ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- smoothSheafCommRing.forgetStalk_hom_comp_evalHom
@[simp, reassoc, elementwise] lemma smoothSheafCommRing.forgetStalk_hom_comp_evalHom
    (x : TopCat.of M) :
    (smoothSheafCommRing.forgetStalk IM I M R x).hom ≫ (smoothSheaf.evalHom IM I R x) =
    (forget _).map (smoothSheafCommRing.evalHom _ _ _ _ _) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u) [TopologicalSpace M] [ChartedSpace HM M] (R : Type u) [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (x : ↑(TopCat.of M))
Goal: CategoryTheory.CategoryStruct.comp (smoothSheafCommRing.forgetStalk IM I M R x).hom (smoothSheaf.evalHom IM I R x) =
  (CategoryTheory.forget CommRingCat).map (smoothSheafCommRing.evalHom IM I M R x)
-- smoothSheafCommRing.eval_surjective
lemma smoothSheafCommRing.eval_surjective (x) :
    Function.Surjective (smoothSheafCommRing.eval IM I M R x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u) [TopologicalSpace M] [ChartedSpace HM M] (R : Type u) [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (x : M)
Goal: Function.Surjective ⇑(smoothSheafCommRing.eval IM I M R x)
-- smoothSheafCommRing.eval_germ
@[simp] lemma smoothSheafCommRing.eval_germ (U : Opens M) (x : U)
    (f : (smoothSheafCommRing IM I M R).presheaf.obj (op U)) :
    smoothSheafCommRing.eval IM I M R x ((smoothSheafCommRing IM I M R).presheaf.germ x f)
    = f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] {R : Type u} [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (U : TopologicalSpace.Opens M) (x : ↥U) (f : ↑((smoothSheafCommRing IM I M R).presheaf.obj (Opposite.op U)))
Goal: (smoothSheafCommRing.eval IM I M R ↑x) (((smoothSheafCommRing IM I M R).presheaf.germ x) f) = ↑f x
-- ContMDiffWithinAt.comp
theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
    (hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
    (st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {n : ℕ∞} {t : Set M'} {g : M' → M''} (x : M) (hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x) (st : Set.MapsTo f s t)
Goal: ContMDiffWithinAt I I'' n (g ∘ f) s x
-- ContMDiffWithinAt.comp_of_eq
theorem ContMDiffWithinAt.comp_of_eq {t : Set M'} {g : M' → M''} {x : M} {y : M'}
    (hg : ContMDiffWithinAt I' I'' n g t y) (hf : ContMDiffWithinAt I I' n f s x)
    (st : MapsTo f s t) (hx : f x = y) : ContMDiffWithinAt I I'' n (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {n : ℕ∞} {t : Set M'} {g : M' → M''} {x : M} {y : M'} (hg : ContMDiffWithinAt I' I'' n g t y) (hf : ContMDiffWithinAt I I' n f s x) (st : Set.MapsTo f s t) (hx : f x = y)
Goal: ContMDiffWithinAt I I'' n (g ∘ f) s x
-- SmoothWithinAt.comp
nonrec theorem SmoothWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
    (hg : SmoothWithinAt I' I'' g t (f x)) (hf : SmoothWithinAt I I' f s x) (st : MapsTo f s t) :
    SmoothWithinAt I I'' (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {t : Set M'} {g : M' → M''} (x : M) (hg : SmoothWithinAt I' I'' g t (f x)) (hf : SmoothWithinAt I I' f s x) (st : Set.MapsTo f s t)
Goal: SmoothWithinAt I I'' (g ∘ f) s x
-- ContMDiffOn.comp
theorem ContMDiffOn.comp {t : Set M'} {g : M' → M''} (hg : ContMDiffOn I' I'' n g t)
    (hf : ContMDiffOn I I' n f s) (st : s ⊆ f ⁻¹' t) : ContMDiffOn I I'' n (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {n : ℕ∞} {t : Set M'} {g : M' → M''} (hg : ContMDiffOn I' I'' n g t) (hf : ContMDiffOn I I' n f s) (st : s ⊆ f ⁻¹' t)
Goal: ContMDiffOn I I'' n (g ∘ f) s
-- SmoothOn.comp
nonrec theorem SmoothOn.comp {t : Set M'} {g : M' → M''} (hg : SmoothOn I' I'' g t)
    (hf : SmoothOn I I' f s) (st : s ⊆ f ⁻¹' t) : SmoothOn I I'' (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {t : Set M'} {g : M' → M''} (hg : SmoothOn I' I'' g t) (hf : SmoothOn I I' f s) (st : s ⊆ f ⁻¹' t)
Goal: SmoothOn I I'' (g ∘ f) s
-- ContMDiffOn.comp'
theorem ContMDiffOn.comp' {t : Set M'} {g : M' → M''} (hg : ContMDiffOn I' I'' n g t)
    (hf : ContMDiffOn I I' n f s) : ContMDiffOn I I'' n (g ∘ f) (s ∩ f ⁻¹' t) 
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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {n : ℕ∞} {t : Set M'} {g : M' → M''} (hg : ContMDiffOn I' I'' n g t) (hf : ContMDiffOn I I' n f s)
Goal: ContMDiffOn I I'' n (g ∘ f) (s ∩ f ⁻¹' t)
-- SmoothOn.comp'
nonrec theorem SmoothOn.comp' {t : Set M'} {g : M' → M''} (hg : SmoothOn I' I'' g t)
    (hf : SmoothOn I I' f s) : SmoothOn I I'' (g ∘ f) (s ∩ f ⁻¹' t) 
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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {t : Set M'} {g : M' → M''} (hg : SmoothOn I' I'' g t) (hf : SmoothOn I I' f s)
Goal: SmoothOn I I'' (g ∘ f) (s ∩ f ⁻¹' t)
-- ContMDiff.comp
theorem ContMDiff.comp {g : M' → M''} (hg : ContMDiff I' I'' n g) (hf : ContMDiff I I' n f) :
    ContMDiff I I'' n (g ∘ f) 
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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {n : ℕ∞} {g : M' → M''} (hg : ContMDiff I' I'' n g) (hf : ContMDiff I I' n f)
Goal: ContMDiff I I'' n (g ∘ f)
-- Smooth.comp
nonrec theorem Smooth.comp {g : M' → M''} (hg : Smooth I' I'' g) (hf : Smooth I I' f) :
    Smooth I I'' (g ∘ f) 
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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M' → M''} (hg : Smooth I' I'' g) (hf : Smooth I I' f)
Goal: Smooth I I'' (g ∘ f)
-- ContMDiffWithinAt.comp'
theorem ContMDiffWithinAt.comp' {t : Set M'} {g : M' → M''} (x : M)
    (hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x) :
    ContMDiffWithinAt I I'' n (g ∘ f) (s ∩ f ⁻¹' t) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {n : ℕ∞} {t : Set M'} {g : M' → M''} (x : M) (hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
Goal: ContMDiffWithinAt I I'' n (g ∘ f) (s ∩ f ⁻¹' t) x
-- SmoothWithinAt.comp'
nonrec theorem SmoothWithinAt.comp' {t : Set M'} {g : M' → M''} (x : M)
    (hg : SmoothWithinAt I' I'' g t (f x)) (hf : SmoothWithinAt I I' f s x) :
    SmoothWithinAt I I'' (g ∘ f) (s ∩ f ⁻¹' t) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {t : Set M'} {g : M' → M''} (x : M) (hg : SmoothWithinAt I' I'' g t (f x)) (hf : SmoothWithinAt I I' f s x)
Goal: SmoothWithinAt I I'' (g ∘ f) (s ∩ f ⁻¹' t) x
-- ContMDiffAt.comp_contMDiffWithinAt
theorem ContMDiffAt.comp_contMDiffWithinAt {g : M' → M''} (x : M)
    (hg : ContMDiffAt I' I'' n g (f x)) (hf : ContMDiffWithinAt I I' n f s x) :
    ContMDiffWithinAt I I'' n (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {n : ℕ∞} {g : M' → M''} (x : M) (hg : ContMDiffAt I' I'' n g (f x)) (hf : ContMDiffWithinAt I I' n f s x)
Goal: ContMDiffWithinAt I I'' n (g ∘ f) s x
-- SmoothAt.comp_smoothWithinAt
theorem SmoothAt.comp_smoothWithinAt {g : M' → M''} (x : M) (hg : SmoothAt I' I'' g (f x))
    (hf : SmoothWithinAt I I' f s x) : SmoothWithinAt I I'' (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {g : M' → M''} (x : M) (hg : SmoothAt I' I'' g (f x)) (hf : SmoothWithinAt I I' f s x)
Goal: SmoothWithinAt I I'' (g ∘ f) s x
-- ContMDiffAt.comp
nonrec theorem ContMDiffAt.comp {g : M' → M''} (x : M) (hg : ContMDiffAt I' I'' n g (f x))
    (hf : ContMDiffAt I I' n f x) : ContMDiffAt I I'' n (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {n : ℕ∞} {g : M' → M''} (x : M) (hg : ContMDiffAt I' I'' n g (f x)) (hf : ContMDiffAt I I' n f x)
Goal: ContMDiffAt I I'' n (g ∘ f) x
-- ContMDiffAt.comp_of_eq
theorem ContMDiffAt.comp_of_eq {g : M' → M''} {x : M} {y : M'} (hg : ContMDiffAt I' I'' n g y)
    (hf : ContMDiffAt I I' n f x) (hx : f x = y) : ContMDiffAt I I'' n (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {n : ℕ∞} {g : M' → M''} {x : M} {y : M'} (hg : ContMDiffAt I' I'' n g y) (hf : ContMDiffAt I I' n f x) (hx : f x = y)
Goal: ContMDiffAt I I'' n (g ∘ f) x
-- SmoothAt.comp
nonrec theorem SmoothAt.comp {g : M' → M''} (x : M) (hg : SmoothAt I' I'' g (f x))
    (hf : SmoothAt I I' f x) : SmoothAt I I'' (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M' → M''} (x : M) (hg : SmoothAt I' I'' g (f x)) (hf : SmoothAt I I' f x)
Goal: SmoothAt I I'' (g ∘ f) x
-- ContMDiff.comp_contMDiffOn
theorem ContMDiff.comp_contMDiffOn {f : M → M'} {g : M' → M''} {s : Set M}
    (hg : ContMDiff I' I'' n g) (hf : ContMDiffOn I I' n f s) : ContMDiffOn I I'' n (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {n : ℕ∞} {f : M → M'} {g : M' → M''} {s : Set M} (hg : ContMDiff I' I'' n g) (hf : ContMDiffOn I I' n f s)
Goal: ContMDiffOn I I'' n (g ∘ f) s
-- Smooth.comp_smoothOn
theorem Smooth.comp_smoothOn {f : M → M'} {g : M' → M''} {s : Set M} (hg : Smooth I' I'' g)
    (hf : SmoothOn I I' f s) : SmoothOn I I'' (g ∘ f) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M' → M''} {s : Set M} (hg : Smooth I' I'' g) (hf : SmoothOn I I' f s)
Goal: SmoothOn I I'' (g ∘ f) s
-- ContMDiffOn.comp_contMDiff
theorem ContMDiffOn.comp_contMDiff {t : Set M'} {g : M' → M''} (hg : ContMDiffOn I' I'' n g t)
    (hf : ContMDiff I I' n f) (ht : ∀ x, f x ∈ t) : ContMDiff I I'' n (g ∘ f) 
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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {n : ℕ∞} {t : Set M'} {g : M' → M''} (hg : ContMDiffOn I' I'' n g t) (hf : ContMDiff I I' n f) (ht : ∀ (x : M), f x ∈ t)
Goal: ContMDiff I I'' n (g ∘ f)
-- SmoothOn.comp_smooth
theorem SmoothOn.comp_smooth {t : Set M'} {g : M' → M''} (hg : SmoothOn I' I'' g t)
    (hf : Smooth I I' f) (ht : ∀ x, f x ∈ t) : Smooth I I'' (g ∘ f) 
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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {t : Set M'} {g : M' → M''} (hg : SmoothOn I' I'' g t) (hf : Smooth I I' f) (ht : ∀ (x : M), f x ∈ t)
Goal: Smooth I I'' (g ∘ f)
-- contMDiff_id
theorem contMDiff_id : ContMDiff I I n (id : M → M) 
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] {n : ℕ∞}
Goal: ContMDiff I I n id
-- smooth_id
theorem smooth_id : Smooth I I (id : M → M) 
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]
Goal: Smooth I I id
-- contMDiffOn_id
theorem contMDiffOn_id : ContMDiffOn I I n (id : M → M) 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] {s : Set M} {n : ℕ∞}
Goal: ContMDiffOn I I n id s
-- smoothOn_id
theorem smoothOn_id : SmoothOn I I (id : M → M) 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] {s : Set M}
Goal: SmoothOn I I id s
-- contMDiffAt_id
theorem contMDiffAt_id : ContMDiffAt I I n (id : M → M) 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] {x : M} {n : ℕ∞}
Goal: ContMDiffAt I I n id x
-- smoothAt_id
theorem smoothAt_id : SmoothAt I I (id : M → M) 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] {x : M}
Goal: SmoothAt I I id x