ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiffOn.continuousLinearMap_comp
theorem ContDiffOn.continuousLinearMap_comp (g : F →L[𝕜] G) (hf : ContDiffOn 𝕜 n f s) :
    ContDiffOn 𝕜 n (g ∘ f) s 
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] {s : Set E} {f : E → F} {n : ℕ∞} (g : F →L[𝕜] G) (hf : ContDiffOn 𝕜 n f s)
Goal: ContDiffOn 𝕜 n (⇑g ∘ f) s
-- ContDiff.continuousLinearMap_comp
theorem ContDiff.continuousLinearMap_comp {f : E → F} (g : F →L[𝕜] G) (hf : ContDiff 𝕜 n f) :
    ContDiff 𝕜 n fun x => g (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] {n : ℕ∞} {f : E → F} (g : F →L[𝕜] G) (hf : ContDiff 𝕜 n f)
Goal: ContDiff 𝕜 n fun x => g (f x)
-- ContinuousLinearMap.iteratedFDerivWithin_comp_left
theorem ContinuousLinearMap.iteratedFDerivWithin_comp_left {f : E → F} (g : F →L[𝕜] G)
    (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
    iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
      g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i 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] {s : Set E} {x : E} {n : ℕ∞} {f : E → F} (g : F →L[𝕜] G) (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : ↑i ≤ n)
Goal: iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x = g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x)
-- ContinuousLinearMap.iteratedFDeriv_comp_left
theorem ContinuousLinearMap.iteratedFDeriv_comp_left {f : E → F} (g : F →L[𝕜] G)
    (hf : ContDiff 𝕜 n f) (x : E) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
    iteratedFDeriv 𝕜 i (g ∘ f) x = g.compContinuousMultilinearMap (iteratedFDeriv 𝕜 i 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] {n : ℕ∞} {f : E → F} (g : F →L[𝕜] G) (hf : ContDiff 𝕜 n f) (x : E) {i : ℕ} (hi : ↑i ≤ n)
Goal: iteratedFDeriv 𝕜 i (⇑g ∘ f) x = g.compContinuousMultilinearMap (iteratedFDeriv 𝕜 i f x)
-- ContinuousLinearEquiv.iteratedFDerivWithin_comp_left
theorem ContinuousLinearEquiv.iteratedFDerivWithin_comp_left (g : F ≃L[𝕜] G) (f : E → F)
    (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (i : ℕ) :
    iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
      (g : F →L[𝕜] G).compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i 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] {s : Set E} {x : E} (g : F ≃L[𝕜] G) (f : E → F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (i : ℕ)
Goal: iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x = (↑g).compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x)
-- LinearIsometry.norm_iteratedFDerivWithin_comp_left
theorem LinearIsometry.norm_iteratedFDerivWithin_comp_left {f : E → F} (g : F →ₗᵢ[𝕜] G)
    (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
    ‖iteratedFDerivWithin 𝕜 i (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i 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] {s : Set E} {x : E} {n : ℕ∞} {f : E → F} (g : F →ₗᵢ[𝕜] G) (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : ↑i ≤ n)
Goal: ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖
-- LinearIsometry.norm_iteratedFDeriv_comp_left
theorem LinearIsometry.norm_iteratedFDeriv_comp_left {f : E → F} (g : F →ₗᵢ[𝕜] G)
    (hf : ContDiff 𝕜 n f) (x : E) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
    ‖iteratedFDeriv 𝕜 i (g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i 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] {n : ℕ∞} {f : E → F} (g : F →ₗᵢ[𝕜] G) (hf : ContDiff 𝕜 n f) (x : E) {i : ℕ} (hi : ↑i ≤ n)
Goal: ‖iteratedFDeriv 𝕜 i (⇑g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i f x‖
-- LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left
theorem LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left (g : F ≃ₗᵢ[𝕜] G) (f : E → F)
    (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (i : ℕ) :
    ‖iteratedFDerivWithin 𝕜 i (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i 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] {s : Set E} {x : E} (g : F ≃ₗᵢ[𝕜] G) (f : E → F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (i : ℕ)
Goal: ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖
-- LinearIsometryEquiv.norm_iteratedFDeriv_comp_left
theorem LinearIsometryEquiv.norm_iteratedFDeriv_comp_left (g : F ≃ₗᵢ[𝕜] G) (f : E → F) (x : E)
    (i : ℕ) : ‖iteratedFDeriv 𝕜 i (g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i 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] (g : F ≃ₗᵢ[𝕜] G) (f : E → F) (x : E) (i : ℕ)
Goal: ‖iteratedFDeriv 𝕜 i (⇑g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i f x‖
-- ContinuousLinearEquiv.comp_contDiffWithinAt_iff
theorem ContinuousLinearEquiv.comp_contDiffWithinAt_iff (e : F ≃L[𝕜] G) :
    ContDiffWithinAt 𝕜 n (e ∘ f) s x ↔ ContDiffWithinAt 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] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} (e : F ≃L[𝕜] G)
Goal: ContDiffWithinAt 𝕜 n (⇑e ∘ f) s x ↔ ContDiffWithinAt 𝕜 n f s x
-- ContinuousLinearEquiv.comp_contDiffAt_iff
theorem ContinuousLinearEquiv.comp_contDiffAt_iff (e : F ≃L[𝕜] G) :
    ContDiffAt 𝕜 n (e ∘ f) x ↔ ContDiffAt 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} {x : E} {n : ℕ∞} (e : F ≃L[𝕜] G)
Goal: ContDiffAt 𝕜 n (⇑e ∘ f) x ↔ ContDiffAt 𝕜 n f x
-- ContinuousLinearEquiv.comp_contDiffOn_iff
theorem ContinuousLinearEquiv.comp_contDiffOn_iff (e : F ≃L[𝕜] G) :
    ContDiffOn 𝕜 n (e ∘ f) s ↔ ContDiffOn 𝕜 n f s 
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] {s : Set E} {f : E → F} {n : ℕ∞} (e : F ≃L[𝕜] G)
Goal: ContDiffOn 𝕜 n (⇑e ∘ f) s ↔ ContDiffOn 𝕜 n f s
-- ContinuousLinearEquiv.comp_contDiff_iff
theorem ContinuousLinearEquiv.comp_contDiff_iff (e : F ≃L[𝕜] G) :
    ContDiff 𝕜 n (e ∘ f) ↔ ContDiff 𝕜 n f 
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} {n : ℕ∞} (e : F ≃L[𝕜] G)
Goal: ContDiff 𝕜 n (⇑e ∘ f) ↔ ContDiff 𝕜 n f
-- HasFTaylorSeriesUpToOn.compContinuousLinearMap
theorem HasFTaylorSeriesUpToOn.compContinuousLinearMap (hf : HasFTaylorSeriesUpToOn n f p s)
    (g : G →L[𝕜] E) :
    HasFTaylorSeriesUpToOn n (f ∘ g) (fun x k => (p (g x) k).compContinuousLinearMap fun _ => g)
      (g ⁻¹' s) 
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] {s : Set E} {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (hf : HasFTaylorSeriesUpToOn n f p s) (g : G →L[𝕜] E)
Goal: HasFTaylorSeriesUpToOn n (f ∘ ⇑g) (fun x k => (p (g x) k).compContinuousLinearMap fun x => g) (⇑g ⁻¹' s)
-- ContDiffWithinAt.comp_continuousLinearMap
theorem ContDiffWithinAt.comp_continuousLinearMap {x : G} (g : G →L[𝕜] E)
    (hf : ContDiffWithinAt 𝕜 n f s (g x)) : ContDiffWithinAt 𝕜 n (f ∘ g) (g ⁻¹' 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] {s : Set E} {f : E → F} {n : ℕ∞} {x : G} (g : G →L[𝕜] E) (hf : ContDiffWithinAt 𝕜 n f s (g x))
Goal: ContDiffWithinAt 𝕜 n (f ∘ ⇑g) (⇑g ⁻¹' s) x
-- ContDiffOn.comp_continuousLinearMap
theorem ContDiffOn.comp_continuousLinearMap (hf : ContDiffOn 𝕜 n f s) (g : G →L[𝕜] E) :
    ContDiffOn 𝕜 n (f ∘ g) (g ⁻¹' s) 
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] {s : Set E} {f : E → F} {n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (g : G →L[𝕜] E)
Goal: ContDiffOn 𝕜 n (f ∘ ⇑g) (⇑g ⁻¹' s)
-- ContDiff.comp_continuousLinearMap
theorem ContDiff.comp_continuousLinearMap {f : E → F} {g : G →L[𝕜] E} (hf : ContDiff 𝕜 n f) :
    ContDiff 𝕜 n (f ∘ g) 
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] {n : ℕ∞} {f : E → F} {g : G →L[𝕜] E} (hf : ContDiff 𝕜 n f)
Goal: ContDiff 𝕜 n (f ∘ ⇑g)
-- ContinuousLinearMap.iteratedFDerivWithin_comp_right
theorem ContinuousLinearMap.iteratedFDerivWithin_comp_right {f : E → F} (g : G →L[𝕜] E)
    (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (h's : UniqueDiffOn 𝕜 (g ⁻¹' s)) {x : G}
    (hx : g x ∈ s) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
    iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x =
      (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun _ => g 
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] {s : Set E} {n : ℕ∞} {f : E → F} (g : G →L[𝕜] E) (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (h's : UniqueDiffOn 𝕜 (⇑g ⁻¹' s)) {x : G} (hx : g x ∈ s) {i : ℕ} (hi : ↑i ≤ n)
Goal: iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x = (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun x => g
-- ContinuousLinearEquiv.iteratedFDerivWithin_comp_right
theorem ContinuousLinearEquiv.iteratedFDerivWithin_comp_right (g : G ≃L[𝕜] E) (f : E → F)
    (hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) :
    iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x =
      (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun _ => g 
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] {s : Set E} (g : G ≃L[𝕜] E) (f : E → F) (hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ)
Goal: iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x =
  (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun x => ↑g
-- ContinuousLinearMap.iteratedFDeriv_comp_right
theorem ContinuousLinearMap.iteratedFDeriv_comp_right (g : G →L[𝕜] E) {f : E → F}
    (hf : ContDiff 𝕜 n f) (x : G) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
    iteratedFDeriv 𝕜 i (f ∘ g) x =
      (iteratedFDeriv 𝕜 i f (g x)).compContinuousLinearMap fun _ => g 
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] {n : ℕ∞} (g : G →L[𝕜] E) {f : E → F} (hf : ContDiff 𝕜 n f) (x : G) {i : ℕ} (hi : ↑i ≤ n)
Goal: iteratedFDeriv 𝕜 i (f ∘ ⇑g) x = (iteratedFDeriv 𝕜 i f (g x)).compContinuousLinearMap fun x => g
-- LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right
theorem LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right (g : G ≃ₗᵢ[𝕜] E) (f : E → F)
    (hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) :
    ‖iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x‖ = ‖iteratedFDerivWithin 𝕜 i f s (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] {s : Set E} (g : G ≃ₗᵢ[𝕜] E) (f : E → F) (hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ)
Goal: ‖iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x‖ = ‖iteratedFDerivWithin 𝕜 i f s (g x)‖
-- LinearIsometryEquiv.norm_iteratedFDeriv_comp_right
theorem LinearIsometryEquiv.norm_iteratedFDeriv_comp_right (g : G ≃ₗᵢ[𝕜] E) (f : E → F) (x : G)
    (i : ℕ) : ‖iteratedFDeriv 𝕜 i (f ∘ g) x‖ = ‖iteratedFDeriv 𝕜 i f (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] (g : G ≃ₗᵢ[𝕜] E) (f : E → F) (x : G) (i : ℕ)
Goal: ‖iteratedFDeriv 𝕜 i (f ∘ ⇑g) x‖ = ‖iteratedFDeriv 𝕜 i f (g x)‖
-- ContinuousLinearEquiv.contDiffWithinAt_comp_iff
theorem ContinuousLinearEquiv.contDiffWithinAt_comp_iff (e : G ≃L[𝕜] E) :
    ContDiffWithinAt 𝕜 n (f ∘ e) (e ⁻¹' s) (e.symm x) ↔ ContDiffWithinAt 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] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} (e : G ≃L[𝕜] E)
Goal: ContDiffWithinAt 𝕜 n (f ∘ ⇑e) (⇑e ⁻¹' s) (e.symm x) ↔ ContDiffWithinAt 𝕜 n f s x
-- ContinuousLinearEquiv.contDiffAt_comp_iff
theorem ContinuousLinearEquiv.contDiffAt_comp_iff (e : G ≃L[𝕜] E) :
    ContDiffAt 𝕜 n (f ∘ e) (e.symm x) ↔ ContDiffAt 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} {x : E} {n : ℕ∞} (e : G ≃L[𝕜] E)
Goal: ContDiffAt 𝕜 n (f ∘ ⇑e) (e.symm x) ↔ ContDiffAt 𝕜 n f x
-- ContinuousLinearEquiv.contDiffOn_comp_iff
theorem ContinuousLinearEquiv.contDiffOn_comp_iff (e : G ≃L[𝕜] E) :
    ContDiffOn 𝕜 n (f ∘ e) (e ⁻¹' s) ↔ ContDiffOn 𝕜 n f s 
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] {s : Set E} {f : E → F} {n : ℕ∞} (e : G ≃L[𝕜] E)
Goal: ContDiffOn 𝕜 n (f ∘ ⇑e) (⇑e ⁻¹' s) ↔ ContDiffOn 𝕜 n f s
-- ContinuousLinearEquiv.contDiff_comp_iff
theorem ContinuousLinearEquiv.contDiff_comp_iff (e : G ≃L[𝕜] E) :
    ContDiff 𝕜 n (f ∘ e) ↔ ContDiff 𝕜 n f 
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} {n : ℕ∞} (e : G ≃L[𝕜] E)
Goal: ContDiff 𝕜 n (f ∘ ⇑e) ↔ ContDiff 𝕜 n f
-- HasFTaylorSeriesUpToOn.prod
theorem HasFTaylorSeriesUpToOn.prod (hf : HasFTaylorSeriesUpToOn n f p s) {g : E → G}
    {q : E → FormalMultilinearSeries 𝕜 E G} (hg : HasFTaylorSeriesUpToOn n g q s) :
    HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => (p y k).prod (q y k)) s 
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] {s : Set E} {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (hf : HasFTaylorSeriesUpToOn n f p s) {g : E → G} {q : E → FormalMultilinearSeries 𝕜 E G} (hg : HasFTaylorSeriesUpToOn n g q s)
Goal: HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => (p y k).prod (q y k)) s
-- ContDiffWithinAt.prod
theorem ContDiffWithinAt.prod {s : Set E} {f : E → F} {g : E → G} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x : E => (f x, g x)) 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] {x : E} {n : ℕ∞} {s : Set E} {f : E → F} {g : E → G} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x)
Goal: ContDiffWithinAt 𝕜 n (fun x => (f x, g x)) s x
-- ContDiffOn.prod
theorem ContDiffOn.prod {s : Set E} {f : E → F} {g : E → G} (hf : ContDiffOn 𝕜 n f s)
    (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x : E => (f x, g x)) s 
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] {n : ℕ∞} {s : Set E} {f : E → F} {g : E → G} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s)
Goal: ContDiffOn 𝕜 n (fun x => (f x, g x)) s
-- ContDiffAt.prod
theorem ContDiffAt.prod {f : E → F} {g : E → G} (hf : ContDiffAt 𝕜 n f x)
    (hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x : E => (f x, g x)) 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] {x : E} {n : ℕ∞} {f : E → F} {g : E → G} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x)
Goal: ContDiffAt 𝕜 n (fun x => (f x, g x)) x
-- ContDiff.prod
theorem ContDiff.prod {f : E → F} {g : E → G} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
    ContDiff 𝕜 n fun x : E => (f x, 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] {n : ℕ∞} {f : E → F} {g : E → G} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g)
Goal: ContDiff 𝕜 n fun x => (f x, g x)
-- _private.Mathlib.Analysis.Calculus.ContDiff.Basic.0.ContDiffOn.comp_same_univ
private theorem ContDiffOn.comp_same_univ {Eu : Type u} [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
    {Fu : Type u} [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] {Gu : Type u} [NormedAddCommGroup Gu]
    [NormedSpace 𝕜 Gu] {s : Set Eu} {t : Set Fu} {g : Fu → Gu} {f : Eu → Fu}
    (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s) (st : s ⊆ f ⁻¹' t) :
    ContDiffOn 𝕜 n (g ∘ f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : ℕ∞} {Eu : Type u} [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu] {Fu : Type u} [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] {Gu : Type u} [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] {s : Set Eu} {t : Set Fu} {g : Fu → Gu} {f : Eu → Fu} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s) (st : s ⊆ f ⁻¹' t)
Goal: ContDiffOn 𝕜 n (g ∘ f) s
-- ContDiffOn.comp
theorem ContDiffOn.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : ContDiffOn 𝕜 n g t)
    (hf : ContDiffOn 𝕜 n f s) (st : s ⊆ f ⁻¹' t) : ContDiffOn 𝕜 n (g ∘ f) s 
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] {n : ℕ∞} {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s) (st : s ⊆ f ⁻¹' t)
Goal: ContDiffOn 𝕜 n (g ∘ f) s
-- ContDiffOn.comp'
theorem ContDiffOn.comp' {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : ContDiffOn 𝕜 n g t)
    (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t) 
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] {n : ℕ∞} {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s)
Goal: ContDiffOn 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t)
-- ContDiff.comp_contDiffOn
theorem ContDiff.comp_contDiffOn {s : Set E} {g : F → G} {f : E → F} (hg : ContDiff 𝕜 n g)
    (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (g ∘ f) s 
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] {n : ℕ∞} {s : Set E} {g : F → G} {f : E → F} (hg : ContDiff 𝕜 n g) (hf : ContDiffOn 𝕜 n f s)
Goal: ContDiffOn 𝕜 n (g ∘ f) s
-- ContDiff.comp
theorem ContDiff.comp {g : F → G} {f : E → F} (hg : ContDiff 𝕜 n g) (hf : ContDiff 𝕜 n f) :
    ContDiff 𝕜 n (g ∘ f) 
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] {n : ℕ∞} {g : F → G} {f : E → F} (hg : ContDiff 𝕜 n g) (hf : ContDiff 𝕜 n f)
Goal: ContDiff 𝕜 n (g ∘ f)
-- ContDiffWithinAt.comp
theorem ContDiffWithinAt.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E)
    (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) (st : s ⊆ f ⁻¹' t) :
    ContDiffWithinAt 𝕜 n (g ∘ 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] {n : ℕ∞} {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E) (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) (st : s ⊆ f ⁻¹' t)
Goal: ContDiffWithinAt 𝕜 n (g ∘ f) s x
-- ContDiffWithinAt.comp_of_mem
theorem ContDiffWithinAt.comp_of_mem {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E)
    (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x)
    (hs : t ∈ 𝓝[f '' s] f x) : ContDiffWithinAt 𝕜 n (g ∘ 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] {n : ℕ∞} {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E) (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) (hs : t ∈ 𝓝[f '' s] f x)
Goal: ContDiffWithinAt 𝕜 n (g ∘ f) s x
-- ContDiffWithinAt.comp'
theorem ContDiffWithinAt.comp' {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E)
    (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) :
    ContDiffWithinAt 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t) 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] {n : ℕ∞} {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E) (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x)
Goal: ContDiffWithinAt 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t) x
-- ContDiffAt.comp_contDiffWithinAt
theorem ContDiffAt.comp_contDiffWithinAt {n} (x : E) (hg : ContDiffAt 𝕜 n g (f x))
    (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (g ∘ 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] {s : Set E} {f : E → F} {g : F → G} {n : ℕ∞} (x : E) (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContDiffWithinAt 𝕜 n f s x)
Goal: ContDiffWithinAt 𝕜 n (g ∘ f) s x