analysis.calculus.cont_diff_def ⟷ Mathlib.Analysis.Calculus.ContDiff.Defs

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1803,7 +1803,7 @@ theorem contDiffAt_one_iff :
   by
   simp_rw [show (1 : ℕ∞) = (0 + 1 : ℕ) from (zero_add 1).symm, contDiffAt_succ_iff_hasFDerivAt,
     show ((0 : ℕ) : ℕ∞) = 0 from rfl, contDiffAt_zero,
-    exists_mem_and_iff antitone_bforall antitone_continuousOn, and_comm']
+    exists_mem_and_iff antitone_bforall antitone_continuousOn, and_comm]
 #align cont_diff_at_one_iff contDiffAt_one_iff
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3,10 +3,10 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Analysis.Calculus.Fderiv.Add
-import Analysis.Calculus.Fderiv.Mul
-import Analysis.Calculus.Fderiv.Equiv
-import Analysis.Calculus.Fderiv.RestrictScalars
+import Analysis.Calculus.FDeriv.Add
+import Analysis.Calculus.FDeriv.Mul
+import Analysis.Calculus.FDeriv.Equiv
+import Analysis.Calculus.FDeriv.RestrictScalars
 import Analysis.Calculus.FormalMultilinearSeries
 
 #align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
@@ -1192,10 +1192,10 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
 #align cont_diff_within_at_zero contDiffWithinAt_zero
 -/
 
-#print HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn /-
+#print HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn /-
 /-- On a set with unique differentiability, any choice of iterated differential has to coincide
 with the one we have chosen in `iterated_fderiv_within 𝕜 m f s`. -/
-theorem HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
+theorem HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
     (h : HasFTaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s)
     (hx : x ∈ s) : p x m = iteratedFDerivWithin 𝕜 m f s x :=
   by
@@ -1209,7 +1209,7 @@ theorem HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
         (IH (le_of_lt A) hx).symm
     rw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderiv_within (hs x hx)]
     exact (ContinuousMultilinearMap.uncurry_curryLeft _).symm
-#align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
+#align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
 -/
 
 #print ContDiffOn.ftaylorSeriesWithin /-
@@ -1231,7 +1231,7 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
       by
       change p x m.succ = iteratedFDerivWithin 𝕜 m.succ f s x
       rw [← iteratedFDerivWithin_inter_open o_open xo]
-      exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩
+      exact (Hp.mono ho).eq_iteratedFDerivWithin_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩
     rw [← this, ← hasFDerivWithinAt_inter (IsOpen.mem_nhds o_open xo)]
     have A : ∀ y ∈ s ∩ o, p y m = ftaylorSeriesWithin 𝕜 f s y m :=
       by
@@ -1239,7 +1239,7 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
       change p y m = iteratedFDerivWithin 𝕜 m f s y
       rw [← iteratedFDerivWithin_inter_open o_open yo]
       exact
-        (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn (WithTop.coe_le_coe.2 (Nat.le_succ m))
+        (Hp.mono ho).eq_iteratedFDerivWithin_of_uniqueDiffOn (WithTop.coe_le_coe.2 (Nat.le_succ m))
           (hs.inter o_open) ⟨hy, yo⟩
     exact
       ((Hp.mono ho).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m)) x ⟨hx, xo⟩).congr
@@ -1257,7 +1257,7 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
       rintro y ⟨hy, yo⟩
       change p y m = iteratedFDerivWithin 𝕜 m f s y
       rw [← iteratedFDerivWithin_inter_open o_open yo]
-      exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hy, yo⟩
+      exact (Hp.mono ho).eq_iteratedFDerivWithin_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hy, yo⟩
     exact ((Hp.mono ho).cont m le_rfl).congr fun y hy => (A y hy).symm
 #align cont_diff_on.ftaylor_series_within ContDiffOn.ftaylorSeriesWithin
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -235,7 +235,7 @@ theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s)
     ContinuousOn f s :=
   by
   have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
-  rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this 
+  rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this
 #align has_ftaylor_series_up_to_on.continuous_on HasFTaylorSeriesUpToOn.continuousOn
 -/
 
@@ -426,7 +426,7 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ)
             ((p x).shift m.succ).curryLeft s x :=
           Htaylor.fderiv_within _ A x hx
-        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this 
+        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this
         convert this
         ext y v
         change
@@ -443,7 +443,7 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
           ContinuousOn
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ) s :=
           Htaylor.cont _ A
-        rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this 
+        rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this
 #align has_ftaylor_series_up_to_on_succ_iff_right hasFTaylorSeriesUpToOn_succ_iff_right
 -/
 
@@ -503,7 +503,7 @@ theorem ContDiffWithinAt.continuousWithinAt (h : ContDiffWithinAt 𝕜 n f s x)
     ContinuousWithinAt f s x :=
   by
   rcases h 0 bot_le with ⟨u, hu, p, H⟩
-  rw [mem_nhdsWithin_insert] at hu 
+  rw [mem_nhdsWithin_insert] at hu
   exact (H.continuous_on.continuous_within_at hu.1).mono_of_mem hu.2
 #align cont_diff_within_at.continuous_within_at ContDiffWithinAt.continuousWithinAt
 -/
@@ -629,7 +629,7 @@ theorem ContDiffWithinAt.differentiable_within_at' (h : ContDiffWithinAt 𝕜 n
   by
   rcases h 1 hn with ⟨u, hu, p, H⟩
   rcases mem_nhdsWithin.1 hu with ⟨t, t_open, xt, tu⟩
-  rw [inter_comm] at tu 
+  rw [inter_comm] at tu
   have := ((H.mono tu).DifferentiableOn le_rfl) x ⟨mem_insert x s, xt⟩
   exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 this
 #align cont_diff_within_at.differentiable_within_at' ContDiffWithinAt.differentiable_within_at'
@@ -661,7 +661,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
     · convert self_mem_nhdsWithin
       have : x ∈ insert x s := by simp
       exact insert_eq_of_mem (mem_of_mem_nhdsWithin this hu)
-    · rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp 
+    · rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
       exact Hp.2.2.of_le hm
   · rintro ⟨u, hu, f', f'_eq_deriv, Hf'⟩
     rw [contDiffWithinAt_nat]
@@ -711,7 +711,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : ℕ} :
   refine' ⟨fun hf => _, _⟩
   · obtain ⟨u, hu, f', huf', hf'⟩ := cont_diff_within_at_succ_iff_has_fderiv_within_at.mp hf
     obtain ⟨w, hw, hxw, hwu⟩ := mem_nhds_within.mp hu
-    rw [inter_comm] at hwu 
+    rw [inter_comm] at hwu
     refine'
       ⟨insert x s ∩ w, inter_mem_nhdsWithin _ (hw.mem_nhds hxw), inter_subset_left _ _, f',
         fun y hy => _, _⟩
@@ -769,7 +769,7 @@ theorem ContDiffWithinAt.contDiffOn' {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : Co
   by
   rcases h m hm with ⟨t, ht, p, hp⟩
   rcases mem_nhdsWithin.1 ht with ⟨u, huo, hxu, hut⟩
-  rw [inter_comm] at hut 
+  rw [inter_comm] at hut
   exact ⟨u, huo, hxu, (hp.mono hut).ContDiffOn⟩
 #align cont_diff_within_at.cont_diff_on' ContDiffWithinAt.contDiffOn'
 -/
@@ -897,7 +897,7 @@ theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
     refine'
       ⟨u, hu, fun y => (continuousMultilinearCurryFin1 𝕜 E F) (p y 1), fun y hy =>
         Hp.has_fderiv_within_at (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _⟩
-    rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp 
+    rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
     intro z hz m hm
     refine' ⟨u, _, fun x : E => (p x).shift, Hp.2.2.of_le hm⟩
     convert self_mem_nhdsWithin
@@ -1183,7 +1183,7 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
     obtain ⟨u, H, p, hp⟩ := h 0 (by norm_num)
     refine' ⟨u, _, _⟩
     · simpa [hx] using H
-    · simp only [WithTop.coe_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp 
+    · simp only [WithTop.coe_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
       exact hp.1.mono (inter_subset_right s u)
   · rintro ⟨u, H, hu⟩
     rw [← contDiffWithinAt_inter' H]
@@ -1224,9 +1224,9 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
       iteratedFDerivWithin_zero_apply]
   · intro m hm x hx
     rcases(h x hx) m.succ (ENat.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩
-    rw [insert_eq_of_mem hx] at hu 
+    rw [insert_eq_of_mem hx] at hu
     rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
-    rw [inter_comm] at ho 
+    rw [inter_comm] at ho
     have : p x m.succ = ftaylorSeriesWithin 𝕜 f s x m.succ :=
       by
       change p x m.succ = iteratedFDerivWithin 𝕜 m.succ f s x
@@ -1249,8 +1249,8 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
     intro x hx
     rcases h x hx m hm with ⟨u, hu, p, Hp⟩
     rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
-    rw [insert_eq_of_mem hx] at ho 
-    rw [inter_comm] at ho 
+    rw [insert_eq_of_mem hx] at ho
+    rw [inter_comm] at ho
     refine' ⟨o, o_open, xo, _⟩
     have A : ∀ y ∈ s ∩ o, p y m = ftaylorSeriesWithin 𝕜 f s y m :=
       by
@@ -1327,7 +1327,7 @@ theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
   have C : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x :=
     hu.differentiable_on_iterated_fderiv_within (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x
       ⟨mem_insert _ _, xu⟩
-  rw [differentiableWithinAt_congr_set' _ A] at C 
+  rw [differentiableWithinAt_congr_set' _ A] at C
   exact C.congr_of_eventually_eq (B.filter_mono inf_le_left) B.self_of_nhds
 #align cont_diff_within_at.differentiable_within_at_iterated_fderiv_within ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin
 -/
@@ -1366,16 +1366,16 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
   refine' ⟨H.differentiable_on (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)), fun x hx => _⟩
   rcases contDiffWithinAt_succ_iff_hasFDerivWithinAt.1 (H x hx) with ⟨u, hu, f', hff', hf'⟩
   rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
-  rw [inter_comm, insert_eq_of_mem hx] at ho 
+  rw [inter_comm, insert_eq_of_mem hx] at ho
   have := hf'.mono ho
-  rw [contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds o_open xo))] at this 
+  rw [contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds o_open xo))] at this
   apply this.congr_of_eventually_eq' _ hx
   have : o ∩ s ∈ 𝓝[s] x := mem_nhdsWithin.2 ⟨o, o_open, xo, subset.refl _⟩
-  rw [inter_comm] at this 
+  rw [inter_comm] at this
   apply Filter.eventuallyEq_of_mem this fun y hy => _
   have A : fderivWithin 𝕜 f (s ∩ o) y = f' y :=
     ((hff' y (ho hy)).mono ho).fderivWithin (hs.inter o_open y hy)
-  rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A 
+  rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A
 #align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin
 -/
 
@@ -1450,11 +1450,11 @@ theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn
     (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s :=
   by
   cases m
-  · change ∞ + 1 ≤ n at hmn 
+  · change ∞ + 1 ≤ n at hmn
     have : n = ∞ := by simpa using hmn
-    rw [this] at hf 
+    rw [this] at hf
     exact ((contDiffOn_top_iff_fderivWithin hs).1 hf).2
-  · change (m.succ : ℕ∞) ≤ n at hmn 
+  · change (m.succ : ℕ∞) ≤ n at hmn
     exact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2
 #align cont_diff_on.fderiv_within ContDiffOn.fderivWithin
 -/
@@ -1545,7 +1545,7 @@ theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤
 #print HasFTaylorSeriesUpTo.continuous /-
 theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f :=
   by
-  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h 
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
   rw [continuous_iff_continuousOn_univ]
   exact h.continuous_on
 #align has_ftaylor_series_up_to.continuous HasFTaylorSeriesUpTo.continuous

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1062,7 +1062,7 @@ theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (
     iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t :=
   by
   induction' n with n ihn
-  · exact h.mono fun y hy => FunLike.ext _ _ fun _ => hy
+  · exact h.mono fun y hy => DFunLike.ext _ _ fun _ => hy
   · have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderiv_within' ht
     apply this.mono
     intro y hy

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1057,8 +1057,8 @@ theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fi
 #align iterated_fderiv_within_one_apply iteratedFDerivWithin_one_apply
 -/
 
-#print Filter.EventuallyEq.iterated_fderiv_within' /-
-theorem Filter.EventuallyEq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
+#print Filter.EventuallyEq.iteratedFDerivWithin' /-
+theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
     iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t :=
   by
   induction' n with n ihn
@@ -1067,13 +1067,13 @@ theorem Filter.EventuallyEq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f)
     apply this.mono
     intro y hy
     simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (· ∘ ·)]
-#align filter.eventually_eq.iterated_fderiv_within' Filter.EventuallyEq.iterated_fderiv_within'
+#align filter.eventually_eq.iterated_fderiv_within' Filter.EventuallyEq.iteratedFDerivWithin'
 -/
 
 #print Filter.EventuallyEq.iteratedFDerivWithin /-
 protected theorem Filter.EventuallyEq.iteratedFDerivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
     iteratedFDerivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s :=
-  h.iterated_fderiv_within' Subset.rfl n
+  h.iteratedFDerivWithin' Subset.rfl n
 #align filter.eventually_eq.iterated_fderiv_within Filter.EventuallyEq.iteratedFDerivWithin
 -/
 
@@ -1083,7 +1083,7 @@ iterated differentials within this set at `x` coincide. -/
 theorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
     (n : ℕ) : iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
   have : f₁ =ᶠ[𝓝[insert x s] x] f := by simpa [eventually_eq, hx]
-  (this.iterated_fderiv_within' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
+  (this.iteratedFDerivWithin' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
 #align filter.eventually_eq.iterated_fderiv_within_eq Filter.EventuallyEq.iteratedFDerivWithin_eq
 -/
 
@@ -1379,8 +1379,8 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
 #align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin
 -/
 
-#print contDiffOn_succ_iff_has_fderiv_within /-
-theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
+#print contDiffOn_succ_iff_hasFDerivWithin /-
+theorem contDiffOn_succ_iff_hasFDerivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       ∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFDerivWithinAt f (f' x) s x :=
   by
@@ -1389,7 +1389,7 @@ theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜
   rcases h with ⟨f', h1, h2⟩
   refine' ⟨fun x hx => (h2 x hx).DifferentiableWithinAt, fun x hx => _⟩
   exact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) hx
-#align cont_diff_on_succ_iff_has_fderiv_within contDiffOn_succ_iff_has_fderiv_within
+#align cont_diff_on_succ_iff_has_fderiv_within contDiffOn_succ_iff_hasFDerivWithin
 -/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]] -/
@@ -1844,15 +1844,15 @@ theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤
 #align cont_diff_iff_forall_nat_le contDiff_iff_forall_nat_le
 -/
 
-#print contDiff_succ_iff_has_fderiv /-
+#print contDiff_succ_iff_hasFDerivAt /-
 /-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/
-theorem contDiff_succ_iff_has_fderiv {n : ℕ} :
+theorem contDiff_succ_iff_hasFDerivAt {n : ℕ} :
     ContDiff 𝕜 (n + 1 : ℕ) f ↔
       ∃ f' : E → E →L[𝕜] F, ContDiff 𝕜 n f' ∧ ∀ x, HasFDerivAt f (f' x) x :=
   by
   simp only [← contDiffOn_univ, ← hasFDerivWithinAt_univ,
-    contDiffOn_succ_iff_has_fderiv_within uniqueDiffOn_univ, Set.mem_univ, forall_true_left]
-#align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_has_fderiv
+    contDiffOn_succ_iff_hasFDerivWithin uniqueDiffOn_univ, Set.mem_univ, forall_true_left]
+#align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_hasFDerivAt
 -/
 
 /-! ### Iterated derivative -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1393,10 +1393,10 @@ theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜
 -/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]] -/
-#print contDiffOn_succ_iff_fderiv_of_open /-
+#print contDiffOn_succ_iff_fderiv_of_isOpen /-
 /-- A function is `C^(n + 1)` on an open domain if and only if it is
 differentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -/
-theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
+theorem contDiffOn_succ_iff_fderiv_of_isOpen {n : ℕ} (hs : IsOpen s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 n (fun y => fderiv 𝕜 f y) s :=
   by
@@ -1406,7 +1406,7 @@ theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
   apply contDiffOn_congr
   intro x hx
   exact fderivWithin_of_isOpen hs hx
-#align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_open
+#align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_isOpen
 -/
 
 #print contDiffOn_top_iff_fderivWithin /-
@@ -1430,10 +1430,10 @@ theorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn 𝕜 s) :
 -/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]] -/
-#print contDiffOn_top_iff_fderiv_of_open /-
+#print contDiffOn_top_iff_fderiv_of_isOpen /-
 /-- A function is `C^∞` on an open domain if and only if it is differentiable there, and its
 derivative (expressed with `fderiv`) is `C^∞`. -/
-theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
+theorem contDiffOn_top_iff_fderiv_of_isOpen (hs : IsOpen s) :
     ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fun y => fderiv 𝕜 f y) s :=
   by
   rw [contDiffOn_top_iff_fderivWithin hs.unique_diff_on]
@@ -1442,7 +1442,7 @@ theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
   apply contDiffOn_congr
   intro x hx
   exact fderivWithin_of_isOpen hs hx
-#align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_open
+#align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_isOpen
 -/
 
 #print ContDiffOn.fderivWithin /-
@@ -1459,11 +1459,11 @@ theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn
 #align cont_diff_on.fderiv_within ContDiffOn.fderivWithin
 -/
 
-#print ContDiffOn.fderiv_of_open /-
-theorem ContDiffOn.fderiv_of_open (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
+#print ContDiffOn.fderiv_of_isOpen /-
+theorem ContDiffOn.fderiv_of_isOpen (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
     ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s :=
   (hf.fderivWithin hs.UniqueDiffOn hmn).congr fun x hx => (fderivWithin_of_isOpen hs hx).symm
-#align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_open
+#align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_isOpen
 -/
 
 #print ContDiffOn.continuousOn_fderivWithin /-
@@ -1473,11 +1473,11 @@ theorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn 𝕜 n f s) (hs : U
 #align cont_diff_on.continuous_on_fderiv_within ContDiffOn.continuousOn_fderivWithin
 -/
 
-#print ContDiffOn.continuousOn_fderiv_of_open /-
-theorem ContDiffOn.continuousOn_fderiv_of_open (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
+#print ContDiffOn.continuousOn_fderiv_of_isOpen /-
+theorem ContDiffOn.continuousOn_fderiv_of_isOpen (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderiv 𝕜 f x) s :=
-  ((contDiffOn_succ_iff_fderiv_of_open hs).1 (h.of_le hn)).2.ContinuousOn
-#align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_open
+  ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 (h.of_le hn)).2.ContinuousOn
+#align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_isOpen
 -/
 
 /-! ### Functions with a Taylor series on the whole space -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1405,7 +1405,7 @@ theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
     "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]]"
   apply contDiffOn_congr
   intro x hx
-  exact fderivWithin_of_open hs hx
+  exact fderivWithin_of_isOpen hs hx
 #align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_open
 -/
 
@@ -1441,7 +1441,7 @@ theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
     "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]]"
   apply contDiffOn_congr
   intro x hx
-  exact fderivWithin_of_open hs hx
+  exact fderivWithin_of_isOpen hs hx
 #align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_open
 -/
 
@@ -1462,7 +1462,7 @@ theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn
 #print ContDiffOn.fderiv_of_open /-
 theorem ContDiffOn.fderiv_of_open (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
     ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s :=
-  (hf.fderivWithin hs.UniqueDiffOn hmn).congr fun x hx => (fderivWithin_of_open hs hx).symm
+  (hf.fderivWithin hs.UniqueDiffOn hmn).congr fun x hx => (fderivWithin_of_isOpen hs hx).symm
 #align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_open
 -/
 
@@ -1984,7 +1984,7 @@ theorem iteratedFDerivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :
     rw [iteratedFDeriv_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]
     dsimp
     congr 1
-    rw [fderivWithin_of_open hs hx]
+    rw [fderivWithin_of_isOpen hs hx]
     apply Filter.EventuallyEq.fderiv_eq
     filter_upwards [hs.mem_nhds hx]
     exact IH

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,11 +3,11 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Analysis.Calculus.Fderiv.Add
-import Mathbin.Analysis.Calculus.Fderiv.Mul
-import Mathbin.Analysis.Calculus.Fderiv.Equiv
-import Mathbin.Analysis.Calculus.Fderiv.RestrictScalars
-import Mathbin.Analysis.Calculus.FormalMultilinearSeries
+import Analysis.Calculus.Fderiv.Add
+import Analysis.Calculus.Fderiv.Mul
+import Analysis.Calculus.Fderiv.Equiv
+import Analysis.Calculus.Fderiv.RestrictScalars
+import Analysis.Calculus.FormalMultilinearSeries
 
 #align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -2123,7 +2123,7 @@ theorem ContDiff.continuous_fderiv (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
 continuous. -/
 theorem ContDiff.continuous_fderiv_apply (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
     Continuous fun p : E × E => (fderiv 𝕜 f p.1 : E → F) p.2 :=
-  have A : Continuous fun q : (E →L[𝕜] F) × E => q.1 q.2 := isBoundedBilinearMapApply.Continuous
+  have A : Continuous fun q : (E →L[𝕜] F) × E => q.1 q.2 := isBoundedBilinearMap_apply.Continuous
   have B : Continuous fun p : E × E => (fderiv 𝕜 f p.1, p.2) :=
     ((h.continuous_fderiv hn).comp continuous_fst).prod_mk continuous_snd
   A.comp B

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -610,7 +610,7 @@ theorem contDiffWithinAt_insert {y : E} :
 #align cont_diff_within_at_insert contDiffWithinAt_insert
 -/
 
-alias contDiffWithinAt_insert ↔ ContDiffWithinAt.of_insert ContDiffWithinAt.insert'
+alias ⟨ContDiffWithinAt.of_insert, ContDiffWithinAt.insert'⟩ := contDiffWithinAt_insert
 #align cont_diff_within_at.of_insert ContDiffWithinAt.of_insert
 #align cont_diff_within_at.insert' ContDiffWithinAt.insert'
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Calculus.Fderiv.Add
 import Mathbin.Analysis.Calculus.Fderiv.Mul
@@ -14,6 +9,8 @@ import Mathbin.Analysis.Calculus.Fderiv.Equiv
 import Mathbin.Analysis.Calculus.Fderiv.RestrictScalars
 import Mathbin.Analysis.Calculus.FormalMultilinearSeries
 
+#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
+
 /-!
 # Higher differentiability
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -305,7 +305,7 @@ theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f
   rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
   have : ((0 : ℕ) : ℕ∞) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 Nat.zero_lt_one) hn
   convert h.fderiv_within _ this x hx
-  ext (y v)
+  ext y v
   change (p x 1) (snoc 0 y) = (p x 1) (cons y v)
   unfold_coes
   congr with i
@@ -404,7 +404,7 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
           (p x m.succ.succ).curryRight.curryLeft s x
       rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
       convert H.fderiv_within _ A x hx
-      ext (y v)
+      ext y v
       change
         (p x m.succ.succ) (snoc (cons y (init v)) (v (last _))) =
           (p x (Nat.succ (Nat.succ m))) (cons y v)
@@ -431,7 +431,7 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
           Htaylor.fderiv_within _ A x hx
         rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this 
         convert this
-        ext (y v)
+        ext y v
         change
           (p x (Nat.succ (Nat.succ m))) (cons y v) =
             (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))
@@ -689,10 +689,10 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
         congr
         decide
       · convert (Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1
-        · ext (x y)
+        · ext x y
           change p' x 0 (init (@snoc 0 (fun i : Fin 1 => E) 0 y)) y = p' x 0 0 y
           rw [init_snoc]
-        · ext (x k v y)
+        · ext x k v y
           change
             p' x k (init (@snoc k (fun i : Fin k.succ => E) v y))
                 (@snoc k (fun i : Fin k.succ => E) v y (last k)) =
@@ -1968,7 +1968,7 @@ theorem iteratedFDerivWithin_univ {n : ℕ} :
   by
   induction' n with n IH
   · ext x; simp
-  · ext (x m)
+  · ext x m
     rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]
 #align iterated_fderiv_within_univ iteratedFDerivWithin_univ
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -168,7 +168,6 @@ noncomputable section
 
 open scoped Classical BigOperators NNReal Topology Filter
 
--- mathport name: «expr∞»
 local notation "∞" => (⊤ : ℕ∞)
 
 attribute [local instance 1001] NormedAddCommGroup.toAddCommGroup NormedSpace.toModule'
@@ -199,10 +198,12 @@ structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMul
 #align has_ftaylor_series_up_to_on HasFTaylorSeriesUpToOn
 -/
 
+#print HasFTaylorSeriesUpToOn.zero_eq' /-
 theorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s) :
     p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by rw [← h.zero_eq x hx]; symm;
   exact ContinuousMultilinearMap.uncurry0_curry0 _
 #align has_ftaylor_series_up_to_on.zero_eq' HasFTaylorSeriesUpToOn.zero_eq'
+-/
 
 #print HasFTaylorSeriesUpToOn.congr /-
 /-- If two functions coincide on a set `s`, then a Taylor series for the first one is as well a
@@ -224,11 +225,13 @@ theorem HasFTaylorSeriesUpToOn.mono (h : HasFTaylorSeriesUpToOn n f p s) {t : Se
 #align has_ftaylor_series_up_to_on.mono HasFTaylorSeriesUpToOn.mono
 -/
 
+#print HasFTaylorSeriesUpToOn.of_le /-
 theorem HasFTaylorSeriesUpToOn.of_le (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m ≤ n) :
     HasFTaylorSeriesUpToOn m f p s :=
   ⟨h.zero_eq, fun k hk x hx => h.fderivWithin k (lt_of_lt_of_le hk hmn) x hx, fun k hk =>
     h.cont k (le_trans hk hmn)⟩
 #align has_ftaylor_series_up_to_on.of_le HasFTaylorSeriesUpToOn.of_le
+-/
 
 #print HasFTaylorSeriesUpToOn.continuousOn /-
 theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
@@ -287,6 +290,7 @@ theorem hasFTaylorSeriesUpToOn_top_iff' :
 #align has_ftaylor_series_up_to_on_top_iff' hasFTaylorSeriesUpToOn_top_iff'
 -/
 
+#print HasFTaylorSeriesUpToOn.hasFDerivWithinAt /-
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
 theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
@@ -308,18 +312,24 @@ theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f
   rw [Unique.eq_default i]
   rfl
 #align has_ftaylor_series_up_to_on.has_fderiv_within_at HasFTaylorSeriesUpToOn.hasFDerivWithinAt
+-/
 
+#print HasFTaylorSeriesUpToOn.differentiableOn /-
 theorem HasFTaylorSeriesUpToOn.differentiableOn (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) :
     DifferentiableOn 𝕜 f s := fun x hx => (h.HasFDerivWithinAt hn hx).DifferentiableWithinAt
 #align has_ftaylor_series_up_to_on.differentiable_on HasFTaylorSeriesUpToOn.differentiableOn
+-/
 
+#print HasFTaylorSeriesUpToOn.hasFDerivAt /-
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term
 of order `1` of this series is a derivative of `f` at `x`. -/
 theorem HasFTaylorSeriesUpToOn.hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
     (hx : s ∈ 𝓝 x) : HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
   (h.HasFDerivWithinAt hn (mem_of_mem_nhds hx)).HasFDerivAt hx
 #align has_ftaylor_series_up_to_on.has_fderiv_at HasFTaylorSeriesUpToOn.hasFDerivAt
+-/
 
+#print HasFTaylorSeriesUpToOn.eventually_hasFDerivAt /-
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
 in a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. -/
 theorem HasFTaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s)
@@ -327,14 +337,18 @@ theorem HasFTaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFTaylorSeriesUpToO
     ∀ᶠ y in 𝓝 x, HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p y 1)) y :=
   (eventually_eventually_nhds.2 hx).mono fun y hy => h.HasFDerivAt hn hy
 #align has_ftaylor_series_up_to_on.eventually_has_fderiv_at HasFTaylorSeriesUpToOn.eventually_hasFDerivAt
+-/
 
+#print HasFTaylorSeriesUpToOn.differentiableAt /-
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
 it is differentiable at `x`. -/
 theorem HasFTaylorSeriesUpToOn.differentiableAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
     (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x :=
   (h.HasFDerivAt hn hx).DifferentiableAt
 #align has_ftaylor_series_up_to_on.differentiable_at HasFTaylorSeriesUpToOn.differentiableAt
+-/
 
+#print hasFTaylorSeriesUpToOn_succ_iff_left /-
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and
 `p (n + 1)` is a derivative of `p n`. -/
 theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
@@ -364,7 +378,9 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
         rw [this]
         exact h.2.2
 #align has_ftaylor_series_up_to_on_succ_iff_left hasFTaylorSeriesUpToOn_succ_iff_left
+-/
 
+#print hasFTaylorSeriesUpToOn_succ_iff_right /-
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
 for `p 1`, which is a derivative of `f`. -/
 theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
@@ -432,6 +448,7 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
           Htaylor.cont _ A
         rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this 
 #align has_ftaylor_series_up_to_on_succ_iff_right hasFTaylorSeriesUpToOn_succ_iff_right
+-/
 
 /-! ### Smooth functions within a set around a point -/
 
@@ -456,21 +473,27 @@ def ContDiffWithinAt (n : ℕ∞) (f : E → F) (s : Set E) (x : E) : Prop :=
 
 variable {𝕜}
 
+#print contDiffWithinAt_nat /-
 theorem contDiffWithinAt_nat {n : ℕ} :
     ContDiffWithinAt 𝕜 n f s x ↔
       ∃ u ∈ 𝓝[insert x s] x,
         ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFTaylorSeriesUpToOn n f p u :=
   ⟨fun H => H n le_rfl, fun ⟨u, hu, p, hp⟩ m hm => ⟨u, hu, p, hp.of_le hm⟩⟩
 #align cont_diff_within_at_nat contDiffWithinAt_nat
+-/
 
+#print ContDiffWithinAt.of_le /-
 theorem ContDiffWithinAt.of_le (h : ContDiffWithinAt 𝕜 n f s x) (hmn : m ≤ n) :
     ContDiffWithinAt 𝕜 m f s x := fun k hk => h k (le_trans hk hmn)
 #align cont_diff_within_at.of_le ContDiffWithinAt.of_le
+-/
 
+#print contDiffWithinAt_iff_forall_nat_le /-
 theorem contDiffWithinAt_iff_forall_nat_le :
     ContDiffWithinAt 𝕜 n f s x ↔ ∀ m : ℕ, ↑m ≤ n → ContDiffWithinAt 𝕜 m f s x :=
   ⟨fun H m hm => H.of_le hm, fun H m hm => H m hm _ le_rfl⟩
 #align cont_diff_within_at_iff_forall_nat_le contDiffWithinAt_iff_forall_nat_le
+-/
 
 #print contDiffWithinAt_top /-
 theorem contDiffWithinAt_top : ContDiffWithinAt 𝕜 ∞ f s x ↔ ∀ n : ℕ, ContDiffWithinAt 𝕜 n f s x :=
@@ -565,15 +588,19 @@ theorem contDiffWithinAt_congr_nhds {t : Set E} (hst : 𝓝[s] x = 𝓝[t] x) :
 #align cont_diff_within_at_congr_nhds contDiffWithinAt_congr_nhds
 -/
 
+#print contDiffWithinAt_inter' /-
 theorem contDiffWithinAt_inter' (h : t ∈ 𝓝[s] x) :
     ContDiffWithinAt 𝕜 n f (s ∩ t) x ↔ ContDiffWithinAt 𝕜 n f s x :=
   contDiffWithinAt_congr_nhds <| Eq.symm <| nhdsWithin_restrict'' _ h
 #align cont_diff_within_at_inter' contDiffWithinAt_inter'
+-/
 
+#print contDiffWithinAt_inter /-
 theorem contDiffWithinAt_inter (h : t ∈ 𝓝 x) :
     ContDiffWithinAt 𝕜 n f (s ∩ t) x ↔ ContDiffWithinAt 𝕜 n f s x :=
   contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds h)
 #align cont_diff_within_at_inter contDiffWithinAt_inter
+-/
 
 #print contDiffWithinAt_insert /-
 theorem contDiffWithinAt_insert {y : E} :
@@ -597,6 +624,7 @@ theorem ContDiffWithinAt.insert (h : ContDiffWithinAt 𝕜 n f s x) :
 #align cont_diff_within_at.insert ContDiffWithinAt.insert
 -/
 
+#print ContDiffWithinAt.differentiable_within_at' /-
 /-- If a function is `C^n` within a set at a point, with `n ≥ 1`, then it is differentiable
 within this set at this point. -/
 theorem ContDiffWithinAt.differentiable_within_at' (h : ContDiffWithinAt 𝕜 n f s x) (hn : 1 ≤ n) :
@@ -608,12 +636,16 @@ theorem ContDiffWithinAt.differentiable_within_at' (h : ContDiffWithinAt 𝕜 n
   have := ((H.mono tu).DifferentiableOn le_rfl) x ⟨mem_insert x s, xt⟩
   exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 this
 #align cont_diff_within_at.differentiable_within_at' ContDiffWithinAt.differentiable_within_at'
+-/
 
+#print ContDiffWithinAt.differentiableWithinAt /-
 theorem ContDiffWithinAt.differentiableWithinAt (h : ContDiffWithinAt 𝕜 n f s x) (hn : 1 ≤ n) :
     DifferentiableWithinAt 𝕜 f s x :=
   (h.differentiable_within_at' hn).mono (subset_insert x s)
 #align cont_diff_within_at.differentiable_within_at ContDiffWithinAt.differentiableWithinAt
+-/
 
+#print contDiffWithinAt_succ_iff_hasFDerivWithinAt /-
 /-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/
 theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
     ContDiffWithinAt 𝕜 (n + 1 : ℕ) f s x ↔
@@ -667,7 +699,9 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
               p' x k v y
           rw [snoc_last, init_snoc]
 #align cont_diff_within_at_succ_iff_has_fderiv_within_at contDiffWithinAt_succ_iff_hasFDerivWithinAt
+-/
 
+#print contDiffWithinAt_succ_iff_hasFDerivWithinAt' /-
 /-- A version of `cont_diff_within_at_succ_iff_has_fderiv_within_at` where all derivatives
   are taken within the same set. -/
 theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : ℕ} :
@@ -693,6 +727,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : ℕ} :
     rintro ⟨u, hu, hus, f', huf', hf'⟩
     refine' ⟨u, hu, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus⟩
 #align cont_diff_within_at_succ_iff_has_fderiv_within_at' contDiffWithinAt_succ_iff_hasFDerivWithinAt'
+-/
 
 /-! ### Smooth functions within a set -/
 
@@ -731,6 +766,7 @@ theorem ContDiffOn.contDiffWithinAt (h : ContDiffOn 𝕜 n f s) (hx : x ∈ s) :
 #align cont_diff_on.cont_diff_within_at ContDiffOn.contDiffWithinAt
 -/
 
+#print ContDiffWithinAt.contDiffOn' /-
 theorem ContDiffWithinAt.contDiffOn' {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : ContDiffWithinAt 𝕜 n f s x) :
     ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u) :=
   by
@@ -739,12 +775,15 @@ theorem ContDiffWithinAt.contDiffOn' {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : Co
   rw [inter_comm] at hut 
   exact ⟨u, huo, hxu, (hp.mono hut).ContDiffOn⟩
 #align cont_diff_within_at.cont_diff_on' ContDiffWithinAt.contDiffOn'
+-/
 
+#print ContDiffWithinAt.contDiffOn /-
 theorem ContDiffWithinAt.contDiffOn {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : ContDiffWithinAt 𝕜 n f s x) :
     ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ContDiffOn 𝕜 m f u :=
   let ⟨u, uo, xu, h⟩ := h.contDiffOn' hm
   ⟨_, inter_mem_nhdsWithin _ (uo.mem_nhds xu), inter_subset_left _ _, h⟩
 #align cont_diff_within_at.cont_diff_on ContDiffWithinAt.contDiffOn
+-/
 
 #print ContDiffWithinAt.eventually /-
 protected theorem ContDiffWithinAt.eventually {n : ℕ} (h : ContDiffWithinAt 𝕜 n f s x) :
@@ -758,21 +797,29 @@ protected theorem ContDiffWithinAt.eventually {n : ℕ} (h : ContDiffWithinAt 
 #align cont_diff_within_at.eventually ContDiffWithinAt.eventually
 -/
 
+#print ContDiffOn.of_le /-
 theorem ContDiffOn.of_le (h : ContDiffOn 𝕜 n f s) (hmn : m ≤ n) : ContDiffOn 𝕜 m f s := fun x hx =>
   (h x hx).of_le hmn
 #align cont_diff_on.of_le ContDiffOn.of_le
+-/
 
+#print ContDiffOn.of_succ /-
 theorem ContDiffOn.of_succ {n : ℕ} (h : ContDiffOn 𝕜 (n + 1) f s) : ContDiffOn 𝕜 n f s :=
   h.of_le <| WithTop.coe_le_coe.mpr le_self_add
 #align cont_diff_on.of_succ ContDiffOn.of_succ
+-/
 
+#print ContDiffOn.one_of_succ /-
 theorem ContDiffOn.one_of_succ {n : ℕ} (h : ContDiffOn 𝕜 (n + 1) f s) : ContDiffOn 𝕜 1 f s :=
   h.of_le <| WithTop.coe_le_coe.mpr le_add_self
 #align cont_diff_on.one_of_succ ContDiffOn.one_of_succ
+-/
 
+#print contDiffOn_iff_forall_nat_le /-
 theorem contDiffOn_iff_forall_nat_le : ContDiffOn 𝕜 n f s ↔ ∀ m : ℕ, ↑m ≤ n → ContDiffOn 𝕜 m f s :=
   ⟨fun H m hm => H.of_le hm, fun H x hx m hm => H m hm x hx m le_rfl⟩
 #align cont_diff_on_iff_forall_nat_le contDiffOn_iff_forall_nat_le
+-/
 
 #print contDiffOn_top /-
 theorem contDiffOn_top : ContDiffOn 𝕜 ∞ f s ↔ ∀ n : ℕ, ContDiffOn 𝕜 n f s :=
@@ -820,11 +867,14 @@ theorem ContDiffOn.congr_mono (hf : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s
 #align cont_diff_on.congr_mono ContDiffOn.congr_mono
 -/
 
+#print ContDiffOn.differentiableOn /-
 /-- If a function is `C^n` on a set with `n ≥ 1`, then it is differentiable there. -/
 theorem ContDiffOn.differentiableOn (h : ContDiffOn 𝕜 n f s) (hn : 1 ≤ n) :
     DifferentiableOn 𝕜 f s := fun x hx => (h x hx).DifferentiableWithinAt hn
 #align cont_diff_on.differentiable_on ContDiffOn.differentiableOn
+-/
 
+#print contDiffOn_of_locally_contDiffOn /-
 /-- If a function is `C^n` around each point in a set, then it is `C^n` on the set. -/
 theorem contDiffOn_of_locally_contDiffOn
     (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)) : ContDiffOn 𝕜 n f s :=
@@ -834,7 +884,9 @@ theorem contDiffOn_of_locally_contDiffOn
   apply (contDiffWithinAt_inter _).1 (hu x ⟨xs, xu⟩)
   exact IsOpen.mem_nhds u_open xu
 #align cont_diff_on_of_locally_cont_diff_on contDiffOn_of_locally_contDiffOn
+-/
 
+#print contDiffOn_succ_iff_hasFDerivWithinAt /-
 /-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/
 theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
@@ -859,6 +911,7 @@ theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
     have : x ∈ u := mem_of_mem_nhdsWithin (mem_insert _ _) u_nhbd
     exact ⟨u, u_nhbd, f', hu, hf' x this⟩
 #align cont_diff_on_succ_iff_has_fderiv_within_at contDiffOn_succ_iff_hasFDerivWithinAt
+-/
 
 /-! ### Iterated derivative within a set -/
 
@@ -893,10 +946,12 @@ theorem iteratedFDerivWithin_zero_apply (m : Fin 0 → E) :
 #align iterated_fderiv_within_zero_apply iteratedFDerivWithin_zero_apply
 -/
 
+#print iteratedFDerivWithin_zero_eq_comp /-
 theorem iteratedFDerivWithin_zero_eq_comp :
     iteratedFDerivWithin 𝕜 0 f s = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
   rfl
 #align iterated_fderiv_within_zero_eq_comp iteratedFDerivWithin_zero_eq_comp
+-/
 
 #print norm_iteratedFDerivWithin_zero /-
 @[simp]
@@ -905,12 +960,15 @@ theorem norm_iteratedFDerivWithin_zero : ‖iteratedFDerivWithin 𝕜 0 f s x‖
 #align norm_iterated_fderiv_within_zero norm_iteratedFDerivWithin_zero
 -/
 
+#print iteratedFDerivWithin_succ_apply_left /-
 theorem iteratedFDerivWithin_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
     (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
       (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
   rfl
 #align iterated_fderiv_within_succ_apply_left iteratedFDerivWithin_succ_apply_left
+-/
 
+#print iteratedFDerivWithin_succ_eq_comp_left /-
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
 and the derivative of the `n`-th derivative. -/
 theorem iteratedFDerivWithin_succ_eq_comp_left {n : ℕ} :
@@ -919,6 +977,7 @@ theorem iteratedFDerivWithin_succ_eq_comp_left {n : ℕ} :
         fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s :=
   rfl
 #align iterated_fderiv_within_succ_eq_comp_left iteratedFDerivWithin_succ_eq_comp_left
+-/
 
 #print norm_fderivWithin_iteratedFDerivWithin /-
 theorem norm_fderivWithin_iteratedFDerivWithin {n : ℕ} :
@@ -927,6 +986,7 @@ theorem norm_fderivWithin_iteratedFDerivWithin {n : ℕ} :
 #align norm_fderiv_within_iterated_fderiv_within norm_fderivWithin_iteratedFDerivWithin
 -/
 
+#print iteratedFDerivWithin_succ_apply_right /-
 theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
     (m : Fin (n + 1) → E) :
     (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
@@ -968,6 +1028,7 @@ theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜
             (m (last (n + 1))) :=
         by rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]; rfl
 #align iterated_fderiv_within_succ_apply_right iteratedFDerivWithin_succ_apply_right
+-/
 
 #print iteratedFDerivWithin_succ_eq_comp_right /-
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
@@ -988,6 +1049,7 @@ theorem norm_iteratedFDerivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜
 #align norm_iterated_fderiv_within_fderiv_within norm_iteratedFDerivWithin_fderivWithin
 -/
 
+#print iteratedFDerivWithin_one_apply /-
 @[simp]
 theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E) :
     (iteratedFDerivWithin 𝕜 1 f s x : (Fin 1 → E) → F) m = (fderivWithin 𝕜 f s x : E → F) (m 0) :=
@@ -996,6 +1058,7 @@ theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fi
     (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_fderivWithin h]
   rfl
 #align iterated_fderiv_within_one_apply iteratedFDerivWithin_one_apply
+-/
 
 #print Filter.EventuallyEq.iterated_fderiv_within' /-
 theorem Filter.EventuallyEq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
@@ -1047,6 +1110,7 @@ protected theorem Set.EqOn.iteratedFDerivWithin (hs : EqOn f₁ f s) (n : ℕ) :
 #align set.eq_on.iterated_fderiv_within Set.EqOn.iteratedFDerivWithin
 -/
 
+#print iteratedFDerivWithin_eventually_congr_set' /-
 theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
     iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t :=
   by
@@ -1056,6 +1120,7 @@ theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}
     simp only [iteratedFDerivWithin_succ_eq_comp_left, (· ∘ ·)]
     rw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]
 #align iterated_fderiv_within_eventually_congr_set' iteratedFDerivWithin_eventually_congr_set'
+-/
 
 #print iteratedFDerivWithin_eventually_congr_set /-
 theorem iteratedFDerivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
@@ -1112,6 +1177,7 @@ theorem contDiffOn_zero : ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s :=
 #align cont_diff_on_zero contDiffOn_zero
 -/
 
+#print contDiffWithinAt_zero /-
 theorem contDiffWithinAt_zero (hx : x ∈ s) :
     ContDiffWithinAt 𝕜 0 f s x ↔ ∃ u ∈ 𝓝[s] x, ContinuousOn f (s ∩ u) :=
   by
@@ -1127,7 +1193,9 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
     have h' : x ∈ s ∩ u := ⟨hx, mem_of_mem_nhdsWithin hx H⟩
     exact (cont_diff_on_zero.mpr hu).ContDiffWithinAt h'
 #align cont_diff_within_at_zero contDiffWithinAt_zero
+-/
 
+#print HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn /-
 /-- On a set with unique differentiability, any choice of iterated differential has to coincide
 with the one we have chosen in `iterated_fderiv_within 𝕜 m f s`. -/
 theorem HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
@@ -1145,6 +1213,7 @@ theorem HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
     rw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderiv_within (hs x hx)]
     exact (ContinuousMultilinearMap.uncurry_curryLeft _).symm
 #align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
+-/
 
 #print ContDiffOn.ftaylorSeriesWithin /-
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
@@ -1196,6 +1265,7 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
 #align cont_diff_on.ftaylor_series_within ContDiffOn.ftaylorSeriesWithin
 -/
 
+#print contDiffOn_of_continuousOn_differentiableOn /-
 theorem contDiffOn_of_continuousOn_differentiableOn
     (Hcont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s)
     (Hdiff :
@@ -1216,25 +1286,33 @@ theorem contDiffOn_of_continuousOn_differentiableOn
   · intro k hk
     exact Hcont k (le_trans hk hm)
 #align cont_diff_on_of_continuous_on_differentiable_on contDiffOn_of_continuousOn_differentiableOn
+-/
 
+#print contDiffOn_of_differentiableOn /-
 theorem contDiffOn_of_differentiableOn
     (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s) :
     ContDiffOn 𝕜 n f s :=
   contDiffOn_of_continuousOn_differentiableOn (fun m hm => (h m hm).ContinuousOn) fun m hm =>
     h m (le_of_lt hm)
 #align cont_diff_on_of_differentiable_on contDiffOn_of_differentiableOn
+-/
 
+#print ContDiffOn.continuousOn_iteratedFDerivWithin /-
 theorem ContDiffOn.continuousOn_iteratedFDerivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
     (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s) : ContinuousOn (iteratedFDerivWithin 𝕜 m f s) s :=
   (h.ftaylorSeriesWithin hs).cont m hmn
 #align cont_diff_on.continuous_on_iterated_fderiv_within ContDiffOn.continuousOn_iteratedFDerivWithin
+-/
 
+#print ContDiffOn.differentiableOn_iteratedFDerivWithin /-
 theorem ContDiffOn.differentiableOn_iteratedFDerivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
     (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 s) :
     DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s := fun x hx =>
   ((h.ftaylorSeriesWithin hs).fderivWithin m hmn x hx).DifferentiableWithinAt
 #align cont_diff_on.differentiable_on_iterated_fderiv_within ContDiffOn.differentiableOn_iteratedFDerivWithin
+-/
 
+#print ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin /-
 theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
     (h : ContDiffWithinAt 𝕜 n f s x) (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
     DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x :=
@@ -1255,7 +1333,9 @@ theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
   rw [differentiableWithinAt_congr_set' _ A] at C 
   exact C.congr_of_eventually_eq (B.filter_mono inf_le_left) B.self_of_nhds
 #align cont_diff_within_at.differentiable_within_at_iterated_fderiv_within ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin
+-/
 
+#print contDiffOn_iff_continuousOn_differentiableOn /-
 theorem contDiffOn_iff_continuousOn_differentiableOn (hs : UniqueDiffOn 𝕜 s) :
     ContDiffOn 𝕜 n f s ↔
       (∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s) ∧
@@ -1265,6 +1345,7 @@ theorem contDiffOn_iff_continuousOn_differentiableOn (hs : UniqueDiffOn 𝕜 s)
       h.differentiableOn_iteratedFDerivWithin hm hs⟩,
     fun h => contDiffOn_of_continuousOn_differentiableOn h.1 h.2⟩
 #align cont_diff_on_iff_continuous_on_differentiable_on contDiffOn_iff_continuousOn_differentiableOn
+-/
 
 #print contDiffOn_succ_of_fderivWithin /-
 theorem contDiffOn_succ_of_fderivWithin {n : ℕ} (hf : DifferentiableOn 𝕜 f s)
@@ -1367,6 +1448,7 @@ theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
 #align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_open
 -/
 
+#print ContDiffOn.fderivWithin /-
 theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
     (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s :=
   by
@@ -1378,21 +1460,28 @@ theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn
   · change (m.succ : ℕ∞) ≤ n at hmn 
     exact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2
 #align cont_diff_on.fderiv_within ContDiffOn.fderivWithin
+-/
 
+#print ContDiffOn.fderiv_of_open /-
 theorem ContDiffOn.fderiv_of_open (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
     ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s :=
   (hf.fderivWithin hs.UniqueDiffOn hmn).congr fun x hx => (fderivWithin_of_open hs hx).symm
 #align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_open
+-/
 
+#print ContDiffOn.continuousOn_fderivWithin /-
 theorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderivWithin 𝕜 f s x) s :=
   ((contDiffOn_succ_iff_fderivWithin hs).1 (h.of_le hn)).2.ContinuousOn
 #align cont_diff_on.continuous_on_fderiv_within ContDiffOn.continuousOn_fderivWithin
+-/
 
+#print ContDiffOn.continuousOn_fderiv_of_open /-
 theorem ContDiffOn.continuousOn_fderiv_of_open (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderiv 𝕜 f x) s :=
   ((contDiffOn_succ_iff_fderiv_of_open hs).1 (h.of_le hn)).2.ContinuousOn
 #align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_open
+-/
 
 /-! ### Functions with a Taylor series on the whole space -/
 
@@ -1409,10 +1498,12 @@ structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMulti
 #align has_ftaylor_series_up_to HasFTaylorSeriesUpTo
 -/
 
+#print HasFTaylorSeriesUpTo.zero_eq' /-
 theorem HasFTaylorSeriesUpTo.zero_eq' (h : HasFTaylorSeriesUpTo n f p) (x : E) :
     p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by rw [← h.zero_eq x]; symm;
   exact ContinuousMultilinearMap.uncurry0_curry0 _
 #align has_ftaylor_series_up_to.zero_eq' HasFTaylorSeriesUpTo.zero_eq'
+-/
 
 #print hasFTaylorSeriesUpToOn_univ_iff /-
 theorem hasFTaylorSeriesUpToOn_univ_iff :
@@ -1447,10 +1538,12 @@ theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n
 #align has_ftaylor_series_up_to.has_ftaylor_series_up_to_on HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn
 -/
 
+#print HasFTaylorSeriesUpTo.ofLe /-
 theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
     HasFTaylorSeriesUpTo m f p := by rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢;
   exact h.of_le hmn
 #align has_ftaylor_series_up_to.of_le HasFTaylorSeriesUpTo.ofLe
+-/
 
 #print HasFTaylorSeriesUpTo.continuous /-
 theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f :=
@@ -1489,6 +1582,7 @@ theorem hasFTaylorSeriesUpTo_top_iff' :
 #align has_ftaylor_series_up_to_top_iff' hasFTaylorSeriesUpTo_top_iff'
 -/
 
+#print HasFTaylorSeriesUpTo.hasFDerivAt /-
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
 theorem HasFTaylorSeriesUpTo.hasFDerivAt (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
@@ -1497,11 +1591,15 @@ theorem HasFTaylorSeriesUpTo.hasFDerivAt (h : HasFTaylorSeriesUpTo n f p) (hn :
   rw [← hasFDerivWithinAt_univ]
   exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).HasFDerivWithinAt hn (mem_univ _)
 #align has_ftaylor_series_up_to.has_fderiv_at HasFTaylorSeriesUpTo.hasFDerivAt
+-/
 
+#print HasFTaylorSeriesUpTo.differentiable /-
 theorem HasFTaylorSeriesUpTo.differentiable (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) :
     Differentiable 𝕜 f := fun x => (h.HasFDerivAt hn x).DifferentiableAt
 #align has_ftaylor_series_up_to.differentiable HasFTaylorSeriesUpTo.differentiable
+-/
 
+#print hasFTaylorSeriesUpTo_succ_iff_right /-
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
 for `p 1`, which is a derivative of `f`. -/
 theorem hasFTaylorSeriesUpTo_succ_iff_right {n : ℕ} :
@@ -1514,6 +1612,7 @@ theorem hasFTaylorSeriesUpTo_succ_iff_right {n : ℕ} :
   simp only [hasFTaylorSeriesUpToOn_succ_iff_right, ← hasFTaylorSeriesUpToOn_univ_iff, mem_univ,
     forall_true_left, hasFDerivWithinAt_univ]
 #align has_ftaylor_series_up_to_succ_iff_right hasFTaylorSeriesUpTo_succ_iff_right
+-/
 
 /-! ### Smooth functions at a point -/
 
@@ -1562,9 +1661,11 @@ theorem ContDiffAt.congr_of_eventuallyEq (h : ContDiffAt 𝕜 n f x) (hg : f₁
 #align cont_diff_at.congr_of_eventually_eq ContDiffAt.congr_of_eventuallyEq
 -/
 
+#print ContDiffAt.of_le /-
 theorem ContDiffAt.of_le (h : ContDiffAt 𝕜 n f x) (hmn : m ≤ n) : ContDiffAt 𝕜 m f x :=
   h.of_le hmn
 #align cont_diff_at.of_le ContDiffAt.of_le
+-/
 
 #print ContDiffAt.continuousAt /-
 theorem ContDiffAt.continuousAt (h : ContDiffAt 𝕜 n f x) : ContinuousAt f x := by
@@ -1572,12 +1673,15 @@ theorem ContDiffAt.continuousAt (h : ContDiffAt 𝕜 n f x) : ContinuousAt f x :
 #align cont_diff_at.continuous_at ContDiffAt.continuousAt
 -/
 
+#print ContDiffAt.differentiableAt /-
 /-- If a function is `C^n` with `n ≥ 1` at a point, then it is differentiable there. -/
 theorem ContDiffAt.differentiableAt (h : ContDiffAt 𝕜 n f x) (hn : 1 ≤ n) :
     DifferentiableAt 𝕜 f x := by
   simpa [hn, differentiableWithinAt_univ] using h.differentiable_within_at
 #align cont_diff_at.differentiable_at ContDiffAt.differentiableAt
+-/
 
+#print contDiffAt_succ_iff_hasFDerivAt /-
 /-- A function is `C^(n + 1)` at a point iff locally, it has a derivative which is `C^n`. -/
 theorem contDiffAt_succ_iff_hasFDerivAt {n : ℕ} :
     ContDiffAt 𝕜 (n + 1 : ℕ) f x ↔
@@ -1598,6 +1702,7 @@ theorem contDiffAt_succ_iff_hasFDerivAt {n : ℕ} :
     intro x hxu
     exact (h_fderiv x hxu).HasFDerivWithinAt
 #align cont_diff_at_succ_iff_has_fderiv_at contDiffAt_succ_iff_hasFDerivAt
+-/
 
 #print ContDiffAt.eventually /-
 protected theorem ContDiffAt.eventually {n : ℕ} (h : ContDiffAt 𝕜 n f x) :
@@ -1688,10 +1793,13 @@ theorem contDiff_zero : ContDiff 𝕜 0 f ↔ Continuous f :=
 #align cont_diff_zero contDiff_zero
 -/
 
+#print contDiffAt_zero /-
 theorem contDiffAt_zero : ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u := by
   rw [← contDiffWithinAt_univ]; simp [contDiffWithinAt_zero, nhdsWithin_univ]
 #align cont_diff_at_zero contDiffAt_zero
+-/
 
+#print contDiffAt_one_iff /-
 theorem contDiffAt_one_iff :
     ContDiffAt 𝕜 1 f x ↔
       ∃ f' : E → E →L[𝕜] F, ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFDerivAt f (f' x) x :=
@@ -1700,18 +1808,25 @@ theorem contDiffAt_one_iff :
     show ((0 : ℕ) : ℕ∞) = 0 from rfl, contDiffAt_zero,
     exists_mem_and_iff antitone_bforall antitone_continuousOn, and_comm']
 #align cont_diff_at_one_iff contDiffAt_one_iff
+-/
 
+#print ContDiff.of_le /-
 theorem ContDiff.of_le (h : ContDiff 𝕜 n f) (hmn : m ≤ n) : ContDiff 𝕜 m f :=
   contDiffOn_univ.1 <| (contDiffOn_univ.2 h).of_le hmn
 #align cont_diff.of_le ContDiff.of_le
+-/
 
+#print ContDiff.of_succ /-
 theorem ContDiff.of_succ {n : ℕ} (h : ContDiff 𝕜 (n + 1) f) : ContDiff 𝕜 n f :=
   h.of_le <| WithTop.coe_le_coe.mpr le_self_add
 #align cont_diff.of_succ ContDiff.of_succ
+-/
 
+#print ContDiff.one_of_succ /-
 theorem ContDiff.one_of_succ {n : ℕ} (h : ContDiff 𝕜 (n + 1) f) : ContDiff 𝕜 1 f :=
   h.of_le <| WithTop.coe_le_coe.mpr le_add_self
 #align cont_diff.one_of_succ ContDiff.one_of_succ
+-/
 
 #print ContDiff.continuous /-
 theorem ContDiff.continuous (h : ContDiff 𝕜 n f) : Continuous f :=
@@ -1719,14 +1834,18 @@ theorem ContDiff.continuous (h : ContDiff 𝕜 n f) : Continuous f :=
 #align cont_diff.continuous ContDiff.continuous
 -/
 
+#print ContDiff.differentiable /-
 /-- If a function is `C^n` with `n ≥ 1`, then it is differentiable. -/
 theorem ContDiff.differentiable (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) : Differentiable 𝕜 f :=
   differentiableOn_univ.1 <| (contDiffOn_univ.2 h).DifferentiableOn hn
 #align cont_diff.differentiable ContDiff.differentiable
+-/
 
+#print contDiff_iff_forall_nat_le /-
 theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤ n → ContDiff 𝕜 m f := by
   simp_rw [← contDiffOn_univ]; exact contDiffOn_iff_forall_nat_le
 #align cont_diff_iff_forall_nat_le contDiff_iff_forall_nat_le
+-/
 
 #print contDiff_succ_iff_has_fderiv /-
 /-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/
@@ -1769,10 +1888,12 @@ theorem iteratedFDeriv_zero_apply (m : Fin 0 → E) :
 #align iterated_fderiv_zero_apply iteratedFDeriv_zero_apply
 -/
 
+#print iteratedFDeriv_zero_eq_comp /-
 theorem iteratedFDeriv_zero_eq_comp :
     iteratedFDeriv 𝕜 0 f = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
   rfl
 #align iterated_fderiv_zero_eq_comp iteratedFDeriv_zero_eq_comp
+-/
 
 #print norm_iteratedFDeriv_zero /-
 @[simp]
@@ -1787,12 +1908,15 @@ theorem iteratedFDeriv_with_zero_eq : iteratedFDerivWithin 𝕜 0 f s = iterated
 #align iterated_fderiv_with_zero_eq iteratedFDeriv_with_zero_eq
 -/
 
+#print iteratedFDeriv_succ_apply_left /-
 theorem iteratedFDeriv_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
     (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
       (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
   rfl
 #align iterated_fderiv_succ_apply_left iteratedFDeriv_succ_apply_left
+-/
 
+#print iteratedFDeriv_succ_eq_comp_left /-
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
 and the derivative of the `n`-th derivative. -/
 theorem iteratedFDeriv_succ_eq_comp_left {n : ℕ} :
@@ -1801,7 +1925,9 @@ theorem iteratedFDeriv_succ_eq_comp_left {n : ℕ} :
         fderiv 𝕜 (iteratedFDeriv 𝕜 n f) :=
   rfl
 #align iterated_fderiv_succ_eq_comp_left iteratedFDeriv_succ_eq_comp_left
+-/
 
+#print fderiv_iteratedFDeriv /-
 /-- Writing explicitly the derivative of the `n`-th derivative as the composition of a currying
 linear equiv, and the `n + 1`-th derivative. -/
 theorem fderiv_iteratedFDeriv {n : ℕ} :
@@ -1813,7 +1939,9 @@ theorem fderiv_iteratedFDeriv {n : ℕ} :
   ext1 x
   simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]
 #align fderiv_iterated_fderiv fderiv_iteratedFDeriv
+-/
 
+#print HasCompactSupport.iteratedFDeriv /-
 theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
     HasCompactSupport (iteratedFDeriv 𝕜 n f) :=
   by
@@ -1825,6 +1953,7 @@ theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
     apply (IH.fderiv 𝕜).compLeft
     exact LinearIsometryEquiv.map_zero _
 #align has_compact_support.iterated_fderiv HasCompactSupport.iteratedFDeriv
+-/
 
 #print norm_fderiv_iteratedFDeriv /-
 theorem norm_fderiv_iteratedFDeriv {n : ℕ} :
@@ -1874,6 +2003,7 @@ theorem ftaylorSeriesWithin_univ : ftaylorSeriesWithin 𝕜 f univ = ftaylorSeri
 #align ftaylor_series_within_univ ftaylorSeriesWithin_univ
 -/
 
+#print iteratedFDeriv_succ_apply_right /-
 theorem iteratedFDeriv_succ_apply_right {n : ℕ} (m : Fin (n + 1) → E) :
     (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
       iteratedFDeriv 𝕜 n (fun y => fderiv 𝕜 f y) x (init m) (m (last n)) :=
@@ -1881,6 +2011,7 @@ theorem iteratedFDeriv_succ_apply_right {n : ℕ} (m : Fin (n + 1) → E) :
   rw [← iteratedFDerivWithin_univ, ← iteratedFDerivWithin_univ, ← fderivWithin_univ]
   exact iteratedFDerivWithin_succ_apply_right uniqueDiffOn_univ (mem_univ _) _
 #align iterated_fderiv_succ_apply_right iteratedFDeriv_succ_apply_right
+-/
 
 #print iteratedFDeriv_succ_eq_comp_right /-
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
@@ -1900,11 +2031,13 @@ theorem norm_iteratedFDeriv_fderiv {n : ℕ} :
 #align norm_iterated_fderiv_fderiv norm_iteratedFDeriv_fderiv
 -/
 
+#print iteratedFDeriv_one_apply /-
 @[simp]
 theorem iteratedFDeriv_one_apply (m : Fin 1 → E) :
     (iteratedFDeriv 𝕜 1 f x : (Fin 1 → E) → F) m = (fderiv 𝕜 f x : E → F) (m 0) := by
   rw [iteratedFDeriv_succ_apply_right, iteratedFDeriv_zero_apply]; rfl
 #align iterated_fderiv_one_apply iteratedFDeriv_one_apply
+-/
 
 #print contDiff_iff_ftaylorSeries /-
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
@@ -1919,6 +2052,7 @@ theorem contDiff_iff_ftaylorSeries :
 #align cont_diff_iff_ftaylor_series contDiff_iff_ftaylorSeries
 -/
 
+#print contDiff_iff_continuous_differentiable /-
 theorem contDiff_iff_continuous_differentiable :
     ContDiff 𝕜 n f ↔
       (∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧
@@ -1927,24 +2061,31 @@ theorem contDiff_iff_continuous_differentiable :
   simp [cont_diff_on_univ.symm, continuous_iff_continuousOn_univ, differentiable_on_univ.symm,
     iteratedFDerivWithin_univ, contDiffOn_iff_continuousOn_differentiableOn uniqueDiffOn_univ]
 #align cont_diff_iff_continuous_differentiable contDiff_iff_continuous_differentiable
+-/
 
+#print ContDiff.continuous_iteratedFDeriv /-
 /-- If `f` is `C^n` then its `m`-times iterated derivative is continuous for `m ≤ n`. -/
 theorem ContDiff.continuous_iteratedFDeriv {m : ℕ} (hm : (m : ℕ∞) ≤ n) (hf : ContDiff 𝕜 n f) :
     Continuous fun x => iteratedFDeriv 𝕜 m f x :=
   (contDiff_iff_continuous_differentiable.mp hf).1 m hm
 #align cont_diff.continuous_iterated_fderiv ContDiff.continuous_iteratedFDeriv
+-/
 
+#print ContDiff.differentiable_iteratedFDeriv /-
 /-- If `f` is `C^n` then its `m`-times iterated derivative is differentiable for `m < n`. -/
 theorem ContDiff.differentiable_iteratedFDeriv {m : ℕ} (hm : (m : ℕ∞) < n) (hf : ContDiff 𝕜 n f) :
     Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x :=
   (contDiff_iff_continuous_differentiable.mp hf).2 m hm
 #align cont_diff.differentiable_iterated_fderiv ContDiff.differentiable_iteratedFDeriv
+-/
 
+#print contDiff_of_differentiable_iteratedFDeriv /-
 theorem contDiff_of_differentiable_iteratedFDeriv
     (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → Differentiable 𝕜 (iteratedFDeriv 𝕜 m f)) : ContDiff 𝕜 n f :=
   contDiff_iff_continuous_differentiable.2
     ⟨fun m hm => (h m hm).Continuous, fun m hm => h m (le_of_lt hm)⟩
 #align cont_diff_of_differentiable_iterated_fderiv contDiff_of_differentiable_iteratedFDeriv
+-/
 
 #print contDiff_succ_iff_fderiv /-
 /-- A function is `C^(n + 1)` if and only if it is differentiable,
@@ -1973,11 +2114,14 @@ theorem contDiff_top_iff_fderiv :
 #align cont_diff_top_iff_fderiv contDiff_top_iff_fderiv
 -/
 
+#print ContDiff.continuous_fderiv /-
 theorem ContDiff.continuous_fderiv (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
     Continuous fun x => fderiv 𝕜 f x :=
   (contDiff_succ_iff_fderiv.1 (h.of_le hn)).2.Continuous
 #align cont_diff.continuous_fderiv ContDiff.continuous_fderiv
+-/
 
+#print ContDiff.continuous_fderiv_apply /-
 /-- If a function is at least `C^1`, its bundled derivative (mapping `(x, v)` to `Df(x) v`) is
 continuous. -/
 theorem ContDiff.continuous_fderiv_apply (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
@@ -1987,4 +2131,5 @@ theorem ContDiff.continuous_fderiv_apply (h : ContDiff 𝕜 n f) (hn : 1 ≤ n)
     ((h.continuous_fderiv hn).comp continuous_fst).prod_mk continuous_snd
   A.comp B
 #align cont_diff.continuous_fderiv_apply ContDiff.continuous_fderiv_apply
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -967,7 +967,6 @@ theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜
           iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x (init m)
             (m (last (n + 1))) :=
         by rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]; rfl
-      
 #align iterated_fderiv_within_succ_apply_right iteratedFDerivWithin_succ_apply_right
 
 #print iteratedFDerivWithin_succ_eq_comp_right /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit 3a69562db5a458db8322b190ec8d9a8bbd8a5b14
+! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.Analysis.Calculus.FormalMultilinearSeries
 /-!
 # Higher differentiability
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A function is `C^1` on a domain if it is differentiable there, and its derivative is continuous.
 By induction, it is `C^n` if it is `C^{n-1}` and its (n-1)-th derivative is `C^1` there or,
 equivalently, if it is `C^1` and its derivative is `C^{n-1}`.
@@ -489,7 +492,7 @@ theorem ContDiffWithinAt.continuousWithinAt (h : ContDiffWithinAt 𝕜 n f s x)
 theorem ContDiffWithinAt.congr_of_eventuallyEq (h : ContDiffWithinAt 𝕜 n f s x)
     (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : ContDiffWithinAt 𝕜 n f₁ s x := fun m hm =>
   let ⟨u, hu, p, H⟩ := h m hm
-  ⟨{ x ∈ u | f₁ x = f x }, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
+  ⟨{x ∈ u | f₁ x = f x}, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
     (H.mono (sep_subset _ _)).congr fun _ => And.right⟩
 #align cont_diff_within_at.congr_of_eventually_eq ContDiffWithinAt.congr_of_eventuallyEq
 -/
@@ -644,7 +647,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
           HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 𝕜 E F).symm (f z))
             (FormalMultilinearSeries.unshift (p' y) (f y) 1).curryLeft (v ∩ u) y
         rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
-        convert(f'_eq_deriv y hy.2).mono (inter_subset_right v u)
+        convert (f'_eq_deriv y hy.2).mono (inter_subset_right v u)
         rw [← Hp'.zero_eq y hy.1]
         ext z
         change
@@ -653,7 +656,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
         unfold_coes
         congr
         decide
-      · convert(Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1
+      · convert (Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1
         · ext (x y)
           change p' x 0 (init (@snoc 0 (fun i : Fin 1 => E) 0 y)) y = p' x 0 0 y
           rw [init_snoc]
@@ -1207,7 +1210,7 @@ theorem contDiffOn_of_continuousOn_differentiableOn
     simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,
       iteratedFDerivWithin_zero_apply]
   · intro k hk y hy
-    convert(Hdiff k (lt_of_lt_of_le hk hm) y hy).HasFDerivWithinAt
+    convert (Hdiff k (lt_of_lt_of_le hk hm) y hy).HasFDerivWithinAt
     simp only [ftaylorSeriesWithin, iteratedFDerivWithin_succ_eq_comp_left,
       ContinuousLinearEquiv.coe_apply, Function.comp_apply, coeFn_coeBase]
     exact ContinuousLinearMap.curry_uncurryLeft _

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -168,8 +168,8 @@ open scoped Classical BigOperators NNReal Topology Filter
 -- mathport name: «expr∞»
 local notation "∞" => (⊤ : ℕ∞)
 
-attribute [local instance 1001]
-  NormedAddCommGroup.toAddCommGroup NormedSpace.toModule' AddCommGroup.toAddCommMonoid
+attribute [local instance 1001] NormedAddCommGroup.toAddCommGroup NormedSpace.toModule'
+  AddCommGroup.toAddCommMonoid
 
 open Set Fin Filter Function
 
@@ -187,7 +187,7 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddC
 derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
 `has_fderiv_within_at` but for higher order derivatives. -/
 structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
-  (s : Set E) : Prop where
+    (s : Set E) : Prop where
   zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
   fderivWithin :
     ∀ (m : ℕ) (hm : (m : ℕ∞) < n),
@@ -232,7 +232,7 @@ theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s)
     ContinuousOn f s :=
   by
   have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
-  rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this
+  rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this 
 #align has_ftaylor_series_up_to_on.continuous_on HasFTaylorSeriesUpToOn.continuousOn
 -/
 
@@ -376,8 +376,8 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
     refine' ⟨H.zero_eq, H.fderiv_within 0 (WithTop.coe_lt_coe.2 (Nat.succ_pos n)), _⟩
     constructor
     · intro x hx; rfl
-    · intro m(hm : (m : ℕ∞) < n)x(hx : x ∈ s)
-      have A : (m.succ : ℕ∞) < n.succ := by rw [WithTop.coe_lt_coe] at hm⊢;
+    · intro m (hm : (m : ℕ∞) < n) x (hx : x ∈ s)
+      have A : (m.succ : ℕ∞) < n.succ := by rw [WithTop.coe_lt_coe] at hm ⊢;
         exact nat.lt_succ_iff.mpr hm
       change
         HasFDerivWithinAt
@@ -390,8 +390,8 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
         (p x m.succ.succ) (snoc (cons y (init v)) (v (last _))) =
           (p x (Nat.succ (Nat.succ m))) (cons y v)
       rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
-    · intro m(hm : (m : ℕ∞) ≤ n)
-      have A : (m.succ : ℕ∞) ≤ n.succ := by rw [WithTop.coe_le_coe] at hm⊢;
+    · intro m (hm : (m : ℕ∞) ≤ n)
+      have A : (m.succ : ℕ∞) ≤ n.succ := by rw [WithTop.coe_le_coe] at hm ⊢;
         exact nat.pred_le_iff.mp hm
       change
         ContinuousOn
@@ -401,33 +401,33 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
   · rintro ⟨Hzero_eq, Hfderiv_zero, Htaylor⟩
     constructor
     · exact Hzero_eq
-    · intro m(hm : (m : ℕ∞) < n.succ)x(hx : x ∈ s)
+    · intro m (hm : (m : ℕ∞) < n.succ) x (hx : x ∈ s)
       cases m
       · exact Hfderiv_zero x hx
-      · have A : (m : ℕ∞) < n := by rw [WithTop.coe_lt_coe] at hm⊢; exact Nat.lt_of_succ_lt_succ hm
+      · have A : (m : ℕ∞) < n := by rw [WithTop.coe_lt_coe] at hm ⊢; exact Nat.lt_of_succ_lt_succ hm
         have :
           HasFDerivWithinAt
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ)
             ((p x).shift m.succ).curryLeft s x :=
           Htaylor.fderiv_within _ A x hx
-        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this
+        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this 
         convert this
         ext (y v)
         change
           (p x (Nat.succ (Nat.succ m))) (cons y v) =
             (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))
         rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
-    · intro m(hm : (m : ℕ∞) ≤ n.succ)
+    · intro m (hm : (m : ℕ∞) ≤ n.succ)
       cases m
       · have : DifferentiableOn 𝕜 (fun x => p x 0) s := fun x hx =>
           (Hfderiv_zero x hx).DifferentiableWithinAt
         exact this.continuous_on
-      · have A : (m : ℕ∞) ≤ n := by rw [WithTop.coe_le_coe] at hm⊢; exact nat.lt_succ_iff.mp hm
+      · have A : (m : ℕ∞) ≤ n := by rw [WithTop.coe_le_coe] at hm ⊢; exact nat.lt_succ_iff.mp hm
         have :
           ContinuousOn
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ) s :=
           Htaylor.cont _ A
-        rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this
+        rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this 
 #align has_ftaylor_series_up_to_on_succ_iff_right hasFTaylorSeriesUpToOn_succ_iff_right
 
 /-! ### Smooth functions within a set around a point -/
@@ -480,7 +480,7 @@ theorem ContDiffWithinAt.continuousWithinAt (h : ContDiffWithinAt 𝕜 n f s x)
     ContinuousWithinAt f s x :=
   by
   rcases h 0 bot_le with ⟨u, hu, p, H⟩
-  rw [mem_nhdsWithin_insert] at hu
+  rw [mem_nhdsWithin_insert] at hu 
   exact (H.continuous_on.continuous_within_at hu.1).mono_of_mem hu.2
 #align cont_diff_within_at.continuous_within_at ContDiffWithinAt.continuousWithinAt
 -/
@@ -601,7 +601,7 @@ theorem ContDiffWithinAt.differentiable_within_at' (h : ContDiffWithinAt 𝕜 n
   by
   rcases h 1 hn with ⟨u, hu, p, H⟩
   rcases mem_nhdsWithin.1 hu with ⟨t, t_open, xt, tu⟩
-  rw [inter_comm] at tu
+  rw [inter_comm] at tu 
   have := ((H.mono tu).DifferentiableOn le_rfl) x ⟨mem_insert x s, xt⟩
   exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 this
 #align cont_diff_within_at.differentiable_within_at' ContDiffWithinAt.differentiable_within_at'
@@ -629,7 +629,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
     · convert self_mem_nhdsWithin
       have : x ∈ insert x s := by simp
       exact insert_eq_of_mem (mem_of_mem_nhdsWithin this hu)
-    · rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
+    · rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp 
       exact Hp.2.2.of_le hm
   · rintro ⟨u, hu, f', f'_eq_deriv, Hf'⟩
     rw [contDiffWithinAt_nat]
@@ -677,7 +677,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : ℕ} :
   refine' ⟨fun hf => _, _⟩
   · obtain ⟨u, hu, f', huf', hf'⟩ := cont_diff_within_at_succ_iff_has_fderiv_within_at.mp hf
     obtain ⟨w, hw, hxw, hwu⟩ := mem_nhds_within.mp hu
-    rw [inter_comm] at hwu
+    rw [inter_comm] at hwu 
     refine'
       ⟨insert x s ∩ w, inter_mem_nhdsWithin _ (hw.mem_nhds hxw), inter_subset_left _ _, f',
         fun y hy => _, _⟩
@@ -733,7 +733,7 @@ theorem ContDiffWithinAt.contDiffOn' {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : Co
   by
   rcases h m hm with ⟨t, ht, p, hp⟩
   rcases mem_nhdsWithin.1 ht with ⟨u, huo, hxu, hut⟩
-  rw [inter_comm] at hut
+  rw [inter_comm] at hut 
   exact ⟨u, huo, hxu, (hp.mono hut).ContDiffOn⟩
 #align cont_diff_within_at.cont_diff_on' ContDiffWithinAt.contDiffOn'
 
@@ -782,7 +782,7 @@ theorem contDiffOn_all_iff_nat : (∀ n, ContDiffOn 𝕜 n f s) ↔ ∀ n : ℕ,
   by
   refine' ⟨fun H n => H n, _⟩
   rintro H (_ | n)
-  exacts[contDiffOn_top.2 H, H n]
+  exacts [contDiffOn_top.2 H, H n]
 #align cont_diff_on_all_iff_nat contDiffOn_all_iff_nat
 -/
 
@@ -845,7 +845,7 @@ theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
     refine'
       ⟨u, hu, fun y => (continuousMultilinearCurryFin1 𝕜 E F) (p y 1), fun y hy =>
         Hp.has_fderiv_within_at (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _⟩
-    rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
+    rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp 
     intro z hz m hm
     refine' ⟨u, _, fun x : E => (p x).shift, Hp.2.2.of_le hm⟩
     convert self_mem_nhdsWithin
@@ -1118,7 +1118,7 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
     obtain ⟨u, H, p, hp⟩ := h 0 (by norm_num)
     refine' ⟨u, _, _⟩
     · simpa [hx] using H
-    · simp only [WithTop.coe_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
+    · simp only [WithTop.coe_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp 
       exact hp.1.mono (inter_subset_right s u)
   · rintro ⟨u, H, hu⟩
     rw [← contDiffWithinAt_inter' H]
@@ -1156,9 +1156,9 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
       iteratedFDerivWithin_zero_apply]
   · intro m hm x hx
     rcases(h x hx) m.succ (ENat.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩
-    rw [insert_eq_of_mem hx] at hu
+    rw [insert_eq_of_mem hx] at hu 
     rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
-    rw [inter_comm] at ho
+    rw [inter_comm] at ho 
     have : p x m.succ = ftaylorSeriesWithin 𝕜 f s x m.succ :=
       by
       change p x m.succ = iteratedFDerivWithin 𝕜 m.succ f s x
@@ -1181,8 +1181,8 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
     intro x hx
     rcases h x hx m hm with ⟨u, hu, p, Hp⟩
     rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
-    rw [insert_eq_of_mem hx] at ho
-    rw [inter_comm] at ho
+    rw [insert_eq_of_mem hx] at ho 
+    rw [inter_comm] at ho 
     refine' ⟨o, o_open, xo, _⟩
     have A : ∀ y ∈ s ∩ o, p y m = ftaylorSeriesWithin 𝕜 f s y m :=
       by
@@ -1244,13 +1244,13 @@ theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
     simp only [set_eventually_eq_iff_inf_principal, ← nhdsWithin_inter']
     rw [← inter_assoc, nhdsWithin_inter_of_mem', ← diff_eq_compl_inter, insert_diff_of_mem,
       diff_eq_compl_inter]
-    exacts[rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]
+    exacts [rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]
   have B : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t :=
     iteratedFDerivWithin_eventually_congr_set' _ A.symm _
   have C : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x :=
     hu.differentiable_on_iterated_fderiv_within (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x
       ⟨mem_insert _ _, xu⟩
-  rw [differentiableWithinAt_congr_set' _ A] at C
+  rw [differentiableWithinAt_congr_set' _ A] at C 
   exact C.congr_of_eventually_eq (B.filter_mono inf_le_left) B.self_of_nhds
 #align cont_diff_within_at.differentiable_within_at_iterated_fderiv_within ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin
 
@@ -1286,16 +1286,16 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
   refine' ⟨H.differentiable_on (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)), fun x hx => _⟩
   rcases contDiffWithinAt_succ_iff_hasFDerivWithinAt.1 (H x hx) with ⟨u, hu, f', hff', hf'⟩
   rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
-  rw [inter_comm, insert_eq_of_mem hx] at ho
+  rw [inter_comm, insert_eq_of_mem hx] at ho 
   have := hf'.mono ho
-  rw [contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds o_open xo))] at this
+  rw [contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds o_open xo))] at this 
   apply this.congr_of_eventually_eq' _ hx
   have : o ∩ s ∈ 𝓝[s] x := mem_nhdsWithin.2 ⟨o, o_open, xo, subset.refl _⟩
-  rw [inter_comm] at this
+  rw [inter_comm] at this 
   apply Filter.eventuallyEq_of_mem this fun y hy => _
   have A : fderivWithin 𝕜 f (s ∩ o) y = f' y :=
     ((hff' y (ho hy)).mono ho).fderivWithin (hs.inter o_open y hy)
-  rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A
+  rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A 
 #align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin
 -/
 
@@ -1369,11 +1369,11 @@ theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn
     (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s :=
   by
   cases m
-  · change ∞ + 1 ≤ n at hmn
+  · change ∞ + 1 ≤ n at hmn 
     have : n = ∞ := by simpa using hmn
-    rw [this] at hf
+    rw [this] at hf 
     exact ((contDiffOn_top_iff_fderivWithin hs).1 hf).2
-  · change (m.succ : ℕ∞) ≤ n at hmn
+  · change (m.succ : ℕ∞) ≤ n at hmn 
     exact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2
 #align cont_diff_on.fderiv_within ContDiffOn.fderivWithin
 
@@ -1400,7 +1400,7 @@ theorem ContDiffOn.continuousOn_fderiv_of_open (h : ContDiffOn 𝕜 n f s) (hs :
 derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
 `has_fderiv_at` but for higher order derivatives. -/
 structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
-  Prop where
+    Prop where
   zero_eq : ∀ x, (p x 0).uncurry0 = f x
   fderiv : ∀ (m : ℕ) (hm : (m : ℕ∞) < n), ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
   cont : ∀ (m : ℕ) (hm : (m : ℕ∞) ≤ n), Continuous fun x => p x m
@@ -1446,13 +1446,14 @@ theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n
 -/
 
 theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
-    HasFTaylorSeriesUpTo m f p := by rw [← hasFTaylorSeriesUpToOn_univ_iff] at h⊢; exact h.of_le hmn
+    HasFTaylorSeriesUpTo m f p := by rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢;
+  exact h.of_le hmn
 #align has_ftaylor_series_up_to.of_le HasFTaylorSeriesUpTo.ofLe
 
 #print HasFTaylorSeriesUpTo.continuous /-
 theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f :=
   by
-  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h 
   rw [continuous_iff_continuousOn_univ]
   exact h.continuous_on
 #align has_ftaylor_series_up_to.continuous HasFTaylorSeriesUpTo.continuous

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -182,54 +182,63 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddC
 /-! ### Functions with a Taylor series on a domain -/
 
 
+#print HasFTaylorSeriesUpToOn /-
 /-- `has_ftaylor_series_up_to_on n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a
 derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
 `has_fderiv_within_at` but for higher order derivatives. -/
-structure HasFtaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
+structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
   (s : Set E) : Prop where
   zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
   fderivWithin :
     ∀ (m : ℕ) (hm : (m : ℕ∞) < n),
       ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x
   cont : ∀ (m : ℕ) (hm : (m : ℕ∞) ≤ n), ContinuousOn (fun x => p x m) s
-#align has_ftaylor_series_up_to_on HasFtaylorSeriesUpToOn
+#align has_ftaylor_series_up_to_on HasFTaylorSeriesUpToOn
+-/
 
-theorem HasFtaylorSeriesUpToOn.zero_eq' (h : HasFtaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s) :
+theorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s) :
     p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by rw [← h.zero_eq x hx]; symm;
   exact ContinuousMultilinearMap.uncurry0_curry0 _
-#align has_ftaylor_series_up_to_on.zero_eq' HasFtaylorSeriesUpToOn.zero_eq'
+#align has_ftaylor_series_up_to_on.zero_eq' HasFTaylorSeriesUpToOn.zero_eq'
 
+#print HasFTaylorSeriesUpToOn.congr /-
 /-- If two functions coincide on a set `s`, then a Taylor series for the first one is as well a
 Taylor series for the second one. -/
-theorem HasFtaylorSeriesUpToOn.congr (h : HasFtaylorSeriesUpToOn n f p s)
-    (h₁ : ∀ x ∈ s, f₁ x = f x) : HasFtaylorSeriesUpToOn n f₁ p s :=
+theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
+    (h₁ : ∀ x ∈ s, f₁ x = f x) : HasFTaylorSeriesUpToOn n f₁ p s :=
   by
   refine' ⟨fun x hx => _, h.fderiv_within, h.cont⟩
   rw [h₁ x hx]
   exact h.zero_eq x hx
-#align has_ftaylor_series_up_to_on.congr HasFtaylorSeriesUpToOn.congr
+#align has_ftaylor_series_up_to_on.congr HasFTaylorSeriesUpToOn.congr
+-/
 
-theorem HasFtaylorSeriesUpToOn.mono (h : HasFtaylorSeriesUpToOn n f p s) {t : Set E} (hst : t ⊆ s) :
-    HasFtaylorSeriesUpToOn n f p t :=
+#print HasFTaylorSeriesUpToOn.mono /-
+theorem HasFTaylorSeriesUpToOn.mono (h : HasFTaylorSeriesUpToOn n f p s) {t : Set E} (hst : t ⊆ s) :
+    HasFTaylorSeriesUpToOn n f p t :=
   ⟨fun x hx => h.zero_eq x (hst hx), fun m hm x hx => (h.fderivWithin m hm x (hst hx)).mono hst,
     fun m hm => (h.cont m hm).mono hst⟩
-#align has_ftaylor_series_up_to_on.mono HasFtaylorSeriesUpToOn.mono
+#align has_ftaylor_series_up_to_on.mono HasFTaylorSeriesUpToOn.mono
+-/
 
-theorem HasFtaylorSeriesUpToOn.ofLe (h : HasFtaylorSeriesUpToOn n f p s) (hmn : m ≤ n) :
-    HasFtaylorSeriesUpToOn m f p s :=
+theorem HasFTaylorSeriesUpToOn.of_le (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m ≤ n) :
+    HasFTaylorSeriesUpToOn m f p s :=
   ⟨h.zero_eq, fun k hk x hx => h.fderivWithin k (lt_of_lt_of_le hk hmn) x hx, fun k hk =>
     h.cont k (le_trans hk hmn)⟩
-#align has_ftaylor_series_up_to_on.of_le HasFtaylorSeriesUpToOn.ofLe
+#align has_ftaylor_series_up_to_on.of_le HasFTaylorSeriesUpToOn.of_le
 
-theorem HasFtaylorSeriesUpToOn.continuousOn (h : HasFtaylorSeriesUpToOn n f p s) :
+#print HasFTaylorSeriesUpToOn.continuousOn /-
+theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
     ContinuousOn f s :=
   by
   have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
   rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this
-#align has_ftaylor_series_up_to_on.continuous_on HasFtaylorSeriesUpToOn.continuousOn
+#align has_ftaylor_series_up_to_on.continuous_on HasFTaylorSeriesUpToOn.continuousOn
+-/
 
-theorem hasFtaylorSeriesUpToOn_zero_iff :
-    HasFtaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).uncurry0 = f x :=
+#print hasFTaylorSeriesUpToOn_zero_iff /-
+theorem hasFTaylorSeriesUpToOn_zero_iff :
+    HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).uncurry0 = f x :=
   by
   refine'
     ⟨fun H => ⟨H.ContinuousOn, H.zero_eq⟩, fun H =>
@@ -240,10 +249,12 @@ theorem hasFtaylorSeriesUpToOn_zero_iff :
     rw [← H.2 x hx]; symm; exact ContinuousMultilinearMap.uncurry0_curry0 _
   rw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]
   exact H.1
-#align has_ftaylor_series_up_to_on_zero_iff hasFtaylorSeriesUpToOn_zero_iff
+#align has_ftaylor_series_up_to_on_zero_iff hasFTaylorSeriesUpToOn_zero_iff
+-/
 
-theorem hasFtaylorSeriesUpToOn_top_iff :
-    HasFtaylorSeriesUpToOn ∞ f p s ↔ ∀ n : ℕ, HasFtaylorSeriesUpToOn n f p s :=
+#print hasFTaylorSeriesUpToOn_top_iff /-
+theorem hasFTaylorSeriesUpToOn_top_iff :
+    HasFTaylorSeriesUpToOn ∞ f p s ↔ ∀ n : ℕ, HasFTaylorSeriesUpToOn n f p s :=
   by
   constructor
   · intro H n; exact H.of_le le_top
@@ -254,12 +265,14 @@ theorem hasFtaylorSeriesUpToOn_top_iff :
       apply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))
     · intro m hm
       apply (H m).cont m le_rfl
-#align has_ftaylor_series_up_to_on_top_iff hasFtaylorSeriesUpToOn_top_iff
+#align has_ftaylor_series_up_to_on_top_iff hasFTaylorSeriesUpToOn_top_iff
+-/
 
+#print hasFTaylorSeriesUpToOn_top_iff' /-
 /-- In the case that `n = ∞` we don't need the continuity assumption in
 `has_ftaylor_series_up_to_on`. -/
-theorem hasFtaylorSeriesUpToOn_top_iff' :
-    HasFtaylorSeriesUpToOn ∞ f p s ↔
+theorem hasFTaylorSeriesUpToOn_top_iff' :
+    HasFTaylorSeriesUpToOn ∞ f p s ↔
       (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
         ∀ m : ℕ, ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x :=
   ⟨-- Everything except for the continuity is trivial:
@@ -268,11 +281,12 @@ theorem hasFtaylorSeriesUpToOn_top_iff' :
       (-- The continuity follows from the existence of a derivative:
             h.2
           m x hx).ContinuousWithinAt⟩⟩
-#align has_ftaylor_series_up_to_on_top_iff' hasFtaylorSeriesUpToOn_top_iff'
+#align has_ftaylor_series_up_to_on_top_iff' hasFTaylorSeriesUpToOn_top_iff'
+-/
 
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
-theorem HasFtaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
     (hx : x ∈ s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x :=
   by
   have A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0) := by intro y hy;
@@ -290,39 +304,39 @@ theorem HasFtaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFtaylorSeriesUpToOn n f
   congr with i
   rw [Unique.eq_default i]
   rfl
-#align has_ftaylor_series_up_to_on.has_fderiv_within_at HasFtaylorSeriesUpToOn.hasFDerivWithinAt
+#align has_ftaylor_series_up_to_on.has_fderiv_within_at HasFTaylorSeriesUpToOn.hasFDerivWithinAt
 
-theorem HasFtaylorSeriesUpToOn.differentiableOn (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) :
+theorem HasFTaylorSeriesUpToOn.differentiableOn (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) :
     DifferentiableOn 𝕜 f s := fun x hx => (h.HasFDerivWithinAt hn hx).DifferentiableWithinAt
-#align has_ftaylor_series_up_to_on.differentiable_on HasFtaylorSeriesUpToOn.differentiableOn
+#align has_ftaylor_series_up_to_on.differentiable_on HasFTaylorSeriesUpToOn.differentiableOn
 
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term
 of order `1` of this series is a derivative of `f` at `x`. -/
-theorem HasFtaylorSeriesUpToOn.hasFDerivAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+theorem HasFTaylorSeriesUpToOn.hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
     (hx : s ∈ 𝓝 x) : HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
   (h.HasFDerivWithinAt hn (mem_of_mem_nhds hx)).HasFDerivAt hx
-#align has_ftaylor_series_up_to_on.has_fderiv_at HasFtaylorSeriesUpToOn.hasFDerivAt
+#align has_ftaylor_series_up_to_on.has_fderiv_at HasFTaylorSeriesUpToOn.hasFDerivAt
 
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
 in a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. -/
-theorem HasFtaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFtaylorSeriesUpToOn n f p s)
+theorem HasFTaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s)
     (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
     ∀ᶠ y in 𝓝 x, HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p y 1)) y :=
   (eventually_eventually_nhds.2 hx).mono fun y hy => h.HasFDerivAt hn hy
-#align has_ftaylor_series_up_to_on.eventually_has_fderiv_at HasFtaylorSeriesUpToOn.eventually_hasFDerivAt
+#align has_ftaylor_series_up_to_on.eventually_has_fderiv_at HasFTaylorSeriesUpToOn.eventually_hasFDerivAt
 
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
 it is differentiable at `x`. -/
-theorem HasFtaylorSeriesUpToOn.differentiableAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+theorem HasFTaylorSeriesUpToOn.differentiableAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
     (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x :=
   (h.HasFDerivAt hn hx).DifferentiableAt
-#align has_ftaylor_series_up_to_on.differentiable_at HasFtaylorSeriesUpToOn.differentiableAt
+#align has_ftaylor_series_up_to_on.differentiable_at HasFTaylorSeriesUpToOn.differentiableAt
 
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and
 `p (n + 1)` is a derivative of `p n`. -/
-theorem hasFtaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
-    HasFtaylorSeriesUpToOn (n + 1) f p s ↔
-      HasFtaylorSeriesUpToOn n f p s ∧
+theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
+    HasFTaylorSeriesUpToOn (n + 1) f p s ↔
+      HasFTaylorSeriesUpToOn n f p s ∧
         (∀ x ∈ s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧
           ContinuousOn (fun x => p x (n + 1)) s :=
   by
@@ -346,15 +360,15 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
       · have : m = n + 1 := le_antisymm (WithTop.coe_le_coe.1 hm) (not_le.1 h')
         rw [this]
         exact h.2.2
-#align has_ftaylor_series_up_to_on_succ_iff_left hasFtaylorSeriesUpToOn_succ_iff_left
+#align has_ftaylor_series_up_to_on_succ_iff_left hasFTaylorSeriesUpToOn_succ_iff_left
 
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
 for `p 1`, which is a derivative of `f`. -/
-theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
-    HasFtaylorSeriesUpToOn (n + 1 : ℕ) f p s ↔
+theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
+    HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s ↔
       (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
         (∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
-          HasFtaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
+          HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
             (fun x => (p x).shift) s :=
   by
   constructor
@@ -414,13 +428,14 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ) s :=
           Htaylor.cont _ A
         rwa [LinearIsometryEquiv.comp_continuousOn_iff] at this
-#align has_ftaylor_series_up_to_on_succ_iff_right hasFtaylorSeriesUpToOn_succ_iff_right
+#align has_ftaylor_series_up_to_on_succ_iff_right hasFTaylorSeriesUpToOn_succ_iff_right
 
 /-! ### Smooth functions within a set around a point -/
 
 
 variable (𝕜)
 
+#print ContDiffWithinAt /-
 /-- A function is continuously differentiable up to order `n` within a set `s` at a point `x` if
 it admits continuous derivatives up to order `n` in a neighborhood of `x` in `s ∪ {x}`.
 For `n = ∞`, we only require that this holds up to any finite order (where the neighborhood may
@@ -432,15 +447,16 @@ better, is `C^∞` at `0` within `univ`.
 def ContDiffWithinAt (n : ℕ∞) (f : E → F) (s : Set E) (x : E) : Prop :=
   ∀ m : ℕ,
     (m : ℕ∞) ≤ n →
-      ∃ u ∈ 𝓝[insert x s] x, ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFtaylorSeriesUpToOn m f p u
+      ∃ u ∈ 𝓝[insert x s] x, ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFTaylorSeriesUpToOn m f p u
 #align cont_diff_within_at ContDiffWithinAt
+-/
 
 variable {𝕜}
 
 theorem contDiffWithinAt_nat {n : ℕ} :
     ContDiffWithinAt 𝕜 n f s x ↔
       ∃ u ∈ 𝓝[insert x s] x,
-        ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFtaylorSeriesUpToOn n f p u :=
+        ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFTaylorSeriesUpToOn n f p u :=
   ⟨fun H => H n le_rfl, fun ⟨u, hu, p, hp⟩ m hm => ⟨u, hu, p, hp.of_le hm⟩⟩
 #align cont_diff_within_at_nat contDiffWithinAt_nat
 
@@ -453,10 +469,13 @@ theorem contDiffWithinAt_iff_forall_nat_le :
   ⟨fun H m hm => H.of_le hm, fun H m hm => H m hm _ le_rfl⟩
 #align cont_diff_within_at_iff_forall_nat_le contDiffWithinAt_iff_forall_nat_le
 
+#print contDiffWithinAt_top /-
 theorem contDiffWithinAt_top : ContDiffWithinAt 𝕜 ∞ f s x ↔ ∀ n : ℕ, ContDiffWithinAt 𝕜 n f s x :=
   contDiffWithinAt_iff_forall_nat_le.trans <| by simp only [forall_prop_of_true, le_top]
 #align cont_diff_within_at_top contDiffWithinAt_top
+-/
 
+#print ContDiffWithinAt.continuousWithinAt /-
 theorem ContDiffWithinAt.continuousWithinAt (h : ContDiffWithinAt 𝕜 n f s x) :
     ContinuousWithinAt f s x :=
   by
@@ -464,41 +483,55 @@ theorem ContDiffWithinAt.continuousWithinAt (h : ContDiffWithinAt 𝕜 n f s x)
   rw [mem_nhdsWithin_insert] at hu
   exact (H.continuous_on.continuous_within_at hu.1).mono_of_mem hu.2
 #align cont_diff_within_at.continuous_within_at ContDiffWithinAt.continuousWithinAt
+-/
 
+#print ContDiffWithinAt.congr_of_eventuallyEq /-
 theorem ContDiffWithinAt.congr_of_eventuallyEq (h : ContDiffWithinAt 𝕜 n f s x)
     (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : ContDiffWithinAt 𝕜 n f₁ s x := fun m hm =>
   let ⟨u, hu, p, H⟩ := h m hm
   ⟨{ x ∈ u | f₁ x = f x }, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
     (H.mono (sep_subset _ _)).congr fun _ => And.right⟩
 #align cont_diff_within_at.congr_of_eventually_eq ContDiffWithinAt.congr_of_eventuallyEq
+-/
 
+#print ContDiffWithinAt.congr_of_eventuallyEq_insert /-
 theorem ContDiffWithinAt.congr_of_eventuallyEq_insert (h : ContDiffWithinAt 𝕜 n f s x)
     (h₁ : f₁ =ᶠ[𝓝[insert x s] x] f) : ContDiffWithinAt 𝕜 n f₁ s x :=
   h.congr_of_eventuallyEq (nhdsWithin_mono x (subset_insert x s) h₁)
     (mem_of_mem_nhdsWithin (mem_insert x s) h₁ : _)
 #align cont_diff_within_at.congr_of_eventually_eq_insert ContDiffWithinAt.congr_of_eventuallyEq_insert
+-/
 
+#print ContDiffWithinAt.congr_of_eventually_eq' /-
 theorem ContDiffWithinAt.congr_of_eventually_eq' (h : ContDiffWithinAt 𝕜 n f s x)
     (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) : ContDiffWithinAt 𝕜 n f₁ s x :=
   h.congr_of_eventuallyEq h₁ <| h₁.self_of_nhdsWithin hx
 #align cont_diff_within_at.congr_of_eventually_eq' ContDiffWithinAt.congr_of_eventually_eq'
+-/
 
+#print Filter.EventuallyEq.contDiffWithinAt_iff /-
 theorem Filter.EventuallyEq.contDiffWithinAt_iff (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
     ContDiffWithinAt 𝕜 n f₁ s x ↔ ContDiffWithinAt 𝕜 n f s x :=
   ⟨fun H => ContDiffWithinAt.congr_of_eventuallyEq H h₁.symm hx.symm, fun H =>
     H.congr_of_eventuallyEq h₁ hx⟩
 #align filter.eventually_eq.cont_diff_within_at_iff Filter.EventuallyEq.contDiffWithinAt_iff
+-/
 
+#print ContDiffWithinAt.congr /-
 theorem ContDiffWithinAt.congr (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y)
     (hx : f₁ x = f x) : ContDiffWithinAt 𝕜 n f₁ s x :=
   h.congr_of_eventuallyEq (Filter.eventuallyEq_of_mem self_mem_nhdsWithin h₁) hx
 #align cont_diff_within_at.congr ContDiffWithinAt.congr
+-/
 
+#print ContDiffWithinAt.congr' /-
 theorem ContDiffWithinAt.congr' (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y)
     (hx : x ∈ s) : ContDiffWithinAt 𝕜 n f₁ s x :=
   h.congr h₁ (h₁ _ hx)
 #align cont_diff_within_at.congr' ContDiffWithinAt.congr'
+-/
 
+#print ContDiffWithinAt.mono_of_mem /-
 theorem ContDiffWithinAt.mono_of_mem (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E}
     (hst : s ∈ 𝓝[t] x) : ContDiffWithinAt 𝕜 n f t x :=
   by
@@ -506,21 +539,28 @@ theorem ContDiffWithinAt.mono_of_mem (h : ContDiffWithinAt 𝕜 n f s x) {t : Se
   rcases h m hm with ⟨u, hu, p, H⟩
   exact ⟨u, nhdsWithin_le_of_mem (insert_mem_nhdsWithin_insert hst) hu, p, H⟩
 #align cont_diff_within_at.mono_of_mem ContDiffWithinAt.mono_of_mem
+-/
 
+#print ContDiffWithinAt.mono /-
 theorem ContDiffWithinAt.mono (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E} (hst : t ⊆ s) :
     ContDiffWithinAt 𝕜 n f t x :=
   h.mono_of_mem <| Filter.mem_of_superset self_mem_nhdsWithin hst
 #align cont_diff_within_at.mono ContDiffWithinAt.mono
+-/
 
+#print ContDiffWithinAt.congr_nhds /-
 theorem ContDiffWithinAt.congr_nhds (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E}
     (hst : 𝓝[s] x = 𝓝[t] x) : ContDiffWithinAt 𝕜 n f t x :=
   h.mono_of_mem <| hst ▸ self_mem_nhdsWithin
 #align cont_diff_within_at.congr_nhds ContDiffWithinAt.congr_nhds
+-/
 
+#print contDiffWithinAt_congr_nhds /-
 theorem contDiffWithinAt_congr_nhds {t : Set E} (hst : 𝓝[s] x = 𝓝[t] x) :
     ContDiffWithinAt 𝕜 n f s x ↔ ContDiffWithinAt 𝕜 n f t x :=
   ⟨fun h => h.congr_nhds hst, fun h => h.congr_nhds hst.symm⟩
 #align cont_diff_within_at_congr_nhds contDiffWithinAt_congr_nhds
+-/
 
 theorem contDiffWithinAt_inter' (h : t ∈ 𝓝[s] x) :
     ContDiffWithinAt 𝕜 n f (s ∩ t) x ↔ ContDiffWithinAt 𝕜 n f s x :=
@@ -532,6 +572,7 @@ theorem contDiffWithinAt_inter (h : t ∈ 𝓝 x) :
   contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds h)
 #align cont_diff_within_at_inter contDiffWithinAt_inter
 
+#print contDiffWithinAt_insert /-
 theorem contDiffWithinAt_insert {y : E} :
     ContDiffWithinAt 𝕜 n f (insert y s) x ↔ ContDiffWithinAt 𝕜 n f s x :=
   by
@@ -540,15 +581,18 @@ theorem contDiffWithinAt_insert {y : E} :
   · simp_rw [insert_eq_of_mem (mem_insert _ _)]
   simp_rw [insert_comm x y, nhdsWithin_insert_of_ne h]
 #align cont_diff_within_at_insert contDiffWithinAt_insert
+-/
 
 alias contDiffWithinAt_insert ↔ ContDiffWithinAt.of_insert ContDiffWithinAt.insert'
 #align cont_diff_within_at.of_insert ContDiffWithinAt.of_insert
 #align cont_diff_within_at.insert' ContDiffWithinAt.insert'
 
+#print ContDiffWithinAt.insert /-
 theorem ContDiffWithinAt.insert (h : ContDiffWithinAt 𝕜 n f s x) :
     ContDiffWithinAt 𝕜 n f (insert x s) x :=
   h.insert'
 #align cont_diff_within_at.insert ContDiffWithinAt.insert
+-/
 
 /-- If a function is `C^n` within a set at a point, with `n ≥ 1`, then it is differentiable
 within this set at this point. -/
@@ -585,7 +629,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
     · convert self_mem_nhdsWithin
       have : x ∈ insert x s := by simp
       exact insert_eq_of_mem (mem_of_mem_nhdsWithin this hu)
-    · rw [hasFtaylorSeriesUpToOn_succ_iff_right] at Hp
+    · rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
       exact Hp.2.2.of_le hm
   · rintro ⟨u, hu, f', f'_eq_deriv, Hf'⟩
     rw [contDiffWithinAt_nat]
@@ -594,7 +638,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
     · apply Filter.inter_mem _ hu
       apply nhdsWithin_le_of_mem hu
       exact nhdsWithin_mono _ (subset_insert x u) hv
-    · rw [hasFtaylorSeriesUpToOn_succ_iff_right]
+    · rw [hasFTaylorSeriesUpToOn_succ_iff_right]
       refine' ⟨fun y hy => rfl, fun y hy => _, _⟩
       · change
           HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 𝕜 E F).symm (f z))
@@ -623,7 +667,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
 
 /-- A version of `cont_diff_within_at_succ_iff_has_fderiv_within_at` where all derivatives
   are taken within the same set. -/
-theorem contDiffWithinAt_succ_iff_has_fderiv_within_at' {n : ℕ} :
+theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : ℕ} :
     ContDiffWithinAt 𝕜 (n + 1 : ℕ) f s x ↔
       ∃ u ∈ 𝓝[insert x s] x,
         u ⊆ insert x s ∧
@@ -645,13 +689,14 @@ theorem contDiffWithinAt_succ_iff_has_fderiv_within_at' {n : ℕ} :
       insert_eq_of_mem (mem_insert _ _)]
     rintro ⟨u, hu, hus, f', huf', hf'⟩
     refine' ⟨u, hu, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus⟩
-#align cont_diff_within_at_succ_iff_has_fderiv_within_at' contDiffWithinAt_succ_iff_has_fderiv_within_at'
+#align cont_diff_within_at_succ_iff_has_fderiv_within_at' contDiffWithinAt_succ_iff_hasFDerivWithinAt'
 
 /-! ### Smooth functions within a set -/
 
 
 variable (𝕜)
 
+#print ContDiffOn /-
 /-- A function is continuously differentiable up to `n` on `s` if, for any point `x` in `s`, it
 admits continuous derivatives up to order `n` on a neighborhood of `x` in `s`.
 
@@ -661,38 +706,44 @@ depend on the finite order we consider).
 def ContDiffOn (n : ℕ∞) (f : E → F) (s : Set E) : Prop :=
   ∀ x ∈ s, ContDiffWithinAt 𝕜 n f s x
 #align cont_diff_on ContDiffOn
+-/
 
 variable {𝕜}
 
-theorem HasFtaylorSeriesUpToOn.contDiffOn {f' : E → FormalMultilinearSeries 𝕜 E F}
-    (hf : HasFtaylorSeriesUpToOn n f f' s) : ContDiffOn 𝕜 n f s :=
+#print HasFTaylorSeriesUpToOn.contDiffOn /-
+theorem HasFTaylorSeriesUpToOn.contDiffOn {f' : E → FormalMultilinearSeries 𝕜 E F}
+    (hf : HasFTaylorSeriesUpToOn n f f' s) : ContDiffOn 𝕜 n f s :=
   by
   intro x hx m hm
   use s
   simp only [Set.insert_eq_of_mem hx, self_mem_nhdsWithin, true_and_iff]
   exact ⟨f', hf.of_le hm⟩
-#align has_ftaylor_series_up_to_on.cont_diff_on HasFtaylorSeriesUpToOn.contDiffOn
+#align has_ftaylor_series_up_to_on.cont_diff_on HasFTaylorSeriesUpToOn.contDiffOn
+-/
 
+#print ContDiffOn.contDiffWithinAt /-
 theorem ContDiffOn.contDiffWithinAt (h : ContDiffOn 𝕜 n f s) (hx : x ∈ s) :
     ContDiffWithinAt 𝕜 n f s x :=
   h x hx
 #align cont_diff_on.cont_diff_within_at ContDiffOn.contDiffWithinAt
+-/
 
-theorem ContDiffWithinAt.cont_diff_on' {m : ℕ} (hm : (m : ℕ∞) ≤ n)
-    (h : ContDiffWithinAt 𝕜 n f s x) : ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u) :=
+theorem ContDiffWithinAt.contDiffOn' {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : ContDiffWithinAt 𝕜 n f s x) :
+    ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u) :=
   by
   rcases h m hm with ⟨t, ht, p, hp⟩
   rcases mem_nhdsWithin.1 ht with ⟨u, huo, hxu, hut⟩
   rw [inter_comm] at hut
   exact ⟨u, huo, hxu, (hp.mono hut).ContDiffOn⟩
-#align cont_diff_within_at.cont_diff_on' ContDiffWithinAt.cont_diff_on'
+#align cont_diff_within_at.cont_diff_on' ContDiffWithinAt.contDiffOn'
 
 theorem ContDiffWithinAt.contDiffOn {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : ContDiffWithinAt 𝕜 n f s x) :
     ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ContDiffOn 𝕜 m f u :=
-  let ⟨u, uo, xu, h⟩ := h.cont_diff_on' hm
+  let ⟨u, uo, xu, h⟩ := h.contDiffOn' hm
   ⟨_, inter_mem_nhdsWithin _ (uo.mem_nhds xu), inter_subset_left _ _, h⟩
 #align cont_diff_within_at.cont_diff_on ContDiffWithinAt.contDiffOn
 
+#print ContDiffWithinAt.eventually /-
 protected theorem ContDiffWithinAt.eventually {n : ℕ} (h : ContDiffWithinAt 𝕜 n f s x) :
     ∀ᶠ y in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y :=
   by
@@ -702,6 +753,7 @@ protected theorem ContDiffWithinAt.eventually {n : ℕ} (h : ContDiffWithinAt 
   refine' this.mono fun y hy => (hd y hy.2).mono_of_mem _
   exact nhdsWithin_mono y (subset_insert _ _) hy.1
 #align cont_diff_within_at.eventually ContDiffWithinAt.eventually
+-/
 
 theorem ContDiffOn.of_le (h : ContDiffOn 𝕜 n f s) (hmn : m ≤ n) : ContDiffOn 𝕜 m f s := fun x hx =>
   (h x hx).of_le hmn
@@ -719,37 +771,51 @@ theorem contDiffOn_iff_forall_nat_le : ContDiffOn 𝕜 n f s ↔ ∀ m : ℕ, 
   ⟨fun H m hm => H.of_le hm, fun H x hx m hm => H m hm x hx m le_rfl⟩
 #align cont_diff_on_iff_forall_nat_le contDiffOn_iff_forall_nat_le
 
+#print contDiffOn_top /-
 theorem contDiffOn_top : ContDiffOn 𝕜 ∞ f s ↔ ∀ n : ℕ, ContDiffOn 𝕜 n f s :=
   contDiffOn_iff_forall_nat_le.trans <| by simp only [le_top, forall_prop_of_true]
 #align cont_diff_on_top contDiffOn_top
+-/
 
+#print contDiffOn_all_iff_nat /-
 theorem contDiffOn_all_iff_nat : (∀ n, ContDiffOn 𝕜 n f s) ↔ ∀ n : ℕ, ContDiffOn 𝕜 n f s :=
   by
   refine' ⟨fun H n => H n, _⟩
   rintro H (_ | n)
   exacts[contDiffOn_top.2 H, H n]
 #align cont_diff_on_all_iff_nat contDiffOn_all_iff_nat
+-/
 
+#print ContDiffOn.continuousOn /-
 theorem ContDiffOn.continuousOn (h : ContDiffOn 𝕜 n f s) : ContinuousOn f s := fun x hx =>
   (h x hx).ContinuousWithinAt
 #align cont_diff_on.continuous_on ContDiffOn.continuousOn
+-/
 
+#print ContDiffOn.congr /-
 theorem ContDiffOn.congr (h : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s, f₁ x = f x) :
     ContDiffOn 𝕜 n f₁ s := fun x hx => (h x hx).congr h₁ (h₁ x hx)
 #align cont_diff_on.congr ContDiffOn.congr
+-/
 
+#print contDiffOn_congr /-
 theorem contDiffOn_congr (h₁ : ∀ x ∈ s, f₁ x = f x) : ContDiffOn 𝕜 n f₁ s ↔ ContDiffOn 𝕜 n f s :=
   ⟨fun H => H.congr fun x hx => (h₁ x hx).symm, fun H => H.congr h₁⟩
 #align cont_diff_on_congr contDiffOn_congr
+-/
 
+#print ContDiffOn.mono /-
 theorem ContDiffOn.mono (h : ContDiffOn 𝕜 n f s) {t : Set E} (hst : t ⊆ s) : ContDiffOn 𝕜 n f t :=
   fun x hx => (h x (hst hx)).mono hst
 #align cont_diff_on.mono ContDiffOn.mono
+-/
 
+#print ContDiffOn.congr_mono /-
 theorem ContDiffOn.congr_mono (hf : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s₁, f₁ x = f x) (hs : s₁ ⊆ s) :
     ContDiffOn 𝕜 n f₁ s₁ :=
   (hf.mono hs).congr h₁
 #align cont_diff_on.congr_mono ContDiffOn.congr_mono
+-/
 
 /-- If a function is `C^n` on a set with `n ≥ 1`, then it is differentiable there. -/
 theorem ContDiffOn.differentiableOn (h : ContDiffOn 𝕜 n f s) (hn : 1 ≤ n) :
@@ -779,7 +845,7 @@ theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
     refine'
       ⟨u, hu, fun y => (continuousMultilinearCurryFin1 𝕜 E F) (p y 1), fun y hy =>
         Hp.has_fderiv_within_at (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _⟩
-    rw [hasFtaylorSeriesUpToOn_succ_iff_right] at Hp
+    rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
     intro z hz m hm
     refine' ⟨u, _, fun x : E => (p x).shift, Hp.2.2.of_le hm⟩
     convert self_mem_nhdsWithin
@@ -796,206 +862,241 @@ theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
 
 variable (𝕜)
 
+#print iteratedFDerivWithin /-
 /-- The `n`-th derivative of a function along a set, defined inductively by saying that the `n+1`-th
 derivative of `f` is the derivative of the `n`-th derivative of `f` along this set, together with
 an uncurrying step to see it as a multilinear map in `n+1` variables..
 -/
-noncomputable def iteratedFderivWithin (n : ℕ) (f : E → F) (s : Set E) : E → E[×n]→L[𝕜] F :=
+noncomputable def iteratedFDerivWithin (n : ℕ) (f : E → F) (s : Set E) : E → E[×n]→L[𝕜] F :=
   Nat.recOn n (fun x => ContinuousMultilinearMap.curry0 𝕜 E (f x)) fun n rec x =>
     ContinuousLinearMap.uncurryLeft (fderivWithin 𝕜 rec s x)
-#align iterated_fderiv_within iteratedFderivWithin
+#align iterated_fderiv_within iteratedFDerivWithin
+-/
 
+#print ftaylorSeriesWithin /-
 /-- Formal Taylor series associated to a function within a set. -/
 def ftaylorSeriesWithin (f : E → F) (s : Set E) (x : E) : FormalMultilinearSeries 𝕜 E F := fun n =>
-  iteratedFderivWithin 𝕜 n f s x
+  iteratedFDerivWithin 𝕜 n f s x
 #align ftaylor_series_within ftaylorSeriesWithin
+-/
 
 variable {𝕜}
 
+#print iteratedFDerivWithin_zero_apply /-
 @[simp]
-theorem iteratedFderivWithin_zero_apply (m : Fin 0 → E) :
-    (iteratedFderivWithin 𝕜 0 f s x : (Fin 0 → E) → F) m = f x :=
+theorem iteratedFDerivWithin_zero_apply (m : Fin 0 → E) :
+    (iteratedFDerivWithin 𝕜 0 f s x : (Fin 0 → E) → F) m = f x :=
   rfl
-#align iterated_fderiv_within_zero_apply iteratedFderivWithin_zero_apply
+#align iterated_fderiv_within_zero_apply iteratedFDerivWithin_zero_apply
+-/
 
-theorem iteratedFderivWithin_zero_eq_comp :
-    iteratedFderivWithin 𝕜 0 f s = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
+theorem iteratedFDerivWithin_zero_eq_comp :
+    iteratedFDerivWithin 𝕜 0 f s = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
   rfl
-#align iterated_fderiv_within_zero_eq_comp iteratedFderivWithin_zero_eq_comp
+#align iterated_fderiv_within_zero_eq_comp iteratedFDerivWithin_zero_eq_comp
 
+#print norm_iteratedFDerivWithin_zero /-
 @[simp]
-theorem norm_iteratedFderivWithin_zero : ‖iteratedFderivWithin 𝕜 0 f s x‖ = ‖f x‖ := by
-  rw [iteratedFderivWithin_zero_eq_comp, LinearIsometryEquiv.norm_map]
-#align norm_iterated_fderiv_within_zero norm_iteratedFderivWithin_zero
+theorem norm_iteratedFDerivWithin_zero : ‖iteratedFDerivWithin 𝕜 0 f s x‖ = ‖f x‖ := by
+  rw [iteratedFDerivWithin_zero_eq_comp, LinearIsometryEquiv.norm_map]
+#align norm_iterated_fderiv_within_zero norm_iteratedFDerivWithin_zero
+-/
 
-theorem iteratedFderivWithin_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
-    (iteratedFderivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
-      (fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n f s) s x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
+theorem iteratedFDerivWithin_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
+    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
+      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
   rfl
-#align iterated_fderiv_within_succ_apply_left iteratedFderivWithin_succ_apply_left
+#align iterated_fderiv_within_succ_apply_left iteratedFDerivWithin_succ_apply_left
 
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
 and the derivative of the `n`-th derivative. -/
-theorem iteratedFderivWithin_succ_eq_comp_left {n : ℕ} :
-    iteratedFderivWithin 𝕜 (n + 1) f s =
+theorem iteratedFDerivWithin_succ_eq_comp_left {n : ℕ} :
+    iteratedFDerivWithin 𝕜 (n + 1) f s =
       continuousMultilinearCurryLeftEquiv 𝕜 (fun i : Fin (n + 1) => E) F ∘
-        fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n f s) s :=
+        fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s :=
   rfl
-#align iterated_fderiv_within_succ_eq_comp_left iteratedFderivWithin_succ_eq_comp_left
+#align iterated_fderiv_within_succ_eq_comp_left iteratedFDerivWithin_succ_eq_comp_left
 
-theorem norm_fderivWithin_iteratedFderivWithin {n : ℕ} :
-    ‖fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n f s) s x‖ = ‖iteratedFderivWithin 𝕜 (n + 1) f s x‖ :=
-  by rw [iteratedFderivWithin_succ_eq_comp_left, LinearIsometryEquiv.norm_map]
-#align norm_fderiv_within_iterated_fderiv_within norm_fderivWithin_iteratedFderivWithin
+#print norm_fderivWithin_iteratedFDerivWithin /-
+theorem norm_fderivWithin_iteratedFDerivWithin {n : ℕ} :
+    ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ = ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖ :=
+  by rw [iteratedFDerivWithin_succ_eq_comp_left, LinearIsometryEquiv.norm_map]
+#align norm_fderiv_within_iterated_fderiv_within norm_fderivWithin_iteratedFDerivWithin
+-/
 
-theorem iteratedFderivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
+theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
     (m : Fin (n + 1) → E) :
-    (iteratedFderivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
-      iteratedFderivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x (init m) (m (last n)) :=
+    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
+      iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x (init m) (m (last n)) :=
   by
   induction' n with n IH generalizing x
-  · rw [iteratedFderivWithin_succ_eq_comp_left, iteratedFderivWithin_zero_eq_comp,
-      iteratedFderivWithin_zero_apply, Function.comp_apply,
+  · rw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp,
+      iteratedFDerivWithin_zero_apply, Function.comp_apply,
       LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]
     rfl
   · let I := continuousMultilinearCurryRightEquiv' 𝕜 n E F
     have A :
       ∀ y ∈ s,
-        iteratedFderivWithin 𝕜 n.succ f s y =
-          (I ∘ iteratedFderivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y :=
+        iteratedFDerivWithin 𝕜 n.succ f s y =
+          (I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y :=
       by intro y hy; ext m; rw [@IH m y hy]; rfl
     calc
-      (iteratedFderivWithin 𝕜 (n + 2) f s x : (Fin (n + 2) → E) → F) m =
-          (fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n.succ f s) s x : E → E[×n + 1]→L[𝕜] F) (m 0)
+      (iteratedFDerivWithin 𝕜 (n + 2) f s x : (Fin (n + 2) → E) → F) m =
+          (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n.succ f s) s x : E → E[×n + 1]→L[𝕜] F) (m 0)
             (tail m) :=
         rfl
       _ =
-          (fderivWithin 𝕜 (I ∘ iteratedFderivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
+          (fderivWithin 𝕜 (I ∘ iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
               E → E[×n + 1]→L[𝕜] F)
             (m 0) (tail m) :=
         by rw [fderivWithin_congr A (A x hx)]
       _ =
-          (I ∘ fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
+          (I ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
               E → E[×n + 1]→L[𝕜] F)
             (m 0) (tail m) :=
         by rw [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]; rfl
       _ =
-          (fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x :
+          (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x :
               E → E[×n]→L[𝕜] E →L[𝕜] F)
             (m 0) (init (tail m)) ((tail m) (last n)) :=
         rfl
       _ =
-          iteratedFderivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x (init m)
+          iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x (init m)
             (m (last (n + 1))) :=
-        by rw [iteratedFderivWithin_succ_apply_left, tail_init_eq_init_tail]; rfl
+        by rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]; rfl
       
-#align iterated_fderiv_within_succ_apply_right iteratedFderivWithin_succ_apply_right
+#align iterated_fderiv_within_succ_apply_right iteratedFDerivWithin_succ_apply_right
 
+#print iteratedFDerivWithin_succ_eq_comp_right /-
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
 and the `n`-th derivative of the derivative. -/
-theorem iteratedFderivWithin_succ_eq_comp_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
-    iteratedFderivWithin 𝕜 (n + 1) f s x =
+theorem iteratedFDerivWithin_succ_eq_comp_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
+    iteratedFDerivWithin 𝕜 (n + 1) f s x =
       (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘
-          iteratedFderivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s)
+          iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s)
         x :=
-  by ext m; rw [iteratedFderivWithin_succ_apply_right hs hx]; rfl
-#align iterated_fderiv_within_succ_eq_comp_right iteratedFderivWithin_succ_eq_comp_right
+  by ext m; rw [iteratedFDerivWithin_succ_apply_right hs hx]; rfl
+#align iterated_fderiv_within_succ_eq_comp_right iteratedFDerivWithin_succ_eq_comp_right
+-/
 
-theorem norm_iteratedFderivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
-    ‖iteratedFderivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ = ‖iteratedFderivWithin 𝕜 (n + 1) f s x‖ :=
-  by rw [iteratedFderivWithin_succ_eq_comp_right hs hx, LinearIsometryEquiv.norm_map]
-#align norm_iterated_fderiv_within_fderiv_within norm_iteratedFderivWithin_fderivWithin
+#print norm_iteratedFDerivWithin_fderivWithin /-
+theorem norm_iteratedFDerivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
+    ‖iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ = ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖ :=
+  by rw [iteratedFDerivWithin_succ_eq_comp_right hs hx, LinearIsometryEquiv.norm_map]
+#align norm_iterated_fderiv_within_fderiv_within norm_iteratedFDerivWithin_fderivWithin
+-/
 
 @[simp]
-theorem iteratedFderivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E) :
-    (iteratedFderivWithin 𝕜 1 f s x : (Fin 1 → E) → F) m = (fderivWithin 𝕜 f s x : E → F) (m 0) :=
+theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E) :
+    (iteratedFDerivWithin 𝕜 1 f s x : (Fin 1 → E) → F) m = (fderivWithin 𝕜 f s x : E → F) (m 0) :=
   by
-  simp only [iteratedFderivWithin_succ_apply_left, iteratedFderivWithin_zero_eq_comp,
+  simp only [iteratedFDerivWithin_succ_apply_left, iteratedFDerivWithin_zero_eq_comp,
     (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_fderivWithin h]
   rfl
-#align iterated_fderiv_within_one_apply iteratedFderivWithin_one_apply
+#align iterated_fderiv_within_one_apply iteratedFDerivWithin_one_apply
 
+#print Filter.EventuallyEq.iterated_fderiv_within' /-
 theorem Filter.EventuallyEq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
-    iteratedFderivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFderivWithin 𝕜 n f t :=
+    iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t :=
   by
   induction' n with n ihn
   · exact h.mono fun y hy => FunLike.ext _ _ fun _ => hy
   · have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderiv_within' ht
     apply this.mono
     intro y hy
-    simp only [iteratedFderivWithin_succ_eq_comp_left, hy, (· ∘ ·)]
+    simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (· ∘ ·)]
 #align filter.eventually_eq.iterated_fderiv_within' Filter.EventuallyEq.iterated_fderiv_within'
+-/
 
-protected theorem Filter.EventuallyEq.iteratedFderivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
-    iteratedFderivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFderivWithin 𝕜 n f s :=
+#print Filter.EventuallyEq.iteratedFDerivWithin /-
+protected theorem Filter.EventuallyEq.iteratedFDerivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s :=
   h.iterated_fderiv_within' Subset.rfl n
-#align filter.eventually_eq.iterated_fderiv_within Filter.EventuallyEq.iteratedFderivWithin
+#align filter.eventually_eq.iterated_fderiv_within Filter.EventuallyEq.iteratedFDerivWithin
+-/
 
+#print Filter.EventuallyEq.iteratedFDerivWithin_eq /-
 /-- If two functions coincide in a neighborhood of `x` within a set `s` and at `x`, then their
 iterated differentials within this set at `x` coincide. -/
-theorem Filter.EventuallyEq.iteratedFderivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
-    (n : ℕ) : iteratedFderivWithin 𝕜 n f₁ s x = iteratedFderivWithin 𝕜 n f s x :=
+theorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
+    (n : ℕ) : iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
   have : f₁ =ᶠ[𝓝[insert x s] x] f := by simpa [eventually_eq, hx]
   (this.iterated_fderiv_within' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
-#align filter.eventually_eq.iterated_fderiv_within_eq Filter.EventuallyEq.iteratedFderivWithin_eq
+#align filter.eventually_eq.iterated_fderiv_within_eq Filter.EventuallyEq.iteratedFDerivWithin_eq
+-/
 
+#print iteratedFDerivWithin_congr /-
 /-- If two functions coincide on a set `s`, then their iterated differentials within this set
 coincide. See also `filter.eventually_eq.iterated_fderiv_within_eq` and
 `filter.eventually_eq.iterated_fderiv_within`. -/
-theorem iteratedFderivWithin_congr (hs : EqOn f₁ f s) (hx : x ∈ s) (n : ℕ) :
-    iteratedFderivWithin 𝕜 n f₁ s x = iteratedFderivWithin 𝕜 n f s x :=
-  (hs.EventuallyEq.filter_mono inf_le_right).iteratedFderivWithin_eq (hs hx) _
-#align iterated_fderiv_within_congr iteratedFderivWithin_congr
+theorem iteratedFDerivWithin_congr (hs : EqOn f₁ f s) (hx : x ∈ s) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
+  (hs.EventuallyEq.filter_mono inf_le_right).iteratedFDerivWithin_eq (hs hx) _
+#align iterated_fderiv_within_congr iteratedFDerivWithin_congr
+-/
 
+#print Set.EqOn.iteratedFDerivWithin /-
 /-- If two functions coincide on a set `s`, then their iterated differentials within this set
 coincide. See also `filter.eventually_eq.iterated_fderiv_within_eq` and
 `filter.eventually_eq.iterated_fderiv_within`. -/
-protected theorem Set.EqOn.iteratedFderivWithin (hs : EqOn f₁ f s) (n : ℕ) :
-    EqOn (iteratedFderivWithin 𝕜 n f₁ s) (iteratedFderivWithin 𝕜 n f s) s := fun x hx =>
-  iteratedFderivWithin_congr hs hx n
-#align set.eq_on.iterated_fderiv_within Set.EqOn.iteratedFderivWithin
+protected theorem Set.EqOn.iteratedFDerivWithin (hs : EqOn f₁ f s) (n : ℕ) :
+    EqOn (iteratedFDerivWithin 𝕜 n f₁ s) (iteratedFDerivWithin 𝕜 n f s) s := fun x hx =>
+  iteratedFDerivWithin_congr hs hx n
+#align set.eq_on.iterated_fderiv_within Set.EqOn.iteratedFDerivWithin
+-/
 
-theorem iteratedFderivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
-    iteratedFderivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFderivWithin 𝕜 n f t :=
+theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t :=
   by
   induction' n with n ihn generalizing x
   · rfl
   · refine' (eventually_nhds_nhdsWithin.2 h).mono fun y hy => _
-    simp only [iteratedFderivWithin_succ_eq_comp_left, (· ∘ ·)]
+    simp only [iteratedFDerivWithin_succ_eq_comp_left, (· ∘ ·)]
     rw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]
-#align iterated_fderiv_within_eventually_congr_set' iteratedFderivWithin_eventually_congr_set'
+#align iterated_fderiv_within_eventually_congr_set' iteratedFDerivWithin_eventually_congr_set'
 
-theorem iteratedFderivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
-    iteratedFderivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFderivWithin 𝕜 n f t :=
-  iteratedFderivWithin_eventually_congr_set' x (h.filter_mono inf_le_left) n
-#align iterated_fderiv_within_eventually_congr_set iteratedFderivWithin_eventually_congr_set
+#print iteratedFDerivWithin_eventually_congr_set /-
+theorem iteratedFDerivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t :=
+  iteratedFDerivWithin_eventually_congr_set' x (h.filter_mono inf_le_left) n
+#align iterated_fderiv_within_eventually_congr_set iteratedFDerivWithin_eventually_congr_set
+-/
 
-theorem iteratedFderivWithin_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
-    iteratedFderivWithin 𝕜 n f s x = iteratedFderivWithin 𝕜 n f t x :=
-  (iteratedFderivWithin_eventually_congr_set h n).self_of_nhds
-#align iterated_fderiv_within_congr_set iteratedFderivWithin_congr_set
+#print iteratedFDerivWithin_congr_set /-
+theorem iteratedFDerivWithin_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f t x :=
+  (iteratedFDerivWithin_eventually_congr_set h n).self_of_nhds
+#align iterated_fderiv_within_congr_set iteratedFDerivWithin_congr_set
+-/
 
+#print iteratedFDerivWithin_inter' /-
 /-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
 `s` with a neighborhood of `x` within `s`. -/
-theorem iteratedFderivWithin_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) :
-    iteratedFderivWithin 𝕜 n f (s ∩ u) x = iteratedFderivWithin 𝕜 n f s x :=
-  iteratedFderivWithin_congr_set (nhdsWithin_eq_iff_eventuallyEq.1 <| nhdsWithin_inter_of_mem' hu) _
-#align iterated_fderiv_within_inter' iteratedFderivWithin_inter'
+theorem iteratedFDerivWithin_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) :
+    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
+  iteratedFDerivWithin_congr_set (nhdsWithin_eq_iff_eventuallyEq.1 <| nhdsWithin_inter_of_mem' hu) _
+#align iterated_fderiv_within_inter' iteratedFDerivWithin_inter'
+-/
 
+#print iteratedFDerivWithin_inter /-
 /-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
 `s` with a neighborhood of `x`. -/
-theorem iteratedFderivWithin_inter {n : ℕ} (hu : u ∈ 𝓝 x) :
-    iteratedFderivWithin 𝕜 n f (s ∩ u) x = iteratedFderivWithin 𝕜 n f s x :=
-  iteratedFderivWithin_inter' (mem_nhdsWithin_of_mem_nhds hu)
-#align iterated_fderiv_within_inter iteratedFderivWithin_inter
+theorem iteratedFDerivWithin_inter {n : ℕ} (hu : u ∈ 𝓝 x) :
+    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
+  iteratedFDerivWithin_inter' (mem_nhdsWithin_of_mem_nhds hu)
+#align iterated_fderiv_within_inter iteratedFDerivWithin_inter
+-/
 
+#print iteratedFDerivWithin_inter_open /-
 /-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
 `s` with an open set containing `x`. -/
-theorem iteratedFderivWithin_inter_open {n : ℕ} (hu : IsOpen u) (hx : x ∈ u) :
-    iteratedFderivWithin 𝕜 n f (s ∩ u) x = iteratedFderivWithin 𝕜 n f s x :=
-  iteratedFderivWithin_inter (hu.mem_nhds hx)
-#align iterated_fderiv_within_inter_open iteratedFderivWithin_inter_open
+theorem iteratedFDerivWithin_inter_open {n : ℕ} (hu : IsOpen u) (hx : x ∈ u) :
+    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
+  iteratedFDerivWithin_inter (hu.mem_nhds hx)
+#align iterated_fderiv_within_inter_open iteratedFDerivWithin_inter_open
+-/
 
+#print contDiffOn_zero /-
 @[simp]
 theorem contDiffOn_zero : ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s :=
   by
@@ -1004,9 +1105,10 @@ theorem contDiffOn_zero : ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s :=
   have : (m : ℕ∞) = 0 := le_antisymm hm bot_le
   rw [this]
   refine' ⟨insert x s, self_mem_nhdsWithin, ftaylorSeriesWithin 𝕜 f s, _⟩
-  rw [hasFtaylorSeriesUpToOn_zero_iff]
+  rw [hasFTaylorSeriesUpToOn_zero_iff]
   exact ⟨by rwa [insert_eq_of_mem hx], fun x hx => by simp [ftaylorSeriesWithin]⟩
 #align cont_diff_on_zero contDiffOn_zero
+-/
 
 theorem contDiffWithinAt_zero (hx : x ∈ s) :
     ContDiffWithinAt 𝕜 0 f s x ↔ ∃ u ∈ 𝓝[s] x, ContinuousOn f (s ∩ u) :=
@@ -1016,7 +1118,7 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
     obtain ⟨u, H, p, hp⟩ := h 0 (by norm_num)
     refine' ⟨u, _, _⟩
     · simpa [hx] using H
-    · simp only [WithTop.coe_zero, hasFtaylorSeriesUpToOn_zero_iff] at hp
+    · simp only [WithTop.coe_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
       exact hp.1.mono (inter_subset_right s u)
   · rintro ⟨u, H, hu⟩
     rw [← contDiffWithinAt_inter' H]
@@ -1026,31 +1128,32 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
 
 /-- On a set with unique differentiability, any choice of iterated differential has to coincide
 with the one we have chosen in `iterated_fderiv_within 𝕜 m f s`. -/
-theorem HasFtaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
-    (h : HasFtaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s)
-    (hx : x ∈ s) : p x m = iteratedFderivWithin 𝕜 m f s x :=
+theorem HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
+    (h : HasFTaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s)
+    (hx : x ∈ s) : p x m = iteratedFDerivWithin 𝕜 m f s x :=
   by
   induction' m with m IH generalizing x
-  · rw [h.zero_eq' hx, iteratedFderivWithin_zero_eq_comp]
+  · rw [h.zero_eq' hx, iteratedFDerivWithin_zero_eq_comp]
   · have A : (m : ℕ∞) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (lt_add_one m)) hmn
     have :
-      HasFDerivWithinAt (fun y : E => iteratedFderivWithin 𝕜 m f s y)
+      HasFDerivWithinAt (fun y : E => iteratedFDerivWithin 𝕜 m f s y)
         (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x :=
       (h.fderiv_within m A x hx).congr (fun y hy => (IH (le_of_lt A) hy).symm)
         (IH (le_of_lt A) hx).symm
-    rw [iteratedFderivWithin_succ_eq_comp_left, Function.comp_apply, this.fderiv_within (hs x hx)]
+    rw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderiv_within (hs x hx)]
     exact (ContinuousMultilinearMap.uncurry_curryLeft _).symm
-#align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFtaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
+#align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
 
+#print ContDiffOn.ftaylorSeriesWithin /-
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
 `ftaylor_series_within 𝕜 f s` as a Taylor series up to order `n` in `s`. -/
 theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) :
-    HasFtaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s :=
+    HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s :=
   by
   constructor
   · intro x hx
     simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,
-      iteratedFderivWithin_zero_apply]
+      iteratedFDerivWithin_zero_apply]
   · intro m hm x hx
     rcases(h x hx) m.succ (ENat.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩
     rw [insert_eq_of_mem hx] at hu
@@ -1058,15 +1161,15 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
     rw [inter_comm] at ho
     have : p x m.succ = ftaylorSeriesWithin 𝕜 f s x m.succ :=
       by
-      change p x m.succ = iteratedFderivWithin 𝕜 m.succ f s x
-      rw [← iteratedFderivWithin_inter_open o_open xo]
+      change p x m.succ = iteratedFDerivWithin 𝕜 m.succ f s x
+      rw [← iteratedFDerivWithin_inter_open o_open xo]
       exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩
     rw [← this, ← hasFDerivWithinAt_inter (IsOpen.mem_nhds o_open xo)]
     have A : ∀ y ∈ s ∩ o, p y m = ftaylorSeriesWithin 𝕜 f s y m :=
       by
       rintro y ⟨hy, yo⟩
-      change p y m = iteratedFderivWithin 𝕜 m f s y
-      rw [← iteratedFderivWithin_inter_open o_open yo]
+      change p y m = iteratedFDerivWithin 𝕜 m f s y
+      rw [← iteratedFDerivWithin_inter_open o_open yo]
       exact
         (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn (WithTop.coe_le_coe.2 (Nat.le_succ m))
           (hs.inter o_open) ⟨hy, yo⟩
@@ -1084,16 +1187,17 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
     have A : ∀ y ∈ s ∩ o, p y m = ftaylorSeriesWithin 𝕜 f s y m :=
       by
       rintro y ⟨hy, yo⟩
-      change p y m = iteratedFderivWithin 𝕜 m f s y
-      rw [← iteratedFderivWithin_inter_open o_open yo]
+      change p y m = iteratedFDerivWithin 𝕜 m f s y
+      rw [← iteratedFDerivWithin_inter_open o_open yo]
       exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hy, yo⟩
     exact ((Hp.mono ho).cont m le_rfl).congr fun y hy => (A y hy).symm
 #align cont_diff_on.ftaylor_series_within ContDiffOn.ftaylorSeriesWithin
+-/
 
 theorem contDiffOn_of_continuousOn_differentiableOn
-    (Hcont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFderivWithin 𝕜 m f s x) s)
+    (Hcont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s)
     (Hdiff :
-      ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedFderivWithin 𝕜 m f s x) s) :
+      ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s) :
     ContDiffOn 𝕜 n f s := by
   intro x hx m hm
   rw [insert_eq_of_mem hx]
@@ -1101,10 +1205,10 @@ theorem contDiffOn_of_continuousOn_differentiableOn
   constructor
   · intro y hy
     simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,
-      iteratedFderivWithin_zero_apply]
+      iteratedFDerivWithin_zero_apply]
   · intro k hk y hy
     convert(Hdiff k (lt_of_lt_of_le hk hm) y hy).HasFDerivWithinAt
-    simp only [ftaylorSeriesWithin, iteratedFderivWithin_succ_eq_comp_left,
+    simp only [ftaylorSeriesWithin, iteratedFDerivWithin_succ_eq_comp_left,
       ContinuousLinearEquiv.coe_apply, Function.comp_apply, coeFn_coeBase]
     exact ContinuousLinearMap.curry_uncurryLeft _
   · intro k hk
@@ -1112,26 +1216,26 @@ theorem contDiffOn_of_continuousOn_differentiableOn
 #align cont_diff_on_of_continuous_on_differentiable_on contDiffOn_of_continuousOn_differentiableOn
 
 theorem contDiffOn_of_differentiableOn
-    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedFderivWithin 𝕜 m f s) s) :
+    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s) :
     ContDiffOn 𝕜 n f s :=
   contDiffOn_of_continuousOn_differentiableOn (fun m hm => (h m hm).ContinuousOn) fun m hm =>
     h m (le_of_lt hm)
 #align cont_diff_on_of_differentiable_on contDiffOn_of_differentiableOn
 
-theorem ContDiffOn.continuousOn_iteratedFderivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
-    (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s) : ContinuousOn (iteratedFderivWithin 𝕜 m f s) s :=
+theorem ContDiffOn.continuousOn_iteratedFDerivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
+    (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s) : ContinuousOn (iteratedFDerivWithin 𝕜 m f s) s :=
   (h.ftaylorSeriesWithin hs).cont m hmn
-#align cont_diff_on.continuous_on_iterated_fderiv_within ContDiffOn.continuousOn_iteratedFderivWithin
+#align cont_diff_on.continuous_on_iterated_fderiv_within ContDiffOn.continuousOn_iteratedFDerivWithin
 
-theorem ContDiffOn.differentiableOn_iteratedFderivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
+theorem ContDiffOn.differentiableOn_iteratedFDerivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
     (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 s) :
-    DifferentiableOn 𝕜 (iteratedFderivWithin 𝕜 m f s) s := fun x hx =>
+    DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s := fun x hx =>
   ((h.ftaylorSeriesWithin hs).fderivWithin m hmn x hx).DifferentiableWithinAt
-#align cont_diff_on.differentiable_on_iterated_fderiv_within ContDiffOn.differentiableOn_iteratedFderivWithin
+#align cont_diff_on.differentiable_on_iterated_fderiv_within ContDiffOn.differentiableOn_iteratedFDerivWithin
 
-theorem ContDiffWithinAt.differentiableWithinAt_iteratedFderivWithin {m : ℕ}
+theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
     (h : ContDiffWithinAt 𝕜 n f s x) (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
-    DifferentiableWithinAt 𝕜 (iteratedFderivWithin 𝕜 m f s) s x :=
+    DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x :=
   by
   rcases h.cont_diff_on' (ENat.add_one_le_of_lt hmn) with ⟨u, uo, xu, hu⟩
   set t := insert x s ∩ u
@@ -1141,25 +1245,26 @@ theorem ContDiffWithinAt.differentiableWithinAt_iteratedFderivWithin {m : ℕ}
     rw [← inter_assoc, nhdsWithin_inter_of_mem', ← diff_eq_compl_inter, insert_diff_of_mem,
       diff_eq_compl_inter]
     exacts[rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]
-  have B : iteratedFderivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFderivWithin 𝕜 m f t :=
-    iteratedFderivWithin_eventually_congr_set' _ A.symm _
-  have C : DifferentiableWithinAt 𝕜 (iteratedFderivWithin 𝕜 m f t) t x :=
+  have B : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t :=
+    iteratedFDerivWithin_eventually_congr_set' _ A.symm _
+  have C : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x :=
     hu.differentiable_on_iterated_fderiv_within (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x
       ⟨mem_insert _ _, xu⟩
   rw [differentiableWithinAt_congr_set' _ A] at C
   exact C.congr_of_eventually_eq (B.filter_mono inf_le_left) B.self_of_nhds
-#align cont_diff_within_at.differentiable_within_at_iterated_fderiv_within ContDiffWithinAt.differentiableWithinAt_iteratedFderivWithin
+#align cont_diff_within_at.differentiable_within_at_iterated_fderiv_within ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin
 
 theorem contDiffOn_iff_continuousOn_differentiableOn (hs : UniqueDiffOn 𝕜 s) :
     ContDiffOn 𝕜 n f s ↔
-      (∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFderivWithin 𝕜 m f s x) s) ∧
-        ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedFderivWithin 𝕜 m f s x) s :=
+      (∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s) ∧
+        ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s :=
   ⟨fun h =>
-    ⟨fun m hm => h.continuousOn_iteratedFderivWithin hm hs, fun m hm =>
-      h.differentiableOn_iteratedFderivWithin hm hs⟩,
+    ⟨fun m hm => h.continuousOn_iteratedFDerivWithin hm hs, fun m hm =>
+      h.differentiableOn_iteratedFDerivWithin hm hs⟩,
     fun h => contDiffOn_of_continuousOn_differentiableOn h.1 h.2⟩
 #align cont_diff_on_iff_continuous_on_differentiable_on contDiffOn_iff_continuousOn_differentiableOn
 
+#print contDiffOn_succ_of_fderivWithin /-
 theorem contDiffOn_succ_of_fderivWithin {n : ℕ} (hf : DifferentiableOn 𝕜 f s)
     (h : ContDiffOn 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) : ContDiffOn 𝕜 (n + 1 : ℕ) f s :=
   by
@@ -1168,7 +1273,9 @@ theorem contDiffOn_succ_of_fderivWithin {n : ℕ} (hf : DifferentiableOn 𝕜 f
   exact
     ⟨s, self_mem_nhdsWithin, fderivWithin 𝕜 f s, fun y hy => (hf y hy).HasFDerivWithinAt, h x hx⟩
 #align cont_diff_on_succ_of_fderiv_within contDiffOn_succ_of_fderivWithin
+-/
 
+#print contDiffOn_succ_iff_fderivWithin /-
 /-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is
 differentiable there, and its derivative (expressed with `fderiv_within`) is `C^n`. -/
 theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
@@ -1190,7 +1297,9 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
     ((hff' y (ho hy)).mono ho).fderivWithin (hs.inter o_open y hy)
   rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A
 #align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin
+-/
 
+#print contDiffOn_succ_iff_has_fderiv_within /-
 theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       ∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFDerivWithinAt f (f' x) s x :=
@@ -1201,8 +1310,10 @@ theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜
   refine' ⟨fun x hx => (h2 x hx).DifferentiableWithinAt, fun x hx => _⟩
   exact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) hx
 #align cont_diff_on_succ_iff_has_fderiv_within contDiffOn_succ_iff_has_fderiv_within
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]] -/
+#print contDiffOn_succ_iff_fderiv_of_open /-
 /-- A function is `C^(n + 1)` on an open domain if and only if it is
 differentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -/
 theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
@@ -1216,7 +1327,9 @@ theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
   intro x hx
   exact fderivWithin_of_open hs hx
 #align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_open
+-/
 
+#print contDiffOn_top_iff_fderivWithin /-
 /-- A function is `C^∞` on a domain with unique derivatives if and only if it is differentiable
 there, and its derivative (expressed with `fderiv_within`) is `C^∞`. -/
 theorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn 𝕜 s) :
@@ -1234,8 +1347,10 @@ theorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn 𝕜 s) :
     apply ((contDiffOn_succ_iff_fderivWithin hs).2 ⟨h.1, h.2.of_le A⟩).of_le
     exact WithTop.coe_le_coe.2 (Nat.le_succ n)
 #align cont_diff_on_top_iff_fderiv_within contDiffOn_top_iff_fderivWithin
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]] -/
+#print contDiffOn_top_iff_fderiv_of_open /-
 /-- A function is `C^∞` on an open domain if and only if it is differentiable there, and its
 derivative (expressed with `fderiv`) is `C^∞`. -/
 theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
@@ -1248,6 +1363,7 @@ theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
   intro x hx
   exact fderivWithin_of_open hs hx
 #align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_open
+-/
 
 theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
     (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s :=
@@ -1279,23 +1395,26 @@ theorem ContDiffOn.continuousOn_fderiv_of_open (h : ContDiffOn 𝕜 n f s) (hs :
 /-! ### Functions with a Taylor series on the whole space -/
 
 
+#print HasFTaylorSeriesUpTo /-
 /-- `has_ftaylor_series_up_to n f p` registers the fact that `p 0 = f` and `p (m+1)` is a
 derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
 `has_fderiv_at` but for higher order derivatives. -/
-structure HasFtaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
+structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
   Prop where
   zero_eq : ∀ x, (p x 0).uncurry0 = f x
   fderiv : ∀ (m : ℕ) (hm : (m : ℕ∞) < n), ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
   cont : ∀ (m : ℕ) (hm : (m : ℕ∞) ≤ n), Continuous fun x => p x m
-#align has_ftaylor_series_up_to HasFtaylorSeriesUpTo
+#align has_ftaylor_series_up_to HasFTaylorSeriesUpTo
+-/
 
-theorem HasFtaylorSeriesUpTo.zero_eq' (h : HasFtaylorSeriesUpTo n f p) (x : E) :
+theorem HasFTaylorSeriesUpTo.zero_eq' (h : HasFTaylorSeriesUpTo n f p) (x : E) :
     p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by rw [← h.zero_eq x]; symm;
   exact ContinuousMultilinearMap.uncurry0_curry0 _
-#align has_ftaylor_series_up_to.zero_eq' HasFtaylorSeriesUpTo.zero_eq'
+#align has_ftaylor_series_up_to.zero_eq' HasFTaylorSeriesUpTo.zero_eq'
 
-theorem hasFtaylorSeriesUpToOn_univ_iff :
-    HasFtaylorSeriesUpToOn n f p univ ↔ HasFtaylorSeriesUpTo n f p :=
+#print hasFTaylorSeriesUpToOn_univ_iff /-
+theorem hasFTaylorSeriesUpToOn_univ_iff :
+    HasFTaylorSeriesUpToOn n f p univ ↔ HasFTaylorSeriesUpTo n f p :=
   by
   constructor
   · intro H
@@ -1316,114 +1435,139 @@ theorem hasFtaylorSeriesUpToOn_univ_iff :
     · intro m hm
       rw [← continuous_iff_continuousOn_univ]
       exact H.cont m hm
-#align has_ftaylor_series_up_to_on_univ_iff hasFtaylorSeriesUpToOn_univ_iff
+#align has_ftaylor_series_up_to_on_univ_iff hasFTaylorSeriesUpToOn_univ_iff
+-/
 
-theorem HasFtaylorSeriesUpTo.hasFtaylorSeriesUpToOn (h : HasFtaylorSeriesUpTo n f p) (s : Set E) :
-    HasFtaylorSeriesUpToOn n f p s :=
-  (hasFtaylorSeriesUpToOn_univ_iff.2 h).mono (subset_univ _)
-#align has_ftaylor_series_up_to.has_ftaylor_series_up_to_on HasFtaylorSeriesUpTo.hasFtaylorSeriesUpToOn
+#print HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn /-
+theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n f p) (s : Set E) :
+    HasFTaylorSeriesUpToOn n f p s :=
+  (hasFTaylorSeriesUpToOn_univ_iff.2 h).mono (subset_univ _)
+#align has_ftaylor_series_up_to.has_ftaylor_series_up_to_on HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn
+-/
 
-theorem HasFtaylorSeriesUpTo.ofLe (h : HasFtaylorSeriesUpTo n f p) (hmn : m ≤ n) :
-    HasFtaylorSeriesUpTo m f p := by rw [← hasFtaylorSeriesUpToOn_univ_iff] at h⊢; exact h.of_le hmn
-#align has_ftaylor_series_up_to.of_le HasFtaylorSeriesUpTo.ofLe
+theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
+    HasFTaylorSeriesUpTo m f p := by rw [← hasFTaylorSeriesUpToOn_univ_iff] at h⊢; exact h.of_le hmn
+#align has_ftaylor_series_up_to.of_le HasFTaylorSeriesUpTo.ofLe
 
-theorem HasFtaylorSeriesUpTo.continuous (h : HasFtaylorSeriesUpTo n f p) : Continuous f :=
+#print HasFTaylorSeriesUpTo.continuous /-
+theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f :=
   by
-  rw [← hasFtaylorSeriesUpToOn_univ_iff] at h
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
   rw [continuous_iff_continuousOn_univ]
   exact h.continuous_on
-#align has_ftaylor_series_up_to.continuous HasFtaylorSeriesUpTo.continuous
+#align has_ftaylor_series_up_to.continuous HasFTaylorSeriesUpTo.continuous
+-/
 
-theorem hasFtaylorSeriesUpTo_zero_iff :
-    HasFtaylorSeriesUpTo 0 f p ↔ Continuous f ∧ ∀ x, (p x 0).uncurry0 = f x := by
+#print hasFTaylorSeriesUpTo_zero_iff /-
+theorem hasFTaylorSeriesUpTo_zero_iff :
+    HasFTaylorSeriesUpTo 0 f p ↔ Continuous f ∧ ∀ x, (p x 0).uncurry0 = f x := by
   simp [has_ftaylor_series_up_to_on_univ_iff.symm, continuous_iff_continuousOn_univ,
-    hasFtaylorSeriesUpToOn_zero_iff]
-#align has_ftaylor_series_up_to_zero_iff hasFtaylorSeriesUpTo_zero_iff
+    hasFTaylorSeriesUpToOn_zero_iff]
+#align has_ftaylor_series_up_to_zero_iff hasFTaylorSeriesUpTo_zero_iff
+-/
 
-theorem hasFtaylorSeriesUpTo_top_iff :
-    HasFtaylorSeriesUpTo ∞ f p ↔ ∀ n : ℕ, HasFtaylorSeriesUpTo n f p := by
-  simp only [← hasFtaylorSeriesUpToOn_univ_iff, hasFtaylorSeriesUpToOn_top_iff]
-#align has_ftaylor_series_up_to_top_iff hasFtaylorSeriesUpTo_top_iff
+#print hasFTaylorSeriesUpTo_top_iff /-
+theorem hasFTaylorSeriesUpTo_top_iff :
+    HasFTaylorSeriesUpTo ∞ f p ↔ ∀ n : ℕ, HasFTaylorSeriesUpTo n f p := by
+  simp only [← hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff]
+#align has_ftaylor_series_up_to_top_iff hasFTaylorSeriesUpTo_top_iff
+-/
 
+#print hasFTaylorSeriesUpTo_top_iff' /-
 /-- In the case that `n = ∞` we don't need the continuity assumption in
 `has_ftaylor_series_up_to`. -/
-theorem hasFtaylorSeriesUpTo_top_iff' :
-    HasFtaylorSeriesUpTo ∞ f p ↔
+theorem hasFTaylorSeriesUpTo_top_iff' :
+    HasFTaylorSeriesUpTo ∞ f p ↔
       (∀ x, (p x 0).uncurry0 = f x) ∧
         ∀ (m : ℕ) (x), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x :=
   by
-  simp only [← hasFtaylorSeriesUpToOn_univ_iff, hasFtaylorSeriesUpToOn_top_iff', mem_univ,
+  simp only [← hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff', mem_univ,
     forall_true_left, hasFDerivWithinAt_univ]
-#align has_ftaylor_series_up_to_top_iff' hasFtaylorSeriesUpTo_top_iff'
+#align has_ftaylor_series_up_to_top_iff' hasFTaylorSeriesUpTo_top_iff'
+-/
 
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
-theorem HasFtaylorSeriesUpTo.hasFDerivAt (h : HasFtaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
+theorem HasFTaylorSeriesUpTo.hasFDerivAt (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
     HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
   by
   rw [← hasFDerivWithinAt_univ]
-  exact (hasFtaylorSeriesUpToOn_univ_iff.2 h).HasFDerivWithinAt hn (mem_univ _)
-#align has_ftaylor_series_up_to.has_fderiv_at HasFtaylorSeriesUpTo.hasFDerivAt
+  exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).HasFDerivWithinAt hn (mem_univ _)
+#align has_ftaylor_series_up_to.has_fderiv_at HasFTaylorSeriesUpTo.hasFDerivAt
 
-theorem HasFtaylorSeriesUpTo.differentiable (h : HasFtaylorSeriesUpTo n f p) (hn : 1 ≤ n) :
+theorem HasFTaylorSeriesUpTo.differentiable (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) :
     Differentiable 𝕜 f := fun x => (h.HasFDerivAt hn x).DifferentiableAt
-#align has_ftaylor_series_up_to.differentiable HasFtaylorSeriesUpTo.differentiable
+#align has_ftaylor_series_up_to.differentiable HasFTaylorSeriesUpTo.differentiable
 
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
 for `p 1`, which is a derivative of `f`. -/
-theorem hasFtaylorSeriesUpTo_succ_iff_right {n : ℕ} :
-    HasFtaylorSeriesUpTo (n + 1 : ℕ) f p ↔
+theorem hasFTaylorSeriesUpTo_succ_iff_right {n : ℕ} :
+    HasFTaylorSeriesUpTo (n + 1 : ℕ) f p ↔
       (∀ x, (p x 0).uncurry0 = f x) ∧
         (∀ x, HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
-          HasFtaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1)) fun x =>
+          HasFTaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1)) fun x =>
             (p x).shift :=
   by
-  simp only [hasFtaylorSeriesUpToOn_succ_iff_right, ← hasFtaylorSeriesUpToOn_univ_iff, mem_univ,
+  simp only [hasFTaylorSeriesUpToOn_succ_iff_right, ← hasFTaylorSeriesUpToOn_univ_iff, mem_univ,
     forall_true_left, hasFDerivWithinAt_univ]
-#align has_ftaylor_series_up_to_succ_iff_right hasFtaylorSeriesUpTo_succ_iff_right
+#align has_ftaylor_series_up_to_succ_iff_right hasFTaylorSeriesUpTo_succ_iff_right
 
 /-! ### Smooth functions at a point -/
 
 
 variable (𝕜)
 
+#print ContDiffAt /-
 /-- A function is continuously differentiable up to `n` at a point `x` if, for any integer `k ≤ n`,
 there is a neighborhood of `x` where `f` admits derivatives up to order `n`, which are continuous.
 -/
 def ContDiffAt (n : ℕ∞) (f : E → F) (x : E) : Prop :=
   ContDiffWithinAt 𝕜 n f univ x
 #align cont_diff_at ContDiffAt
+-/
 
 variable {𝕜}
 
+#print contDiffWithinAt_univ /-
 theorem contDiffWithinAt_univ : ContDiffWithinAt 𝕜 n f univ x ↔ ContDiffAt 𝕜 n f x :=
   Iff.rfl
 #align cont_diff_within_at_univ contDiffWithinAt_univ
+-/
 
+#print contDiffAt_top /-
 theorem contDiffAt_top : ContDiffAt 𝕜 ∞ f x ↔ ∀ n : ℕ, ContDiffAt 𝕜 n f x := by
   simp [← contDiffWithinAt_univ, contDiffWithinAt_top]
 #align cont_diff_at_top contDiffAt_top
+-/
 
+#print ContDiffAt.contDiffWithinAt /-
 theorem ContDiffAt.contDiffWithinAt (h : ContDiffAt 𝕜 n f x) : ContDiffWithinAt 𝕜 n f s x :=
   h.mono (subset_univ _)
 #align cont_diff_at.cont_diff_within_at ContDiffAt.contDiffWithinAt
+-/
 
+#print ContDiffWithinAt.contDiffAt /-
 theorem ContDiffWithinAt.contDiffAt (h : ContDiffWithinAt 𝕜 n f s x) (hx : s ∈ 𝓝 x) :
     ContDiffAt 𝕜 n f x := by rwa [ContDiffAt, ← contDiffWithinAt_inter hx, univ_inter]
 #align cont_diff_within_at.cont_diff_at ContDiffWithinAt.contDiffAt
+-/
 
+#print ContDiffAt.congr_of_eventuallyEq /-
 theorem ContDiffAt.congr_of_eventuallyEq (h : ContDiffAt 𝕜 n f x) (hg : f₁ =ᶠ[𝓝 x] f) :
     ContDiffAt 𝕜 n f₁ x :=
   h.congr_of_eventually_eq' (by rwa [nhdsWithin_univ]) (mem_univ x)
 #align cont_diff_at.congr_of_eventually_eq ContDiffAt.congr_of_eventuallyEq
+-/
 
 theorem ContDiffAt.of_le (h : ContDiffAt 𝕜 n f x) (hmn : m ≤ n) : ContDiffAt 𝕜 m f x :=
   h.of_le hmn
 #align cont_diff_at.of_le ContDiffAt.of_le
 
+#print ContDiffAt.continuousAt /-
 theorem ContDiffAt.continuousAt (h : ContDiffAt 𝕜 n f x) : ContinuousAt f x := by
   simpa [continuousWithinAt_univ] using h.continuous_within_at
 #align cont_diff_at.continuous_at ContDiffAt.continuousAt
+-/
 
 /-- If a function is `C^n` with `n ≥ 1` at a point, then it is differentiable there. -/
 theorem ContDiffAt.differentiableAt (h : ContDiffAt 𝕜 n f x) (hn : 1 ≤ n) :
@@ -1452,72 +1596,94 @@ theorem contDiffAt_succ_iff_hasFDerivAt {n : ℕ} :
     exact (h_fderiv x hxu).HasFDerivWithinAt
 #align cont_diff_at_succ_iff_has_fderiv_at contDiffAt_succ_iff_hasFDerivAt
 
+#print ContDiffAt.eventually /-
 protected theorem ContDiffAt.eventually {n : ℕ} (h : ContDiffAt 𝕜 n f x) :
     ∀ᶠ y in 𝓝 x, ContDiffAt 𝕜 n f y := by simpa [nhdsWithin_univ] using h.eventually
 #align cont_diff_at.eventually ContDiffAt.eventually
+-/
 
 /-! ### Smooth functions -/
 
 
 variable (𝕜)
 
+#print ContDiff /-
 /-- A function is continuously differentiable up to `n` if it admits derivatives up to
 order `n`, which are continuous. Contrary to the case of definitions in domains (where derivatives
 might not be unique) we do not need to localize the definition in space or time.
 -/
 def ContDiff (n : ℕ∞) (f : E → F) : Prop :=
-  ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFtaylorSeriesUpTo n f p
+  ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFTaylorSeriesUpTo n f p
 #align cont_diff ContDiff
+-/
 
 variable {𝕜}
 
+#print HasFTaylorSeriesUpTo.contDiff /-
 /-- If `f` has a Taylor series up to `n`, then it is `C^n`. -/
-theorem HasFtaylorSeriesUpTo.contDiff {f' : E → FormalMultilinearSeries 𝕜 E F}
-    (hf : HasFtaylorSeriesUpTo n f f') : ContDiff 𝕜 n f :=
+theorem HasFTaylorSeriesUpTo.contDiff {f' : E → FormalMultilinearSeries 𝕜 E F}
+    (hf : HasFTaylorSeriesUpTo n f f') : ContDiff 𝕜 n f :=
   ⟨f', hf⟩
-#align has_ftaylor_series_up_to.cont_diff HasFtaylorSeriesUpTo.contDiff
+#align has_ftaylor_series_up_to.cont_diff HasFTaylorSeriesUpTo.contDiff
+-/
 
+#print contDiffOn_univ /-
 theorem contDiffOn_univ : ContDiffOn 𝕜 n f univ ↔ ContDiff 𝕜 n f :=
   by
   constructor
   · intro H
     use ftaylorSeriesWithin 𝕜 f univ
-    rw [← hasFtaylorSeriesUpToOn_univ_iff]
+    rw [← hasFTaylorSeriesUpToOn_univ_iff]
     exact H.ftaylor_series_within uniqueDiffOn_univ
   · rintro ⟨p, hp⟩ x hx m hm
     exact ⟨univ, Filter.univ_sets _, p, (hp.has_ftaylor_series_up_to_on univ).of_le hm⟩
 #align cont_diff_on_univ contDiffOn_univ
+-/
 
+#print contDiff_iff_contDiffAt /-
 theorem contDiff_iff_contDiffAt : ContDiff 𝕜 n f ↔ ∀ x, ContDiffAt 𝕜 n f x := by
   simp [← contDiffOn_univ, ContDiffOn, ContDiffAt]
 #align cont_diff_iff_cont_diff_at contDiff_iff_contDiffAt
+-/
 
+#print ContDiff.contDiffAt /-
 theorem ContDiff.contDiffAt (h : ContDiff 𝕜 n f) : ContDiffAt 𝕜 n f x :=
   contDiff_iff_contDiffAt.1 h x
 #align cont_diff.cont_diff_at ContDiff.contDiffAt
+-/
 
+#print ContDiff.contDiffWithinAt /-
 theorem ContDiff.contDiffWithinAt (h : ContDiff 𝕜 n f) : ContDiffWithinAt 𝕜 n f s x :=
   h.ContDiffAt.ContDiffWithinAt
 #align cont_diff.cont_diff_within_at ContDiff.contDiffWithinAt
+-/
 
+#print contDiff_top /-
 theorem contDiff_top : ContDiff 𝕜 ∞ f ↔ ∀ n : ℕ, ContDiff 𝕜 n f := by
   simp [cont_diff_on_univ.symm, contDiffOn_top]
 #align cont_diff_top contDiff_top
+-/
 
+#print contDiff_all_iff_nat /-
 theorem contDiff_all_iff_nat : (∀ n, ContDiff 𝕜 n f) ↔ ∀ n : ℕ, ContDiff 𝕜 n f := by
   simp only [← contDiffOn_univ, contDiffOn_all_iff_nat]
 #align cont_diff_all_iff_nat contDiff_all_iff_nat
+-/
 
+#print ContDiff.contDiffOn /-
 theorem ContDiff.contDiffOn (h : ContDiff 𝕜 n f) : ContDiffOn 𝕜 n f s :=
   (contDiffOn_univ.2 h).mono (subset_univ _)
 #align cont_diff.cont_diff_on ContDiff.contDiffOn
+-/
 
+#print contDiff_zero /-
 @[simp]
 theorem contDiff_zero : ContDiff 𝕜 0 f ↔ Continuous f :=
   by
   rw [← contDiffOn_univ, continuous_iff_continuousOn_univ]
   exact contDiffOn_zero
 #align cont_diff_zero contDiff_zero
+-/
 
 theorem contDiffAt_zero : ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u := by
   rw [← contDiffWithinAt_univ]; simp [contDiffWithinAt_zero, nhdsWithin_univ]
@@ -1544,9 +1710,11 @@ theorem ContDiff.one_of_succ {n : ℕ} (h : ContDiff 𝕜 (n + 1) f) : ContDiff
   h.of_le <| WithTop.coe_le_coe.mpr le_add_self
 #align cont_diff.one_of_succ ContDiff.one_of_succ
 
+#print ContDiff.continuous /-
 theorem ContDiff.continuous (h : ContDiff 𝕜 n f) : Continuous f :=
   contDiff_zero.1 (h.of_le bot_le)
 #align cont_diff.continuous ContDiff.continuous
+-/
 
 /-- If a function is `C^n` with `n ≥ 1`, then it is differentiable. -/
 theorem ContDiff.differentiable (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) : Differentiable 𝕜 f :=
@@ -1557,6 +1725,7 @@ theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤
   simp_rw [← contDiffOn_univ]; exact contDiffOn_iff_forall_nat_le
 #align cont_diff_iff_forall_nat_le contDiff_iff_forall_nat_le
 
+#print contDiff_succ_iff_has_fderiv /-
 /-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/
 theorem contDiff_succ_iff_has_fderiv {n : ℕ} :
     ContDiff 𝕜 (n + 1 : ℕ) f ↔
@@ -1565,190 +1734,216 @@ theorem contDiff_succ_iff_has_fderiv {n : ℕ} :
   simp only [← contDiffOn_univ, ← hasFDerivWithinAt_univ,
     contDiffOn_succ_iff_has_fderiv_within uniqueDiffOn_univ, Set.mem_univ, forall_true_left]
 #align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_has_fderiv
+-/
 
 /-! ### Iterated derivative -/
 
 
 variable (𝕜)
 
+#print iteratedFDeriv /-
 /-- The `n`-th derivative of a function, as a multilinear map, defined inductively. -/
-noncomputable def iteratedFderiv (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F :=
+noncomputable def iteratedFDeriv (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F :=
   Nat.recOn n (fun x => ContinuousMultilinearMap.curry0 𝕜 E (f x)) fun n rec x =>
     ContinuousLinearMap.uncurryLeft (fderiv 𝕜 rec x)
-#align iterated_fderiv iteratedFderiv
+#align iterated_fderiv iteratedFDeriv
+-/
 
+#print ftaylorSeries /-
 /-- Formal Taylor series associated to a function within a set. -/
 def ftaylorSeries (f : E → F) (x : E) : FormalMultilinearSeries 𝕜 E F := fun n =>
-  iteratedFderiv 𝕜 n f x
+  iteratedFDeriv 𝕜 n f x
 #align ftaylor_series ftaylorSeries
+-/
 
 variable {𝕜}
 
+#print iteratedFDeriv_zero_apply /-
 @[simp]
-theorem iteratedFderiv_zero_apply (m : Fin 0 → E) :
-    (iteratedFderiv 𝕜 0 f x : (Fin 0 → E) → F) m = f x :=
+theorem iteratedFDeriv_zero_apply (m : Fin 0 → E) :
+    (iteratedFDeriv 𝕜 0 f x : (Fin 0 → E) → F) m = f x :=
   rfl
-#align iterated_fderiv_zero_apply iteratedFderiv_zero_apply
+#align iterated_fderiv_zero_apply iteratedFDeriv_zero_apply
+-/
 
-theorem iteratedFderiv_zero_eq_comp :
-    iteratedFderiv 𝕜 0 f = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
+theorem iteratedFDeriv_zero_eq_comp :
+    iteratedFDeriv 𝕜 0 f = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
   rfl
-#align iterated_fderiv_zero_eq_comp iteratedFderiv_zero_eq_comp
+#align iterated_fderiv_zero_eq_comp iteratedFDeriv_zero_eq_comp
 
+#print norm_iteratedFDeriv_zero /-
 @[simp]
-theorem norm_iteratedFderiv_zero : ‖iteratedFderiv 𝕜 0 f x‖ = ‖f x‖ := by
-  rw [iteratedFderiv_zero_eq_comp, LinearIsometryEquiv.norm_map]
-#align norm_iterated_fderiv_zero norm_iteratedFderiv_zero
+theorem norm_iteratedFDeriv_zero : ‖iteratedFDeriv 𝕜 0 f x‖ = ‖f x‖ := by
+  rw [iteratedFDeriv_zero_eq_comp, LinearIsometryEquiv.norm_map]
+#align norm_iterated_fderiv_zero norm_iteratedFDeriv_zero
+-/
 
-theorem iteratedFderiv_with_zero_eq : iteratedFderivWithin 𝕜 0 f s = iteratedFderiv 𝕜 0 f := by ext;
+#print iteratedFDeriv_with_zero_eq /-
+theorem iteratedFDeriv_with_zero_eq : iteratedFDerivWithin 𝕜 0 f s = iteratedFDeriv 𝕜 0 f := by ext;
   rfl
-#align iterated_fderiv_with_zero_eq iteratedFderiv_with_zero_eq
+#align iterated_fderiv_with_zero_eq iteratedFDeriv_with_zero_eq
+-/
 
-theorem iteratedFderiv_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
-    (iteratedFderiv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
-      (fderiv 𝕜 (iteratedFderiv 𝕜 n f) x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
+theorem iteratedFDeriv_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
+    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
+      (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
   rfl
-#align iterated_fderiv_succ_apply_left iteratedFderiv_succ_apply_left
+#align iterated_fderiv_succ_apply_left iteratedFDeriv_succ_apply_left
 
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
 and the derivative of the `n`-th derivative. -/
-theorem iteratedFderiv_succ_eq_comp_left {n : ℕ} :
-    iteratedFderiv 𝕜 (n + 1) f =
+theorem iteratedFDeriv_succ_eq_comp_left {n : ℕ} :
+    iteratedFDeriv 𝕜 (n + 1) f =
       continuousMultilinearCurryLeftEquiv 𝕜 (fun i : Fin (n + 1) => E) F ∘
-        fderiv 𝕜 (iteratedFderiv 𝕜 n f) :=
+        fderiv 𝕜 (iteratedFDeriv 𝕜 n f) :=
   rfl
-#align iterated_fderiv_succ_eq_comp_left iteratedFderiv_succ_eq_comp_left
+#align iterated_fderiv_succ_eq_comp_left iteratedFDeriv_succ_eq_comp_left
 
 /-- Writing explicitly the derivative of the `n`-th derivative as the composition of a currying
 linear equiv, and the `n + 1`-th derivative. -/
-theorem fderiv_iteratedFderiv {n : ℕ} :
-    fderiv 𝕜 (iteratedFderiv 𝕜 n f) =
+theorem fderiv_iteratedFDeriv {n : ℕ} :
+    fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =
       (continuousMultilinearCurryLeftEquiv 𝕜 (fun i : Fin (n + 1) => E) F).symm ∘
-        iteratedFderiv 𝕜 (n + 1) f :=
+        iteratedFDeriv 𝕜 (n + 1) f :=
   by
-  rw [iteratedFderiv_succ_eq_comp_left]
+  rw [iteratedFDeriv_succ_eq_comp_left]
   ext1 x
   simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]
-#align fderiv_iterated_fderiv fderiv_iteratedFderiv
+#align fderiv_iterated_fderiv fderiv_iteratedFDeriv
 
-theorem HasCompactSupport.iteratedFderiv (hf : HasCompactSupport f) (n : ℕ) :
-    HasCompactSupport (iteratedFderiv 𝕜 n f) :=
+theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
+    HasCompactSupport (iteratedFDeriv 𝕜 n f) :=
   by
   induction' n with n IH
-  · rw [iteratedFderiv_zero_eq_comp]
+  · rw [iteratedFDeriv_zero_eq_comp]
     apply hf.comp_left
     exact LinearIsometryEquiv.map_zero _
-  · rw [iteratedFderiv_succ_eq_comp_left]
+  · rw [iteratedFDeriv_succ_eq_comp_left]
     apply (IH.fderiv 𝕜).compLeft
     exact LinearIsometryEquiv.map_zero _
-#align has_compact_support.iterated_fderiv HasCompactSupport.iteratedFderiv
+#align has_compact_support.iterated_fderiv HasCompactSupport.iteratedFDeriv
 
-theorem norm_fderiv_iteratedFderiv {n : ℕ} :
-    ‖fderiv 𝕜 (iteratedFderiv 𝕜 n f) x‖ = ‖iteratedFderiv 𝕜 (n + 1) f x‖ := by
-  rw [iteratedFderiv_succ_eq_comp_left, LinearIsometryEquiv.norm_map]
-#align norm_fderiv_iterated_fderiv norm_fderiv_iteratedFderiv
+#print norm_fderiv_iteratedFDeriv /-
+theorem norm_fderiv_iteratedFDeriv {n : ℕ} :
+    ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖ := by
+  rw [iteratedFDeriv_succ_eq_comp_left, LinearIsometryEquiv.norm_map]
+#align norm_fderiv_iterated_fderiv norm_fderiv_iteratedFDeriv
+-/
 
-theorem iteratedFderivWithin_univ {n : ℕ} :
-    iteratedFderivWithin 𝕜 n f univ = iteratedFderiv 𝕜 n f :=
+#print iteratedFDerivWithin_univ /-
+theorem iteratedFDerivWithin_univ {n : ℕ} :
+    iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f :=
   by
   induction' n with n IH
   · ext x; simp
   · ext (x m)
-    rw [iteratedFderiv_succ_apply_left, iteratedFderivWithin_succ_apply_left, IH, fderivWithin_univ]
-#align iterated_fderiv_within_univ iteratedFderivWithin_univ
+    rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]
+#align iterated_fderiv_within_univ iteratedFDerivWithin_univ
+-/
 
+#print iteratedFDerivWithin_of_isOpen /-
 /-- In an open set, the iterated derivative within this set coincides with the global iterated
 derivative. -/
-theorem iteratedFderivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :
-    EqOn (iteratedFderivWithin 𝕜 n f s) (iteratedFderiv 𝕜 n f) s :=
+theorem iteratedFDerivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :
+    EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s :=
   by
   induction' n with n IH
   · intro x hx
     ext1 m
-    simp only [iteratedFderivWithin_zero_apply, iteratedFderiv_zero_apply]
+    simp only [iteratedFDerivWithin_zero_apply, iteratedFDeriv_zero_apply]
   · intro x hx
-    rw [iteratedFderiv_succ_eq_comp_left, iteratedFderivWithin_succ_eq_comp_left]
+    rw [iteratedFDeriv_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]
     dsimp
     congr 1
     rw [fderivWithin_of_open hs hx]
     apply Filter.EventuallyEq.fderiv_eq
     filter_upwards [hs.mem_nhds hx]
     exact IH
-#align iterated_fderiv_within_of_is_open iteratedFderivWithin_of_isOpen
+#align iterated_fderiv_within_of_is_open iteratedFDerivWithin_of_isOpen
+-/
 
+#print ftaylorSeriesWithin_univ /-
 theorem ftaylorSeriesWithin_univ : ftaylorSeriesWithin 𝕜 f univ = ftaylorSeries 𝕜 f :=
   by
   ext1 x; ext1 n
-  change iteratedFderivWithin 𝕜 n f univ x = iteratedFderiv 𝕜 n f x
-  rw [iteratedFderivWithin_univ]
+  change iteratedFDerivWithin 𝕜 n f univ x = iteratedFDeriv 𝕜 n f x
+  rw [iteratedFDerivWithin_univ]
 #align ftaylor_series_within_univ ftaylorSeriesWithin_univ
+-/
 
-theorem iteratedFderiv_succ_apply_right {n : ℕ} (m : Fin (n + 1) → E) :
-    (iteratedFderiv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
-      iteratedFderiv 𝕜 n (fun y => fderiv 𝕜 f y) x (init m) (m (last n)) :=
+theorem iteratedFDeriv_succ_apply_right {n : ℕ} (m : Fin (n + 1) → E) :
+    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
+      iteratedFDeriv 𝕜 n (fun y => fderiv 𝕜 f y) x (init m) (m (last n)) :=
   by
-  rw [← iteratedFderivWithin_univ, ← iteratedFderivWithin_univ, ← fderivWithin_univ]
-  exact iteratedFderivWithin_succ_apply_right uniqueDiffOn_univ (mem_univ _) _
-#align iterated_fderiv_succ_apply_right iteratedFderiv_succ_apply_right
+  rw [← iteratedFDerivWithin_univ, ← iteratedFDerivWithin_univ, ← fderivWithin_univ]
+  exact iteratedFDerivWithin_succ_apply_right uniqueDiffOn_univ (mem_univ _) _
+#align iterated_fderiv_succ_apply_right iteratedFDeriv_succ_apply_right
 
+#print iteratedFDeriv_succ_eq_comp_right /-
 /-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
 and the `n`-th derivative of the derivative. -/
-theorem iteratedFderiv_succ_eq_comp_right {n : ℕ} :
-    iteratedFderiv 𝕜 (n + 1) f x =
-      (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘ iteratedFderiv 𝕜 n fun y => fderiv 𝕜 f y)
+theorem iteratedFDeriv_succ_eq_comp_right {n : ℕ} :
+    iteratedFDeriv 𝕜 (n + 1) f x =
+      (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘ iteratedFDeriv 𝕜 n fun y => fderiv 𝕜 f y)
         x :=
-  by ext m; rw [iteratedFderiv_succ_apply_right]; rfl
-#align iterated_fderiv_succ_eq_comp_right iteratedFderiv_succ_eq_comp_right
+  by ext m; rw [iteratedFDeriv_succ_apply_right]; rfl
+#align iterated_fderiv_succ_eq_comp_right iteratedFDeriv_succ_eq_comp_right
+-/
 
-theorem norm_iteratedFderiv_fderiv {n : ℕ} :
-    ‖iteratedFderiv 𝕜 n (fderiv 𝕜 f) x‖ = ‖iteratedFderiv 𝕜 (n + 1) f x‖ := by
-  rw [iteratedFderiv_succ_eq_comp_right, LinearIsometryEquiv.norm_map]
-#align norm_iterated_fderiv_fderiv norm_iteratedFderiv_fderiv
+#print norm_iteratedFDeriv_fderiv /-
+theorem norm_iteratedFDeriv_fderiv {n : ℕ} :
+    ‖iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖ := by
+  rw [iteratedFDeriv_succ_eq_comp_right, LinearIsometryEquiv.norm_map]
+#align norm_iterated_fderiv_fderiv norm_iteratedFDeriv_fderiv
+-/
 
 @[simp]
-theorem iteratedFderiv_one_apply (m : Fin 1 → E) :
-    (iteratedFderiv 𝕜 1 f x : (Fin 1 → E) → F) m = (fderiv 𝕜 f x : E → F) (m 0) := by
-  rw [iteratedFderiv_succ_apply_right, iteratedFderiv_zero_apply]; rfl
-#align iterated_fderiv_one_apply iteratedFderiv_one_apply
+theorem iteratedFDeriv_one_apply (m : Fin 1 → E) :
+    (iteratedFDeriv 𝕜 1 f x : (Fin 1 → E) → F) m = (fderiv 𝕜 f x : E → F) (m 0) := by
+  rw [iteratedFDeriv_succ_apply_right, iteratedFDeriv_zero_apply]; rfl
+#align iterated_fderiv_one_apply iteratedFDeriv_one_apply
 
+#print contDiff_iff_ftaylorSeries /-
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
 `ftaylor_series_within 𝕜 f s` as a Taylor series up to order `n` in `s`. -/
 theorem contDiff_iff_ftaylorSeries :
-    ContDiff 𝕜 n f ↔ HasFtaylorSeriesUpTo n f (ftaylorSeries 𝕜 f) :=
+    ContDiff 𝕜 n f ↔ HasFTaylorSeriesUpTo n f (ftaylorSeries 𝕜 f) :=
   by
   constructor
-  · rw [← contDiffOn_univ, ← hasFtaylorSeriesUpToOn_univ_iff, ← ftaylorSeriesWithin_univ]
+  · rw [← contDiffOn_univ, ← hasFTaylorSeriesUpToOn_univ_iff, ← ftaylorSeriesWithin_univ]
     exact fun h => ContDiffOn.ftaylorSeriesWithin h uniqueDiffOn_univ
   · intro h; exact ⟨ftaylorSeries 𝕜 f, h⟩
 #align cont_diff_iff_ftaylor_series contDiff_iff_ftaylorSeries
+-/
 
 theorem contDiff_iff_continuous_differentiable :
     ContDiff 𝕜 n f ↔
-      (∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => iteratedFderiv 𝕜 m f x) ∧
-        ∀ m : ℕ, (m : ℕ∞) < n → Differentiable 𝕜 fun x => iteratedFderiv 𝕜 m f x :=
+      (∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧
+        ∀ m : ℕ, (m : ℕ∞) < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x :=
   by
   simp [cont_diff_on_univ.symm, continuous_iff_continuousOn_univ, differentiable_on_univ.symm,
-    iteratedFderivWithin_univ, contDiffOn_iff_continuousOn_differentiableOn uniqueDiffOn_univ]
+    iteratedFDerivWithin_univ, contDiffOn_iff_continuousOn_differentiableOn uniqueDiffOn_univ]
 #align cont_diff_iff_continuous_differentiable contDiff_iff_continuous_differentiable
 
 /-- If `f` is `C^n` then its `m`-times iterated derivative is continuous for `m ≤ n`. -/
-theorem ContDiff.continuous_iteratedFderiv {m : ℕ} (hm : (m : ℕ∞) ≤ n) (hf : ContDiff 𝕜 n f) :
-    Continuous fun x => iteratedFderiv 𝕜 m f x :=
+theorem ContDiff.continuous_iteratedFDeriv {m : ℕ} (hm : (m : ℕ∞) ≤ n) (hf : ContDiff 𝕜 n f) :
+    Continuous fun x => iteratedFDeriv 𝕜 m f x :=
   (contDiff_iff_continuous_differentiable.mp hf).1 m hm
-#align cont_diff.continuous_iterated_fderiv ContDiff.continuous_iteratedFderiv
+#align cont_diff.continuous_iterated_fderiv ContDiff.continuous_iteratedFDeriv
 
 /-- If `f` is `C^n` then its `m`-times iterated derivative is differentiable for `m < n`. -/
-theorem ContDiff.differentiable_iteratedFderiv {m : ℕ} (hm : (m : ℕ∞) < n) (hf : ContDiff 𝕜 n f) :
-    Differentiable 𝕜 fun x => iteratedFderiv 𝕜 m f x :=
+theorem ContDiff.differentiable_iteratedFDeriv {m : ℕ} (hm : (m : ℕ∞) < n) (hf : ContDiff 𝕜 n f) :
+    Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x :=
   (contDiff_iff_continuous_differentiable.mp hf).2 m hm
-#align cont_diff.differentiable_iterated_fderiv ContDiff.differentiable_iteratedFderiv
+#align cont_diff.differentiable_iterated_fderiv ContDiff.differentiable_iteratedFDeriv
 
-theorem contDiff_of_differentiable_iteratedFderiv
-    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → Differentiable 𝕜 (iteratedFderiv 𝕜 m f)) : ContDiff 𝕜 n f :=
+theorem contDiff_of_differentiable_iteratedFDeriv
+    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → Differentiable 𝕜 (iteratedFDeriv 𝕜 m f)) : ContDiff 𝕜 n f :=
   contDiff_iff_continuous_differentiable.2
     ⟨fun m hm => (h m hm).Continuous, fun m hm => h m (le_of_lt hm)⟩
-#align cont_diff_of_differentiable_iterated_fderiv contDiff_of_differentiable_iteratedFderiv
+#align cont_diff_of_differentiable_iterated_fderiv contDiff_of_differentiable_iteratedFDeriv
 
+#print contDiff_succ_iff_fderiv /-
 /-- A function is `C^(n + 1)` if and only if it is differentiable,
 and its derivative (formulated in terms of `fderiv`) is `C^n`. -/
 theorem contDiff_succ_iff_fderiv {n : ℕ} :
@@ -1756,11 +1951,15 @@ theorem contDiff_succ_iff_fderiv {n : ℕ} :
   simp only [← contDiffOn_univ, ← differentiableOn_univ, ← fderivWithin_univ,
     contDiffOn_succ_iff_fderivWithin uniqueDiffOn_univ]
 #align cont_diff_succ_iff_fderiv contDiff_succ_iff_fderiv
+-/
 
+#print contDiff_one_iff_fderiv /-
 theorem contDiff_one_iff_fderiv : ContDiff 𝕜 1 f ↔ Differentiable 𝕜 f ∧ Continuous (fderiv 𝕜 f) :=
   contDiff_succ_iff_fderiv.trans <| Iff.rfl.And contDiff_zero
 #align cont_diff_one_iff_fderiv contDiff_one_iff_fderiv
+-/
 
+#print contDiff_top_iff_fderiv /-
 /-- A function is `C^∞` if and only if it is differentiable,
 and its derivative (formulated in terms of `fderiv`) is `C^∞`. -/
 theorem contDiff_top_iff_fderiv :
@@ -1769,6 +1968,7 @@ theorem contDiff_top_iff_fderiv :
   simp only [← contDiffOn_univ, ← differentiableOn_univ, ← fderivWithin_univ]
   rw [contDiffOn_top_iff_fderivWithin uniqueDiffOn_univ]
 #align cont_diff_top_iff_fderiv contDiff_top_iff_fderiv
+-/
 
 theorem ContDiff.continuous_fderiv (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
     Continuous fun x => fderiv 𝕜 f x :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -163,7 +163,7 @@ derivative, differentiability, higher derivative, `C^n`, multilinear, Taylor ser
 
 noncomputable section
 
-open Classical BigOperators NNReal Topology Filter
+open scoped Classical BigOperators NNReal Topology Filter
 
 -- mathport name: «expr∞»
 local notation "∞" => (⊤ : ℕ∞)

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -195,10 +195,7 @@ structure HasFtaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMul
 #align has_ftaylor_series_up_to_on HasFtaylorSeriesUpToOn
 
 theorem HasFtaylorSeriesUpToOn.zero_eq' (h : HasFtaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s) :
-    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) :=
-  by
-  rw [← h.zero_eq x hx]
-  symm
+    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by rw [← h.zero_eq x hx]; symm;
   exact ContinuousMultilinearMap.uncurry0_curry0 _
 #align has_ftaylor_series_up_to_on.zero_eq' HasFtaylorSeriesUpToOn.zero_eq'
 
@@ -239,12 +236,8 @@ theorem hasFtaylorSeriesUpToOn_zero_iff :
       ⟨H.2, fun m hm => False.elim (not_le.2 hm bot_le), _⟩⟩
   intro m hm
   obtain rfl : m = 0 := by exact_mod_cast hm.antisymm (zero_le _)
-  have : ∀ x ∈ s, p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) :=
-    by
-    intro x hx
-    rw [← H.2 x hx]
-    symm
-    exact ContinuousMultilinearMap.uncurry0_curry0 _
+  have : ∀ x ∈ s, p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by intro x hx;
+    rw [← H.2 x hx]; symm; exact ContinuousMultilinearMap.uncurry0_curry0 _
   rw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]
   exact H.1
 #align has_ftaylor_series_up_to_on_zero_iff hasFtaylorSeriesUpToOn_zero_iff
@@ -253,8 +246,7 @@ theorem hasFtaylorSeriesUpToOn_top_iff :
     HasFtaylorSeriesUpToOn ∞ f p s ↔ ∀ n : ℕ, HasFtaylorSeriesUpToOn n f p s :=
   by
   constructor
-  · intro H n
-    exact H.of_le le_top
+  · intro H n; exact H.of_le le_top
   · intro H
     constructor
     · exact (H 0).zero_eq
@@ -283,11 +275,8 @@ series is a derivative of `f`. -/
 theorem HasFtaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
     (hx : x ∈ s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x :=
   by
-  have A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0) :=
-    by
-    intro y hy
-    rw [← h.zero_eq y hy]
-    rfl
+  have A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0) := by intro y hy;
+    rw [← h.zero_eq y hy]; rfl
   suffices H :
     HasFDerivWithinAt (fun y => continuousMultilinearCurryFin0 𝕜 E F (p y 0))
       (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x
@@ -372,12 +361,9 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
   · intro H
     refine' ⟨H.zero_eq, H.fderiv_within 0 (WithTop.coe_lt_coe.2 (Nat.succ_pos n)), _⟩
     constructor
-    · intro x hx
-      rfl
+    · intro x hx; rfl
     · intro m(hm : (m : ℕ∞) < n)x(hx : x ∈ s)
-      have A : (m.succ : ℕ∞) < n.succ :=
-        by
-        rw [WithTop.coe_lt_coe] at hm⊢
+      have A : (m.succ : ℕ∞) < n.succ := by rw [WithTop.coe_lt_coe] at hm⊢;
         exact nat.lt_succ_iff.mpr hm
       change
         HasFDerivWithinAt
@@ -391,9 +377,7 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
           (p x (Nat.succ (Nat.succ m))) (cons y v)
       rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
     · intro m(hm : (m : ℕ∞) ≤ n)
-      have A : (m.succ : ℕ∞) ≤ n.succ :=
-        by
-        rw [WithTop.coe_le_coe] at hm⊢
+      have A : (m.succ : ℕ∞) ≤ n.succ := by rw [WithTop.coe_le_coe] at hm⊢;
         exact nat.pred_le_iff.mp hm
       change
         ContinuousOn
@@ -406,9 +390,7 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
     · intro m(hm : (m : ℕ∞) < n.succ)x(hx : x ∈ s)
       cases m
       · exact Hfderiv_zero x hx
-      · have A : (m : ℕ∞) < n := by
-          rw [WithTop.coe_lt_coe] at hm⊢
-          exact Nat.lt_of_succ_lt_succ hm
+      · have A : (m : ℕ∞) < n := by rw [WithTop.coe_lt_coe] at hm⊢; exact Nat.lt_of_succ_lt_succ hm
         have :
           HasFDerivWithinAt
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ)
@@ -426,9 +408,7 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
       · have : DifferentiableOn 𝕜 (fun x => p x 0) s := fun x hx =>
           (Hfderiv_zero x hx).DifferentiableWithinAt
         exact this.continuous_on
-      · have A : (m : ℕ∞) ≤ n := by
-          rw [WithTop.coe_le_coe] at hm⊢
-          exact nat.lt_succ_iff.mp hm
+      · have A : (m : ℕ∞) ≤ n := by rw [WithTop.coe_le_coe] at hm⊢; exact nat.lt_succ_iff.mp hm
         have :
           ContinuousOn
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ) s :=
@@ -883,11 +863,7 @@ theorem iteratedFderivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜
       ∀ y ∈ s,
         iteratedFderivWithin 𝕜 n.succ f s y =
           (I ∘ iteratedFderivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y :=
-      by
-      intro y hy
-      ext m
-      rw [@IH m y hy]
-      rfl
+      by intro y hy; ext m; rw [@IH m y hy]; rfl
     calc
       (iteratedFderivWithin 𝕜 (n + 2) f s x : (Fin (n + 2) → E) → F) m =
           (fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n.succ f s) s x : E → E[×n + 1]→L[𝕜] F) (m 0)
@@ -902,9 +878,7 @@ theorem iteratedFderivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜
           (I ∘ fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
               E → E[×n + 1]→L[𝕜] F)
             (m 0) (tail m) :=
-        by
-        rw [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]
-        rfl
+        by rw [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]; rfl
       _ =
           (fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x :
               E → E[×n]→L[𝕜] E →L[𝕜] F)
@@ -913,9 +887,7 @@ theorem iteratedFderivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜
       _ =
           iteratedFderivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x (init m)
             (m (last (n + 1))) :=
-        by
-        rw [iteratedFderivWithin_succ_apply_left, tail_init_eq_init_tail]
-        rfl
+        by rw [iteratedFderivWithin_succ_apply_left, tail_init_eq_init_tail]; rfl
       
 #align iterated_fderiv_within_succ_apply_right iteratedFderivWithin_succ_apply_right
 
@@ -926,10 +898,7 @@ theorem iteratedFderivWithin_succ_eq_comp_right {n : ℕ} (hs : UniqueDiffOn 
       (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘
           iteratedFderivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s)
         x :=
-  by
-  ext m
-  rw [iteratedFderivWithin_succ_apply_right hs hx]
-  rfl
+  by ext m; rw [iteratedFderivWithin_succ_apply_right hs hx]; rfl
 #align iterated_fderiv_within_succ_eq_comp_right iteratedFderivWithin_succ_eq_comp_right
 
 theorem norm_iteratedFderivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
@@ -1321,10 +1290,7 @@ structure HasFtaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMulti
 #align has_ftaylor_series_up_to HasFtaylorSeriesUpTo
 
 theorem HasFtaylorSeriesUpTo.zero_eq' (h : HasFtaylorSeriesUpTo n f p) (x : E) :
-    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) :=
-  by
-  rw [← h.zero_eq x]
-  symm
+    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by rw [← h.zero_eq x]; symm;
   exact ContinuousMultilinearMap.uncurry0_curry0 _
 #align has_ftaylor_series_up_to.zero_eq' HasFtaylorSeriesUpTo.zero_eq'
 
@@ -1358,10 +1324,7 @@ theorem HasFtaylorSeriesUpTo.hasFtaylorSeriesUpToOn (h : HasFtaylorSeriesUpTo n
 #align has_ftaylor_series_up_to.has_ftaylor_series_up_to_on HasFtaylorSeriesUpTo.hasFtaylorSeriesUpToOn
 
 theorem HasFtaylorSeriesUpTo.ofLe (h : HasFtaylorSeriesUpTo n f p) (hmn : m ≤ n) :
-    HasFtaylorSeriesUpTo m f p :=
-  by
-  rw [← hasFtaylorSeriesUpToOn_univ_iff] at h⊢
-  exact h.of_le hmn
+    HasFtaylorSeriesUpTo m f p := by rw [← hasFtaylorSeriesUpToOn_univ_iff] at h⊢; exact h.of_le hmn
 #align has_ftaylor_series_up_to.of_le HasFtaylorSeriesUpTo.ofLe
 
 theorem HasFtaylorSeriesUpTo.continuous (h : HasFtaylorSeriesUpTo n f p) : Continuous f :=
@@ -1556,10 +1519,8 @@ theorem contDiff_zero : ContDiff 𝕜 0 f ↔ Continuous f :=
   exact contDiffOn_zero
 #align cont_diff_zero contDiff_zero
 
-theorem contDiffAt_zero : ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u :=
-  by
-  rw [← contDiffWithinAt_univ]
-  simp [contDiffWithinAt_zero, nhdsWithin_univ]
+theorem contDiffAt_zero : ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u := by
+  rw [← contDiffWithinAt_univ]; simp [contDiffWithinAt_zero, nhdsWithin_univ]
 #align cont_diff_at_zero contDiffAt_zero
 
 theorem contDiffAt_one_iff :
@@ -1592,10 +1553,8 @@ theorem ContDiff.differentiable (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) : Differe
   differentiableOn_univ.1 <| (contDiffOn_univ.2 h).DifferentiableOn hn
 #align cont_diff.differentiable ContDiff.differentiable
 
-theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤ n → ContDiff 𝕜 m f :=
-  by
-  simp_rw [← contDiffOn_univ]
-  exact contDiffOn_iff_forall_nat_le
+theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤ n → ContDiff 𝕜 m f := by
+  simp_rw [← contDiffOn_univ]; exact contDiffOn_iff_forall_nat_le
 #align cont_diff_iff_forall_nat_le contDiff_iff_forall_nat_le
 
 /-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/
@@ -1641,9 +1600,7 @@ theorem norm_iteratedFderiv_zero : ‖iteratedFderiv 𝕜 0 f x‖ = ‖f x‖ :
   rw [iteratedFderiv_zero_eq_comp, LinearIsometryEquiv.norm_map]
 #align norm_iterated_fderiv_zero norm_iteratedFderiv_zero
 
-theorem iteratedFderiv_with_zero_eq : iteratedFderivWithin 𝕜 0 f s = iteratedFderiv 𝕜 0 f :=
-  by
-  ext
+theorem iteratedFderiv_with_zero_eq : iteratedFderivWithin 𝕜 0 f s = iteratedFderiv 𝕜 0 f := by ext;
   rfl
 #align iterated_fderiv_with_zero_eq iteratedFderiv_with_zero_eq
 
@@ -1695,8 +1652,7 @@ theorem iteratedFderivWithin_univ {n : ℕ} :
     iteratedFderivWithin 𝕜 n f univ = iteratedFderiv 𝕜 n f :=
   by
   induction' n with n IH
-  · ext x
-    simp
+  · ext x; simp
   · ext (x m)
     rw [iteratedFderiv_succ_apply_left, iteratedFderivWithin_succ_apply_left, IH, fderivWithin_univ]
 #align iterated_fderiv_within_univ iteratedFderivWithin_univ
@@ -1741,10 +1697,7 @@ theorem iteratedFderiv_succ_eq_comp_right {n : ℕ} :
     iteratedFderiv 𝕜 (n + 1) f x =
       (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘ iteratedFderiv 𝕜 n fun y => fderiv 𝕜 f y)
         x :=
-  by
-  ext m
-  rw [iteratedFderiv_succ_apply_right]
-  rfl
+  by ext m; rw [iteratedFderiv_succ_apply_right]; rfl
 #align iterated_fderiv_succ_eq_comp_right iteratedFderiv_succ_eq_comp_right
 
 theorem norm_iteratedFderiv_fderiv {n : ℕ} :
@@ -1754,10 +1707,8 @@ theorem norm_iteratedFderiv_fderiv {n : ℕ} :
 
 @[simp]
 theorem iteratedFderiv_one_apply (m : Fin 1 → E) :
-    (iteratedFderiv 𝕜 1 f x : (Fin 1 → E) → F) m = (fderiv 𝕜 f x : E → F) (m 0) :=
-  by
-  rw [iteratedFderiv_succ_apply_right, iteratedFderiv_zero_apply]
-  rfl
+    (iteratedFderiv 𝕜 1 f x : (Fin 1 → E) → F) m = (fderiv 𝕜 f x : E → F) (m 0) := by
+  rw [iteratedFderiv_succ_apply_right, iteratedFderiv_zero_apply]; rfl
 #align iterated_fderiv_one_apply iteratedFderiv_one_apply
 
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
@@ -1768,8 +1719,7 @@ theorem contDiff_iff_ftaylorSeries :
   constructor
   · rw [← contDiffOn_univ, ← hasFtaylorSeriesUpToOn_univ_iff, ← ftaylorSeriesWithin_univ]
     exact fun h => ContDiffOn.ftaylorSeriesWithin h uniqueDiffOn_univ
-  · intro h
-    exact ⟨ftaylorSeries 𝕜 f, h⟩
+  · intro h; exact ⟨ftaylorSeries 𝕜 f, h⟩
 #align cont_diff_iff_ftaylor_series contDiff_iff_ftaylorSeries
 
 theorem contDiff_iff_continuous_differentiable :

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit e3fb84046afd187b710170887195d50bada934ee
+! leanprover-community/mathlib commit 3a69562db5a458db8322b190ec8d9a8bbd8a5b14
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -163,7 +163,7 @@ derivative, differentiability, higher derivative, `C^n`, multilinear, Taylor ser
 
 noncomputable section
 
-open Classical BigOperators NNReal Topology
+open Classical BigOperators NNReal Topology Filter
 
 -- mathport name: «expr∞»
 local notation "∞" => (⊤ : ℕ∞)
@@ -698,15 +698,19 @@ theorem ContDiffOn.contDiffWithinAt (h : ContDiffOn 𝕜 n f s) (hx : x ∈ s) :
   h x hx
 #align cont_diff_on.cont_diff_within_at ContDiffOn.contDiffWithinAt
 
+theorem ContDiffWithinAt.cont_diff_on' {m : ℕ} (hm : (m : ℕ∞) ≤ n)
+    (h : ContDiffWithinAt 𝕜 n f s x) : ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u) :=
+  by
+  rcases h m hm with ⟨t, ht, p, hp⟩
+  rcases mem_nhdsWithin.1 ht with ⟨u, huo, hxu, hut⟩
+  rw [inter_comm] at hut
+  exact ⟨u, huo, hxu, (hp.mono hut).ContDiffOn⟩
+#align cont_diff_within_at.cont_diff_on' ContDiffWithinAt.cont_diff_on'
+
 theorem ContDiffWithinAt.contDiffOn {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : ContDiffWithinAt 𝕜 n f s x) :
     ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ContDiffOn 𝕜 m f u :=
-  by
-  rcases h m hm with ⟨u, u_nhd, p, hp⟩
-  refine' ⟨u ∩ insert x s, Filter.inter_mem u_nhd self_mem_nhdsWithin, inter_subset_right _ _, _⟩
-  intro y hy m' hm'
-  refine' ⟨u ∩ insert x s, _, p, (hp.mono (inter_subset_left _ _)).of_le hm'⟩
-  convert self_mem_nhdsWithin
-  exact insert_eq_of_mem hy
+  let ⟨u, uo, xu, h⟩ := h.cont_diff_on' hm
+  ⟨_, inter_mem_nhdsWithin _ (uo.mem_nhds xu), inter_subset_left _ _, h⟩
 #align cont_diff_within_at.cont_diff_on ContDiffWithinAt.contDiffOn
 
 protected theorem ContDiffWithinAt.eventually {n : ℕ} (h : ContDiffWithinAt 𝕜 n f s x) :
@@ -893,7 +897,7 @@ theorem iteratedFderivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜
           (fderivWithin 𝕜 (I ∘ iteratedFderivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
               E → E[×n + 1]→L[𝕜] F)
             (m 0) (tail m) :=
-        by rw [fderivWithin_congr (hs x hx) A (A x hx)]
+        by rw [fderivWithin_congr A (A x hx)]
       _ =
           (I ∘ fderivWithin 𝕜 (iteratedFderivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
               E → E[×n + 1]→L[𝕜] F)
@@ -934,79 +938,95 @@ theorem norm_iteratedFderivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜
 #align norm_iterated_fderiv_within_fderiv_within norm_iteratedFderivWithin_fderivWithin
 
 @[simp]
-theorem iteratedFderivWithin_one_apply (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (m : Fin 1 → E) :
+theorem iteratedFderivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E) :
     (iteratedFderivWithin 𝕜 1 f s x : (Fin 1 → E) → F) m = (fderivWithin 𝕜 f s x : E → F) (m 0) :=
   by
-  rw [iteratedFderivWithin_succ_apply_right hs hx, iteratedFderivWithin_zero_apply]
+  simp only [iteratedFderivWithin_succ_apply_left, iteratedFderivWithin_zero_eq_comp,
+    (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_fderivWithin h]
   rfl
 #align iterated_fderiv_within_one_apply iteratedFderivWithin_one_apply
 
-/-- If two functions coincide on a set `s` of unique differentiability, then their iterated
-differentials within this set coincide. -/
-theorem iteratedFderivWithin_congr {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hL : ∀ y ∈ s, f₁ y = f y)
-    (hx : x ∈ s) : iteratedFderivWithin 𝕜 n f₁ s x = iteratedFderivWithin 𝕜 n f s x :=
+theorem Filter.EventuallyEq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
+    iteratedFderivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFderivWithin 𝕜 n f t :=
   by
-  induction' n with n IH generalizing x
-  · ext m
-    simp [hL x hx]
-  · have :
-      fderivWithin 𝕜 (fun y => iteratedFderivWithin 𝕜 n f₁ s y) s x =
-        fderivWithin 𝕜 (fun y => iteratedFderivWithin 𝕜 n f s y) s x :=
-      fderivWithin_congr (hs x hx) (fun y hy => IH hy) (IH hx)
-    ext m
-    rw [iteratedFderivWithin_succ_apply_left, iteratedFderivWithin_succ_apply_left, this]
+  induction' n with n ihn
+  · exact h.mono fun y hy => FunLike.ext _ _ fun _ => hy
+  · have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderiv_within' ht
+    apply this.mono
+    intro y hy
+    simp only [iteratedFderivWithin_succ_eq_comp_left, hy, (· ∘ ·)]
+#align filter.eventually_eq.iterated_fderiv_within' Filter.EventuallyEq.iterated_fderiv_within'
+
+protected theorem Filter.EventuallyEq.iteratedFderivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
+    iteratedFderivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFderivWithin 𝕜 n f s :=
+  h.iterated_fderiv_within' Subset.rfl n
+#align filter.eventually_eq.iterated_fderiv_within Filter.EventuallyEq.iteratedFderivWithin
+
+/-- If two functions coincide in a neighborhood of `x` within a set `s` and at `x`, then their
+iterated differentials within this set at `x` coincide. -/
+theorem Filter.EventuallyEq.iteratedFderivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
+    (n : ℕ) : iteratedFderivWithin 𝕜 n f₁ s x = iteratedFderivWithin 𝕜 n f s x :=
+  have : f₁ =ᶠ[𝓝[insert x s] x] f := by simpa [eventually_eq, hx]
+  (this.iterated_fderiv_within' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
+#align filter.eventually_eq.iterated_fderiv_within_eq Filter.EventuallyEq.iteratedFderivWithin_eq
+
+/-- If two functions coincide on a set `s`, then their iterated differentials within this set
+coincide. See also `filter.eventually_eq.iterated_fderiv_within_eq` and
+`filter.eventually_eq.iterated_fderiv_within`. -/
+theorem iteratedFderivWithin_congr (hs : EqOn f₁ f s) (hx : x ∈ s) (n : ℕ) :
+    iteratedFderivWithin 𝕜 n f₁ s x = iteratedFderivWithin 𝕜 n f s x :=
+  (hs.EventuallyEq.filter_mono inf_le_right).iteratedFderivWithin_eq (hs hx) _
 #align iterated_fderiv_within_congr iteratedFderivWithin_congr
 
-/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
-`s` with an open set containing `x`. -/
-theorem iteratedFderivWithin_inter_open {n : ℕ} (hu : IsOpen u) (hs : UniqueDiffOn 𝕜 (s ∩ u))
-    (hx : x ∈ s ∩ u) : iteratedFderivWithin 𝕜 n f (s ∩ u) x = iteratedFderivWithin 𝕜 n f s x :=
-  by
-  induction' n with n IH generalizing x
-  · ext m
-    simp
-  · have A :
-      fderivWithin 𝕜 (fun y => iteratedFderivWithin 𝕜 n f (s ∩ u) y) (s ∩ u) x =
-        fderivWithin 𝕜 (fun y => iteratedFderivWithin 𝕜 n f s y) (s ∩ u) x :=
-      fderivWithin_congr (hs x hx) (fun y hy => IH hy) (IH hx)
-    have B :
-      fderivWithin 𝕜 (fun y => iteratedFderivWithin 𝕜 n f s y) (s ∩ u) x =
-        fderivWithin 𝕜 (fun y => iteratedFderivWithin 𝕜 n f s y) s x :=
-      fderivWithin_inter (IsOpen.mem_nhds hu hx.2)
-        ((uniqueDiffWithinAt_inter (IsOpen.mem_nhds hu hx.2)).1 (hs x hx))
-    ext m
-    rw [iteratedFderivWithin_succ_apply_left, iteratedFderivWithin_succ_apply_left, A, B]
-#align iterated_fderiv_within_inter_open iteratedFderivWithin_inter_open
+/-- If two functions coincide on a set `s`, then their iterated differentials within this set
+coincide. See also `filter.eventually_eq.iterated_fderiv_within_eq` and
+`filter.eventually_eq.iterated_fderiv_within`. -/
+protected theorem Set.EqOn.iteratedFderivWithin (hs : EqOn f₁ f s) (n : ℕ) :
+    EqOn (iteratedFderivWithin 𝕜 n f₁ s) (iteratedFderivWithin 𝕜 n f s) s := fun x hx =>
+  iteratedFderivWithin_congr hs hx n
+#align set.eq_on.iterated_fderiv_within Set.EqOn.iteratedFderivWithin
+
+theorem iteratedFderivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
+    iteratedFderivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFderivWithin 𝕜 n f t :=
+  by
+  induction' n with n ihn generalizing x
+  · rfl
+  · refine' (eventually_nhds_nhdsWithin.2 h).mono fun y hy => _
+    simp only [iteratedFderivWithin_succ_eq_comp_left, (· ∘ ·)]
+    rw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]
+#align iterated_fderiv_within_eventually_congr_set' iteratedFderivWithin_eventually_congr_set'
+
+theorem iteratedFderivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
+    iteratedFderivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFderivWithin 𝕜 n f t :=
+  iteratedFderivWithin_eventually_congr_set' x (h.filter_mono inf_le_left) n
+#align iterated_fderiv_within_eventually_congr_set iteratedFderivWithin_eventually_congr_set
+
+theorem iteratedFderivWithin_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
+    iteratedFderivWithin 𝕜 n f s x = iteratedFderivWithin 𝕜 n f t x :=
+  (iteratedFderivWithin_eventually_congr_set h n).self_of_nhds
+#align iterated_fderiv_within_congr_set iteratedFderivWithin_congr_set
 
 /-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
 `s` with a neighborhood of `x` within `s`. -/
-theorem iteratedFderivWithin_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) (hs : UniqueDiffOn 𝕜 s) (xs : x ∈ s) :
+theorem iteratedFderivWithin_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) :
     iteratedFderivWithin 𝕜 n f (s ∩ u) x = iteratedFderivWithin 𝕜 n f s x :=
-  by
-  obtain ⟨v, v_open, xv, vu⟩ : ∃ v, IsOpen v ∧ x ∈ v ∧ v ∩ s ⊆ u := mem_nhdsWithin.1 hu
-  have A : s ∩ u ∩ v = s ∩ v :=
-    by
-    apply subset.antisymm (inter_subset_inter (inter_subset_left _ _) (subset.refl _))
-    exact fun y ⟨ys, yv⟩ => ⟨⟨ys, vu ⟨yv, ys⟩⟩, yv⟩
-  have : iteratedFderivWithin 𝕜 n f (s ∩ v) x = iteratedFderivWithin 𝕜 n f s x :=
-    iteratedFderivWithin_inter_open v_open (hs.inter v_open) ⟨xs, xv⟩
-  rw [← this]
-  have : iteratedFderivWithin 𝕜 n f (s ∩ u ∩ v) x = iteratedFderivWithin 𝕜 n f (s ∩ u) x :=
-    by
-    refine' iteratedFderivWithin_inter_open v_open _ ⟨⟨xs, vu ⟨xv, xs⟩⟩, xv⟩
-    rw [A]
-    exact hs.inter v_open
-  rw [A] at this
-  rw [← this]
+  iteratedFderivWithin_congr_set (nhdsWithin_eq_iff_eventuallyEq.1 <| nhdsWithin_inter_of_mem' hu) _
 #align iterated_fderiv_within_inter' iteratedFderivWithin_inter'
 
 /-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
 `s` with a neighborhood of `x`. -/
-theorem iteratedFderivWithin_inter {n : ℕ} (hu : u ∈ 𝓝 x) (hs : UniqueDiffOn 𝕜 s) (xs : x ∈ s) :
+theorem iteratedFderivWithin_inter {n : ℕ} (hu : u ∈ 𝓝 x) :
     iteratedFderivWithin 𝕜 n f (s ∩ u) x = iteratedFderivWithin 𝕜 n f s x :=
-  iteratedFderivWithin_inter' (mem_nhdsWithin_of_mem_nhds hu) hs xs
+  iteratedFderivWithin_inter' (mem_nhdsWithin_of_mem_nhds hu)
 #align iterated_fderiv_within_inter iteratedFderivWithin_inter
 
+/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
+`s` with an open set containing `x`. -/
+theorem iteratedFderivWithin_inter_open {n : ℕ} (hu : IsOpen u) (hx : x ∈ u) :
+    iteratedFderivWithin 𝕜 n f (s ∩ u) x = iteratedFderivWithin 𝕜 n f s x :=
+  iteratedFderivWithin_inter (hu.mem_nhds hx)
+#align iterated_fderiv_within_inter_open iteratedFderivWithin_inter_open
+
 @[simp]
 theorem contDiffOn_zero : ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s :=
   by
@@ -1070,14 +1090,14 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
     have : p x m.succ = ftaylorSeriesWithin 𝕜 f s x m.succ :=
       by
       change p x m.succ = iteratedFderivWithin 𝕜 m.succ f s x
-      rw [← iteratedFderivWithin_inter (IsOpen.mem_nhds o_open xo) hs hx]
+      rw [← iteratedFderivWithin_inter_open o_open xo]
       exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩
     rw [← this, ← hasFDerivWithinAt_inter (IsOpen.mem_nhds o_open xo)]
     have A : ∀ y ∈ s ∩ o, p y m = ftaylorSeriesWithin 𝕜 f s y m :=
       by
       rintro y ⟨hy, yo⟩
       change p y m = iteratedFderivWithin 𝕜 m f s y
-      rw [← iteratedFderivWithin_inter (IsOpen.mem_nhds o_open yo) hs hy]
+      rw [← iteratedFderivWithin_inter_open o_open yo]
       exact
         (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn (WithTop.coe_le_coe.2 (Nat.le_succ m))
           (hs.inter o_open) ⟨hy, yo⟩
@@ -1096,7 +1116,7 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
       by
       rintro y ⟨hy, yo⟩
       change p y m = iteratedFderivWithin 𝕜 m f s y
-      rw [← iteratedFderivWithin_inter (IsOpen.mem_nhds o_open yo) hs hy]
+      rw [← iteratedFderivWithin_inter_open o_open yo]
       exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hy, yo⟩
     exact ((Hp.mono ho).cont m le_rfl).congr fun y hy => (A y hy).symm
 #align cont_diff_on.ftaylor_series_within ContDiffOn.ftaylorSeriesWithin
@@ -1140,20 +1160,35 @@ theorem ContDiffOn.differentiableOn_iteratedFderivWithin {m : ℕ} (h : ContDiff
   ((h.ftaylorSeriesWithin hs).fderivWithin m hmn x hx).DifferentiableWithinAt
 #align cont_diff_on.differentiable_on_iterated_fderiv_within ContDiffOn.differentiableOn_iteratedFderivWithin
 
+theorem ContDiffWithinAt.differentiableWithinAt_iteratedFderivWithin {m : ℕ}
+    (h : ContDiffWithinAt 𝕜 n f s x) (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
+    DifferentiableWithinAt 𝕜 (iteratedFderivWithin 𝕜 m f s) s x :=
+  by
+  rcases h.cont_diff_on' (ENat.add_one_le_of_lt hmn) with ⟨u, uo, xu, hu⟩
+  set t := insert x s ∩ u
+  have A : t =ᶠ[𝓝[≠] x] s :=
+    by
+    simp only [set_eventually_eq_iff_inf_principal, ← nhdsWithin_inter']
+    rw [← inter_assoc, nhdsWithin_inter_of_mem', ← diff_eq_compl_inter, insert_diff_of_mem,
+      diff_eq_compl_inter]
+    exacts[rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]
+  have B : iteratedFderivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFderivWithin 𝕜 m f t :=
+    iteratedFderivWithin_eventually_congr_set' _ A.symm _
+  have C : DifferentiableWithinAt 𝕜 (iteratedFderivWithin 𝕜 m f t) t x :=
+    hu.differentiable_on_iterated_fderiv_within (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x
+      ⟨mem_insert _ _, xu⟩
+  rw [differentiableWithinAt_congr_set' _ A] at C
+  exact C.congr_of_eventually_eq (B.filter_mono inf_le_left) B.self_of_nhds
+#align cont_diff_within_at.differentiable_within_at_iterated_fderiv_within ContDiffWithinAt.differentiableWithinAt_iteratedFderivWithin
+
 theorem contDiffOn_iff_continuousOn_differentiableOn (hs : UniqueDiffOn 𝕜 s) :
     ContDiffOn 𝕜 n f s ↔
       (∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFderivWithin 𝕜 m f s x) s) ∧
         ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedFderivWithin 𝕜 m f s x) s :=
-  by
-  constructor
-  · intro h
-    constructor
-    · intro m hm
-      exact h.continuous_on_iterated_fderiv_within hm hs
-    · intro m hm
-      exact h.differentiable_on_iterated_fderiv_within hm hs
-  · intro h
-    exact contDiffOn_of_continuousOn_differentiableOn h.1 h.2
+  ⟨fun h =>
+    ⟨fun m hm => h.continuousOn_iteratedFderivWithin hm hs, fun m hm =>
+      h.differentiableOn_iteratedFderivWithin hm hs⟩,
+    fun h => contDiffOn_of_continuousOn_differentiableOn h.1 h.2⟩
 #align cont_diff_on_iff_continuous_on_differentiable_on contDiffOn_iff_continuousOn_differentiableOn
 
 theorem contDiffOn_succ_of_fderivWithin {n : ℕ} (hf : DifferentiableOn 𝕜 f s)
@@ -1184,7 +1219,7 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
   apply Filter.eventuallyEq_of_mem this fun y hy => _
   have A : fderivWithin 𝕜 f (s ∩ o) y = f' y :=
     ((hff' y (ho hy)).mono ho).fderivWithin (hs.inter o_open y hy)
-  rwa [fderivWithin_inter (IsOpen.mem_nhds o_open hy.2) (hs y hy.1)] at A
+  rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A
 #align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin
 
 theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -190,7 +190,7 @@ structure HasFtaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMul
   zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
   fderivWithin :
     ∀ (m : ℕ) (hm : (m : ℕ∞) < n),
-      ∀ x ∈ s, HasFderivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x
+      ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x
   cont : ∀ (m : ℕ) (hm : (m : ℕ∞) ≤ n), ContinuousOn (fun x => p x m) s
 #align has_ftaylor_series_up_to_on HasFtaylorSeriesUpToOn
 
@@ -269,7 +269,7 @@ theorem hasFtaylorSeriesUpToOn_top_iff :
 theorem hasFtaylorSeriesUpToOn_top_iff' :
     HasFtaylorSeriesUpToOn ∞ f p s ↔
       (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
-        ∀ m : ℕ, ∀ x ∈ s, HasFderivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x :=
+        ∀ m : ℕ, ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x :=
   ⟨-- Everything except for the continuity is trivial:
   fun h => ⟨h.1, fun m => h.2 m (WithTop.coe_lt_top m)⟩, fun h =>
     ⟨h.1, fun m _ => h.2 m, fun m _ x hx =>
@@ -280,8 +280,8 @@ theorem hasFtaylorSeriesUpToOn_top_iff' :
 
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
-theorem HasFtaylorSeriesUpToOn.hasFderivWithinAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
-    (hx : x ∈ s) : HasFderivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x :=
+theorem HasFtaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+    (hx : x ∈ s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x :=
   by
   have A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0) :=
     by
@@ -289,10 +289,10 @@ theorem HasFtaylorSeriesUpToOn.hasFderivWithinAt (h : HasFtaylorSeriesUpToOn n f
     rw [← h.zero_eq y hy]
     rfl
   suffices H :
-    HasFderivWithinAt (fun y => continuousMultilinearCurryFin0 𝕜 E F (p y 0))
+    HasFDerivWithinAt (fun y => continuousMultilinearCurryFin0 𝕜 E F (p y 0))
       (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x
   · exact H.congr A (A x hx)
-  rw [LinearIsometryEquiv.comp_hasFderivWithinAt_iff']
+  rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
   have : ((0 : ℕ) : ℕ∞) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 Nat.zero_lt_one) hn
   convert h.fderiv_within _ this x hx
   ext (y v)
@@ -301,32 +301,32 @@ theorem HasFtaylorSeriesUpToOn.hasFderivWithinAt (h : HasFtaylorSeriesUpToOn n f
   congr with i
   rw [Unique.eq_default i]
   rfl
-#align has_ftaylor_series_up_to_on.has_fderiv_within_at HasFtaylorSeriesUpToOn.hasFderivWithinAt
+#align has_ftaylor_series_up_to_on.has_fderiv_within_at HasFtaylorSeriesUpToOn.hasFDerivWithinAt
 
 theorem HasFtaylorSeriesUpToOn.differentiableOn (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) :
-    DifferentiableOn 𝕜 f s := fun x hx => (h.HasFderivWithinAt hn hx).DifferentiableWithinAt
+    DifferentiableOn 𝕜 f s := fun x hx => (h.HasFDerivWithinAt hn hx).DifferentiableWithinAt
 #align has_ftaylor_series_up_to_on.differentiable_on HasFtaylorSeriesUpToOn.differentiableOn
 
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term
 of order `1` of this series is a derivative of `f` at `x`. -/
-theorem HasFtaylorSeriesUpToOn.hasFderivAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
-    (hx : s ∈ 𝓝 x) : HasFderivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
-  (h.HasFderivWithinAt hn (mem_of_mem_nhds hx)).HasFderivAt hx
-#align has_ftaylor_series_up_to_on.has_fderiv_at HasFtaylorSeriesUpToOn.hasFderivAt
+theorem HasFtaylorSeriesUpToOn.hasFDerivAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+    (hx : s ∈ 𝓝 x) : HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
+  (h.HasFDerivWithinAt hn (mem_of_mem_nhds hx)).HasFDerivAt hx
+#align has_ftaylor_series_up_to_on.has_fderiv_at HasFtaylorSeriesUpToOn.hasFDerivAt
 
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
 in a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. -/
-theorem HasFtaylorSeriesUpToOn.eventually_hasFderivAt (h : HasFtaylorSeriesUpToOn n f p s)
+theorem HasFtaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFtaylorSeriesUpToOn n f p s)
     (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
-    ∀ᶠ y in 𝓝 x, HasFderivAt f (continuousMultilinearCurryFin1 𝕜 E F (p y 1)) y :=
-  (eventually_eventually_nhds.2 hx).mono fun y hy => h.HasFderivAt hn hy
-#align has_ftaylor_series_up_to_on.eventually_has_fderiv_at HasFtaylorSeriesUpToOn.eventually_hasFderivAt
+    ∀ᶠ y in 𝓝 x, HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p y 1)) y :=
+  (eventually_eventually_nhds.2 hx).mono fun y hy => h.HasFDerivAt hn hy
+#align has_ftaylor_series_up_to_on.eventually_has_fderiv_at HasFtaylorSeriesUpToOn.eventually_hasFDerivAt
 
 /-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
 it is differentiable at `x`. -/
 theorem HasFtaylorSeriesUpToOn.differentiableAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
     (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x :=
-  (h.HasFderivAt hn hx).DifferentiableAt
+  (h.HasFDerivAt hn hx).DifferentiableAt
 #align has_ftaylor_series_up_to_on.differentiable_at HasFtaylorSeriesUpToOn.differentiableAt
 
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and
@@ -334,7 +334,7 @@ theorem HasFtaylorSeriesUpToOn.differentiableAt (h : HasFtaylorSeriesUpToOn n f
 theorem hasFtaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
     HasFtaylorSeriesUpToOn (n + 1) f p s ↔
       HasFtaylorSeriesUpToOn n f p s ∧
-        (∀ x ∈ s, HasFderivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧
+        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧
           ContinuousOn (fun x => p x (n + 1)) s :=
   by
   constructor
@@ -364,7 +364,7 @@ for `p 1`, which is a derivative of `f`. -/
 theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
     HasFtaylorSeriesUpToOn (n + 1 : ℕ) f p s ↔
       (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
-        (∀ x ∈ s, HasFderivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
+        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
           HasFtaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
             (fun x => (p x).shift) s :=
   by
@@ -380,10 +380,10 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
         rw [WithTop.coe_lt_coe] at hm⊢
         exact nat.lt_succ_iff.mpr hm
       change
-        HasFderivWithinAt
+        HasFDerivWithinAt
           ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ)
           (p x m.succ.succ).curryRight.curryLeft s x
-      rw [LinearIsometryEquiv.comp_hasFderivWithinAt_iff']
+      rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
       convert H.fderiv_within _ A x hx
       ext (y v)
       change
@@ -410,11 +410,11 @@ theorem hasFtaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
           rw [WithTop.coe_lt_coe] at hm⊢
           exact Nat.lt_of_succ_lt_succ hm
         have :
-          HasFderivWithinAt
+          HasFDerivWithinAt
             ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ fun y : E => p y m.succ)
             ((p x).shift m.succ).curryLeft s x :=
           Htaylor.fderiv_within _ A x hx
-        rw [LinearIsometryEquiv.comp_hasFderivWithinAt_iff'] at this
+        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this
         convert this
         ext (y v)
         change
@@ -588,11 +588,11 @@ theorem ContDiffWithinAt.differentiableWithinAt (h : ContDiffWithinAt 𝕜 n f s
 #align cont_diff_within_at.differentiable_within_at ContDiffWithinAt.differentiableWithinAt
 
 /-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/
-theorem contDiffWithinAt_succ_iff_hasFderivWithinAt {n : ℕ} :
+theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
     ContDiffWithinAt 𝕜 (n + 1 : ℕ) f s x ↔
       ∃ u ∈ 𝓝[insert x s] x,
         ∃ f' : E → E →L[𝕜] F,
-          (∀ x ∈ u, HasFderivWithinAt f (f' x) u x) ∧ ContDiffWithinAt 𝕜 n f' u x :=
+          (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffWithinAt 𝕜 n f' u x :=
   by
   constructor
   · intro h
@@ -617,9 +617,9 @@ theorem contDiffWithinAt_succ_iff_hasFderivWithinAt {n : ℕ} :
     · rw [hasFtaylorSeriesUpToOn_succ_iff_right]
       refine' ⟨fun y hy => rfl, fun y hy => _, _⟩
       · change
-          HasFderivWithinAt (fun z => (continuousMultilinearCurryFin0 𝕜 E F).symm (f z))
+          HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 𝕜 E F).symm (f z))
             (FormalMultilinearSeries.unshift (p' y) (f y) 1).curryLeft (v ∩ u) y
-        rw [LinearIsometryEquiv.comp_hasFderivWithinAt_iff']
+        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
         convert(f'_eq_deriv y hy.2).mono (inter_subset_right v u)
         rw [← Hp'.zero_eq y hy.1]
         ext z
@@ -639,7 +639,7 @@ theorem contDiffWithinAt_succ_iff_hasFderivWithinAt {n : ℕ} :
                 (@snoc k (fun i : Fin k.succ => E) v y (last k)) =
               p' x k v y
           rw [snoc_last, init_snoc]
-#align cont_diff_within_at_succ_iff_has_fderiv_within_at contDiffWithinAt_succ_iff_hasFderivWithinAt
+#align cont_diff_within_at_succ_iff_has_fderiv_within_at contDiffWithinAt_succ_iff_hasFDerivWithinAt
 
 /-- A version of `cont_diff_within_at_succ_iff_has_fderiv_within_at` where all derivatives
   are taken within the same set. -/
@@ -648,7 +648,7 @@ theorem contDiffWithinAt_succ_iff_has_fderiv_within_at' {n : ℕ} :
       ∃ u ∈ 𝓝[insert x s] x,
         u ⊆ insert x s ∧
           ∃ f' : E → E →L[𝕜] F,
-            (∀ x ∈ u, HasFderivWithinAt f (f' x) s x) ∧ ContDiffWithinAt 𝕜 n f' s x :=
+            (∀ x ∈ u, HasFDerivWithinAt f (f' x) s x) ∧ ContDiffWithinAt 𝕜 n f' s x :=
   by
   refine' ⟨fun hf => _, _⟩
   · obtain ⟨u, hu, f', huf', hf'⟩ := cont_diff_within_at_succ_iff_has_fderiv_within_at.mp hf
@@ -661,7 +661,7 @@ theorem contDiffWithinAt_succ_iff_has_fderiv_within_at' {n : ℕ} :
       refine' mem_of_superset _ (inter_subset_inter_left _ (subset_insert _ _))
       refine' inter_mem_nhdsWithin _ (hw.mem_nhds hy.2)
     · exact hf'.mono_of_mem (nhdsWithin_mono _ (subset_insert _ _) hu)
-  · rw [← contDiffWithinAt_insert, contDiffWithinAt_succ_iff_hasFderivWithinAt,
+  · rw [← contDiffWithinAt_insert, contDiffWithinAt_succ_iff_hasFDerivWithinAt,
       insert_eq_of_mem (mem_insert _ _)]
     rintro ⟨u, hu, hus, f', huf', hf'⟩
     refine' ⟨u, hu, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus⟩
@@ -783,11 +783,11 @@ theorem contDiffOn_of_locally_contDiffOn
 #align cont_diff_on_of_locally_cont_diff_on contDiffOn_of_locally_contDiffOn
 
 /-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/
-theorem contDiffOn_succ_iff_hasFderivWithinAt {n : ℕ} :
+theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       ∀ x ∈ s,
         ∃ u ∈ 𝓝[insert x s] x,
-          ∃ f' : E → E →L[𝕜] F, (∀ x ∈ u, HasFderivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u :=
+          ∃ f' : E → E →L[𝕜] F, (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u :=
   by
   constructor
   · intro h x hx
@@ -801,11 +801,11 @@ theorem contDiffOn_succ_iff_hasFderivWithinAt {n : ℕ} :
     convert self_mem_nhdsWithin
     exact insert_eq_of_mem hz
   · intro h x hx
-    rw [contDiffWithinAt_succ_iff_hasFderivWithinAt]
+    rw [contDiffWithinAt_succ_iff_hasFDerivWithinAt]
     rcases h x hx with ⟨u, u_nhbd, f', hu, hf'⟩
     have : x ∈ u := mem_of_mem_nhdsWithin (mem_insert _ _) u_nhbd
     exact ⟨u, u_nhbd, f', hu, hf' x this⟩
-#align cont_diff_on_succ_iff_has_fderiv_within_at contDiffOn_succ_iff_hasFderivWithinAt
+#align cont_diff_on_succ_iff_has_fderiv_within_at contDiffOn_succ_iff_hasFDerivWithinAt
 
 /-! ### Iterated derivative within a set -/
 
@@ -1045,7 +1045,7 @@ theorem HasFtaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
   · rw [h.zero_eq' hx, iteratedFderivWithin_zero_eq_comp]
   · have A : (m : ℕ∞) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (lt_add_one m)) hmn
     have :
-      HasFderivWithinAt (fun y : E => iteratedFderivWithin 𝕜 m f s y)
+      HasFDerivWithinAt (fun y : E => iteratedFderivWithin 𝕜 m f s y)
         (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x :=
       (h.fderiv_within m A x hx).congr (fun y hy => (IH (le_of_lt A) hy).symm)
         (IH (le_of_lt A) hx).symm
@@ -1072,7 +1072,7 @@ theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueD
       change p x m.succ = iteratedFderivWithin 𝕜 m.succ f s x
       rw [← iteratedFderivWithin_inter (IsOpen.mem_nhds o_open xo) hs hx]
       exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩
-    rw [← this, ← hasFderivWithinAt_inter (IsOpen.mem_nhds o_open xo)]
+    rw [← this, ← hasFDerivWithinAt_inter (IsOpen.mem_nhds o_open xo)]
     have A : ∀ y ∈ s ∩ o, p y m = ftaylorSeriesWithin 𝕜 f s y m :=
       by
       rintro y ⟨hy, yo⟩
@@ -1114,7 +1114,7 @@ theorem contDiffOn_of_continuousOn_differentiableOn
     simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,
       iteratedFderivWithin_zero_apply]
   · intro k hk y hy
-    convert(Hdiff k (lt_of_lt_of_le hk hm) y hy).HasFderivWithinAt
+    convert(Hdiff k (lt_of_lt_of_le hk hm) y hy).HasFDerivWithinAt
     simp only [ftaylorSeriesWithin, iteratedFderivWithin_succ_eq_comp_left,
       ContinuousLinearEquiv.coe_apply, Function.comp_apply, coeFn_coeBase]
     exact ContinuousLinearMap.curry_uncurryLeft _
@@ -1160,9 +1160,9 @@ theorem contDiffOn_succ_of_fderivWithin {n : ℕ} (hf : DifferentiableOn 𝕜 f
     (h : ContDiffOn 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) : ContDiffOn 𝕜 (n + 1 : ℕ) f s :=
   by
   intro x hx
-  rw [contDiffWithinAt_succ_iff_hasFderivWithinAt, insert_eq_of_mem hx]
+  rw [contDiffWithinAt_succ_iff_hasFDerivWithinAt, insert_eq_of_mem hx]
   exact
-    ⟨s, self_mem_nhdsWithin, fderivWithin 𝕜 f s, fun y hy => (hf y hy).HasFderivWithinAt, h x hx⟩
+    ⟨s, self_mem_nhdsWithin, fderivWithin 𝕜 f s, fun y hy => (hf y hy).HasFDerivWithinAt, h x hx⟩
 #align cont_diff_on_succ_of_fderiv_within contDiffOn_succ_of_fderivWithin
 
 /-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is
@@ -1173,7 +1173,7 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
   by
   refine' ⟨fun H => _, fun h => contDiffOn_succ_of_fderivWithin h.1 h.2⟩
   refine' ⟨H.differentiable_on (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)), fun x hx => _⟩
-  rcases contDiffWithinAt_succ_iff_hasFderivWithinAt.1 (H x hx) with ⟨u, hu, f', hff', hf'⟩
+  rcases contDiffWithinAt_succ_iff_hasFDerivWithinAt.1 (H x hx) with ⟨u, hu, f', hff', hf'⟩
   rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
   rw [inter_comm, insert_eq_of_mem hx] at ho
   have := hf'.mono ho
@@ -1189,10 +1189,10 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
 
 theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
-      ∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFderivWithinAt f (f' x) s x :=
+      ∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFDerivWithinAt f (f' x) s x :=
   by
   rw [contDiffOn_succ_iff_fderivWithin hs]
-  refine' ⟨fun h => ⟨fderivWithin 𝕜 f s, h.2, fun x hx => (h.1 x hx).HasFderivWithinAt⟩, fun h => _⟩
+  refine' ⟨fun h => ⟨fderivWithin 𝕜 f s, h.2, fun x hx => (h.1 x hx).HasFDerivWithinAt⟩, fun h => _⟩
   rcases h with ⟨f', h1, h2⟩
   refine' ⟨fun x hx => (h2 x hx).DifferentiableWithinAt, fun x hx => _⟩
   exact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) hx
@@ -1281,7 +1281,7 @@ derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a pred
 structure HasFtaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
   Prop where
   zero_eq : ∀ x, (p x 0).uncurry0 = f x
-  fderiv : ∀ (m : ℕ) (hm : (m : ℕ∞) < n), ∀ x, HasFderivAt (fun y => p y m) (p x m.succ).curryLeft x
+  fderiv : ∀ (m : ℕ) (hm : (m : ℕ∞) < n), ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
   cont : ∀ (m : ℕ) (hm : (m : ℕ∞) ≤ n), Continuous fun x => p x m
 #align has_ftaylor_series_up_to HasFtaylorSeriesUpTo
 
@@ -1301,7 +1301,7 @@ theorem hasFtaylorSeriesUpToOn_univ_iff :
     constructor
     · exact fun x => H.zero_eq x (mem_univ x)
     · intro m hm x
-      rw [← hasFderivWithinAt_univ]
+      rw [← hasFDerivWithinAt_univ]
       exact H.fderiv_within m hm x (mem_univ x)
     · intro m hm
       rw [continuous_iff_continuousOn_univ]
@@ -1310,7 +1310,7 @@ theorem hasFtaylorSeriesUpToOn_univ_iff :
     constructor
     · exact fun x hx => H.zero_eq x
     · intro m hm x hx
-      rw [hasFderivWithinAt_univ]
+      rw [hasFDerivWithinAt_univ]
       exact H.fderiv m hm x
     · intro m hm
       rw [← continuous_iff_continuousOn_univ]
@@ -1352,23 +1352,23 @@ theorem hasFtaylorSeriesUpTo_top_iff :
 theorem hasFtaylorSeriesUpTo_top_iff' :
     HasFtaylorSeriesUpTo ∞ f p ↔
       (∀ x, (p x 0).uncurry0 = f x) ∧
-        ∀ (m : ℕ) (x), HasFderivAt (fun y => p y m) (p x m.succ).curryLeft x :=
+        ∀ (m : ℕ) (x), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x :=
   by
   simp only [← hasFtaylorSeriesUpToOn_univ_iff, hasFtaylorSeriesUpToOn_top_iff', mem_univ,
-    forall_true_left, hasFderivWithinAt_univ]
+    forall_true_left, hasFDerivWithinAt_univ]
 #align has_ftaylor_series_up_to_top_iff' hasFtaylorSeriesUpTo_top_iff'
 
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
-theorem HasFtaylorSeriesUpTo.hasFderivAt (h : HasFtaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
-    HasFderivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
+theorem HasFtaylorSeriesUpTo.hasFDerivAt (h : HasFtaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
+    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
   by
-  rw [← hasFderivWithinAt_univ]
-  exact (hasFtaylorSeriesUpToOn_univ_iff.2 h).HasFderivWithinAt hn (mem_univ _)
-#align has_ftaylor_series_up_to.has_fderiv_at HasFtaylorSeriesUpTo.hasFderivAt
+  rw [← hasFDerivWithinAt_univ]
+  exact (hasFtaylorSeriesUpToOn_univ_iff.2 h).HasFDerivWithinAt hn (mem_univ _)
+#align has_ftaylor_series_up_to.has_fderiv_at HasFtaylorSeriesUpTo.hasFDerivAt
 
 theorem HasFtaylorSeriesUpTo.differentiable (h : HasFtaylorSeriesUpTo n f p) (hn : 1 ≤ n) :
-    Differentiable 𝕜 f := fun x => (h.HasFderivAt hn x).DifferentiableAt
+    Differentiable 𝕜 f := fun x => (h.HasFDerivAt hn x).DifferentiableAt
 #align has_ftaylor_series_up_to.differentiable HasFtaylorSeriesUpTo.differentiable
 
 /-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
@@ -1376,12 +1376,12 @@ for `p 1`, which is a derivative of `f`. -/
 theorem hasFtaylorSeriesUpTo_succ_iff_right {n : ℕ} :
     HasFtaylorSeriesUpTo (n + 1 : ℕ) f p ↔
       (∀ x, (p x 0).uncurry0 = f x) ∧
-        (∀ x, HasFderivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
+        (∀ x, HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
           HasFtaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1)) fun x =>
             (p x).shift :=
   by
   simp only [hasFtaylorSeriesUpToOn_succ_iff_right, ← hasFtaylorSeriesUpToOn_univ_iff, mem_univ,
-    forall_true_left, hasFderivWithinAt_univ]
+    forall_true_left, hasFDerivWithinAt_univ]
 #align has_ftaylor_series_up_to_succ_iff_right hasFtaylorSeriesUpTo_succ_iff_right
 
 /-! ### Smooth functions at a point -/
@@ -1434,11 +1434,11 @@ theorem ContDiffAt.differentiableAt (h : ContDiffAt 𝕜 n f x) (hn : 1 ≤ n) :
 #align cont_diff_at.differentiable_at ContDiffAt.differentiableAt
 
 /-- A function is `C^(n + 1)` at a point iff locally, it has a derivative which is `C^n`. -/
-theorem contDiffAt_succ_iff_hasFderivAt {n : ℕ} :
+theorem contDiffAt_succ_iff_hasFDerivAt {n : ℕ} :
     ContDiffAt 𝕜 (n + 1 : ℕ) f x ↔
-      ∃ f' : E → E →L[𝕜] F, (∃ u ∈ 𝓝 x, ∀ x ∈ u, HasFderivAt f (f' x) x) ∧ ContDiffAt 𝕜 n f' x :=
+      ∃ f' : E → E →L[𝕜] F, (∃ u ∈ 𝓝 x, ∀ x ∈ u, HasFDerivAt f (f' x) x) ∧ ContDiffAt 𝕜 n f' x :=
   by
-  rw [← contDiffWithinAt_univ, contDiffWithinAt_succ_iff_hasFderivWithinAt]
+  rw [← contDiffWithinAt_univ, contDiffWithinAt_succ_iff_hasFDerivWithinAt]
   simp only [nhdsWithin_univ, exists_prop, mem_univ, insert_eq_of_mem]
   constructor
   · rintro ⟨u, H, f', h_fderiv, h_cont_diff⟩
@@ -1446,13 +1446,13 @@ theorem contDiffAt_succ_iff_hasFderivAt {n : ℕ} :
     refine' ⟨f', ⟨t, _⟩, h_cont_diff.cont_diff_at H⟩
     refine' ⟨mem_nhds_iff.mpr ⟨t, subset.rfl, ht, hxt⟩, _⟩
     intro y hyt
-    refine' (h_fderiv y (htu hyt)).HasFderivAt _
+    refine' (h_fderiv y (htu hyt)).HasFDerivAt _
     exact mem_nhds_iff.mpr ⟨t, htu, ht, hyt⟩
   · rintro ⟨f', ⟨u, H, h_fderiv⟩, h_cont_diff⟩
     refine' ⟨u, H, f', _, h_cont_diff.cont_diff_within_at⟩
     intro x hxu
-    exact (h_fderiv x hxu).HasFderivWithinAt
-#align cont_diff_at_succ_iff_has_fderiv_at contDiffAt_succ_iff_hasFderivAt
+    exact (h_fderiv x hxu).HasFDerivWithinAt
+#align cont_diff_at_succ_iff_has_fderiv_at contDiffAt_succ_iff_hasFDerivAt
 
 protected theorem ContDiffAt.eventually {n : ℕ} (h : ContDiffAt 𝕜 n f x) :
     ∀ᶠ y in 𝓝 x, ContDiffAt 𝕜 n f y := by simpa [nhdsWithin_univ] using h.eventually
@@ -1529,9 +1529,9 @@ theorem contDiffAt_zero : ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, Continuous
 
 theorem contDiffAt_one_iff :
     ContDiffAt 𝕜 1 f x ↔
-      ∃ f' : E → E →L[𝕜] F, ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFderivAt f (f' x) x :=
+      ∃ f' : E → E →L[𝕜] F, ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFDerivAt f (f' x) x :=
   by
-  simp_rw [show (1 : ℕ∞) = (0 + 1 : ℕ) from (zero_add 1).symm, contDiffAt_succ_iff_hasFderivAt,
+  simp_rw [show (1 : ℕ∞) = (0 + 1 : ℕ) from (zero_add 1).symm, contDiffAt_succ_iff_hasFDerivAt,
     show ((0 : ℕ) : ℕ∞) = 0 from rfl, contDiffAt_zero,
     exists_mem_and_iff antitone_bforall antitone_continuousOn, and_comm']
 #align cont_diff_at_one_iff contDiffAt_one_iff
@@ -1566,9 +1566,9 @@ theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤
 /-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/
 theorem contDiff_succ_iff_has_fderiv {n : ℕ} :
     ContDiff 𝕜 (n + 1 : ℕ) f ↔
-      ∃ f' : E → E →L[𝕜] F, ContDiff 𝕜 n f' ∧ ∀ x, HasFderivAt f (f' x) x :=
+      ∃ f' : E → E →L[𝕜] F, ContDiff 𝕜 n f' ∧ ∀ x, HasFDerivAt f (f' x) x :=
   by
-  simp only [← contDiffOn_univ, ← hasFderivWithinAt_univ,
+  simp only [← contDiffOn_univ, ← hasFDerivWithinAt_univ,
     contDiffOn_succ_iff_has_fderiv_within uniqueDiffOn_univ, Set.mem_univ, forall_true_left]
 #align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_has_fderiv
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -4,11 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
+! leanprover-community/mathlib commit e3fb84046afd187b710170887195d50bada934ee
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.Analysis.Calculus.Fderiv
+import Mathbin.Analysis.Calculus.Fderiv.Add
+import Mathbin.Analysis.Calculus.Fderiv.Mul
+import Mathbin.Analysis.Calculus.Fderiv.Equiv
+import Mathbin.Analysis.Calculus.Fderiv.RestrictScalars
 import Mathbin.Analysis.Calculus.FormalMultilinearSeries
 
 /-!

mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab

@@ -4,14 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit 2cf3ee29d9361ee308a14a46d820a49b8856d041
+! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Calculus.Fderiv
-import Mathbin.Analysis.NormedSpace.Multilinear
 import Mathbin.Analysis.Calculus.FormalMultilinearSeries
-import Mathbin.Data.Enat.Basic
 
 /-!
 # Higher differentiability

mathlib commit https://github.com/leanprover-community/mathlib/commit/09079525fd01b3dda35e96adaa08d2f943e1648c

@@ -167,7 +167,7 @@ open Classical BigOperators NNReal Topology
 -- mathport name: «expr∞»
 local notation "∞" => (⊤ : ℕ∞)
 
-attribute [local instance]
+attribute [local instance 1001]
   NormedAddCommGroup.toAddCommGroup NormedSpace.toModule' AddCommGroup.toAddCommMonoid
 
 open Set Fin Filter Function

mathlib commit https://github.com/leanprover-community/mathlib/commit/0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit a493616c740a3252e4cd0e4d0851984946b7b268
+! leanprover-community/mathlib commit 2cf3ee29d9361ee308a14a46d820a49b8856d041
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -263,6 +263,20 @@ theorem hasFtaylorSeriesUpToOn_top_iff :
       apply (H m).cont m le_rfl
 #align has_ftaylor_series_up_to_on_top_iff hasFtaylorSeriesUpToOn_top_iff
 
+/-- In the case that `n = ∞` we don't need the continuity assumption in
+`has_ftaylor_series_up_to_on`. -/
+theorem hasFtaylorSeriesUpToOn_top_iff' :
+    HasFtaylorSeriesUpToOn ∞ f p s ↔
+      (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
+        ∀ m : ℕ, ∀ x ∈ s, HasFderivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x :=
+  ⟨-- Everything except for the continuity is trivial:
+  fun h => ⟨h.1, fun m => h.2 m (WithTop.coe_lt_top m)⟩, fun h =>
+    ⟨h.1, fun m _ => h.2 m, fun m _ x hx =>
+      (-- The continuity follows from the existence of a derivative:
+            h.2
+          m x hx).ContinuousWithinAt⟩⟩
+#align has_ftaylor_series_up_to_on_top_iff' hasFtaylorSeriesUpToOn_top_iff'
+
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
 theorem HasFtaylorSeriesUpToOn.hasFderivWithinAt (h : HasFtaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
@@ -1327,6 +1341,22 @@ theorem hasFtaylorSeriesUpTo_zero_iff :
     hasFtaylorSeriesUpToOn_zero_iff]
 #align has_ftaylor_series_up_to_zero_iff hasFtaylorSeriesUpTo_zero_iff
 
+theorem hasFtaylorSeriesUpTo_top_iff :
+    HasFtaylorSeriesUpTo ∞ f p ↔ ∀ n : ℕ, HasFtaylorSeriesUpTo n f p := by
+  simp only [← hasFtaylorSeriesUpToOn_univ_iff, hasFtaylorSeriesUpToOn_top_iff]
+#align has_ftaylor_series_up_to_top_iff hasFtaylorSeriesUpTo_top_iff
+
+/-- In the case that `n = ∞` we don't need the continuity assumption in
+`has_ftaylor_series_up_to`. -/
+theorem hasFtaylorSeriesUpTo_top_iff' :
+    HasFtaylorSeriesUpTo ∞ f p ↔
+      (∀ x, (p x 0).uncurry0 = f x) ∧
+        ∀ (m : ℕ) (x), HasFderivAt (fun y => p y m) (p x m.succ).curryLeft x :=
+  by
+  simp only [← hasFtaylorSeriesUpToOn_univ_iff, hasFtaylorSeriesUpToOn_top_iff', mem_univ,
+    forall_true_left, hasFderivWithinAt_univ]
+#align has_ftaylor_series_up_to_top_iff' hasFtaylorSeriesUpTo_top_iff'
+
 /-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
 series is a derivative of `f`. -/
 theorem HasFtaylorSeriesUpTo.hasFderivAt (h : HasFtaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit 0a0b3b4148b35efb8b8a38118517c5a1d30d0e69
+! leanprover-community/mathlib commit a493616c740a3252e4cd0e4d0851984946b7b268
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1172,6 +1172,17 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
   rwa [fderivWithin_inter (IsOpen.mem_nhds o_open hy.2) (hs y hy.1)] at A
 #align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin
 
+theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
+    ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
+      ∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFderivWithinAt f (f' x) s x :=
+  by
+  rw [contDiffOn_succ_iff_fderivWithin hs]
+  refine' ⟨fun h => ⟨fderivWithin 𝕜 f s, h.2, fun x hx => (h.1 x hx).HasFderivWithinAt⟩, fun h => _⟩
+  rcases h with ⟨f', h1, h2⟩
+  refine' ⟨fun x hx => (h2 x hx).DifferentiableWithinAt, fun x hx => _⟩
+  exact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) hx
+#align cont_diff_on_succ_iff_has_fderiv_within contDiffOn_succ_iff_has_fderiv_within
+
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr «expr ∧ »(_, _)]] -/
 /-- A function is `C^(n + 1)` on an open domain if and only if it is
 differentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -/
@@ -1521,6 +1532,15 @@ theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤
   exact contDiffOn_iff_forall_nat_le
 #align cont_diff_iff_forall_nat_le contDiff_iff_forall_nat_le
 
+/-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/
+theorem contDiff_succ_iff_has_fderiv {n : ℕ} :
+    ContDiff 𝕜 (n + 1 : ℕ) f ↔
+      ∃ f' : E → E →L[𝕜] F, ContDiff 𝕜 n f' ∧ ∀ x, HasFderivAt f (f' x) x :=
+  by
+  simp only [← contDiffOn_univ, ← hasFderivWithinAt_univ,
+    contDiffOn_succ_iff_has_fderiv_within uniqueDiffOn_univ, Set.mem_univ, forall_true_left]
+#align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_has_fderiv
+
 /-! ### Iterated derivative -/
 
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/c9236f47f5b9df573443aa499c0d3968769628b7

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit 284fdd2962e67d2932fa3a79ce19fcf92d38e228
+! leanprover-community/mathlib commit 0a0b3b4148b35efb8b8a38118517c5a1d30d0e69
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -669,6 +669,15 @@ def ContDiffOn (n : ℕ∞) (f : E → F) (s : Set E) : Prop :=
 
 variable {𝕜}
 
+theorem HasFtaylorSeriesUpToOn.contDiffOn {f' : E → FormalMultilinearSeries 𝕜 E F}
+    (hf : HasFtaylorSeriesUpToOn n f f' s) : ContDiffOn 𝕜 n f s :=
+  by
+  intro x hx m hm
+  use s
+  simp only [Set.insert_eq_of_mem hx, self_mem_nhdsWithin, true_and_iff]
+  exact ⟨f', hf.of_le hm⟩
+#align has_ftaylor_series_up_to_on.cont_diff_on HasFtaylorSeriesUpToOn.contDiffOn
+
 theorem ContDiffOn.contDiffWithinAt (h : ContDiffOn 𝕜 n f s) (hx : x ∈ s) :
     ContDiffWithinAt 𝕜 n f s x :=
   h x hx
@@ -1422,6 +1431,12 @@ def ContDiff (n : ℕ∞) (f : E → F) : Prop :=
 
 variable {𝕜}
 
+/-- If `f` has a Taylor series up to `n`, then it is `C^n`. -/
+theorem HasFtaylorSeriesUpTo.contDiff {f' : E → FormalMultilinearSeries 𝕜 E F}
+    (hf : HasFtaylorSeriesUpTo n f f') : ContDiff 𝕜 n f :=
+  ⟨f', hf⟩
+#align has_ftaylor_series_up_to.cont_diff HasFtaylorSeriesUpTo.contDiff
+
 theorem contDiffOn_univ : ContDiffOn 𝕜 n f univ ↔ ContDiff 𝕜 n f :=
   by
   constructor

mathlib commit https://github.com/leanprover-community/mathlib/commit/284fdd2962e67d2932fa3a79ce19fcf92d38e228

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit dd6388c44e6f6b4547070b887c5905d5cfe6c9f8
+! leanprover-community/mathlib commit 284fdd2962e67d2932fa3a79ce19fcf92d38e228
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -613,6 +613,7 @@ theorem contDiffWithinAt_succ_iff_hasFderivWithinAt {n : ℕ} :
             ((p' y 0) 0) z
         unfold_coes
         congr
+        decide
       · convert(Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1
         · ext (x y)
           change p' x 0 (init (@snoc 0 (fun i : Fin 1 => E) 0 y)) y = p' x 0 0 y

mathlib commit https://github.com/leanprover-community/mathlib/commit/dd6388c44e6f6b4547070b887c5905d5cfe6c9f8

Could we have an open linter, that checked for unused opened namespaces?

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -157,7 +157,7 @@ derivative, differentiability, higher derivative, `C^n`, multilinear, Taylor ser
 noncomputable section
 
 open scoped Classical
-open BigOperators NNReal Topology Filter
+open NNReal Topology Filter
 
 local notation "∞" => (⊤ : ℕ∞)
 

We already have the localized version to a set, but not the global version.

Also rename the localized version to a better name.

@@ -995,7 +995,7 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
 
 /-- On a set with unique differentiability, any choice of iterated differential has to coincide
 with the one we have chosen in `iteratedFDerivWithin 𝕜 m f s`. -/
-theorem HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
+theorem HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
     (h : HasFTaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s)
     (hx : x ∈ s) : p x m = iteratedFDerivWithin 𝕜 m f s x := by
   induction' m with m IH generalizing x
@@ -1008,7 +1008,10 @@ theorem HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
         (IH (le_of_lt A) hx).symm
     rw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderivWithin (hs x hx)]
     exact (ContinuousMultilinearMap.uncurry_curryLeft _).symm
-#align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn
+#align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
+
+@[deprecated] alias HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn :=
+  HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn -- 2024-03-28
 
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
 `ftaylorSeriesWithin 𝕜 f s` as a Taylor series up to order `n` in `s`. -/
@@ -1604,6 +1607,13 @@ theorem iteratedFDerivWithin_univ {n : ℕ} :
     rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]
 #align iterated_fderiv_within_univ iteratedFDerivWithin_univ
 
+theorem HasFTaylorSeriesUpTo.eq_iteratedFDeriv
+    (h : HasFTaylorSeriesUpTo n f p) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (x : E) :
+    p x m = iteratedFDeriv 𝕜 m f x := by
+  rw [← iteratedFDerivWithin_univ]
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
+  exact h.eq_iteratedFDerivWithin_of_uniqueDiffOn hmn uniqueDiffOn_univ (mem_univ _)
+
 /-- In an open set, the iterated derivative within this set coincides with the global iterated
 derivative. -/
 theorem iteratedFDerivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :

Classifies by adding issue number #10756 to porting notes claiming anything semantically equivalent to:

• "new lemma"
• "added lemma"

@@ -1351,7 +1351,7 @@ theorem ContDiffWithinAt.contDiffAt (h : ContDiffWithinAt 𝕜 n f s x) (hx : s
     ContDiffAt 𝕜 n f x := by rwa [ContDiffAt, ← contDiffWithinAt_inter hx, univ_inter]
 #align cont_diff_within_at.cont_diff_at ContDiffWithinAt.contDiffAt
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem ContDiffOn.contDiffAt (h : ContDiffOn 𝕜 n f s) (hx : s ∈ 𝓝 x) :
     ContDiffAt 𝕜 n f x :=
   (h _ (mem_of_mem_nhds hx)).contDiffAt hx

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -156,7 +156,8 @@ derivative, differentiability, higher derivative, `C^n`, multilinear, Taylor ser
 
 noncomputable section
 
-open Classical BigOperators NNReal Topology Filter
+open scoped Classical
+open BigOperators NNReal Topology Filter
 
 local notation "∞" => (⊤ : ℕ∞)
 

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -1350,7 +1350,7 @@ theorem ContDiffWithinAt.contDiffAt (h : ContDiffWithinAt 𝕜 n f s x) (hx : s
     ContDiffAt 𝕜 n f x := by rwa [ContDiffAt, ← contDiffWithinAt_inter hx, univ_inter]
 #align cont_diff_within_at.cont_diff_at ContDiffWithinAt.contDiffAt
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem ContDiffOn.contDiffAt (h : ContDiffOn 𝕜 n f s) (hx : s ∈ 𝓝 x) :
     ContDiffAt 𝕜 n f x :=
   (h _ (mem_of_mem_nhds hx)).contDiffAt hx

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

@@ -612,12 +612,12 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : ℕ} :
       fun y hy => _, _⟩
     · refine' ((huf' y <| hwu hy).mono hwu).mono_of_mem _
       refine' mem_of_superset _ (inter_subset_inter_left _ (subset_insert _ _))
-      refine' inter_mem_nhdsWithin _ (hw.mem_nhds hy.2)
+      exact inter_mem_nhdsWithin _ (hw.mem_nhds hy.2)
     · exact hf'.mono_of_mem (nhdsWithin_mono _ (subset_insert _ _) hu)
   · rw [← contDiffWithinAt_insert, contDiffWithinAt_succ_iff_hasFDerivWithinAt,
       insert_eq_of_mem (mem_insert _ _)]
     rintro ⟨u, hu, hus, f', huf', hf'⟩
-    refine' ⟨u, hu, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus⟩
+    exact ⟨u, hu, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus⟩
 #align cont_diff_within_at_succ_iff_has_fderiv_within_at' contDiffWithinAt_succ_iff_hasFDerivWithinAt'
 
 /-! ### Smooth functions within a set -/

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -269,8 +269,7 @@ theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f
   have A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0) := fun y hy ↦
     (h.zero_eq y hy).symm
   suffices H : HasFDerivWithinAt (continuousMultilinearCurryFin0 𝕜 E F ∘ (p · 0))
-    (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x
-  · exact H.congr A (A x hx)
+    (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x from H.congr A (A x hx)
   rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
   have : ((0 : ℕ) : ℕ∞) < n := zero_lt_one.trans_le hn
   convert h.fderivWithin _ this x hx
@@ -343,8 +342,8 @@ theorem HasFTaylorSeriesUpToOn.shift_of_succ
   · intro x _
     rfl
   · intro m (hm : (m : ℕ∞) < n) x (hx : x ∈ s)
-    have A : (m.succ : ℕ∞) < n.succ
-    · rw [Nat.cast_lt] at hm ⊢
+    have A : (m.succ : ℕ∞) < n.succ := by
+      rw [Nat.cast_lt] at hm ⊢
       exact Nat.succ_lt_succ hm
     change HasFDerivWithinAt ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ (p · m.succ))
       (p x m.succ.succ).curryRight.curryLeft s x
@@ -354,8 +353,8 @@ theorem HasFTaylorSeriesUpToOn.shift_of_succ
     change p x (m + 2) (snoc (cons y (init v)) (v (last _))) = p x (m + 2) (cons y v)
     rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
   · intro m (hm : (m : ℕ∞) ≤ n)
-    suffices A : ContinuousOn (p · (m + 1)) s
-    · exact ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm).continuous.comp_continuousOn A
+    suffices A : ContinuousOn (p · (m + 1)) s from
+      ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm).continuous.comp_continuousOn A
     refine H.cont _ ?_
     rw [Nat.cast_le] at hm ⊢
     exact Nat.succ_le_succ hm

See here on Zulip.

This adds some general lemmas due to Junyan Xu, culminating in

theorem HasFPowerSeriesOnBall.hasSum_iteratedFDeriv {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*}
    [ NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} {r : ℝ≥0∞}
    (h : HasFPowerSeriesOnBall f p x r) [CompleteSpace F] [CharZero 𝕜] {y : E} (hy : y ∈ EMetric.ball 0 r) :+1:
    HasSum (fun n ↦ (n ! : 𝕜)⁻¹• (iteratedFDeriv 𝕜 n f x) fun x ↦ y) (f (x + y))

and uses this to show that the Taylor series of a function that is complex differentiable on an open ball in ℂ converges there to the function; similarly for functions that are holomorphic on all of ℂ:

lemma Complex.hasSum_taylorSeries_on_ball {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
    [CompleteSpace E] ⦃f : ℂ → E⦄ ⦃c : ℂ⦄ ⦃r : NNReal⦄ (hf : DifferentiableOn ℂ f (Metric.ball c ↑r)) ⦃z : ℂ⦄
    (hz : z ∈ Metric.ball c ↑r) :
    HasSum (fun n ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)

lemma Complex.taylorSeries_eq_on_ball {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
    [CompleteSpace E] ⦃f : ℂ → E⦄ ⦃c : ℂ⦄ ⦃r : NNReal⦄ (hf : DifferentiableOn ℂ f (Metric.ball c ↑r)) ⦃z : ℂ⦄
    (hz : z ∈ Metric.ball c ↑r) : 
    ∑' (n : ℕ), (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z

lemma Complex.taylorSeries_eq_on_ball' ⦃c : ℂ⦄ ⦃r : NNReal⦄ ⦃z : ℂ⦄ (hz : z ∈ Metric.ball c ↑r) {f : ℂ → ℂ}
    (hf : DifferentiableOn ℂ f (Metric.ball c ↑r)) : 
    ∑' (n : ℕ), (n ! : ℂ)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z

and similar lemmas for EMetric.balls and entire functions.

@@ -1520,7 +1520,7 @@ noncomputable def iteratedFDeriv (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜
     ContinuousLinearMap.uncurryLeft (fderiv 𝕜 rec x)
 #align iterated_fderiv iteratedFDeriv
 
-/-- Formal Taylor series associated to a function within a set. -/
+/-- Formal Taylor series associated to a function. -/
 def ftaylorSeries (f : E → F) (x : E) : FormalMultilinearSeries 𝕜 E F := fun n =>
   iteratedFDeriv 𝕜 n f x
 #align ftaylor_series ftaylorSeries

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -891,7 +891,7 @@ theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fi
 theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
     iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t := by
   induction' n with n ihn
-  · exact h.mono fun y hy => FunLike.ext _ _ fun _ => hy
+  · exact h.mono fun y hy => DFunLike.ext _ _ fun _ => hy
   · have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderivWithin' ht
     apply this.mono
     intro y hy

Grep for ^[^#].*deriv_within and fix all occurrences.

@@ -892,7 +892,7 @@ theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (
     iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t := by
   induction' n with n ihn
   · exact h.mono fun y hy => FunLike.ext _ _ fun _ => hy
-  · have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderiv_within' ht
+  · have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderivWithin' ht
     apply this.mono
     intro y hy
     simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (· ∘ ·)]

Rename

• Filter.EventuallyEq.iterated_fderiv_within' → Filter.EventuallyEq.iteratedFDerivWithin'
• contDiffOn_succ_iff_has_fderiv_within → contDiffOn_succ_iff_hasFDerivWithin
• contDiff_succ_iff_has_fderiv → contDiff_succ_iff_hasFDerivAt

@@ -888,7 +888,7 @@ theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fi
   rfl
 #align iterated_fderiv_within_one_apply iteratedFDerivWithin_one_apply
 
-theorem Filter.EventuallyEq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
+theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
     iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t := by
   induction' n with n ihn
   · exact h.mono fun y hy => FunLike.ext _ _ fun _ => hy
@@ -896,11 +896,11 @@ theorem Filter.EventuallyEq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f)
     apply this.mono
     intro y hy
     simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (· ∘ ·)]
-#align filter.eventually_eq.iterated_fderiv_within' Filter.EventuallyEq.iterated_fderiv_within'
+#align filter.eventually_eq.iterated_fderiv_within' Filter.EventuallyEq.iteratedFDerivWithin'
 
 protected theorem Filter.EventuallyEq.iteratedFDerivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
     iteratedFDerivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s :=
-  h.iterated_fderiv_within' Subset.rfl n
+  h.iteratedFDerivWithin' Subset.rfl n
 #align filter.eventually_eq.iterated_fderiv_within Filter.EventuallyEq.iteratedFDerivWithin
 
 /-- If two functions coincide in a neighborhood of `x` within a set `s` and at `x`, then their
@@ -908,7 +908,7 @@ iterated differentials within this set at `x` coincide. -/
 theorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
     (n : ℕ) : iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
   have : f₁ =ᶠ[𝓝[insert x s] x] f := by simpa [EventuallyEq, hx]
-  (this.iterated_fderiv_within' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
+  (this.iteratedFDerivWithin' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
 #align filter.eventually_eq.iterated_fderiv_within_eq Filter.EventuallyEq.iteratedFDerivWithin_eq
 
 /-- If two functions coincide on a set `s`, then their iterated differentials within this set
@@ -1147,7 +1147,7 @@ theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
   rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A
 #align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin
 
-theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
+theorem contDiffOn_succ_iff_hasFDerivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       ∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFDerivWithinAt f (f' x) s x := by
   rw [contDiffOn_succ_iff_fderivWithin hs]
@@ -1155,7 +1155,7 @@ theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜
   rcases h with ⟨f', h1, h2⟩
   refine' ⟨fun x hx => (h2 x hx).differentiableWithinAt, fun x hx => _⟩
   exact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) hx
-#align cont_diff_on_succ_iff_has_fderiv_within contDiffOn_succ_iff_has_fderiv_within
+#align cont_diff_on_succ_iff_has_fderiv_within contDiffOn_succ_iff_hasFDerivWithin
 
 /-- A function is `C^(n + 1)` on an open domain if and only if it is
 differentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -/
@@ -1502,12 +1502,12 @@ theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤
 #align cont_diff_iff_forall_nat_le contDiff_iff_forall_nat_le
 
 /-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/
-theorem contDiff_succ_iff_has_fderiv {n : ℕ} :
+theorem contDiff_succ_iff_hasFDerivAt {n : ℕ} :
     ContDiff 𝕜 (n + 1 : ℕ) f ↔
       ∃ f' : E → E →L[𝕜] F, ContDiff 𝕜 n f' ∧ ∀ x, HasFDerivAt f (f' x) x := by
   simp only [← contDiffOn_univ, ← hasFDerivWithinAt_univ,
-    contDiffOn_succ_iff_has_fderiv_within uniqueDiffOn_univ, Set.mem_univ, forall_true_left]
-#align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_has_fderiv
+    contDiffOn_succ_iff_hasFDerivWithin uniqueDiffOn_univ, Set.mem_univ, forall_true_left]
+#align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_hasFDerivAt
 
 /-! ### Iterated derivative -/
 

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -180,7 +180,10 @@ variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAdd
 
 /-- `HasFTaylorSeriesUpToOn n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a
 derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
-`HasFDerivWithinAt` but for higher order derivatives. -/
+`HasFDerivWithinAt` but for higher order derivatives.
+
+Notice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if
+`f` is analytic and `n = ∞`: an additional `1/m!` factor on the `m`th term is necessary for that. -/
 structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
   (s : Set E) : Prop where
   zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
@@ -1218,7 +1221,10 @@ theorem ContDiffOn.continuousOn_fderiv_of_isOpen (h : ContDiffOn 𝕜 n f s) (hs
 
 /-- `HasFTaylorSeriesUpTo n f p` registers the fact that `p 0 = f` and `p (m+1)` is a
 derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
-`HasFDerivAt` but for higher order derivatives. -/
+`HasFDerivAt` but for higher order derivatives.
+
+Notice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if
+`f` is analytic and `n = ∞`: an addition `1/m!` factor on the `m`th term is necessary for that. -/
 structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
   Prop where
   zero_eq : ∀ x, (p x 0).uncurry0 = f x

Follow-up #9184

@@ -1222,8 +1222,8 @@ derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a pred
 structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
   Prop where
   zero_eq : ∀ x, (p x 0).uncurry0 = f x
-  fderiv : ∀ (m : ℕ) (_ : (m : ℕ∞) < n), ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
-  cont : ∀ (m : ℕ) (_ : (m : ℕ∞) ≤ n), Continuous fun x => p x m
+  fderiv : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
+  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => p x m
 #align has_ftaylor_series_up_to HasFTaylorSeriesUpTo
 
 theorem HasFTaylorSeriesUpTo.zero_eq' (h : HasFTaylorSeriesUpTo n f p) (x : E) :

Mostly, this means replacing "of_open" by "of_isOpen". A few lemmas names were misleading and are corrected differently. Zulip discussion.

@@ -1156,12 +1156,12 @@ theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜
 
 /-- A function is `C^(n + 1)` on an open domain if and only if it is
 differentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -/
-theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
+theorem contDiffOn_succ_iff_fderiv_of_isOpen {n : ℕ} (hs : IsOpen s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 n (fun y => fderiv 𝕜 f y) s := by
   rw [contDiffOn_succ_iff_fderivWithin hs.uniqueDiffOn]
   exact Iff.rfl.and (contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx)
-#align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_open
+#align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_isOpen
 
 /-- A function is `C^∞` on a domain with unique derivatives if and only if it is differentiable
 there, and its derivative (expressed with `fderivWithin`) is `C^∞`. -/
@@ -1182,11 +1182,11 @@ theorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn 𝕜 s) :
 
 /-- A function is `C^∞` on an open domain if and only if it is differentiable there, and its
 derivative (expressed with `fderiv`) is `C^∞`. -/
-theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
+theorem contDiffOn_top_iff_fderiv_of_isOpen (hs : IsOpen s) :
     ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fun y => fderiv 𝕜 f y) s := by
   rw [contDiffOn_top_iff_fderivWithin hs.uniqueDiffOn]
   exact Iff.rfl.and <| contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx
-#align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_open
+#align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_isOpen
 
 protected theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
     (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s := by
@@ -1199,20 +1199,20 @@ protected theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : Uni
     exact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2
 #align cont_diff_on.fderiv_within ContDiffOn.fderivWithin
 
-theorem ContDiffOn.fderiv_of_open (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
+theorem ContDiffOn.fderiv_of_isOpen (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
     ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s :=
   (hf.fderivWithin hs.uniqueDiffOn hmn).congr fun _ hx => (fderivWithin_of_isOpen hs hx).symm
-#align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_open
+#align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_isOpen
 
 theorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderivWithin 𝕜 f s x) s :=
   ((contDiffOn_succ_iff_fderivWithin hs).1 (h.of_le hn)).2.continuousOn
 #align cont_diff_on.continuous_on_fderiv_within ContDiffOn.continuousOn_fderivWithin
 
-theorem ContDiffOn.continuousOn_fderiv_of_open (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
+theorem ContDiffOn.continuousOn_fderiv_of_isOpen (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderiv 𝕜 f x) s :=
-  ((contDiffOn_succ_iff_fderiv_of_open hs).1 (h.of_le hn)).2.continuousOn
-#align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_open
+  ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 (h.of_le hn)).2.continuousOn
+#align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_isOpen
 
 /-! ### Functions with a Taylor series on the whole space -/
 

@@ -1369,6 +1369,10 @@ theorem ContDiffAt.differentiableAt (h : ContDiffAt 𝕜 n f x) (hn : 1 ≤ n) :
   simpa [hn, differentiableWithinAt_univ] using h.differentiableWithinAt
 #align cont_diff_at.differentiable_at ContDiffAt.differentiableAt
 
+nonrec lemma ContDiffAt.contDiffOn {m : ℕ} (h : ContDiffAt 𝕜 n f x) (hm : m ≤ n) :
+    ∃ u ∈ 𝓝 x, ContDiffOn 𝕜 m f u := by
+  simpa [nhdsWithin_univ] using h.contDiffOn hm
+
 /-- A function is `C^(n + 1)` at a point iff locally, it has a derivative which is `C^n`. -/
 theorem contDiffAt_succ_iff_hasFDerivAt {n : ℕ} :
     ContDiffAt 𝕜 (n + 1 : ℕ) f x ↔

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -226,7 +226,7 @@ theorem hasFTaylorSeriesUpToOn_zero_iff :
     HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).uncurry0 = f x := by
   refine ⟨fun H => ⟨H.continuousOn, H.zero_eq⟩, fun H =>
       ⟨H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm ↦ ?_⟩⟩
-  obtain rfl : m = 0 := by exact_mod_cast hm.antisymm (zero_le _)
+  obtain rfl : m = 0 := mod_cast hm.antisymm (zero_le _)
   have : EqOn (p · 0) ((continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s := fun x hx ↦
     (continuousMultilinearCurryFin0 𝕜 E F).eq_symm_apply.2 (H.2 x hx)
   rw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]

Not moving anything around this time, just postponing imports.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -3,8 +3,6 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.Analysis.Calculus.FDeriv.Add
-import Mathlib.Analysis.Calculus.FDeriv.Mul
 import Mathlib.Analysis.Calculus.FDeriv.Equiv
 import Mathlib.Analysis.Calculus.FormalMultilinearSeries
 

Mathlib.Logic.Unique contains the line attribute [simp] eq_iff_true_of_subsingleton in ...:

https://github.com/leanprover-community/mathlib4/blob/96a11c7aac574c00370c2b3dab483cb676405c5d/Mathlib/Logic/Unique.lean#L255-L256

Despite what the in part may imply, this adds the lemma to the simp set "globally", including for downstream files; it is likely that attribute [local simp] eq_iff_true_of_subsingleton in ... was meant instead (or maybe scoped simp, but I think "scoped" refers to the current namespace). Indeed, the relevant lemma is not marked with @[simp] for possible slowness: https://github.com/leanprover/std4/blob/846e9e1d6bb534774d1acd2dc430e70987da3c18/Std/Logic.lean#L749. Adding it to the simp set causes the example at https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Regression.20in.20simp to slow down.

This PR changes this and fixes the relevant downstream simps. There was also one ocurrence of attribute [simp] FullSubcategory.comp_def FullSubcategory.id_def in in Mathlib.CategoryTheory.Monoidal.Subcategory but that was much easier to fix.

https://github.com/leanprover-community/mathlib4/blob/bc49eb9ba756a233370b4b68bcdedd60402f71ed/Mathlib/CategoryTheory/Monoidal/Subcategory.lean#L118-L119

@@ -587,7 +587,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
         change ((p' y 0) (init (@cons 0 (fun _ => E) z 0))) (@cons 0 (fun _ => E) z 0 (last 0)) =
           ((p' y 0) 0) z
         congr
-        norm_num
+        norm_num [eq_iff_true_of_subsingleton]
       · convert (Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1 using 1
         · ext x y
           change p' x 0 (init (@snoc 0 (fun _ : Fin 1 => E) 0 y)) y = p' x 0 0 y

No changes to content, or splitting, just a rename so there is room for more files.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -6,7 +6,6 @@ Authors: Sébastien Gouëzel
 import Mathlib.Analysis.Calculus.FDeriv.Add
 import Mathlib.Analysis.Calculus.FDeriv.Mul
 import Mathlib.Analysis.Calculus.FDeriv.Equiv
-import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
 import Mathlib.Analysis.Calculus.FormalMultilinearSeries
 
 #align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"

• from the project towards Sard's theorem

Co-authored-by: Michael Rothgang <rothgami@math.hu-berlin.de>

@@ -1163,7 +1163,7 @@ theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 n (fun y => fderiv 𝕜 f y) s := by
   rw [contDiffOn_succ_iff_fderivWithin hs.uniqueDiffOn]
-  exact Iff.rfl.and (contDiffOn_congr fun x hx ↦ fderivWithin_of_open hs hx)
+  exact Iff.rfl.and (contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx)
 #align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_open
 
 /-- A function is `C^∞` on a domain with unique derivatives if and only if it is differentiable
@@ -1188,7 +1188,7 @@ derivative (expressed with `fderiv`) is `C^∞`. -/
 theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
     ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fun y => fderiv 𝕜 f y) s := by
   rw [contDiffOn_top_iff_fderivWithin hs.uniqueDiffOn]
-  exact Iff.rfl.and <| contDiffOn_congr fun x hx ↦ fderivWithin_of_open hs hx
+  exact Iff.rfl.and <| contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx
 #align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_open
 
 protected theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
@@ -1204,7 +1204,7 @@ protected theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : Uni
 
 theorem ContDiffOn.fderiv_of_open (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
     ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s :=
-  (hf.fderivWithin hs.uniqueDiffOn hmn).congr fun _ hx => (fderivWithin_of_open hs hx).symm
+  (hf.fderivWithin hs.uniqueDiffOn hmn).congr fun _ hx => (fderivWithin_of_isOpen hs hx).symm
 #align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_open
 
 theorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
@@ -1609,7 +1609,7 @@ theorem iteratedFDerivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :
     rw [iteratedFDeriv_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]
     dsimp
     congr 1
-    rw [fderivWithin_of_open hs hx]
+    rw [fderivWithin_of_isOpen hs hx]
     apply Filter.EventuallyEq.fderiv_eq
     filter_upwards [hs.mem_nhds hx]
     exact IH

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -742,7 +742,7 @@ theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
         (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u := by
   constructor
   · intro h x hx
-    rcases(h x hx) n.succ le_rfl with ⟨u, hu, p, Hp⟩
+    rcases (h x hx) n.succ le_rfl with ⟨u, hu, p, Hp⟩
     refine'
       ⟨u, hu, fun y => (continuousMultilinearCurryFin1 𝕜 E F) (p y 1), fun y hy =>
         Hp.hasFDerivWithinAt (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _⟩
@@ -1019,7 +1019,7 @@ protected theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs
     simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,
       iteratedFDerivWithin_zero_apply]
   · intro m hm x hx
-    rcases(h x hx) m.succ (ENat.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩
+    rcases (h x hx) m.succ (ENat.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩
     rw [insert_eq_of_mem hx] at hu
     rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
     rw [inter_comm] at ho

As discussed on Zulip.

Co-authored-by: grunweg <grunweg@posteo.de>

@@ -1580,7 +1580,7 @@ theorem support_iteratedFDeriv_subset (n : ℕ) : support (iteratedFDeriv 𝕜 n
 
 theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
     HasCompactSupport (iteratedFDeriv 𝕜 n f) :=
-  isCompact_of_isClosed_subset hf isClosed_closure (tsupport_iteratedFDeriv_subset n)
+  hf.of_isClosed_subset isClosed_closure (tsupport_iteratedFDeriv_subset n)
 #align has_compact_support.iterated_fderiv HasCompactSupport.iteratedFDeriv
 
 theorem norm_fderiv_iteratedFDeriv {n : ℕ} :

@@ -1714,7 +1714,7 @@ theorem ContDiff.continuous_fderiv (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
 continuous. -/
 theorem ContDiff.continuous_fderiv_apply (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
     Continuous fun p : E × E => (fderiv 𝕜 f p.1 : E → F) p.2 :=
-  have A : Continuous fun q : (E →L[𝕜] F) × E => q.1 q.2 := isBoundedBilinearMapApply.continuous
+  have A : Continuous fun q : (E →L[𝕜] F) × E => q.1 q.2 := isBoundedBilinearMap_apply.continuous
   have B : Continuous fun p : E × E => (fderiv 𝕜 f p.1, p.2) :=
     ((h.continuous_fderiv hn).comp continuous_fst).prod_mk continuous_snd
   A.comp B

@@ -524,7 +524,7 @@ theorem contDiffWithinAt_insert {y : E} :
   simp_rw [ContDiffWithinAt, insert_comm x y, nhdsWithin_insert_of_ne h]
 #align cont_diff_within_at_insert contDiffWithinAt_insert
 
-alias contDiffWithinAt_insert ↔ ContDiffWithinAt.of_insert ContDiffWithinAt.insert'
+alias ⟨ContDiffWithinAt.of_insert, ContDiffWithinAt.insert'⟩ := contDiffWithinAt_insert
 #align cont_diff_within_at.of_insert ContDiffWithinAt.of_insert
 #align cont_diff_within_at.insert' ContDiffWithinAt.insert'
 

@@ -813,6 +813,14 @@ theorem iteratedFDerivWithin_succ_eq_comp_left {n : ℕ} :
   rfl
 #align iterated_fderiv_within_succ_eq_comp_left iteratedFDerivWithin_succ_eq_comp_left
 
+theorem fderivWithin_iteratedFDerivWithin {s : Set E} {n : ℕ} :
+    fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s =
+      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
+        iteratedFDerivWithin 𝕜 (n + 1) f s := by
+  rw [iteratedFDerivWithin_succ_eq_comp_left]
+  ext1 x
+  simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]
+
 theorem norm_fderivWithin_iteratedFDerivWithin {n : ℕ} :
     ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ =
       ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖ := by

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.calculus.cont_diff_def
-! leanprover-community/mathlib commit 3a69562db5a458db8322b190ec8d9a8bbd8a5b14
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Calculus.FDeriv.Add
 import Mathlib.Analysis.Calculus.FDeriv.Mul
@@ -14,6 +9,8 @@ import Mathlib.Analysis.Calculus.FDeriv.Equiv
 import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
 import Mathlib.Analysis.Calculus.FormalMultilinearSeries
 
+#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
+
 /-!
 # Higher differentiability
 

We already had that the iterated derivative of a compactly supported function is compactly supported, this just makes it a bit more precise by iterating support_fderiv_subset.

@@ -1561,15 +1561,21 @@ theorem fderiv_iteratedFDeriv {n : ℕ} :
   simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]
 #align fderiv_iterated_fderiv fderiv_iteratedFDeriv
 
-theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
-    HasCompactSupport (iteratedFDeriv 𝕜 n f) := by
+theorem tsupport_iteratedFDeriv_subset (n : ℕ) : tsupport (iteratedFDeriv 𝕜 n f) ⊆ tsupport f := by
   induction' n with n IH
   · rw [iteratedFDeriv_zero_eq_comp]
-    apply hf.comp_left
-    exact LinearIsometryEquiv.map_zero _
+    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans
+      subset_closure) isClosed_closure
   · rw [iteratedFDeriv_succ_eq_comp_left]
-    apply (IH.fderiv 𝕜).comp_left
-    exact LinearIsometryEquiv.map_zero _
+    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans
+      ((support_fderiv_subset 𝕜).trans IH)) isClosed_closure
+
+theorem support_iteratedFDeriv_subset (n : ℕ) : support (iteratedFDeriv 𝕜 n f) ⊆ tsupport f :=
+  subset_closure.trans (tsupport_iteratedFDeriv_subset n)
+
+theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
+    HasCompactSupport (iteratedFDeriv 𝕜 n f) :=
+  isCompact_of_isClosed_subset hf isClosed_closure (tsupport_iteratedFDeriv_subset n)
 #align has_compact_support.iterated_fderiv HasCompactSupport.iteratedFDeriv
 
 theorem norm_fderiv_iteratedFDeriv {n : ℕ} :

• depends on: https://github.com/leanprover/std4/pull/163
• depends on: #5584
• depends on: #5715
• depends on: #5729
• depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -166,8 +166,11 @@ open Classical BigOperators NNReal Topology Filter
 
 local notation "∞" => (⊤ : ℕ∞)
 
+/-
+Porting note: These lines are not required in Mathlib4.
 attribute [local instance 1001]
   NormedAddCommGroup.toAddCommGroup NormedSpace.toModule' AddCommGroup.toAddCommMonoid
+-/
 
 open Set Fin Filter Function
 

Instead of fixing a proof "as is", I'm golfing it and moving parts of it to lemmas.

@@ -513,12 +513,15 @@ theorem contDiffWithinAt_inter (h : t ∈ 𝓝 x) :
   contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds h)
 #align cont_diff_within_at_inter contDiffWithinAt_inter
 
+theorem contDiffWithinAt_insert_self :
+    ContDiffWithinAt 𝕜 n f (insert x s) x ↔ ContDiffWithinAt 𝕜 n f s x := by
+  simp_rw [ContDiffWithinAt, insert_idem]
+
 theorem contDiffWithinAt_insert {y : E} :
     ContDiffWithinAt 𝕜 n f (insert y s) x ↔ ContDiffWithinAt 𝕜 n f s x := by
-  simp_rw [ContDiffWithinAt]
   rcases eq_or_ne x y with (rfl | h)
-  · simp_rw [insert_eq_of_mem (mem_insert _ _)]
-  simp_rw [insert_comm x y, nhdsWithin_insert_of_ne h]
+  · exact contDiffWithinAt_insert_self
+  simp_rw [ContDiffWithinAt, insert_comm x y, nhdsWithin_insert_of_ne h]
 #align cont_diff_within_at_insert contDiffWithinAt_insert
 
 alias contDiffWithinAt_insert ↔ ContDiffWithinAt.of_insert ContDiffWithinAt.insert'

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -1253,7 +1253,8 @@ theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n
 #align has_ftaylor_series_up_to.has_ftaylor_series_up_to_on HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn
 
 theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
-    HasFTaylorSeriesUpTo m f p := by rw [← hasFTaylorSeriesUpToOn_univ_iff] at h⊢; exact h.of_le hmn
+    HasFTaylorSeriesUpTo m f p := by
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢; exact h.of_le hmn
 #align has_ftaylor_series_up_to.of_le HasFTaylorSeriesUpTo.ofLe
 
 theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f := by

Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Pol'tta / Miyahara Kō <pol_tta@outlook.jp> Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -386,7 +386,7 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
             ((p x).shift m.succ).curryLeft s x := Htaylor.fderivWithin _ A x hx
         rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this
         convert this
-        ext (y v)
+        ext y v
         change
           (p x (Nat.succ (Nat.succ m))) (cons y v) =
             (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))
@@ -1575,7 +1575,7 @@ theorem iteratedFDerivWithin_univ {n : ℕ} :
     iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f := by
   induction' n with n IH
   · ext x; simp
-  · ext (x m)
+  · ext x m
     rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]
 #align iterated_fderiv_within_univ iteratedFDerivWithin_univ
 

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -577,6 +577,7 @@ theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
           HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 𝕜 E F).symm (f z))
             (FormalMultilinearSeries.unshift (p' y) (f y) 1).curryLeft (v ∩ u) y
         -- Porting note: needed `erw` here.
+        -- https://github.com/leanprover-community/mathlib4/issues/5164
         erw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
         convert (f'_eq_deriv y hy.2).mono (inter_subset_right v u)
         rw [← Hp'.zero_eq y hy.1]

Too often tempted to change these during other PRs, so doing a mass edit here.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -693,7 +693,7 @@ theorem contDiffOn_top : ContDiffOn 𝕜 ∞ f s ↔ ∀ n : ℕ, ContDiffOn 
 theorem contDiffOn_all_iff_nat : (∀ n, ContDiffOn 𝕜 n f s) ↔ ∀ n : ℕ, ContDiffOn 𝕜 n f s := by
   refine' ⟨fun H n => H n, _⟩
   rintro H (_ | n)
-  exacts[contDiffOn_top.2 H, H n]
+  exacts [contDiffOn_top.2 H, H n]
 #align cont_diff_on_all_iff_nat contDiffOn_all_iff_nat
 
 theorem ContDiffOn.continuousOn (h : ContDiffOn 𝕜 n f s) : ContinuousOn f s := fun x hx =>
@@ -1087,7 +1087,7 @@ theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
     simp only [set_eventuallyEq_iff_inf_principal, ← nhdsWithin_inter']
     rw [← inter_assoc, nhdsWithin_inter_of_mem', ← diff_eq_compl_inter, insert_diff_of_mem,
       diff_eq_compl_inter]
-    exacts[rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]
+    exacts [rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]
   have B : iteratedFDerivWithin 𝕜 m f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 m f t :=
     iteratedFDerivWithin_eventually_congr_set' _ A.symm _
   have C : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x :=

@@ -1335,6 +1335,11 @@ theorem ContDiffWithinAt.contDiffAt (h : ContDiffWithinAt 𝕜 n f s x) (hx : s
     ContDiffAt 𝕜 n f x := by rwa [ContDiffAt, ← contDiffWithinAt_inter hx, univ_inter]
 #align cont_diff_within_at.cont_diff_at ContDiffWithinAt.contDiffAt
 
+-- porting note: new lemma
+theorem ContDiffOn.contDiffAt (h : ContDiffOn 𝕜 n f s) (hx : s ∈ 𝓝 x) :
+    ContDiffAt 𝕜 n f x :=
+  (h _ (mem_of_mem_nhds hx)).contDiffAt hx
+
 theorem ContDiffAt.congr_of_eventuallyEq (h : ContDiffAt 𝕜 n f x) (hg : f₁ =ᶠ[𝓝 x] f) :
     ContDiffAt 𝕜 n f₁ x :=
   h.congr_of_eventually_eq' (by rwa [nhdsWithin_univ]) (mem_univ x)

• Protect HasFTaylorSeriesUpToOn.fderivWithin, IsBoundedBilinearMap.fderiv, and IsBoundedBilinearMap.fderivWithin.

• Rename

  • isBoundedBilinearMapSmul -> isBoundedBilinearMap_smul;
  • isBoundedBilinearMapMul -> isBoundedBilinearMap_mul;
  • isBoundedBilinearMapComp -> isBoundedBilinearMap_comp;
  • isBoundedBilinearMapSmulRight -> isBoundedBilinearMap_smulRight;
  • isBoundedBilinearMapCompMultilinear -> isBoundedBilinearMap_compMultilinear;
  • ContinuousLinearMap.mulLeftRightIsBoundedBilinear -> ContinuousLinearMap.mulLeftRight_isBoundedBilinear;
  • nhdsWithin_eq_nhds_within' -> nhdsWithin_eq_nhdsWithin';
  • ContinuousWithinAt.preimage_mem_nhds_within' -> ContinuousWithinAt.preimage_mem_nhdsWithin'.

Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -187,7 +187,7 @@ derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a pred
 structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
   (s : Set E) : Prop where
   zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
-  fderivWithin : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x ∈ s,
+  protected fderivWithin : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x ∈ s,
     HasFDerivWithinAt (p · m) (p x m.succ).curryLeft s x
   cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (p · m) s
 #align has_ftaylor_series_up_to_on HasFTaylorSeriesUpToOn

Co-authored-by: @semorrison