ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- pathConnectedSpace_iff_zerothHomotopy
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X)
-- isPathConnected_iff_pathConnectedSpace
theorem isPathConnected_iff_pathConnectedSpace : IsPathConnected F ↔ PathConnectedSpace F 
Arguments: {X : Type u_1} [TopologicalSpace X] {F : Set X}
Goal: IsPathConnected F ↔ PathConnectedSpace ↑F
-- pathConnectedSpace_iff_univ
theorem pathConnectedSpace_iff_univ : PathConnectedSpace X ↔ IsPathConnected (univ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: PathConnectedSpace X ↔ IsPathConnected Set.univ
-- isPathConnected_univ
theorem isPathConnected_univ [PathConnectedSpace X] : IsPathConnected (univ : Set X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [PathConnectedSpace X]
Goal: IsPathConnected Set.univ
-- isPathConnected_range
theorem isPathConnected_range [PathConnectedSpace X] {f : X → Y} (hf : Continuous f) :
    IsPathConnected (range f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [PathConnectedSpace X] {f : X → Y} (hf : Continuous f)
Goal: IsPathConnected (Set.range f)
-- Function.Surjective.pathConnectedSpace
theorem Function.Surjective.pathConnectedSpace [PathConnectedSpace X]
    {f : X → Y} (hf : Surjective f) (hf' : Continuous f) : PathConnectedSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [PathConnectedSpace X] {f : X → Y} (hf : Function.Surjective f) (hf' : Continuous f)
Goal: PathConnectedSpace Y
-- pathConnectedSpace_iff_eq
theorem pathConnectedSpace_iff_eq : PathConnectedSpace X ↔ ∃ x : X, pathComponent x = univ 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: PathConnectedSpace X ↔ ∃ x, pathComponent x = Set.univ
-- IsPathConnected.isConnected
theorem IsPathConnected.isConnected (hF : IsPathConnected F) : IsConnected F 
Arguments: {X : Type u_1} [TopologicalSpace X] {F : Set X} (hF : IsPathConnected F)
Goal: IsConnected F
-- PathConnectedSpace.exists_path_through_family
theorem exists_path_through_family {n : ℕ} (p : Fin (n + 1) → X) :
    ∃ γ : Path (p 0) (p n), ∀ i, p i ∈ range γ 
Arguments: {X : Type u_1} [TopologicalSpace X] [PathConnectedSpace X] {n : ℕ} (p : Fin (n + 1) → X)
Goal: ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ Set.range ⇑γ
-- PathConnectedSpace.exists_path_through_family'
theorem exists_path_through_family' {n : ℕ} (p : Fin (n + 1) → X) :
    ∃ (γ : Path (p 0) (p n)) (t : Fin (n + 1) → I), ∀ i, γ (t i) = p i 
Arguments: {X : Type u_1} [TopologicalSpace X] [PathConnectedSpace X] {n : ℕ} (p : Fin (n + 1) → X)
Goal: ∃ γ t, ∀ (i : Fin (n + 1)), γ (t i) = p i
-- LocPathConnectedSpace.path_connected_basis
class LocPathConnectedSpace (X : Type*) [TopologicalSpace X] : Prop where
    path_connected_basis : ∀ x : X, (𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsPathConnected s) id
Arguments: {X : Type u_4} [TopologicalSpace X] [self : LocPathConnectedSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id
-- locPathConnected_of_bases
theorem locPathConnected_of_bases {p : ι → Prop} {s : X → ι → Set X}
    (h : ∀ x, (𝓝 x).HasBasis p (s x)) (h' : ∀ x i, p i → IsPathConnected (s x i)) :
    LocPathConnectedSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Type u_3} {p : ι → Prop} {s : X → ι → Set X} (h : ∀ (x : X), (𝓝 x).HasBasis p (s x)) (h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i))
Goal: LocPathConnectedSpace X
-- pathConnectedSpace_iff_connectedSpace
theorem pathConnectedSpace_iff_connectedSpace [LocPathConnectedSpace X] :
    PathConnectedSpace X ↔ ConnectedSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocPathConnectedSpace X]
Goal: PathConnectedSpace X ↔ ConnectedSpace X
-- pathConnected_subset_basis
theorem pathConnected_subset_basis [LocPathConnectedSpace X] {U : Set X} (h : IsOpen U)
    (hx : x ∈ U) : (𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsPathConnected s ∧ s ⊆ U) id 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X} [LocPathConnectedSpace X] {U : Set X} (h : IsOpen U) (hx : x ∈ U)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsPathConnected s ∧ s ⊆ U) id
-- locPathConnected_of_isOpen
theorem locPathConnected_of_isOpen [LocPathConnectedSpace X] {U : Set X} (h : IsOpen U) :
    LocPathConnectedSpace U 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocPathConnectedSpace X] {U : Set X} (h : IsOpen U)
Goal: LocPathConnectedSpace ↑U
-- IsOpen.isConnected_iff_isPathConnected
theorem IsOpen.isConnected_iff_isPathConnected [LocPathConnectedSpace X] {U : Set X}
    (U_op : IsOpen U) : IsPathConnected U ↔ IsConnected U 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocPathConnectedSpace X] {U : Set X} (U_op : IsOpen U)
Goal: IsPathConnected U ↔ IsConnected U
-- SmoothAdd.smooth_add
class SmoothAdd {𝕜 : Type*} [NontriviallyNormedField 𝕜] {H : Type*} [TopologicalSpace H]
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (I : ModelWithCorners 𝕜 E H) (G : Type*)
    [Add G] [TopologicalSpace G] [ChartedSpace H G] extends SmoothManifoldWithCorners I G :
    Prop where
  smooth_add : Smooth (I.prod I) I fun p : G × G => p.1 + p.2
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} [Add G] [TopologicalSpace G] [ChartedSpace H G] [self : SmoothAdd I G]
Goal: Smooth (I.prod I) I fun p => p.1 + p.2
-- SmoothMul.smooth_mul
@[to_additive]
class SmoothMul {𝕜 : Type*} [NontriviallyNormedField 𝕜] {H : Type*} [TopologicalSpace H]
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (I : ModelWithCorners 𝕜 E H) (G : Type*)
    [Mul G] [TopologicalSpace G] [ChartedSpace H G] extends SmoothManifoldWithCorners I G :
    Prop where
  smooth_mul : Smooth (I.prod I) I fun p : G × G => p.1 * p.2
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} [Mul G] [TopologicalSpace G] [ChartedSpace H G] [self : SmoothMul I G]
Goal: Smooth (I.prod I) I fun p => p.1 * p.2
-- smooth_add
@[to_additive]
theorem smooth_mul : Smooth (I.prod I) I fun p : G × G => p.1 * p.2 
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} [Add G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G]
Goal: Smooth (I.prod I) I fun p => p.1 + p.2
-- smooth_mul
@[to_additive]
theorem smooth_mul : Smooth (I.prod I) I fun p : G × G => p.1 * p.2 
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} [Mul G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G]
Goal: Smooth (I.prod I) I fun p => p.1 * p.2
-- continuousMul_of_smooth
@[to_additive "If the addition is smooth, then it is continuous. This is not an instance for
technical reasons, see note [Design choices about smooth algebraic structures]."]
theorem continuousMul_of_smooth : ContinuousMul G 
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} [Mul G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G]
Goal: ContinuousMul G
-- continuousAdd_of_smooth
@[to_additive "If the addition is smooth, then it is continuous. This is not an instance for
technical reasons, see note [Design choices about smooth algebraic structures]."]
theorem continuousMul_of_smooth : ContinuousMul G 
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} [Add G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G]
Goal: ContinuousAdd G
-- ContMDiffWithinAt.add
@[to_additive]
theorem ContMDiffWithinAt.mul (hf : ContMDiffWithinAt I' I n f s x)
    (hg : ContMDiffWithinAt I' I n g s x) : ContMDiffWithinAt I' I n (f * g) 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} [Add 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 g : M → G} {s : Set M} {x : M} {n : ℕ∞} (hf : ContMDiffWithinAt I' I n f s x) (hg : ContMDiffWithinAt I' I n g s x)
Goal: ContMDiffWithinAt I' I n (f + g) s x
-- ContMDiffWithinAt.mul
@[to_additive]
theorem ContMDiffWithinAt.mul (hf : ContMDiffWithinAt I' I n f s x)
    (hg : ContMDiffWithinAt I' I n g s x) : ContMDiffWithinAt I' I n (f * g) 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} [Mul 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 g : M → G} {s : Set M} {x : M} {n : ℕ∞} (hf : ContMDiffWithinAt I' I n f s x) (hg : ContMDiffWithinAt I' I n g s x)
Goal: ContMDiffWithinAt I' I n (f * g) s x
-- ContMDiffAt.add
@[to_additive]
nonrec theorem ContMDiffAt.mul (hf : ContMDiffAt I' I n f x) (hg : ContMDiffAt I' I n g x) :
    ContMDiffAt I' I n (f * g) 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} [Add 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 g : M → G} {x : M} {n : ℕ∞} (hf : ContMDiffAt I' I n f x) (hg : ContMDiffAt I' I n g x)
Goal: ContMDiffAt I' I n (f + g) x
-- ContMDiffAt.mul
@[to_additive]
nonrec theorem ContMDiffAt.mul (hf : ContMDiffAt I' I n f x) (hg : ContMDiffAt I' I n g x) :
    ContMDiffAt I' I n (f * g) 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} [Mul 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 g : M → G} {x : M} {n : ℕ∞} (hf : ContMDiffAt I' I n f x) (hg : ContMDiffAt I' I n g x)
Goal: ContMDiffAt I' I n (f * g) x
-- ContMDiffOn.add
@[to_additive]
theorem ContMDiffOn.mul (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) :
    ContMDiffOn I' I n (f * g) 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} [Add 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 g : M → G} {s : Set M} {n : ℕ∞} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s)
Goal: ContMDiffOn I' I n (f + g) s
-- ContMDiffOn.mul
@[to_additive]
theorem ContMDiffOn.mul (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) :
    ContMDiffOn I' I n (f * g) 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} [Mul 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 g : M → G} {s : Set M} {n : ℕ∞} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s)
Goal: ContMDiffOn I' I n (f * g) s
-- ContMDiff.mul
@[to_additive]
theorem ContMDiff.mul (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
    ContMDiff I' I n (f * g) 
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} [Mul 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 g : M → G} {n : ℕ∞} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g)
Goal: ContMDiff I' I n (f * g)
-- ContMDiff.add
@[to_additive]
theorem ContMDiff.mul (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
    ContMDiff I' I n (f * g) 
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} [Add 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 g : M → G} {n : ℕ∞} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g)
Goal: ContMDiff I' I n (f + g)
-- SmoothWithinAt.add
@[to_additive]
nonrec theorem SmoothWithinAt.mul (hf : SmoothWithinAt I' I f s x)
    (hg : SmoothWithinAt I' I g s x) : SmoothWithinAt I' I (f * g) 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} [Add 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 g : M → G} {s : Set M} {x : M} (hf : SmoothWithinAt I' I f s x) (hg : SmoothWithinAt I' I g s x)
Goal: SmoothWithinAt I' I (f + g) s x
-- SmoothWithinAt.mul
@[to_additive]
nonrec theorem SmoothWithinAt.mul (hf : SmoothWithinAt I' I f s x)
    (hg : SmoothWithinAt I' I g s x) : SmoothWithinAt I' I (f * g) 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} [Mul 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 g : M → G} {s : Set M} {x : M} (hf : SmoothWithinAt I' I f s x) (hg : SmoothWithinAt I' I g s x)
Goal: SmoothWithinAt I' I (f * g) s x
-- SmoothAt.add
@[to_additive]
nonrec theorem SmoothAt.mul (hf : SmoothAt I' I f x) (hg : SmoothAt I' I g x) :
    SmoothAt I' I (f * g) 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} [Add 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 g : M → G} {x : M} (hf : SmoothAt I' I f x) (hg : SmoothAt I' I g x)
Goal: SmoothAt I' I (f + g) x
-- SmoothAt.mul
@[to_additive]
nonrec theorem SmoothAt.mul (hf : SmoothAt I' I f x) (hg : SmoothAt I' I g x) :
    SmoothAt I' I (f * g) 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} [Mul 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 g : M → G} {x : M} (hf : SmoothAt I' I f x) (hg : SmoothAt I' I g x)
Goal: SmoothAt I' I (f * g) x
-- SmoothOn.add
@[to_additive]
nonrec theorem SmoothOn.mul (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s) :
    SmoothOn I' I (f * g) 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} [Add 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 g : M → G} {s : Set M} (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s)
Goal: SmoothOn I' I (f + g) s
-- SmoothOn.mul
@[to_additive]
nonrec theorem SmoothOn.mul (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s) :
    SmoothOn I' I (f * g) 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} [Mul 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 g : M → G} {s : Set M} (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s)
Goal: SmoothOn I' I (f * g) s
-- Smooth.mul
@[to_additive]
nonrec theorem Smooth.mul (hf : Smooth I' I f) (hg : Smooth I' I g) : Smooth I' I (f * g) 
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} [Mul 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 g : M → G} (hf : Smooth I' I f) (hg : Smooth I' I g)
Goal: Smooth I' I (f * g)
-- Smooth.add
@[to_additive]
nonrec theorem Smooth.mul (hf : Smooth I' I f) (hg : Smooth I' I g) : Smooth I' I (f * g) 
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} [Add 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 g : M → G} (hf : Smooth I' I f) (hg : Smooth I' I g)
Goal: Smooth I' I (f + g)
-- smooth_add_left
@[to_additive]
theorem smooth_mul_left {a : G} : Smooth I I fun b : G => a * b 
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} [Add G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {a : G}
Goal: Smooth I I fun b => a + b
-- smooth_mul_left
@[to_additive]
theorem smooth_mul_left {a : G} : Smooth I I fun b : G => a * b 
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} [Mul G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {a : G}
Goal: Smooth I I fun b => a * b
-- smooth_mul_right
@[to_additive]
theorem smooth_mul_right {a : G} : Smooth I I fun b : G => b * a 
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} [Mul G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {a : G}
Goal: Smooth I I fun b => b * a
-- smooth_add_right
@[to_additive]
theorem smooth_mul_right {a : G} : Smooth I I fun b : G => b * a 
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} [Add G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {a : G}
Goal: Smooth I I fun b => b + a
-- L_apply
@[simp]
theorem L_apply : (𝑳 I g) h = g * h 
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} [Mul G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] (g h : G)
Goal: (smoothLeftMul I g) h = g * h
-- R_apply
@[simp]
theorem R_apply : (𝑹 I g) h = h * g 
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} [Mul G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] (g h : G)
Goal: (smoothRightMul I g) h = h * g
-- L_mul
@[simp]
theorem L_mul {G : Type*} [Semigroup G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G]
    (g h : G) : 𝑳 I (g * h) = (𝑳 I g).comp (𝑳 I h) 
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_8} [Semigroup G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] (g h : G)
Goal: smoothLeftMul I (g * h) = (smoothLeftMul I g).comp (smoothLeftMul I h)
-- R_mul
@[simp]
theorem R_mul {G : Type*} [Semigroup G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G]
    (g h : G) : 𝑹 I (g * h) = (𝑹 I h).comp (𝑹 I g) 
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_8} [Semigroup G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] (g h : G)
Goal: smoothRightMul I (g * h) = (smoothRightMul I h).comp (smoothRightMul I g)
-- smoothLeftMul_one
theorem smoothLeftMul_one : (𝑳 I g') 1 = g' 
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_8} [Monoid G'] [TopologicalSpace G'] [ChartedSpace H G'] [SmoothMul I G'] (g' : G')
Goal: (smoothLeftMul I g') 1 = g'
-- smoothRightMul_one
theorem smoothRightMul_one : (𝑹 I g') 1 = g' 
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_8} [Monoid G'] [TopologicalSpace G'] [ChartedSpace H G'] [SmoothMul I G'] (g' : G')
Goal: (smoothRightMul I g') 1 = g'
-- smooth_nsmul
@[to_additive]
theorem smooth_pow : ∀ n : ℕ, Smooth I I fun a : G => a ^ n
  
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} [AddMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] (n : ℕ)
Goal: Smooth I I fun a => n • a
-- smooth_pow
@[to_additive]
theorem smooth_pow : ∀ n : ℕ, Smooth I I fun a : G => a ^ n
  
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} [Monoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] (n : ℕ)
Goal: Smooth I I fun a => a ^ n