ContMDiffSection.mdifferentiable'
protected theorem mdifferentiable' (s : Cₛ^n⟮I; F, V⟯) (hn : 1 ≤ n) :
    MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => TotalSpace.mk' F x (s x : V x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : Cₛ^n⟮I; F, V⟯) (hn : 1 ≤ n)
Goal: MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => Bundle.TotalSpace.mk' F x (s x)
-- ContMDiffSection.mdifferentiable
protected theorem mdifferentiable (s : Cₛ^∞⟮I; F, V⟯) :
    MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => TotalSpace.mk' F x (s x : V x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : Cₛ^⊤⟮I; F, V⟯)
Goal: MDifferentiable I (I.prod 𝓘(𝕜, F)) fun x => Bundle.TotalSpace.mk' F x (s x)
-- ContMDiffSection.mdifferentiableAt
protected theorem mdifferentiableAt (s : Cₛ^∞⟮I; F, V⟯) {x} :
    MDifferentiableAt I (I.prod 𝓘(𝕜, F)) (fun x => TotalSpace.mk' F x (s x : V x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : Cₛ^⊤⟮I; F, V⟯) {x : M}
Goal: MDifferentiableAt I (I.prod 𝓘(𝕜, F)) (fun x => Bundle.TotalSpace.mk' F x (s x)) x
-- ContMDiffSection.coe_inj
theorem coe_inj ⦃s t : Cₛ^n⟮I; F, V⟯⦄ (h : (s : ∀ x, V x) = t) : s = t 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] ⦃s t : Cₛ^n⟮I; F, V⟯⦄ (h : ⇑s = ⇑t)
Goal: s = t
-- ContMDiffSection.coe_injective
theorem coe_injective : Injective ((↑) : Cₛ^n⟮I; F, V⟯ → ∀ x, V x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V]
Goal: Function.Injective DFunLike.coe
-- ContMDiffSection.ext
@[ext]
theorem ext (h : ∀ x, s x = t x) : s = t 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] {s t : Cₛ^n⟮I; F, V⟯} (h : ∀ (x : M), s x = t x)
Goal: s = t
-- ContMDiffSection.coe_add
@[simp]
theorem coe_add (s t : Cₛ^n⟮I; F, V⟯) : ⇑(s + t) = ⇑s + t 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] (s t : Cₛ^n⟮I; F, V⟯)
Goal: ⇑(s + t) = ⇑s + ⇑t
-- ContMDiffSection.coe_sub
@[simp]
theorem coe_sub (s t : Cₛ^n⟮I; F, V⟯) : ⇑(s - t) = s - t 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] (s t : Cₛ^n⟮I; F, V⟯)
Goal: ⇑(s - t) = ⇑s - ⇑t
-- ContMDiffSection.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : Cₛ^n⟮I; F, V⟯) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V]
Goal: ⇑0 = 0
-- ContMDiffSection.coe_smul
@[simp]
theorem coe_smul (r : 𝕜) (s : Cₛ^n⟮I; F, V⟯) : ⇑(r • s : Cₛ^n⟮I; F, V⟯) = r • ⇑s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] (r : 𝕜) (s : Cₛ^n⟮I; F, V⟯)
Goal: ⇑(r • s) = r • ⇑s
-- ContMDiffSection.coe_neg
@[simp]
theorem coe_neg (s : Cₛ^n⟮I; F, V⟯) : ⇑(-s : Cₛ^n⟮I; F, V⟯) = -s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] (s : Cₛ^n⟮I; F, V⟯)
Goal: ⇑(-s) = -⇑s
-- ContMDiffSection.coe_nsmul
@[simp]
theorem coe_nsmul (s : Cₛ^n⟮I; F, V⟯) (k : ℕ) : ⇑(k • s : Cₛ^n⟮I; F, V⟯) = k • ⇑s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] (s : Cₛ^n⟮I; F, V⟯) (k : ℕ)
Goal: ⇑(k • s) = k • ⇑s
-- ContMDiffSection.coe_zsmul
@[simp]
theorem coe_zsmul (s : Cₛ^n⟮I; F, V⟯) (z : ℤ) : ⇑(z • s : Cₛ^n⟮I; F, V⟯) = z • ⇑s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {V : M → Type u_12} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] (s : Cₛ^n⟮I; F, V⟯) (z : ℤ)
Goal: ⇑(z • s) = z • ⇑s
-- smoothSheafCommRing.isUnit_stalk_iff
theorem smoothSheafCommRing.isUnit_stalk_iff {x : M}
    (f : (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x) :
    IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {EM : Type u_1} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_2} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] {x : M} (f : ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x))
Goal: IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜, 𝕜) M 𝕜 x)
-- smoothSheafCommRing.nonunits_stalk
theorem smoothSheafCommRing.nonunits_stalk (x : M) :
    nonunits ((smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x)
    = RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {EM : Type u_1} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_2} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] (x : M)
Goal: nonunits ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x) =
  ↑(RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜, 𝕜) M 𝕜 x))
-- Path.target'
-- porting note (#5171): removed @[nolint has_nonempty_instance]
structure Path (x y : X) extends C(I, X) where
    source' : toFun 0 = x
    target' : toFun 1 = y
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (self : Path x y)
Goal: self.toFun 1 = y
-- Path.source'
-- porting note (#5171): removed @[nolint has_nonempty_instance]
structure Path (x y : X) extends C(I, X) where
    source' : toFun 0 = x
    target' : toFun 1 = y
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (self : Path x y)
Goal: self.toFun 0 = x
-- Path.ext
@[ext]
protected theorem Path.ext : ∀ {γ₁ γ₂ : Path x y}, (γ₁ : I → X) = γ₂ → γ₁ = γ₂ 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {γ₁ γ₂ : Path x y}
Goal: ⇑γ₁ = ⇑γ₂ → γ₁ = γ₂
-- Path.coe_mk_mk
@[simp]
theorem coe_mk_mk (f : I → X) (h₁) (h₂ : f 0 = x) (h₃ : f 1 = y) :
    ⇑(mk ⟨f, h₁⟩ h₂ h₃ : Path x y) = f 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (f : ↑I → X) (h₁ : Continuous f) (h₂ : f 0 = x) (h₃ : f 1 = y)
Goal: ⇑{ toFun := f, continuous_toFun := h₁, source' := h₂, target' := h₃ } = f
-- Path.continuous
@[continuity]
protected theorem continuous : Continuous γ 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: Continuous ⇑γ
-- Path.source
@[simp]
protected theorem source : γ 0 = x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: γ 0 = x
-- Path.target
@[simp]
protected theorem target : γ 1 = y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: γ 1 = y
-- Path.coe_toContinuousMap
@[simp]
theorem coe_toContinuousMap : ⇑γ.toContinuousMap = γ 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: ⇑γ.toContinuousMap = ⇑γ
-- Path.coe_mk
@[simp]
theorem coe_mk : ⇑(γ : C(I, X)) = γ 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: ⇑↑γ = ⇑γ
-- Path.refl_apply
@[refl, simps]
def refl (x : X) : Path x x 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X) (_t : ↑I)
Goal: (Path.refl x) _t = x
-- Path.refl_range
@[simp]
theorem refl_range {a : X} : range (Path.refl a) = {a} 
Arguments: {X : Type u_1} [TopologicalSpace X] {a : X}
Goal: Set.range ⇑(Path.refl a) = {a}
-- Path.symm_apply
@[symm, simps]
def symm (γ : Path x y) : Path y x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: ∀ (a : ↑I), γ.symm a = (⇑γ ∘ σ) a
-- Path.symm_symm
@[simp]
theorem symm_symm (γ : Path x y) : γ.symm.symm = γ 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: γ.symm.symm = γ
-- Path.symm_bijective
theorem symm_bijective : Function.Bijective (Path.symm : Path x y → Path y x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: Function.Bijective Path.symm
-- Path.refl_symm
@[simp]
theorem refl_symm {a : X} : (Path.refl a).symm = Path.refl a 
Arguments: {X : Type u_1} [TopologicalSpace X] {a : X}
Goal: (Path.refl a).symm = Path.refl a
-- Path.symm_range
@[simp]
theorem symm_range {a b : X} (γ : Path a b) : range γ.symm = range γ 
Arguments: {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b)
Goal: Set.range ⇑γ.symm = Set.range ⇑γ
-- Path.continuous_eval
theorem continuous_eval : Continuous fun p : Path x y × I => p.1 p.2 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: Continuous fun p => p.1 p.2
-- Continuous.path_eval
@[continuity]
theorem _root_.Continuous.path_eval {Y} [TopologicalSpace Y] {f : Y → Path x y} {g : Y → I}
    (hf : Continuous f) (hg : Continuous g) : Continuous fun y => f y (g y) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {Y : Type u_4} [TopologicalSpace Y] {f : Y → Path x y} {g : Y → ↑I} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun y_1 => (f y_1) (g y_1)
-- Path.continuous_uncurry_iff
theorem continuous_uncurry_iff {Y} [TopologicalSpace Y] {g : Y → Path x y} :
    Continuous ↿g ↔ Continuous g 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} {Y : Type u_4} [TopologicalSpace Y] {g : Y → Path x y}
Goal: Continuous ↿g ↔ Continuous g
-- Continuous.path_extend
theorem _root_.Continuous.path_extend {γ : Y → Path x y} {f : Y → ℝ} (hγ : Continuous ↿γ)
    (hf : Continuous f) : Continuous fun t => (γ t).extend (f t) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {γ : Y → Path x y} {f : Y → ℝ} (hγ : Continuous ↿γ) (hf : Continuous f)
Goal: Continuous fun t => (γ t).extend (f t)
-- Path.continuous_extend
@[continuity, fun_prop]
theorem continuous_extend : Continuous γ.extend 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: Continuous γ.extend
-- Filter.Tendsto.path_extend
theorem _root_.Filter.Tendsto.path_extend
    {l r : Y → X} {y : Y} {l₁ : Filter ℝ} {l₂ : Filter X} {γ : ∀ y, Path (l y) (r y)}
    (hγ : Tendsto (↿γ) (𝓝 y ×ˢ l₁.map (projIcc 0 1 zero_le_one)) l₂) :
    Tendsto (↿fun x => (γ x).extend) (𝓝 y ×ˢ l₁) l₂ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {l r : Y → X} {y : Y} {l₁ : Filter ℝ} {l₂ : Filter X} {γ : (y : Y) → Path (l y) (r y)} (hγ : Filter.Tendsto (↿γ) (𝓝 y ×ˢ Filter.map (Set.projIcc 0 1 ⋯) l₁) l₂)
Goal: Filter.Tendsto (↿fun x => (γ x).extend) (𝓝 y ×ˢ l₁) l₂
-- ContinuousAt.path_extend
theorem _root_.ContinuousAt.path_extend {g : Y → ℝ} {l r : Y → X} (γ : ∀ y, Path (l y) (r y))
    {y : Y} (hγ : ContinuousAt (↿γ) (y, projIcc 0 1 zero_le_one (g y))) (hg : ContinuousAt g y) :
    ContinuousAt (fun i => (γ i).extend (g i)) y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {g : Y → ℝ} {l r : Y → X} (γ : (y : Y) → Path (l y) (r y)) {y : Y} (hγ : ContinuousAt (↿γ) (y, Set.projIcc 0 1 ⋯ (g y))) (hg : ContinuousAt g y)
Goal: ContinuousAt (fun i => (γ i).extend (g i)) y
-- Path.extend_extends
@[simp]
theorem extend_extends {a b : X} (γ : Path a b) {t : ℝ}
    (ht : t ∈ (Icc 0 1 : Set ℝ)) : γ.extend t = γ ⟨t, ht⟩ 
Arguments: {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) {t : ℝ} (ht : t ∈ Set.Icc 0 1)
Goal: γ.extend t = γ ⟨t, ht⟩
-- Path.extend_zero
theorem extend_zero : γ.extend 0 = x 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y)
Goal: γ.extend 0 = x