awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ
rw :
  norm (E := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F)
      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) =
    norm (E := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin (n + 1)) ↦ E) F)
      (comp (β := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F)
        (⇑(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv 𝕜 (fun (x : Fin (n + 1)) ↦ E) F) :
              (E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F) ≃ₗᵢ[𝕜]
                ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (x : Fin (n + 1)) ↦ E) F) :
          (a : E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) F) →
            ContinuousMultilinearMap 𝕜 (fun (x : Fin (n + 1)) ↦ E) F)
        (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s) x)
rw₁ :
  norm (E := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F)
      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) =
    norm (E := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin (n + 1)) ↦ E) F)
      (((LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv 𝕜 (fun (x : Fin (n + 1)) ↦ E) F) :
              (E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F) ≃ₗᵢ[𝕜]
                ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (x : Fin (n + 1)) ↦ E) F) :
            (a : E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) F) →
              ContinuousMultilinearMap 𝕜 (fun (x : Fin (n + 1)) ↦ E) F)
          (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) :
        ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (x : Fin (n + 1)) ↦ E) F)
rw₂ rw₃ :
  norm (E := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F)
      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) =
    norm (E := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F)
      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x)
⊢ norm (E := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n) ↦ E) F)
      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) =
    norm (E := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin (n + 1)) ↦ E) F)
      (iteratedFDerivWithin 𝕜 (n + 1) f s x)

Tactic:
rw [iteratedFDerivWithin_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]

Premises used: iteratedFDerivWithin_succ_eq_comp_left, LinearIsometryEquiv.norm_map, Function.comp_apply
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s t : Set E
x : E
f g : E → F
y : E
hsx : UniqueDiffWithinAt 𝕜 s x
hyt : UniqueDiffWithinAt 𝕜 t y
simp :
  dist (α := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 0) ↦ E) F)
      ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
              F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
          F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
        fderivWithin 𝕜 f s x)
      ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
              F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
          F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
        fderivWithin 𝕜 g t y) =
    dist (α := E →L[𝕜] F) (fderivWithin 𝕜 f s x) (fderivWithin 𝕜 g t y)
⊢ dist (α := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 1) ↦ E) F) (iteratedFDerivWithin 𝕜 1 f s x)
      (iteratedFDerivWithin 𝕜 1 g t y) =
    dist (α := E →L[𝕜] F) (fderivWithin 𝕜 f s x) (fderivWithin 𝕜 g t y)

Tactic:
simp only [iteratedFDerivWithin_succ_eq_comp_left, comp_apply,
    LinearIsometryEquiv.dist_map, iteratedFDerivWithin_zero_eq_comp,
    LinearIsometryEquiv.comp_fderivWithin, hsx, hyt]

Premises used: LinearIsometryEquiv.comp_fderivWithin, iteratedFDerivWithin_succ_eq_comp_left, LinearIsometryEquiv.dist_map, iteratedFDerivWithin_zero_eq_comp, Function.comp_apply
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s t : Set E
x : E
f g : E → F
y : E
hsx : UniqueDiffWithinAt 𝕜 s x
hyt : UniqueDiffWithinAt 𝕜 t y
⊢ dist (α := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 0) ↦ E) F)
      ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
              F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
          F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
        fderivWithin 𝕜 f s x)
      ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
              F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
          F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
        fderivWithin 𝕜 g t y) =
    dist (α := E →L[𝕜] F) (fderivWithin 𝕜 f s x) (fderivWithin 𝕜 g t y)

Tactic:
apply (continuousMultilinearCurryFin0 𝕜 E F).symm.toLinearIsometry.postcomp.dist_map

Premises used: LinearIsometryEquiv.toLinearIsometry, LinearIsometryEquiv.symm, continuousMultilinearCurryFin0, LinearIsometry.dist_map, LinearIsometry.postcomp
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
x : E
f : E → F
h : UniqueDiffWithinAt 𝕜 s x
simp :
  norm (E := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 0) ↦ E) F)
      ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
              F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
          F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
        fderivWithin 𝕜 f s x) =
    norm (E := E →L[𝕜] F) (fderivWithin 𝕜 f s x)
⊢ norm (E := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 1) ↦ E) F) (iteratedFDerivWithin 𝕜 1 f s x) =
    norm (E := E →L[𝕜] F) (fderivWithin 𝕜 f s x)

Tactic:
simp only [← norm_fderivWithin_iteratedFDerivWithin,
    iteratedFDerivWithin_zero_eq_comp, LinearIsometryEquiv.comp_fderivWithin _ h]

Premises used: LinearIsometryEquiv.comp_fderivWithin, norm_fderivWithin_iteratedFDerivWithin, iteratedFDerivWithin_zero_eq_comp
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
x : E
f : E → F
h : UniqueDiffWithinAt 𝕜 s x
⊢ norm (E := E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 0) ↦ E) F)
      ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
              F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
          F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
        fderivWithin 𝕜 f s x) =
    norm (E := E →L[𝕜] F) (fderivWithin 𝕜 f s x)

Tactic:
apply (continuousMultilinearCurryFin0 𝕜 E F).symm.toLinearIsometry.norm_toContinuousLinearMap_comp

Premises used: LinearIsometryEquiv.toLinearIsometry, LinearIsometryEquiv.symm, continuousMultilinearCurryFin0, LinearIsometry.norm_toContinuousLinearMap_comp
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
m : Fin (HAdd.hAdd (α := ℕ) 0 1) → E
zero :
  Eq.{uF + 1} (α := F)
    (((((LinearIsometryEquiv.symm
                    (continuousMultilinearCurryLeftEquiv 𝕜 (fun (x : Fin (HAdd.hAdd (α := ℕ) 0 1)) ↦ E) F) :
                  (E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 0) ↦ E) F) ≃ₗᵢ[𝕜]
                    ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (x : Fin (HAdd.hAdd (α := ℕ) 0 1)) ↦ E) F) :
                (a : E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) →
                  ContinuousMultilinearMap 𝕜 (fun (x : Fin (0 + 1)) ↦ E) F)
              ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
                      F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
                  F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
                fderivWithin 𝕜 f s x) :
            ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (x : Fin (HAdd.hAdd (α := ℕ) 0 1)) ↦ E) F) :
          (a : (i : Fin (0 + 1)) → E) → F)
        m :
      F)
    ((fderivWithin 𝕜 f s x : (a : E) → F) (m (last 0)) : F)
⊢ Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 (HAdd.hAdd (α := ℕ) 0 1) f s x : (a : (i : Fin (0 + 1)) → E) → F) m : F)
    ((((iteratedFDerivWithin 𝕜 0 (fun (y : E) ↦ fderivWithin 𝕜 f s y) s x : (a : (i : Fin 0) → E) → E →L[𝕜] F)
              (init (α := fun (i : Fin (HAdd.hAdd (α := ℕ) 0 1)) ↦ E) m) :
            E →L[𝕜] F) :
          (a : E) → F)
        (m (last 0)) :
      F)

Tactic:
rw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp,
      iteratedFDerivWithin_zero_apply, Function.comp_apply,
      LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]

Premises used: LinearIsometryEquiv.comp_fderivWithin, iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp, Function.comp_apply, iteratedFDerivWithin_zero_apply
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
m : Fin (HAdd.hAdd (α := ℕ) 0 1) → E
⊢ Eq.{uF + 1} (α := F)
    (((((LinearIsometryEquiv.symm
                    (continuousMultilinearCurryLeftEquiv 𝕜 (fun (x : Fin (HAdd.hAdd (α := ℕ) 0 1)) ↦ E) F) :
                  (E →L[𝕜] ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin 0) ↦ E) F) ≃ₗᵢ[𝕜]
                    ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (x : Fin (HAdd.hAdd (α := ℕ) 0 1)) ↦ E) F) :
                (a : E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) →
                  ContinuousMultilinearMap 𝕜 (fun (x : Fin (0 + 1)) ↦ E) F)
              ((↑(↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) :
                      F ≃L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) :
                  F →L[𝕜] ContinuousMultilinearMap 𝕜 (fun (i : Fin 0) ↦ E) F) ∘SL
                fderivWithin 𝕜 f s x) :
            ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (x : Fin (HAdd.hAdd (α := ℕ) 0 1)) ↦ E) F) :
          (a : (i : Fin (0 + 1)) → E) → F)
        m :
      F)
    ((fderivWithin 𝕜 f s x : (a : E) → F) (m (last 0)) : F)

Tactic:
simp
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E} (hx : x ∈ s) (m : Fin (n + 1) → E),
    Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 (n + 1) f s x : (a : (i : Fin (n + 1)) → E) → F) m : F)
      ((((iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s x : (a : (i : Fin n) → E) → E →L[𝕜] F)
                (init (α := fun (i : Fin (n + 1)) ↦ E) m) :
              E →L[𝕜] F) :
            (a : E) → F)
          (m (last n)) :
        F)
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F) ≃ₗᵢ[𝕜]
  ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F :=
  LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 n E F)
y : E
hy : y ∈ s
H :
  ∀ (m : (i : Fin n.succ) → E),
    Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 n.succ f s y : (a : (i : Fin n.succ) → E) → F) m : F)
      ((comp (β := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
              ((⇑I :
                  (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
                    ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) :
                (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
                  ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
              (iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s :
                E → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
              y :
            (a : (i : Fin n.succ) → E) → F)
          m :
        F)
⊢ Eq (α := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) (iteratedFDerivWithin 𝕜 n.succ f s y)
    (comp (β := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
      ((⇑I :
          (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
            ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) :
        (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
          ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
      (iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s :
        E → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
      y)

Tactic:
ext m
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E} (hx : x ∈ s) (m : Fin (n + 1) → E),
    Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 (n + 1) f s x : (a : (i : Fin (n + 1)) → E) → F) m : F)
      ((((iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s x : (a : (i : Fin n) → E) → E →L[𝕜] F)
                (init (α := fun (i : Fin (n + 1)) ↦ E) m) :
              E →L[𝕜] F) :
            (a : E) → F)
          (m (last n)) :
        F)
x : E
hx : x ∈ s
m_1 : Fin (n + 1 + 1) → E
I : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F) ≃ₗᵢ[𝕜]
  ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F :=
  LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 n E F)
y : E
hy : y ∈ s
m : (i : Fin n.succ) → E
⊢ Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 n.succ f s y : (a : (i : Fin n.succ) → E) → F) m : F)
    ((comp (β := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
            ((⇑I :
                (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
                  ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) :
              (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
                ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
            (iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s :
              E → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
            y :
          (a : (i : Fin n.succ) → E) → F)
        m :
      F)

Tactic:
simp [IH hy m, I]
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E} (hx : x ∈ s) (m : Fin (n + 1) → E),
    Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 (n + 1) f s x : (a : (i : Fin (n + 1)) → E) → F) m : F)
      ((((iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s x : (a : (i : Fin n) → E) → E →L[𝕜] F)
                (init (α := fun (i : Fin (n + 1)) ↦ E) m) :
              E →L[𝕜] F) :
            (a : E) → F)
          (m (last n)) :
        F)
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F) ≃ₗᵢ[𝕜]
  ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F :=
  LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 n E F)
⊢ ∀ y ∈ s,
    Eq (α := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
      (iteratedFDerivWithin 𝕜 n.succ f s y)
      (comp (β := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
        ((⇑I :
            (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
              ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) :
          (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
            ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
        (iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s :
          E → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
        y)

Tactic:
have A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y =
        (I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y := fun y hy ↦ by
      ext m
      simp [IH hy m, I]
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E} (hx : x ∈ s) (m : Fin (n + 1) → E),
    Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 (n + 1) f s x : (a : (i : Fin (n + 1)) → E) → F) m : F)
      ((((iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s x : (a : (i : Fin n) → E) → E →L[𝕜] F)
                (init (α := fun (i : Fin (n + 1)) ↦ E) m) :
              E →L[𝕜] F) :
            (a : E) → F)
          (m (last n)) :
        F)
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F) ≃ₗᵢ[𝕜]
  ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F :=
  LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 n E F)
A :
  ∀ y ∈ s,
    Eq (α := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
      (iteratedFDerivWithin 𝕜 n.succ f s y)
      (comp (β := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
        ((⇑I :
            (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
              ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) :
          (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
            ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
        (iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s :
          E → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
        y)
⊢ Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 (n + 2) f s x : (a : (i : Fin (n + 2)) → E) → F) m : F)
    ((((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n.succ f s) s x :
                (a : E) → ContinuousMultilinearMap 𝕜 (fun (i : Fin n.succ) ↦ E) F)
              (m 0) :
            ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) :
          (a : (i : Fin n.succ) → E) → F)
        (tail (α := fun (i : Fin (n + 1 + 1)) ↦ E) m) :
      F)

Tactic:
simp [iteratedFDerivWithin_succ_eq_comp_left]

Premises used: iteratedFDerivWithin_succ_eq_comp_left
-- Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
E : Type uE
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type uF
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E} (hx : x ∈ s) (m : Fin (n + 1) → E),
    Eq.{uF + 1} (α := F) ((iteratedFDerivWithin 𝕜 (n + 1) f s x : (a : (i : Fin (n + 1)) → E) → F) m : F)
      ((((iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s x : (a : (i : Fin n) → E) → E →L[𝕜] F)
                (init (α := fun (i : Fin (n + 1)) ↦ E) m) :
              E →L[𝕜] F) :
            (a : E) → F)
          (m (last n)) :
        F)
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F) ≃ₗᵢ[𝕜]
  ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F :=
  LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 n E F)
A :
  ∀ y ∈ s,
    Eq (α := ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
      (iteratedFDerivWithin 𝕜 n.succ f s y)
      (comp (β := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
        ((⇑I :
            (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
              ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F) :
          (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
            ContinuousMultilinearMap.{u, 0, uE, uF} 𝕜 (fun (i : Fin n.succ) ↦ E) F)
        (iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s :
          E → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
        y)
⊢ Eq.{uF + 1} (α := F)
    ((comp (β := ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
            (⇑I :
              (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) →
                ContinuousMultilinearMap 𝕜 (fun (i : Fin n.succ) ↦ E) F)
            (⇑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x) :
              (a : E) → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
            (m 0) :
          (a : (i : Fin (n + 1)) → E) → F)
        (tail (α := fun (i : Fin (n + 1 + 1)) ↦ E) m) :
      F)
    ((((((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun (y : E) ↦ fderivWithin 𝕜 f s y) s) s x :
                      (a : E) → ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F))
                    (m 0) :
                  ContinuousMultilinearMap 𝕜 (fun (i : Fin n) ↦ E) (E →L[𝕜] F)) :
                (a : (i : Fin n) → E) → E →L[𝕜] F)
              (init (α := fun (i : Fin (n + 1)) ↦ E) (tail (α := fun (i : Fin (n + 1 + 1)) ↦ E) m)) :
            E →L[𝕜] F) :
          (a : E) → F)
        (tail (α := fun (i : Fin (n + 1 + 1)) ↦ E) m (last n)) :
      F)

Tactic:
simp [I]