ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- UpperHalfPlane.isometry_real_vadd
theorem isometry_real_vadd (a : ℝ) : Isometry (a +ᵥ · : ℍ → ℍ) 
Arguments: (a : ℝ)
Goal: Isometry fun x => a +ᵥ x
-- UpperHalfPlane.isometry_pos_mul
theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry (a • · : ℍ → ℍ) 
Arguments: (a : { x // 0 < x })
Goal: Isometry fun x => a • x
-- ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux {Du Eu Fu Gu : Type u}
    [NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
    [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu]
    (B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du → Eu} {g : Du → Fu} {n : ℕ} {s : Set Du} {x : Du}
    (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {Du Eu Fu Gu : Type u} [NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu] [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] (B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du → Eu} {g : Du → Fu} {n : ℕ} {s : Set Du} {x : Du} (hf : ContDiffOn 𝕜 (↑n) f s) (hg : ContDiffOn 𝕜 (↑n) g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
  ‖B‖ *
    ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {D : Type uD} [NormedAddCommGroup D] [NormedSpace 𝕜 D] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
  ‖B‖ *
    ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear
theorem ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G) {f : D → E}
    {g : D → F} {N : ℕ∞} (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : D) {n : ℕ}
    (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDeriv 𝕜 n (fun y => B (f y) (g y)) x‖ ≤ ‖B‖ * ∑ i ∈ Finset.range (n + 1),
      (n.choose i : ℝ) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {D : Type uD} [NormedAddCommGroup D] [NormedSpace 𝕜 D] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : D) {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDeriv 𝕜 n (fun y => (B (f y)) (g y)) x‖ ≤
  ‖B‖ * ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖
-- ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_of_le_one
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_of_le_one
    (B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D}
    (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ}
    (hn : (n : ℕ∞) ≤ N) (hB : ‖B‖ ≤ 1) : ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {D : Type uD} [NormedAddCommGroup D] [NormedSpace 𝕜 D] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : ↑n ≤ N) (hB : ‖B‖ ≤ 1)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear_of_le_one
theorem ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear_of_le_one (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : D) {n : ℕ}
    (hn : (n : ℕ∞) ≤ N) (hB : ‖B‖ ≤ 1) : ‖iteratedFDeriv 𝕜 n (fun y => B (f y) (g y)) x‖ ≤
      ∑ i ∈ Finset.range (n + 1),
        (n.choose i : ℝ) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {D : Type uD} [NormedAddCommGroup D] [NormedSpace 𝕜 D] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : D) {n : ℕ} (hn : ↑n ≤ N) (hB : ‖B‖ ≤ 1)
Goal: ‖iteratedFDeriv 𝕜 n (fun y => (B (f y)) (g y)) x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖
-- norm_iteratedFDerivWithin_smul_le
theorem norm_iteratedFDerivWithin_smul_le {f : E → 𝕜'} {g : E → F} {N : ℕ∞}
    (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) {x : E} (hx : x ∈ s)
    {n : ℕ} (hn : (n : ℕ∞) ≤ N) : ‖iteratedFDerivWithin 𝕜 n (fun y => f y • g y) s x‖ ≤
      ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {𝕜' : Type u_2} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → 𝕜'} {g : E → F} {N : ℕ∞} (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun y => f y • g y) s x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- norm_iteratedFDeriv_smul_le
theorem norm_iteratedFDeriv_smul_le {f : E → 𝕜'} {g : E → F} {N : ℕ∞} (hf : ContDiff 𝕜 N f)
    (hg : ContDiff 𝕜 N g) (x : E) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDeriv 𝕜 n (fun y => f y • g y) x‖ ≤ ∑ i ∈ Finset.range (n + 1),
      (n.choose i : ℝ) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {𝕜' : Type u_2} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → 𝕜'} {g : E → F} {N : ℕ∞} (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : E) {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDeriv 𝕜 n (fun y => f y • g y) x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖
-- norm_iteratedFDerivWithin_mul_le
theorem norm_iteratedFDerivWithin_mul_le {f : E → A} {g : E → A} {N : ℕ∞} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ}
    (hn : (n : ℕ∞) ≤ N) : ‖iteratedFDerivWithin 𝕜 n (fun y => f y * g y) s x‖ ≤
      ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {A : Type u_3} [NormedRing A] [NormedAlgebra 𝕜 A] {f g : E → A} {N : ℕ∞} (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun y => f y * g y) s x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- norm_iteratedFDeriv_mul_le
theorem norm_iteratedFDeriv_mul_le {f : E → A} {g : E → A} {N : ℕ∞} (hf : ContDiff 𝕜 N f)
    (hg : ContDiff 𝕜 N g) (x : E) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDeriv 𝕜 n (fun y => f y * g y) x‖ ≤ ∑ i ∈ Finset.range (n + 1),
      (n.choose i : ℝ) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {A : Type u_3} [NormedRing A] [NormedAlgebra 𝕜 A] {f g : E → A} {N : ℕ∞} (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : E) {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDeriv 𝕜 n (fun y => f y * g y) x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖
-- norm_iteratedFDerivWithin_prod_le
theorem norm_iteratedFDerivWithin_prod_le [DecidableEq ι] [NormOneClass A'] {u : Finset ι}
    {f : ι → E → A'} {N : ℕ∞} (hf : ∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) (hs : UniqueDiffOn 𝕜 s) {x : E}
    (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (∏ j ∈ u, f j ·) s x‖ ≤
      ∑ p ∈ u.sym n, (p : Multiset ι).multinomial *
        ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j p) (f j) s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_2} {A' : Type u_4} [NormedCommRing A'] [NormedAlgebra 𝕜 A'] [DecidableEq ι] [NormOneClass A'] {u : Finset ι} {f : ι → E → A'} {N : ℕ∞} (hf : ∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) (hs : UniqueDiffOn 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun x => ∏ j ∈ u, f j x) s x‖ ≤
  ∑ p ∈ u.sym n, ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
-- norm_iteratedFDeriv_prod_le
theorem norm_iteratedFDeriv_prod_le [DecidableEq ι] [NormOneClass A'] {u : Finset ι}
    {f : ι → E → A'} {N : ℕ∞} (hf : ∀ i ∈ u, ContDiff 𝕜 N (f i)) {x : E} {n : ℕ}
    (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDeriv 𝕜 n (∏ j ∈ u, f j ·) x‖ ≤
      ∑ p ∈ u.sym n, (p : Multiset ι).multinomial *
        ∏ j ∈ u, ‖iteratedFDeriv 𝕜 ((p : Multiset ι).count j) (f j) x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_2} {A' : Type u_4} [NormedCommRing A'] [NormedAlgebra 𝕜 A'] [DecidableEq ι] [NormOneClass A'] {u : Finset ι} {f : ι → E → A'} {N : ℕ∞} (hf : ∀ i ∈ u, ContDiff 𝕜 N (f i)) {x : E} {n : ℕ} (hn : ↑n ≤ N)
Goal: ‖iteratedFDeriv 𝕜 n (fun x => ∏ j ∈ u, f j x) x‖ ≤
  ∑ p ∈ u.sym n, ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDeriv 𝕜 (Multiset.count j ↑p) (f j) x‖
-- norm_iteratedFDerivWithin_comp_le_aux
theorem norm_iteratedFDerivWithin_comp_le_aux {Fu Gu : Type u} [NormedAddCommGroup Fu]
    [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] {g : Fu → Gu} {f : E → Fu} {n : ℕ}
    {s : Set E} {t : Set Fu} {x : E} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s)
    (ht : UniqueDiffOn 𝕜 t) (hs : UniqueDiffOn 𝕜 s) (hst : MapsTo f s t) (hx : x ∈ s) {C : ℝ}
    {D : ℝ} (hC : ∀ i, i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C)
    (hD : ∀ i, 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) :
    ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ n ! * C * D ^ n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {Fu Gu : Type u} [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] {g : Fu → Gu} {f : E → Fu} {n : ℕ} {s : Set E} {t : Set Fu} {x : E} (hg : ContDiffOn 𝕜 (↑n) g t) (hf : ContDiffOn 𝕜 (↑n) f s) (ht : UniqueDiffOn 𝕜 t) (hs : UniqueDiffOn 𝕜 s) (hst : Set.MapsTo f s t) (hx : x ∈ s) {C D : ℝ} (hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) (hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i)
Goal: ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n
-- norm_iteratedFDerivWithin_comp_le
theorem norm_iteratedFDerivWithin_comp_le {g : F → G} {f : E → F} {n : ℕ} {s : Set E} {t : Set F}
    {x : E} {N : ℕ∞} (hg : ContDiffOn 𝕜 N g t) (hf : ContDiffOn 𝕜 N f s) (hn : (n : ℕ∞) ≤ N)
    (ht : UniqueDiffOn 𝕜 t) (hs : UniqueDiffOn 𝕜 s) (hst : MapsTo f s t) (hx : x ∈ s) {C : ℝ}
    {D : ℝ} (hC : ∀ i, i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C)
    (hD : ∀ i, 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) :
    ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ n ! * C * D ^ n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : F → G} {f : E → F} {n : ℕ} {s : Set E} {t : Set F} {x : E} {N : ℕ∞} (hg : ContDiffOn 𝕜 N g t) (hf : ContDiffOn 𝕜 N f s) (hn : ↑n ≤ N) (ht : UniqueDiffOn 𝕜 t) (hs : UniqueDiffOn 𝕜 s) (hst : Set.MapsTo f s t) (hx : x ∈ s) {C D : ℝ} (hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) (hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i)
Goal: ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n
-- norm_iteratedFDeriv_comp_le
theorem norm_iteratedFDeriv_comp_le {g : F → G} {f : E → F} {n : ℕ} {N : ℕ∞} (hg : ContDiff 𝕜 N g)
    (hf : ContDiff 𝕜 N f) (hn : (n : ℕ∞) ≤ N) (x : E) {C : ℝ} {D : ℝ}
    (hC : ∀ i, i ≤ n → ‖iteratedFDeriv 𝕜 i g (f x)‖ ≤ C)
    (hD : ∀ i, 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i) :
    ‖iteratedFDeriv 𝕜 n (g ∘ f) x‖ ≤ n ! * C * D ^ n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : F → G} {f : E → F} {n : ℕ} {N : ℕ∞} (hg : ContDiff 𝕜 N g) (hf : ContDiff 𝕜 N f) (hn : ↑n ≤ N) (x : E) {C D : ℝ} (hC : ∀ i ≤ n, ‖iteratedFDeriv 𝕜 i g (f x)‖ ≤ C) (hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i)
Goal: ‖iteratedFDeriv 𝕜 n (g ∘ f) x‖ ≤ ↑n ! * C * D ^ n
-- norm_iteratedFDerivWithin_clm_apply
theorem norm_iteratedFDerivWithin_clm_apply {f : E → F →L[𝕜] G} {g : E → F} {s : Set E} {x : E}
    {N : ℕ∞} {n : ℕ} (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s)
    (hx : x ∈ s) (hn : ↑n ≤ N) : ‖iteratedFDerivWithin 𝕜 n (fun y => (f y) (g y)) s x‖ ≤
      ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F →L[𝕜] G} {g : E → F} {s : Set E} {x : E} {N : ℕ∞} {n : ℕ} (hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (hn : ↑n ≤ N)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun y => (f y) (g y)) s x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- norm_iteratedFDeriv_clm_apply
theorem norm_iteratedFDeriv_clm_apply {f : E → F →L[𝕜] G} {g : E → F} {N : ℕ∞} {n : ℕ}
    (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : E) (hn : ↑n ≤ N) :
    ‖iteratedFDeriv 𝕜 n (fun y : E => (f y) (g y)) x‖ ≤ ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F →L[𝕜] G} {g : E → F} {N : ℕ∞} {n : ℕ} (hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : E) (hn : ↑n ≤ N)
Goal: ‖iteratedFDeriv 𝕜 n (fun y => (f y) (g y)) x‖ ≤
  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖
-- norm_iteratedFDerivWithin_clm_apply_const
theorem norm_iteratedFDerivWithin_clm_apply_const {f : E → F →L[𝕜] G} {c : F} {s : Set E} {x : E}
    {N : ℕ∞} {n : ℕ} (hf : ContDiffOn 𝕜 N f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (hn : ↑n ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y : E => (f y) c) s x‖ ≤
      ‖c‖ * ‖iteratedFDerivWithin 𝕜 n f s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F →L[𝕜] G} {c : F} {s : Set E} {x : E} {N : ℕ∞} {n : ℕ} (hf : ContDiffOn 𝕜 N f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (hn : ↑n ≤ N)
Goal: ‖iteratedFDerivWithin 𝕜 n (fun y => (f y) c) s x‖ ≤ ‖c‖ * ‖iteratedFDerivWithin 𝕜 n f s x‖
-- norm_iteratedFDeriv_clm_apply_const
theorem norm_iteratedFDeriv_clm_apply_const {f : E → F →L[𝕜] G} {c : F} {x : E} {N : ℕ∞} {n : ℕ}
    (hf : ContDiff 𝕜 N f) (hn : ↑n ≤ N) :
    ‖iteratedFDeriv 𝕜 n (fun y : E => (f y) c) x‖ ≤ ‖c‖ * ‖iteratedFDeriv 𝕜 n f x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F →L[𝕜] G} {c : F} {x : E} {N : ℕ∞} {n : ℕ} (hf : ContDiff 𝕜 N f) (hn : ↑n ≤ N)
Goal: ‖iteratedFDeriv 𝕜 n (fun y => (f y) c) x‖ ≤ ‖c‖ * ‖iteratedFDeriv 𝕜 n f x‖
-- SimplicialObject.Splitting.IndexSet.mk_fst
@[simps]
def mk {Δ Δ' : SimplexCategory} (f : Δ ⟶ Δ') [Epi f] : IndexSet (op Δ) 
Arguments: {Δ Δ' : SimplexCategory} (f : Δ ⟶ Δ') [CategoryTheory.Epi f]
Goal: (SimplicialObject.Splitting.IndexSet.mk f).fst = Opposite.op Δ'
-- SimplicialObject.Splitting.IndexSet.mk_snd_coe
@[simps]
def mk {Δ Δ' : SimplexCategory} (f : Δ ⟶ Δ') [Epi f] : IndexSet (op Δ) 
Arguments: {Δ Δ' : SimplexCategory} (f : Δ ⟶ Δ') [CategoryTheory.Epi f]
Goal: ↑(SimplicialObject.Splitting.IndexSet.mk f).snd = f
-- SimplicialObject.Splitting.IndexSet.ext'
theorem ext' : A = ⟨A.1, ⟨A.e, A.2.2⟩⟩ 
Arguments: {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: A = ⟨A.fst, ⟨A.e, ⋯⟩⟩
-- SimplicialObject.Splitting.IndexSet.ext
theorem ext (A₁ A₂ : IndexSet Δ) (h₁ : A₁.1 = A₂.1) (h₂ : A₁.e ≫ eqToHom (by rw [h₁]) = A₂.e) :
    A₁ = A₂ 
Arguments: {Δ : SimplexCategoryᵒᵖ} (A₁ A₂ : SimplicialObject.Splitting.IndexSet Δ) (h₁ : A₁.fst = A₂.fst) (h₂ : A₁.e ≫ CategoryTheory.eqToHom ⋯ = A₂.e)
Goal: A₁ = A₂
-- SimplicialObject.Splitting.IndexSet.id_snd_coe
@[simps]
def id : IndexSet Δ 
Arguments: (Δ : SimplexCategoryᵒᵖ)
Goal: ↑(SimplicialObject.Splitting.IndexSet.id Δ).snd = 𝟙 (Opposite.unop Δ)
-- SimplicialObject.Splitting.IndexSet.id_fst
@[simps]
def id : IndexSet Δ 
Arguments: (Δ : SimplexCategoryᵒᵖ)
Goal: (SimplicialObject.Splitting.IndexSet.id Δ).fst = Δ
-- SimplicialObject.Splitting.IndexSet.eqId_iff_eq
theorem eqId_iff_eq : A.EqId ↔ A.1 = Δ 
Arguments: {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: A.EqId ↔ A.fst = Δ
-- SimplicialObject.Splitting.IndexSet.eqId_iff_len_eq
theorem eqId_iff_len_eq : A.EqId ↔ A.1.unop.len = Δ.unop.len 
Arguments: {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: A.EqId ↔ (Opposite.unop A.fst).len = (Opposite.unop Δ).len
-- SimplicialObject.Splitting.IndexSet.eqId_iff_len_le
theorem eqId_iff_len_le : A.EqId ↔ Δ.unop.len ≤ A.1.unop.len 
Arguments: {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: A.EqId ↔ (Opposite.unop Δ).len ≤ (Opposite.unop A.fst).len
-- SimplicialObject.Splitting.IndexSet.eqId_iff_mono
theorem eqId_iff_mono : A.EqId ↔ Mono A.e 
Arguments: {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: A.EqId ↔ CategoryTheory.Mono A.e
-- SimplicialObject.Splitting.IndexSet.epiComp_fst
@[simps]
def epiComp {Δ₁ Δ₂ : SimplexCategoryᵒᵖ} (A : IndexSet Δ₁) (p : Δ₁ ⟶ Δ₂) [Epi p.unop] :
    IndexSet Δ₂ 
Arguments: {Δ₁ Δ₂ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ₁) (p : Δ₁ ⟶ Δ₂) [CategoryTheory.Epi p.unop]
Goal: (A.epiComp p).fst = A.fst
-- SimplicialObject.Splitting.IndexSet.epiComp_snd_coe
@[simps]
def epiComp {Δ₁ Δ₂ : SimplexCategoryᵒᵖ} (A : IndexSet Δ₁) (p : Δ₁ ⟶ Δ₂) [Epi p.unop] :
    IndexSet Δ₂ 
Arguments: {Δ₁ Δ₂ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ₁) (p : Δ₁ ⟶ Δ₂) [CategoryTheory.Epi p.unop]
Goal: ↑(A.epiComp p).snd = p.unop ≫ A.e
-- SimplicialObject.Splitting.IndexSet.fac_pull
@[reassoc]
theorem fac_pull : (A.pull θ).e ≫ image.ι (θ.unop ≫ A.e) = θ.unop ≫ A.e 
Arguments: {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ) {Δ' : SimplexCategoryᵒᵖ} (θ : Δ ⟶ Δ')
Goal: (A.pull θ).e ≫ CategoryTheory.Limits.image.ι (θ.unop ≫ A.e) = θ.unop ≫ A.e
-- SimplicialObject.Splitting.cofan_inj_eq
@[reassoc]
theorem cofan_inj_eq {Δ : SimplexCategoryᵒᵖ} (A : IndexSet Δ) :
    (s.cofan Δ).inj  A = s.ι A.1.unop.len ≫ X.map A.e.op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject C} (s : SimplicialObject.Splitting X) {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: (s.cofan Δ).inj A = s.ι (Opposite.unop A.fst).len ≫ X.map A.e.op
-- SimplicialObject.Splitting.cofan_inj_id
theorem cofan_inj_id (n : ℕ) : (s.cofan _).inj (IndexSet.id (op [n])) = s.ι n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject C} (s : SimplicialObject.Splitting X) (n : ℕ)
Goal: (s.cofan (Opposite.op [n])).inj (SimplicialObject.Splitting.IndexSet.id (Opposite.op [n])) = s.ι n
-- SimplicialObject.Splitting.cofan_inj_comp_app
@[reassoc (attr := simp)]
theorem cofan_inj_comp_app (f : X ⟶ Y) {Δ : SimplexCategoryᵒᵖ} (A : IndexSet Δ) :
    (s.cofan Δ).inj A ≫ f.app Δ = s.φ f A.1.unop.len ≫ Y.map A.e.op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X Y : CategoryTheory.SimplicialObject C} (s : SimplicialObject.Splitting X) (f : X ⟶ Y) {Δ : SimplexCategoryᵒᵖ} (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: (s.cofan Δ).inj A ≫ f.app Δ = s.φ f (Opposite.unop A.fst).len ≫ Y.map A.e.op
-- SimplicialObject.Splitting.hom_ext'
theorem hom_ext' {Z : C} {Δ : SimplexCategoryᵒᵖ} (f g : X.obj Δ ⟶ Z)
    (h : ∀ A : IndexSet Δ, (s.cofan Δ).inj A ≫ f = (s.cofan Δ).inj A ≫ g) : f = g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject C} (s : SimplicialObject.Splitting X) {Z : C} {Δ : SimplexCategoryᵒᵖ} (f g : X.obj Δ ⟶ Z) (h : ∀ (A : SimplicialObject.Splitting.IndexSet Δ), (s.cofan Δ).inj A ≫ f = (s.cofan Δ).inj A ≫ g)
Goal: f = g
-- SimplicialObject.Splitting.hom_ext
theorem hom_ext (f g : X ⟶ Y) (h : ∀ n : ℕ, s.φ f n = s.φ g n) : f = g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X Y : CategoryTheory.SimplicialObject C} (s : SimplicialObject.Splitting X) (f g : X ⟶ Y) (h : ∀ (n : ℕ), s.φ f n = s.φ g n)
Goal: f = g
-- SimplicialObject.Splitting.ι_desc
@[reassoc (attr := simp)]
theorem ι_desc {Z : C} (Δ : SimplexCategoryᵒᵖ) (F : ∀ A : IndexSet Δ, s.N A.1.unop.len ⟶ Z)
    (A : IndexSet Δ) : (s.cofan Δ).inj A ≫ s.desc Δ F = F A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject C} (s : SimplicialObject.Splitting X) {Z : C} (Δ : SimplexCategoryᵒᵖ) (F : (A : SimplicialObject.Splitting.IndexSet Δ) → s.N (Opposite.unop A.fst).len ⟶ Z) (A : SimplicialObject.Splitting.IndexSet Δ)
Goal: (s.cofan Δ).inj A ≫ s.desc Δ F = F A