ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- exists_contMDiffOn_forall_mem_convex_of_local
theorem exists_contMDiffOn_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
    (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, ContMDiffOn I 𝓘(ℝ, F) n g U ∧ ∀ y ∈ U, g y ∈ t y) :
    ∃ g : C^n⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] [T2Space M] {t : M → Set F} {n : ℕ∞} (ht : ∀ (x : M), Convex ℝ (t x)) (Hloc : ∀ (x : M), ∃ U ∈ 𝓝 x, ∃ g, ContMDiffOn I 𝓘(ℝ, F) n g U ∧ ∀ y ∈ U, g y ∈ t y)
Goal: ∃ g, ∀ (x : M), g x ∈ t x
-- exists_smooth_forall_mem_convex_of_local
theorem exists_smooth_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
    (Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, SmoothOn I 𝓘(ℝ, F) g U ∧ ∀ y ∈ U, g y ∈ t y) :
    ∃ g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] [T2Space M] {t : M → Set F} (ht : ∀ (x : M), Convex ℝ (t x)) (Hloc : ∀ (x : M), ∃ U ∈ 𝓝 x, ∃ g, SmoothOn I 𝓘(ℝ, F) g U ∧ ∀ y ∈ U, g y ∈ t y)
Goal: ∃ g, ∀ (x : M), g x ∈ t x
-- exists_smooth_forall_mem_convex_of_local_const
theorem exists_smooth_forall_mem_convex_of_local_const (ht : ∀ x, Convex ℝ (t x))
    (Hloc : ∀ x : M, ∃ c : F, ∀ᶠ y in 𝓝 x, c ∈ t y) : ∃ g : C^∞⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] [T2Space M] {t : M → Set F} (ht : ∀ (x : M), Convex ℝ (t x)) (Hloc : ∀ (x : M), ∃ c, ∀ᶠ (y : M) in 𝓝 x, c ∈ t y)
Goal: ∃ g, ∀ (x : M), g x ∈ t x
-- Emetric.exists_smooth_forall_closedBall_subset
theorem Emetric.exists_smooth_forall_closedBall_subset {M} [EMetricSpace M] [ChartedSpace H M]
    [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] {K : ι → Set M} {U : ι → Set M}
    (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i)
    (hfin : LocallyFinite K) :
    ∃ δ : C^∞⟮I, M; 𝓘(ℝ, ℝ), ℝ⟯,
      (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type u_1} [EMetricSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] {K U : ι → Set M} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K)
Goal: ∃ δ, (∀ (x : M), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i
-- Metric.exists_smooth_forall_closedBall_subset
theorem Metric.exists_smooth_forall_closedBall_subset {M} [MetricSpace M] [ChartedSpace H M]
    [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] {K : ι → Set M} {U : ι → Set M}
    (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i)
    (hfin : LocallyFinite K) :
    ∃ δ : C^∞⟮I, M; 𝓘(ℝ, ℝ), ℝ⟯,
      (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, Metric.closedBall x (δ x) ⊆ U i 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type u_1} [MetricSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] {K U : ι → Set M} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K)
Goal: ∃ δ, (∀ (x : M), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x (δ x) ⊆ U i
-- IsOpen.exists_msmooth_support_eq_aux
lemma IsOpen.exists_msmooth_support_eq_aux {s : Set H} (hs : IsOpen s) :
    ∃ f : H → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ Set.range f ⊆ Set.Icc 0 1 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {s : Set H} (hs : IsOpen s)
Goal: ∃ f, Function.support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ Set.range f ⊆ Set.Icc 0 1
-- IsOpen.exists_msmooth_support_eq
theorem IsOpen.exists_msmooth_support_eq {s : Set M} (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 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] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] [T2Space M] {s : Set M} (hs : IsOpen s)
Goal: ∃ f, Function.support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ ∀ (x : M), 0 ≤ f x
-- exists_msmooth_support_eq_eq_one_iff
theorem exists_msmooth_support_eq_eq_one_iff
    {s t : Set M} (hs : IsOpen s) (ht : IsClosed t) (h : t ⊆ s) :
    ∃ f : M → ℝ, Smooth I 𝓘(ℝ) f ∧ range f ⊆ Icc 0 1 ∧ support f = s
      ∧ (∀ x, x ∈ t ↔ 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] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] [T2Space M] {s t : Set M} (hs : IsOpen s) (ht : IsClosed t) (h : t ⊆ s)
Goal: ∃ f, Smooth I 𝓘(ℝ, ℝ) f ∧ Set.range f ⊆ Set.Icc 0 1 ∧ Function.support f = s ∧ ∀ (x : M), x ∈ t ↔ f x = 1
-- exists_msmooth_zero_iff_one_iff_of_isClosed
theorem exists_msmooth_zero_iff_one_iff_of_isClosed {s t : Set M}
    (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
    ∃ f : M → ℝ, Smooth I 𝓘(ℝ) f ∧ range f ⊆ Icc 0 1 ∧ (∀ x, x ∈ s ↔ f x = 0)
      ∧ (∀ x, x ∈ t ↔ 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] [SmoothManifoldWithCorners I M] [SigmaCompactSpace M] [T2Space M] {s t : Set M} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: ∃ f, Smooth I 𝓘(ℝ, ℝ) f ∧ Set.range f ⊆ Set.Icc 0 1 ∧ (∀ (x : M), x ∈ s ↔ f x = 0) ∧ ∀ (x : M), x ∈ t ↔ f x = 1
-- AlgebraicGeometry.RingedSpace.isUnit_res_of_isUnit_germ
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) 
Arguments: (X : AlgebraicGeometry.RingedSpace) (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) (f : ↑(X.presheaf.obj (Opposite.op U))) (x : ↥U) (h : IsUnit ((X.presheaf.germ x) f))
Goal: ∃ V i, ∃ (_ : ↑x ∈ V), IsUnit ((X.presheaf.map i.op) f)
-- AlgebraicGeometry.RingedSpace.isUnit_of_isUnit_germ
theorem isUnit_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U))
    (h : ∀ x : U, IsUnit (X.presheaf.germ x f)) : IsUnit f 
Arguments: (X : AlgebraicGeometry.RingedSpace) (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) (f : ↑(X.presheaf.obj (Opposite.op U))) (h : ∀ (x : ↥U), IsUnit ((X.presheaf.germ x) f))
Goal: IsUnit f
-- AlgebraicGeometry.RingedSpace.mem_basicOpen
@[simp]
theorem mem_basicOpen {U : Opens X} (f : X.presheaf.obj (op U)) (x : U) :
    ↑x ∈ X.basicOpen f ↔ IsUnit (X.presheaf.germ x f) 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (f : ↑(X.presheaf.obj (Opposite.op U))) (x : ↥U)
Goal: ↑x ∈ X.basicOpen f ↔ IsUnit ((X.presheaf.germ x) f)
-- AlgebraicGeometry.RingedSpace.mem_top_basicOpen
@[simp]
theorem mem_top_basicOpen (f : X.presheaf.obj (op ⊤)) (x : X) :
    x ∈ X.basicOpen f ↔ IsUnit (X.presheaf.germ ⟨x, show x ∈ (⊤ : Opens X) by trivial⟩ f) 
Arguments: (X : AlgebraicGeometry.RingedSpace) (f : ↑(X.presheaf.obj (Opposite.op ⊤))) (x : ↑↑X.toPresheafedSpace)
Goal: x ∈ X.basicOpen f ↔ IsUnit ((X.presheaf.germ ⟨x, ⋯⟩) f)
-- AlgebraicGeometry.RingedSpace.basicOpen_le
theorem basicOpen_le {U : Opens X} (f : X.presheaf.obj (op U)) : X.basicOpen f ≤ U 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: X.basicOpen f ≤ U
-- AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen
theorem isUnit_res_basicOpen {U : Opens X} (f : X.presheaf.obj (op U)) :
    IsUnit (X.presheaf.map (@homOfLE (Opens X) _ _ _ (X.basicOpen_le f)).op f) 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: IsUnit ((X.presheaf.map (CategoryTheory.homOfLE ⋯).op) f)
-- AlgebraicGeometry.RingedSpace.basicOpen_res
@[simp]
theorem basicOpen_res {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) (f : X.presheaf.obj U) :
    @basicOpen X (unop V) (X.presheaf.map i f) = unop V ⊓ @basicOpen X (unop U) f 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U V : (TopologicalSpace.Opens ↑↑X.toPresheafedSpace)ᵒᵖ} (i : U ⟶ V) (f : ↑(X.presheaf.obj U))
Goal: X.basicOpen ((X.presheaf.map i) f) = Opposite.unop V ⊓ X.basicOpen f
-- AlgebraicGeometry.RingedSpace.basicOpen_res_eq
@[simp (high)]
theorem basicOpen_res_eq {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) [IsIso i] (f : X.presheaf.obj U) :
    @basicOpen X (unop V) (X.presheaf.map i f) = @RingedSpace.basicOpen X (unop U) f 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U V : (TopologicalSpace.Opens ↑↑X.toPresheafedSpace)ᵒᵖ} (i : U ⟶ V) [CategoryTheory.IsIso i] (f : ↑(X.presheaf.obj U))
Goal: X.basicOpen ((X.presheaf.map i) f) = X.basicOpen f
-- AlgebraicGeometry.RingedSpace.basicOpen_mul
@[simp]
theorem basicOpen_mul {U : Opens X} (f g : X.presheaf.obj (op U)) :
    X.basicOpen (f * g) = X.basicOpen f ⊓ X.basicOpen g 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (f g : ↑(X.presheaf.obj (Opposite.op U)))
Goal: X.basicOpen (f * g) = X.basicOpen f ⊓ X.basicOpen g
-- AlgebraicGeometry.RingedSpace.basicOpen_pow
@[simp]
lemma basicOpen_pow {U : Opens X} (f : X.presheaf.obj (op U)) (n : ℕ) (h : 0 < n) :
    X.basicOpen (f ^ n) = X.basicOpen f 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (f : ↑(X.presheaf.obj (Opposite.op U))) (n : ℕ) (h : 0 < n)
Goal: X.basicOpen (f ^ n) = X.basicOpen f
-- AlgebraicGeometry.RingedSpace.basicOpen_of_isUnit
theorem basicOpen_of_isUnit {U : Opens X} {f : X.presheaf.obj (op U)} (hf : IsUnit f) :
    X.basicOpen f = U 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} {f : ↑(X.presheaf.obj (Opposite.op U))} (hf : IsUnit f)
Goal: X.basicOpen f = U
-- AlgebraicGeometry.RingedSpace.zeroLocus_isClosed
lemma zeroLocus_isClosed {U : Opens X} (s : Set (X.presheaf.obj (op U))) :
    IsClosed (X.zeroLocus s) 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (s : Set ↑(X.presheaf.obj (Opposite.op U)))
Goal: IsClosed (X.zeroLocus s)
-- AlgebraicGeometry.RingedSpace.zeroLocus_singleton
lemma zeroLocus_singleton {U : Opens X} (f : X.presheaf.obj (op U)) :
    X.zeroLocus {f} = (X.basicOpen f).carrierᶜ 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: X.zeroLocus {f} = (X.basicOpen f).carrierᶜ
-- AlgebraicGeometry.RingedSpace.zeroLocus_empty_eq_univ
@[simp]
lemma zeroLocus_empty_eq_univ {U : Opens X} :
    X.zeroLocus (∅ : Set (X.presheaf.obj (op U))) = Set.univ 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace}
Goal: X.zeroLocus ∅ = Set.univ
-- AlgebraicGeometry.RingedSpace.mem_zeroLocus_iff
@[simp]
lemma mem_zeroLocus_iff {U : Opens X} (s : Set (X.presheaf.obj (op U))) (x : X) :
    x ∈ X.zeroLocus s ↔ ∀ f ∈ s, x ∉ X.basicOpen f 
Arguments: (X : AlgebraicGeometry.RingedSpace) {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (s : Set ↑(X.presheaf.obj (Opposite.op U))) (x : ↑↑X.toPresheafedSpace)
Goal: x ∈ X.zeroLocus s ↔ ∀ f ∈ s, x ∉ X.basicOpen f
-- ContMDiffWithinAt.prod_mk
theorem ContMDiffWithinAt.prod_mk {f : M → M'} {g : M → N'} (hf : ContMDiffWithinAt I I' n f s x)
    (hg : ContMDiffWithinAt I J' n g s x) :
    ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {s : Set M} {x : M} {n : ℕ∞} {f : M → M'} {g : M → N'} (hf : ContMDiffWithinAt I I' n f s x) (hg : ContMDiffWithinAt I J' n g s x)
Goal: ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x
-- ContMDiffWithinAt.prod_mk_space
theorem ContMDiffWithinAt.prod_mk_space {f : M → E'} {g : M → F'}
    (hf : ContMDiffWithinAt I 𝓘(𝕜, E') n f s x) (hg : ContMDiffWithinAt I 𝓘(𝕜, F') n g s x) :
    ContMDiffWithinAt I 𝓘(𝕜, E' × F') n (fun x => (f x, g x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set M} {x : M} {n : ℕ∞} {f : M → E'} {g : M → F'} (hf : ContMDiffWithinAt I 𝓘(𝕜, E') n f s x) (hg : ContMDiffWithinAt I 𝓘(𝕜, F') n g s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, E' × F') n (fun x => (f x, g x)) s x
-- ContMDiffAt.prod_mk
nonrec theorem ContMDiffAt.prod_mk {f : M → M'} {g : M → N'} (hf : ContMDiffAt I I' n f x)
    (hg : ContMDiffAt I J' n g x) : ContMDiffAt I (I'.prod J') n (fun x => (f x, g 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] {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' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {x : M} {n : ℕ∞} {f : M → M'} {g : M → N'} (hf : ContMDiffAt I I' n f x) (hg : ContMDiffAt I J' n g x)
Goal: ContMDiffAt I (I'.prod J') n (fun x => (f x, g x)) x
-- ContMDiffAt.prod_mk_space
nonrec theorem ContMDiffAt.prod_mk_space {f : M → E'} {g : M → F'}
    (hf : ContMDiffAt I 𝓘(𝕜, E') n f x) (hg : ContMDiffAt I 𝓘(𝕜, F') n g x) :
    ContMDiffAt I 𝓘(𝕜, E' × F') n (fun x => (f x, g 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] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {x : M} {n : ℕ∞} {f : M → E'} {g : M → F'} (hf : ContMDiffAt I 𝓘(𝕜, E') n f x) (hg : ContMDiffAt I 𝓘(𝕜, F') n g x)
Goal: ContMDiffAt I 𝓘(𝕜, E' × F') n (fun x => (f x, g x)) x
-- ContMDiffOn.prod_mk
theorem ContMDiffOn.prod_mk {f : M → M'} {g : M → N'} (hf : ContMDiffOn I I' n f s)
    (hg : ContMDiffOn I J' n g s) : ContMDiffOn I (I'.prod J') n (fun x => (f x, g x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {s : Set M} {n : ℕ∞} {f : M → M'} {g : M → N'} (hf : ContMDiffOn I I' n f s) (hg : ContMDiffOn I J' n g s)
Goal: ContMDiffOn I (I'.prod J') n (fun x => (f x, g x)) s
-- ContMDiffOn.prod_mk_space
theorem ContMDiffOn.prod_mk_space {f : M → E'} {g : M → F'} (hf : ContMDiffOn I 𝓘(𝕜, E') n f s)
    (hg : ContMDiffOn I 𝓘(𝕜, F') n g s) : ContMDiffOn I 𝓘(𝕜, E' × F') n (fun x => (f x, g x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set M} {n : ℕ∞} {f : M → E'} {g : M → F'} (hf : ContMDiffOn I 𝓘(𝕜, E') n f s) (hg : ContMDiffOn I 𝓘(𝕜, F') n g s)
Goal: ContMDiffOn I 𝓘(𝕜, E' × F') n (fun x => (f x, g x)) s
-- ContMDiff.prod_mk
nonrec theorem ContMDiff.prod_mk {f : M → M'} {g : M → N'} (hf : ContMDiff I I' n f)
    (hg : ContMDiff I J' n g) : ContMDiff I (I'.prod J') n fun x => (f x, g x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {n : ℕ∞} {f : M → M'} {g : M → N'} (hf : ContMDiff I I' n f) (hg : ContMDiff I J' n g)
Goal: ContMDiff I (I'.prod J') n fun x => (f x, g x)
-- ContMDiff.prod_mk_space
theorem ContMDiff.prod_mk_space {f : M → E'} {g : M → F'} (hf : ContMDiff I 𝓘(𝕜, E') n f)
    (hg : ContMDiff I 𝓘(𝕜, F') n g) : ContMDiff I 𝓘(𝕜, E' × F') n fun x => (f x, g x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {n : ℕ∞} {f : M → E'} {g : M → F'} (hf : ContMDiff I 𝓘(𝕜, E') n f) (hg : ContMDiff I 𝓘(𝕜, F') n g)
Goal: ContMDiff I 𝓘(𝕜, E' × F') n fun x => (f x, g x)
-- SmoothWithinAt.prod_mk
nonrec theorem SmoothWithinAt.prod_mk {f : M → M'} {g : M → N'} (hf : SmoothWithinAt I I' f s x)
    (hg : SmoothWithinAt I J' g s x) : SmoothWithinAt I (I'.prod J') (fun x => (f x, g x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {s : Set M} {x : M} {f : M → M'} {g : M → N'} (hf : SmoothWithinAt I I' f s x) (hg : SmoothWithinAt I J' g s x)
Goal: SmoothWithinAt I (I'.prod J') (fun x => (f x, g x)) s x
-- SmoothWithinAt.prod_mk_space
nonrec theorem SmoothWithinAt.prod_mk_space {f : M → E'} {g : M → F'}
    (hf : SmoothWithinAt I 𝓘(𝕜, E') f s x) (hg : SmoothWithinAt I 𝓘(𝕜, F') g s x) :
    SmoothWithinAt I 𝓘(𝕜, E' × F') (fun x => (f x, g x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set M} {x : M} {f : M → E'} {g : M → F'} (hf : SmoothWithinAt I 𝓘(𝕜, E') f s x) (hg : SmoothWithinAt I 𝓘(𝕜, F') g s x)
Goal: SmoothWithinAt I 𝓘(𝕜, E' × F') (fun x => (f x, g x)) s x
-- SmoothAt.prod_mk
nonrec theorem SmoothAt.prod_mk {f : M → M'} {g : M → N'} (hf : SmoothAt I I' f x)
    (hg : SmoothAt I J' g x) : SmoothAt I (I'.prod J') (fun x => (f x, g 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] {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' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {x : M} {f : M → M'} {g : M → N'} (hf : SmoothAt I I' f x) (hg : SmoothAt I J' g x)
Goal: SmoothAt I (I'.prod J') (fun x => (f x, g x)) x
-- SmoothAt.prod_mk_space
nonrec theorem SmoothAt.prod_mk_space {f : M → E'} {g : M → F'} (hf : SmoothAt I 𝓘(𝕜, E') f x)
    (hg : SmoothAt I 𝓘(𝕜, F') g x) : SmoothAt I 𝓘(𝕜, E' × F') (fun x => (f x, g 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] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {x : M} {f : M → E'} {g : M → F'} (hf : SmoothAt I 𝓘(𝕜, E') f x) (hg : SmoothAt I 𝓘(𝕜, F') g x)
Goal: SmoothAt I 𝓘(𝕜, E' × F') (fun x => (f x, g x)) x
-- SmoothOn.prod_mk
nonrec theorem SmoothOn.prod_mk {f : M → M'} {g : M → N'} (hf : SmoothOn I I' f s)
    (hg : SmoothOn I J' g s) : SmoothOn I (I'.prod J') (fun x => (f x, g x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {s : Set M} {f : M → M'} {g : M → N'} (hf : SmoothOn I I' f s) (hg : SmoothOn I J' g s)
Goal: SmoothOn I (I'.prod J') (fun x => (f x, g x)) s
-- SmoothOn.prod_mk_space
nonrec theorem SmoothOn.prod_mk_space {f : M → E'} {g : M → F'} (hf : SmoothOn I 𝓘(𝕜, E') f s)
    (hg : SmoothOn I 𝓘(𝕜, F') g s) : SmoothOn I 𝓘(𝕜, E' × F') (fun x => (f x, g x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set M} {f : M → E'} {g : M → F'} (hf : SmoothOn I 𝓘(𝕜, E') f s) (hg : SmoothOn I 𝓘(𝕜, F') g s)
Goal: SmoothOn I 𝓘(𝕜, E' × F') (fun x => (f x, g x)) s