ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasStrictDerivAt.comp_of_eq
theorem HasStrictDerivAt.comp_of_eq
    (hh₂ : HasStrictDerivAt h₂ h₂' y) (hh : HasStrictDerivAt h h' x) (hy : y = h x) :
    HasStrictDerivAt (h₂ ∘ h) (h₂' * h') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' y : 𝕜'} (hh₂ : HasStrictDerivAt h₂ h₂' y) (hh : HasStrictDerivAt h h' x) (hy : y = h x)
Goal: HasStrictDerivAt (h₂ ∘ h) (h₂' * h') x
-- HasDerivAt.comp_hasDerivWithinAt
theorem HasDerivAt.comp_hasDerivWithinAt (hh₂ : HasDerivAt h₂ h₂' (h x))
    (hh : HasDerivWithinAt h h' s x) : HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'} (hh₂ : HasDerivAt h₂ h₂' (h x)) (hh : HasDerivWithinAt h h' s x)
Goal: HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x
-- HasDerivAt.comp_hasDerivWithinAt_of_eq
theorem HasDerivAt.comp_hasDerivWithinAt_of_eq (hh₂ : HasDerivAt h₂ h₂' y)
    (hh : HasDerivWithinAt h h' s x) (hy : y = h x) :
    HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' y : 𝕜'} (hh₂ : HasDerivAt h₂ h₂' y) (hh : HasDerivWithinAt h h' s x) (hy : y = h x)
Goal: HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x
-- derivWithin.comp
theorem derivWithin.comp (hh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x))
    (hh : DifferentiableWithinAt 𝕜 h s x) (hs : MapsTo h s s') (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {s' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} (hh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x)) (hh : DifferentiableWithinAt 𝕜 h s x) (hs : Set.MapsTo h s s') (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x
-- derivWithin.comp_of_eq
theorem derivWithin.comp_of_eq (hh₂ : DifferentiableWithinAt 𝕜' h₂ s' y)
    (hh : DifferentiableWithinAt 𝕜 h s x) (hs : MapsTo h s s') (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hy : y = h x) :
    derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {s' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {y : 𝕜'} (hh₂ : DifferentiableWithinAt 𝕜' h₂ s' y) (hh : DifferentiableWithinAt 𝕜 h s x) (hs : Set.MapsTo h s s') (hxs : UniqueDiffWithinAt 𝕜 s x) (hy : y = h x)
Goal: derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x
-- deriv.comp
theorem deriv.comp (hh₂ : DifferentiableAt 𝕜' h₂ (h x)) (hh : DifferentiableAt 𝕜 h x) :
    deriv (h₂ ∘ h) x = deriv h₂ (h x) * deriv h x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} (hh₂ : DifferentiableAt 𝕜' h₂ (h x)) (hh : DifferentiableAt 𝕜 h x)
Goal: deriv (h₂ ∘ h) x = deriv h₂ (h x) * deriv h x
-- deriv.comp_of_eq
theorem deriv.comp_of_eq (hh₂ : DifferentiableAt 𝕜' h₂ y) (hh : DifferentiableAt 𝕜 h x)
    (hy : y = h x) :
    deriv (h₂ ∘ h) x = deriv h₂ (h x) * deriv h x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {y : 𝕜'} (hh₂ : DifferentiableAt 𝕜' h₂ y) (hh : DifferentiableAt 𝕜 h x) (hy : y = h x)
Goal: deriv (h₂ ∘ h) x = deriv h₂ (h x) * deriv h x
-- HasDerivAtFilter.iterate
protected nonrec theorem HasDerivAtFilter.iterate {f : 𝕜 → 𝕜} {f' : 𝕜}
    (hf : HasDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : ℕ) :
    HasDerivAtFilter f^[n] (f' ^ n) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {L : Filter 𝕜} {f : 𝕜 → 𝕜} {f' : 𝕜} (hf : HasDerivAtFilter f f' x L) (hL : Filter.Tendsto f L L) (hx : f x = x) (n : ℕ)
Goal: HasDerivAtFilter f^[n] (f' ^ n) x L
-- HasDerivAt.iterate
protected nonrec theorem HasDerivAt.iterate {f : 𝕜 → 𝕜} {f' : 𝕜} (hf : HasDerivAt f f' x)
    (hx : f x = x) (n : ℕ) : HasDerivAt f^[n] (f' ^ n) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {f : 𝕜 → 𝕜} {f' : 𝕜} (hf : HasDerivAt f f' x) (hx : f x = x) (n : ℕ)
Goal: HasDerivAt f^[n] (f' ^ n) x
-- HasDerivWithinAt.iterate
protected theorem HasDerivWithinAt.iterate {f : 𝕜 → 𝕜} {f' : 𝕜} (hf : HasDerivWithinAt f f' s x)
    (hx : f x = x) (hs : MapsTo f s s) (n : ℕ) : HasDerivWithinAt f^[n] (f' ^ n) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {s : Set 𝕜} {f : 𝕜 → 𝕜} {f' : 𝕜} (hf : HasDerivWithinAt f f' s x) (hx : f x = x) (hs : Set.MapsTo f s s) (n : ℕ)
Goal: HasDerivWithinAt f^[n] (f' ^ n) s x
-- HasStrictDerivAt.iterate
protected nonrec theorem HasStrictDerivAt.iterate {f : 𝕜 → 𝕜} {f' : 𝕜}
    (hf : HasStrictDerivAt f f' x) (hx : f x = x) (n : ℕ) :
    HasStrictDerivAt f^[n] (f' ^ n) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {f : 𝕜 → 𝕜} {f' : 𝕜} (hf : HasStrictDerivAt f f' x) (hx : f x = x) (n : ℕ)
Goal: HasStrictDerivAt f^[n] (f' ^ n) x
-- HasFDerivWithinAt.comp_hasDerivWithinAt
theorem HasFDerivWithinAt.comp_hasDerivWithinAt {t : Set F} (hl : HasFDerivWithinAt l l' t (f x))
    (hf : HasDerivWithinAt f f' s x) (hst : MapsTo f s t) :
    HasDerivWithinAt (l ∘ f) (l' f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {s : Set 𝕜} {l : F → E} {l' : F →L[𝕜] E} {t : Set F} (hl : HasFDerivWithinAt l l' t (f x)) (hf : HasDerivWithinAt f f' s x) (hst : Set.MapsTo f s t)
Goal: HasDerivWithinAt (l ∘ f) (l' f') s x
-- HasFDerivWithinAt.comp_hasDerivWithinAt_of_eq
theorem HasFDerivWithinAt.comp_hasDerivWithinAt_of_eq {t : Set F}
    (hl : HasFDerivWithinAt l l' t y)
    (hf : HasDerivWithinAt f f' s x) (hst : MapsTo f s t) (hy : y = f x) :
    HasDerivWithinAt (l ∘ f) (l' f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {s : Set 𝕜} {l : F → E} {l' : F →L[𝕜] E} {y : F} {t : Set F} (hl : HasFDerivWithinAt l l' t y) (hf : HasDerivWithinAt f f' s x) (hst : Set.MapsTo f s t) (hy : y = f x)
Goal: HasDerivWithinAt (l ∘ f) (l' f') s x
-- HasFDerivAt.comp_hasDerivWithinAt
theorem HasFDerivAt.comp_hasDerivWithinAt (hl : HasFDerivAt l l' (f x))
    (hf : HasDerivWithinAt f f' s x) : HasDerivWithinAt (l ∘ f) (l' f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {s : Set 𝕜} {l : F → E} {l' : F →L[𝕜] E} (hl : HasFDerivAt l l' (f x)) (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (l ∘ f) (l' f') s x
-- HasFDerivAt.comp_hasDerivWithinAt_of_eq
theorem HasFDerivAt.comp_hasDerivWithinAt_of_eq (hl : HasFDerivAt l l' y)
    (hf : HasDerivWithinAt f f' s x) (hy : y = f x) :
    HasDerivWithinAt (l ∘ f) (l' f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {s : Set 𝕜} {l : F → E} {l' : F →L[𝕜] E} {y : F} (hl : HasFDerivAt l l' y) (hf : HasDerivWithinAt f f' s x) (hy : y = f x)
Goal: HasDerivWithinAt (l ∘ f) (l' f') s x
-- HasFDerivAt.comp_hasDerivAt
theorem HasFDerivAt.comp_hasDerivAt (hl : HasFDerivAt l l' (f x)) (hf : HasDerivAt f f' x) :
    HasDerivAt (l ∘ f) (l' f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {l : F → E} {l' : F →L[𝕜] E} (hl : HasFDerivAt l l' (f x)) (hf : HasDerivAt f f' x)
Goal: HasDerivAt (l ∘ f) (l' f') x
-- HasFDerivAt.comp_hasDerivAt_of_eq
theorem HasFDerivAt.comp_hasDerivAt_of_eq
    (hl : HasFDerivAt l l' y) (hf : HasDerivAt f f' x) (hy : y = f x) :
    HasDerivAt (l ∘ f) (l' f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {l : F → E} {l' : F →L[𝕜] E} {y : F} (hl : HasFDerivAt l l' y) (hf : HasDerivAt f f' x) (hy : y = f x)
Goal: HasDerivAt (l ∘ f) (l' f') x
-- HasStrictFDerivAt.comp_hasStrictDerivAt
theorem HasStrictFDerivAt.comp_hasStrictDerivAt (hl : HasStrictFDerivAt l l' (f x))
    (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (l ∘ f) (l' f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {l : F → E} {l' : F →L[𝕜] E} (hl : HasStrictFDerivAt l l' (f x)) (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (l ∘ f) (l' f') x
-- HasStrictFDerivAt.comp_hasStrictDerivAt_of_eq
theorem HasStrictFDerivAt.comp_hasStrictDerivAt_of_eq (hl : HasStrictFDerivAt l l' y)
    (hf : HasStrictDerivAt f f' x) (hy : y = f x) :
    HasStrictDerivAt (l ∘ f) (l' f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} {f' : F} (x : 𝕜) {l : F → E} {l' : F →L[𝕜] E} {y : F} (hl : HasStrictFDerivAt l l' y) (hf : HasStrictDerivAt f f' x) (hy : y = f x)
Goal: HasStrictDerivAt (l ∘ f) (l' f') x
-- fderivWithin.comp_derivWithin
theorem fderivWithin.comp_derivWithin {t : Set F} (hl : DifferentiableWithinAt 𝕜 l t (f x))
    (hf : DifferentiableWithinAt 𝕜 f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x) : F → E) (derivWithin f s x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} (x : 𝕜) {s : Set 𝕜} {l : F → E} {t : Set F} (hl : DifferentiableWithinAt 𝕜 l t (f x)) (hf : DifferentiableWithinAt 𝕜 f s x) (hs : Set.MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)
-- fderivWithin.comp_derivWithin_of_eq
theorem fderivWithin.comp_derivWithin_of_eq {t : Set F} (hl : DifferentiableWithinAt 𝕜 l t y)
    (hf : DifferentiableWithinAt 𝕜 f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hy : y = f x) :
    derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x) : F → E) (derivWithin f s x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} (x : 𝕜) {s : Set 𝕜} {l : F → E} {y : F} {t : Set F} (hl : DifferentiableWithinAt 𝕜 l t y) (hf : DifferentiableWithinAt 𝕜 f s x) (hs : Set.MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x) (hy : y = f x)
Goal: derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)
-- fderiv.comp_deriv
theorem fderiv.comp_deriv (hl : DifferentiableAt 𝕜 l (f x)) (hf : DifferentiableAt 𝕜 f x) :
    deriv (l ∘ f) x = (fderiv 𝕜 l (f x) : F → E) (deriv f x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} (x : 𝕜) {l : F → E} (hl : DifferentiableAt 𝕜 l (f x)) (hf : DifferentiableAt 𝕜 f x)
Goal: deriv (l ∘ f) x = (fderiv 𝕜 l (f x)) (deriv f x)
-- fderiv.comp_deriv_of_eq
theorem fderiv.comp_deriv_of_eq (hl : DifferentiableAt 𝕜 l y) (hf : DifferentiableAt 𝕜 f x)
    (hy : y = f x) :
    deriv (l ∘ f) x = (fderiv 𝕜 l (f x) : F → E) (deriv f x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → F} (x : 𝕜) {l : F → E} {y : F} (hl : DifferentiableAt 𝕜 l y) (hf : DifferentiableAt 𝕜 f x) (hy : y = f x)
Goal: deriv (l ∘ f) x = (fderiv 𝕜 l (f x)) (deriv f x)
-- TrivSqZeroExt.fst_expSeries
@[simp] theorem fst_expSeries (x : tsze R M) (n : ℕ) :
    fst (expSeries 𝕜 (tsze R M) n fun _ => x) = expSeries 𝕜 R n fun _ => x.fst 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) (n : ℕ)
Goal: ((NormedSpace.expSeries 𝕜 (tsze R M) n) fun x_1 => x).fst = (NormedSpace.expSeries 𝕜 R n) fun x_1 => x.fst
-- TrivSqZeroExt.snd_expSeries_of_smul_comm
theorem snd_expSeries_of_smul_comm
    (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) (n : ℕ) :
    snd (expSeries 𝕜 (tsze R M) (n + 1) fun _ => x) = (expSeries 𝕜 R n fun _ => x.fst) • x.snd 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) (n : ℕ)
Goal: ((NormedSpace.expSeries 𝕜 (tsze R M) (n + 1)) fun x_1 => x).snd =
  ((NormedSpace.expSeries 𝕜 R n) fun x_1 => x.fst) • x.snd
-- TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm
theorem hasSum_snd_expSeries_of_smul_comm (x : tsze R M)
    (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R}
    (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
    HasSum (fun n => snd (expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd) 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R} (h : HasSum (fun n => (NormedSpace.expSeries 𝕜 R n) fun x_1 => x.fst) e)
Goal: HasSum (fun n => ((NormedSpace.expSeries 𝕜 (tsze R M) n) fun x_1 => x).snd) (e • x.snd)
-- TrivSqZeroExt.hasSum_expSeries_of_smul_comm
theorem hasSum_expSeries_of_smul_comm
    (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
    {e : R} (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
    HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R} (h : HasSum (fun n => (NormedSpace.expSeries 𝕜 R n) fun x_1 => x.fst) e)
Goal: HasSum (fun n => (NormedSpace.expSeries 𝕜 (tsze R M) n) fun x_1 => x)
  (TrivSqZeroExt.inl e + TrivSqZeroExt.inr (e • x.snd))
-- TrivSqZeroExt.exp_def_of_smul_comm
theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) :
    exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
Goal: NormedSpace.exp 𝕜 x = TrivSqZeroExt.inl (NormedSpace.exp 𝕜 x.fst) + TrivSqZeroExt.inr (NormedSpace.exp 𝕜 x.fst • x.snd)
-- TrivSqZeroExt.exp_inl
@[simp]
theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (x : R)
Goal: NormedSpace.exp 𝕜 (TrivSqZeroExt.inl x) = TrivSqZeroExt.inl (NormedSpace.exp 𝕜 x)
-- TrivSqZeroExt.exp_inr
@[simp]
theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (m : M)
Goal: NormedSpace.exp 𝕜 (TrivSqZeroExt.inr m) = 1 + TrivSqZeroExt.inr m
-- TrivSqZeroExt.exp_def
theorem exp_def (x : tsze R M) : exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [IsScalarTower 𝕜 R M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (x : tsze R M)
Goal: NormedSpace.exp 𝕜 x = TrivSqZeroExt.inl (NormedSpace.exp 𝕜 x.fst) + TrivSqZeroExt.inr (NormedSpace.exp 𝕜 x.fst • x.snd)
-- TrivSqZeroExt.fst_exp
@[simp]
theorem fst_exp (x : tsze R M) : fst (exp 𝕜 x) = exp 𝕜 x.fst 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [IsScalarTower 𝕜 R M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (x : tsze R M)
Goal: (NormedSpace.exp 𝕜 x).fst = NormedSpace.exp 𝕜 x.fst
-- TrivSqZeroExt.snd_exp
@[simp]
theorem snd_exp (x : tsze R M) : snd (exp 𝕜 x) = exp 𝕜 x.fst • x.snd 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [IsScalarTower 𝕜 R M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (x : tsze R M)
Goal: (NormedSpace.exp 𝕜 x).snd = NormedSpace.exp 𝕜 x.fst • x.snd
-- TrivSqZeroExt.eq_smul_exp_of_invertible
theorem eq_smul_exp_of_invertible (x : tsze R M) [Invertible x.fst] :
    x = x.fst • exp 𝕜 (⅟ x.fst • inr x.snd) 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [IsScalarTower 𝕜 R M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (x : tsze R M) [Invertible x.fst]
Goal: x = x.fst • NormedSpace.exp 𝕜 (⅟x.fst • TrivSqZeroExt.inr x.snd)
-- TrivSqZeroExt.eq_smul_exp_of_ne_zero
theorem eq_smul_exp_of_ne_zero (x : tsze R M) (hx : x.fst ≠ 0) :
    x = x.fst • exp 𝕜 (x.fst⁻¹ • inr x.snd) 
Arguments: (𝕜 : Type u_1) {R : Type u_3} {M : Type u_4} [Field 𝕜] [CharZero 𝕜] [Field R] [AddCommGroup M] [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [IsScalarTower 𝕜 R M] [TopologicalSpace R] [TopologicalSpace M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] [T2Space R] [T2Space M] (x : tsze R M) (hx : x.fst ≠ 0)
Goal: x = x.fst • NormedSpace.exp 𝕜 (x.fst⁻¹ • TrivSqZeroExt.inr x.snd)
-- TrivSqZeroExt.norm_def
theorem norm_def (x : tsze R M) : ‖x‖ = ‖fst x‖ + ‖snd x‖ 
Arguments: {R : Type u_3} {M : Type u_4} [SeminormedRing R] [SeminormedAddCommGroup M] (x : tsze R M)
Goal: ‖x‖ = ‖x.fst‖ + ‖x.snd‖
-- TrivSqZeroExt.nnnorm_def
theorem nnnorm_def (x : tsze R M) : ‖x‖₊ = ‖fst x‖₊ + ‖snd x‖₊ 
Arguments: {R : Type u_3} {M : Type u_4} [SeminormedRing R] [SeminormedAddCommGroup M] (x : tsze R M)
Goal: ‖x‖₊ = ‖x.fst‖₊ + ‖x.snd‖₊
-- TrivSqZeroExt.norm_inl
@[simp] theorem norm_inl (r : R) : ‖(inl r : tsze R M)‖ = ‖r‖ 
Arguments: {R : Type u_3} {M : Type u_4} [SeminormedRing R] [SeminormedAddCommGroup M] (r : R)
Goal: ‖TrivSqZeroExt.inl r‖ = ‖r‖
-- TrivSqZeroExt.norm_inr
@[simp] theorem norm_inr (m : M) : ‖(inr m : tsze R M)‖ = ‖m‖ 
Arguments: {R : Type u_3} {M : Type u_4} [SeminormedRing R] [SeminormedAddCommGroup M] (m : M)
Goal: ‖TrivSqZeroExt.inr m‖ = ‖m‖
-- TrivSqZeroExt.nnnorm_inl
@[simp] theorem nnnorm_inl (r : R) : ‖(inl r : tsze R M)‖₊ = ‖r‖₊ 
Arguments: {R : Type u_3} {M : Type u_4} [SeminormedRing R] [SeminormedAddCommGroup M] (r : R)
Goal: ‖TrivSqZeroExt.inl r‖₊ = ‖r‖₊
-- TrivSqZeroExt.nnnorm_inr
@[simp] theorem nnnorm_inr (m : M) : ‖(inr m : tsze R M)‖₊ = ‖m‖₊ 
Arguments: {R : Type u_3} {M : Type u_4} [SeminormedRing R] [SeminormedAddCommGroup M] (m : M)
Goal: ‖TrivSqZeroExt.inr m‖₊ = ‖m‖₊
-- HasStrictFDerivAt.restrictScalars
@[fun_prop]
theorem HasStrictFDerivAt.restrictScalars (h : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt f (f'.restrictScalars 𝕜) x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {f' : E →L[𝕜'] F} {x : E} (h : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt f (ContinuousLinearMap.restrictScalars 𝕜 f') x
-- HasFDerivAtFilter.restrictScalars
theorem HasFDerivAtFilter.restrictScalars {L} (h : HasFDerivAtFilter f f' x L) :
    HasFDerivAtFilter f (f'.restrictScalars 𝕜) x L 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {f' : E →L[𝕜'] F} {x : E} {L : Filter E} (h : HasFDerivAtFilter f f' x L)
Goal: HasFDerivAtFilter f (ContinuousLinearMap.restrictScalars 𝕜 f') x L
-- HasFDerivAt.restrictScalars
@[fun_prop]
theorem HasFDerivAt.restrictScalars (h : HasFDerivAt f f' x) :
    HasFDerivAt f (f'.restrictScalars 𝕜) x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {f' : E →L[𝕜'] F} {x : E} (h : HasFDerivAt f f' x)
Goal: HasFDerivAt f (ContinuousLinearMap.restrictScalars 𝕜 f') x
-- HasFDerivWithinAt.restrictScalars
@[fun_prop]
theorem HasFDerivWithinAt.restrictScalars (h : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt f (f'.restrictScalars 𝕜) s x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → F} {f' : E →L[𝕜'] F} {s : Set E} {x : E} (h : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt f (ContinuousLinearMap.restrictScalars 𝕜 f') s x