analysis.calculus.mean_value ⟷ Mathlib.Analysis.Calculus.MeanValue

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

@@ -4,10 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 -/
 import Analysis.Calculus.Deriv.Slope
-import Analysis.Calculus.LocalExtr
+import Analysis.Calculus.LocalExtr.Basic
 import Analysis.Convex.Slope
 import Analysis.Convex.Normed
-import Data.IsROrC.Basic
+import Analysis.RCLike.Basic
 import Topology.Instances.RealVectorSpace
 
 #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
@@ -500,7 +500,7 @@ also assume `[normed_space ℝ E]` to have a notion of a `convex` set. -/
 
 section
 
-variable {𝕜 G : Type _} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+variable {𝕜 G : Type _} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
 
 namespace Convex
 
@@ -529,7 +529,7 @@ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDeri
     apply hs.segment_subset xs ys
   have : f x = f (g 0) := by simp only [g]; rw [zero_smul, add_zero]
   rw [this]
-  have : f y = f (g 1) := by simp only [g]; rw [one_smul, add_sub_cancel'_right]
+  have : f y = f (g 1) := by simp only [g]; rw [one_smul, add_sub_cancel]
   rw [this]
   have D2 : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t :=
     by
@@ -885,7 +885,7 @@ theorem exists_hasDerivAt_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b
     ⟨c, cmem, hc⟩
   use c, cmem
   simp only [_root_.id, Pi.one_apply, mul_one] at hc
-  rw [← hc, mul_div_cancel_left]
+  rw [← hc, mul_div_cancel_left₀]
   exact ne_of_gt (sub_pos.2 hab)
 #align exists_has_deriv_at_eq_slope exists_hasDerivAt_eq_slope
 -/
@@ -924,7 +924,7 @@ theorem exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b
 
 end Interval
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.mul_sub_lt_image_sub_of_lt_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
@@ -956,7 +956,7 @@ theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable
 #align mul_sub_lt_image_sub_of_lt_deriv mul_sub_lt_image_sub_of_lt_deriv
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.mul_sub_le_image_sub_of_le_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
@@ -989,7 +989,7 @@ theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable
 #align mul_sub_le_image_sub_of_le_deriv mul_sub_le_image_sub_of_le_deriv
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.image_sub_lt_mul_sub_of_deriv_lt /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
@@ -1021,7 +1021,7 @@ theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable
 #align image_sub_lt_mul_sub_of_deriv_lt image_sub_lt_mul_sub_of_deriv_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.image_sub_le_mul_sub_of_deriv_le /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then
@@ -1578,7 +1578,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
 #align domain_mvt domain_mvt
 -/
 
-section IsROrC
+section RCLike
 
 /-!
 ### Vector-valued functions `f : E → F`.  Strict differentiability.
@@ -1591,7 +1591,7 @@ balls over `ℝ` or `ℂ`. For now, we only include the ones that we need.
 -/
 
 
-variable {𝕜 : Type _} [IsROrC 𝕜] {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type _}
+variable {𝕜 : Type _} [RCLike 𝕜] {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type _}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H] {f : G → H} {f' : G → G →L[𝕜] H} {x : G}
 
 #print hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt /-
@@ -1630,5 +1630,5 @@ theorem hasStrictDerivAt_of_hasDerivAt_of_continuousAt {f f' : 𝕜 → G} {x :
 #align has_strict_deriv_at_of_has_deriv_at_of_continuous_at hasStrictDerivAt_of_hasDerivAt_of_continuousAt
 -/
 
-end IsROrC
+end RCLike
 

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

@@ -125,7 +125,7 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {
     refine' ⟨z, _, hz⟩
     have := (hfz.trans hzB).le
     rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
-      sub_le_sub_iff_right] at this 
+      sub_le_sub_iff_right] at this
 #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary'
 -/
 
@@ -570,7 +570,7 @@ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs
   by
   obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ {y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K}
   exact mem_nhds_within_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))
-  rw [inter_comm] at hε 
+  rw [inter_comm] at hε
   refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩
   exact
     (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
@@ -705,7 +705,7 @@ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜
     (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) :
     s.EqOn f g := by
   intro y hy
-  suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this 
+  suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this
   refine' hs.is_const_of_fderiv_within_eq_zero (hf.sub hg) _ hx hy
   intro z hz
   rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz]
@@ -884,7 +884,7 @@ theorem exists_hasDerivAt_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b
       fun x hx => hasDerivAt_id x with
     ⟨c, cmem, hc⟩
   use c, cmem
-  simp only [_root_.id, Pi.one_apply, mul_one] at hc 
+  simp only [_root_.id, Pi.one_apply, mul_one] at hc
   rw [← hc, mul_div_cancel_left]
   exact ne_of_gt (sub_pos.2 hab)
 #align exists_has_deriv_at_eq_slope exists_hasDerivAt_eq_slope
@@ -1220,7 +1220,7 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
   by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
-  · push_neg at h 
+  · push_neg at h
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
     obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), (f w - f x) / (w - x) < deriv f a :=
       by
@@ -1275,7 +1275,7 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
   by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h
-  · push_neg at h 
+  · push_neg at h
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
     obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), deriv f a < (f w - f x) / (w - x) :=
       by
@@ -1553,7 +1553,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
     intro t ht; exact ⟨t, ht, rfl⟩
   have hseg' : Icc 0 1 ⊆ g ⁻¹' s := by
     rw [← image_subset_iff]; unfold image; change ∀ _, _
-    intro z Hz; rw [mem_set_of_eq] at Hz ; rcases Hz with ⟨t, Ht, hgt⟩
+    intro z Hz; rw [mem_set_of_eq] at Hz; rcases Hz with ⟨t, Ht, hgt⟩
     rw [← hgt]; exact hs.segment_subset xs ys (hseg t Ht)
   -- derivative of pullback of f under parametrization
   have hfg :
@@ -1563,7 +1563,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
     have hg : HasDerivAt g (y - x) t :=
       by
       have := ((hasDerivAt_id t).smul_const (y - x)).const_add x
-      rwa [one_smul] at this 
+      rwa [one_smul] at this
     exact (hf (g t) <| hseg' Ht).comp_hasDerivWithinAt _ hg.has_deriv_within_at hseg'
   -- apply 1-variable mean value theorem to pullback
   have hMVT : ∃ t ∈ Ioo (0 : ℝ) 1, (f' (g t) : E → ℝ) (y - x) = (f (g 1) - f (g 0)) / (1 - 0) :=
@@ -1608,7 +1608,7 @@ theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
   refine' ⟨ε, ε0, _⟩
   -- simplify formulas involving the product E × E
   rintro ⟨a, b⟩ h
-  rw [← ball_prod_same, prod_mk_mem_set_prod_eq] at h 
+  rw [← ball_prod_same, prod_mk_mem_set_prod_eq] at h
   -- exploit the choice of ε as the modulus of continuity of f'
   have hf' : ∀ x' ∈ ball x ε, ‖f' x' - f' x‖ ≤ c := by intro x' H'; rw [← dist_eq_norm];
     exact le_of_lt (hε H').2

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

@@ -1053,20 +1053,20 @@ theorem image_sub_le_mul_sub_of_deriv_le {f : ℝ → ℝ} (hf : Differentiable
 #align image_sub_le_mul_sub_of_deriv_le image_sub_le_mul_sub_of_deriv_le
 -/
 
-#print Convex.strictMonoOn_of_deriv_pos /-
+#print strictMonoOn_of_deriv_pos /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is positive, then
 `f` is a strictly monotone function on `D`.
 Note that we don't require differentiability explicitly as it already implied by the derivative
 being strictly positive. -/
-theorem Convex.strictMonoOn_of_deriv_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, 0 < deriv f x) : StrictMonoOn f D :=
+theorem strictMonoOn_of_deriv_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
+    (hf' : ∀ x ∈ interior D, 0 < deriv f x) : StrictMonoOn f D :=
   by
   rintro x hx y hy
   simpa only [MulZeroClass.zero_mul, sub_pos] using
     hD.mul_sub_lt_image_sub_of_lt_deriv hf _ hf' x hx y hy
   exact fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).ne').DifferentiableWithinAt
-#align convex.strict_mono_on_of_deriv_pos Convex.strictMonoOn_of_deriv_pos
+#align convex.strict_mono_on_of_deriv_pos strictMonoOn_of_deriv_pos
 -/
 
 #print strictMono_of_deriv_pos /-
@@ -1083,16 +1083,16 @@ theorem strictMono_of_deriv_pos {f : ℝ → ℝ} (hf' : ∀ x, 0 < deriv f x) :
 #align strict_mono_of_deriv_pos strictMono_of_deriv_pos
 -/
 
-#print Convex.monotoneOn_of_deriv_nonneg /-
+#print monotoneOn_of_deriv_nonneg /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonnegative, then
 `f` is a monotone function on `D`. -/
-theorem Convex.monotoneOn_of_deriv_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
-    (hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x) : MonotoneOn f D := fun x hx y hy hxy => by
+theorem monotoneOn_of_deriv_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
+    (hf' : DifferentiableOn ℝ f (interior D)) (hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x) :
+    MonotoneOn f D := fun x hx y hy hxy => by
   simpa only [MulZeroClass.zero_mul, sub_nonneg] using
     hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg x hx y hy hxy
-#align convex.monotone_on_of_deriv_nonneg Convex.monotoneOn_of_deriv_nonneg
+#align convex.monotone_on_of_deriv_nonneg monotoneOn_of_deriv_nonneg
 -/
 
 #print monotone_of_deriv_nonneg /-
@@ -1106,18 +1106,17 @@ theorem monotone_of_deriv_nonneg {f : ℝ → ℝ} (hf : Differentiable ℝ f) (
 #align monotone_of_deriv_nonneg monotone_of_deriv_nonneg
 -/
 
-#print Convex.strictAntiOn_of_deriv_neg /-
+#print strictAntiOn_of_deriv_neg /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is negative, then
 `f` is a strictly antitone function on `D`. -/
-theorem Convex.strictAntiOn_of_deriv_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, deriv f x < 0) : StrictAntiOn f D :=
-  fun x hx y => by
+theorem strictAntiOn_of_deriv_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
+    (hf' : ∀ x ∈ interior D, deriv f x < 0) : StrictAntiOn f D := fun x hx y => by
   simpa only [MulZeroClass.zero_mul, sub_lt_zero] using
     hD.image_sub_lt_mul_sub_of_deriv_lt hf
       (fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).Ne).DifferentiableWithinAt) hf' x
       hx y
-#align convex.strict_anti_on_of_deriv_neg Convex.strictAntiOn_of_deriv_neg
+#align convex.strict_anti_on_of_deriv_neg strictAntiOn_of_deriv_neg
 -/
 
 #print strictAnti_of_deriv_neg /-
@@ -1134,16 +1133,16 @@ theorem strictAnti_of_deriv_neg {f : ℝ → ℝ} (hf' : ∀ x, deriv f x < 0) :
 #align strict_anti_of_deriv_neg strictAnti_of_deriv_neg
 -/
 
-#print Convex.antitoneOn_of_deriv_nonpos /-
+#print antitoneOn_of_deriv_nonpos /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonpositive, then
 `f` is an antitone function on `D`. -/
-theorem Convex.antitoneOn_of_deriv_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
-    (hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0) : AntitoneOn f D := fun x hx y hy hxy => by
+theorem antitoneOn_of_deriv_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
+    (hf' : DifferentiableOn ℝ f (interior D)) (hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0) :
+    AntitoneOn f D := fun x hx y hy hxy => by
   simpa only [MulZeroClass.zero_mul, sub_nonpos] using
     hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy
-#align convex.antitone_on_of_deriv_nonpos Convex.antitoneOn_of_deriv_nonpos
+#align convex.antitone_on_of_deriv_nonpos antitoneOn_of_deriv_nonpos
 -/
 
 #print antitone_of_deriv_nonpos /-

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

@@ -3,12 +3,12 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 -/
-import Mathbin.Analysis.Calculus.Deriv.Slope
-import Mathbin.Analysis.Calculus.LocalExtr
-import Mathbin.Analysis.Convex.Slope
-import Mathbin.Analysis.Convex.Normed
-import Mathbin.Data.IsROrC.Basic
-import Mathbin.Topology.Instances.RealVectorSpace
+import Analysis.Calculus.Deriv.Slope
+import Analysis.Calculus.LocalExtr
+import Analysis.Convex.Slope
+import Analysis.Convex.Normed
+import Data.IsROrC.Basic
+import Topology.Instances.RealVectorSpace
 
 #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
 
@@ -924,7 +924,7 @@ theorem exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b
 
 end Interval
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.mul_sub_lt_image_sub_of_lt_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
@@ -956,7 +956,7 @@ theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable
 #align mul_sub_lt_image_sub_of_lt_deriv mul_sub_lt_image_sub_of_lt_deriv
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.mul_sub_le_image_sub_of_le_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
@@ -989,7 +989,7 @@ theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable
 #align mul_sub_le_image_sub_of_le_deriv mul_sub_le_image_sub_of_le_deriv
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.image_sub_lt_mul_sub_of_deriv_lt /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
@@ -1021,7 +1021,7 @@ theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable
 #align image_sub_lt_mul_sub_of_deriv_lt image_sub_lt_mul_sub_of_deriv_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.image_sub_le_mul_sub_of_deriv_le /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then

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

@@ -799,7 +799,7 @@ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, Diff
 then the function is `C`-Lipschitz.  Version with `deriv` and `lipschitz_with`. -/
 theorem lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f)
     (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f :=
-  lipschitz_on_univ.1 <|
+  lipschitzOn_univ.1 <|
     convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x hx => hf x) fun x hx => bound x
 #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le
 -/

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, Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.calculus.mean_value
-! 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.Deriv.Slope
 import Mathbin.Analysis.Calculus.LocalExtr
@@ -15,6 +10,8 @@ import Mathbin.Analysis.Convex.Normed
 import Mathbin.Data.IsROrC.Basic
 import Mathbin.Topology.Instances.RealVectorSpace
 
+#align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
+
 /-!
 # The mean value inequality and equalities
 
@@ -927,7 +924,7 @@ theorem exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b
 
 end Interval
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.mul_sub_lt_image_sub_of_lt_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
@@ -959,7 +956,7 @@ theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable
 #align mul_sub_lt_image_sub_of_lt_deriv mul_sub_lt_image_sub_of_lt_deriv
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.mul_sub_le_image_sub_of_le_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
@@ -992,7 +989,7 @@ theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable
 #align mul_sub_le_image_sub_of_le_deriv mul_sub_le_image_sub_of_le_deriv
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.image_sub_lt_mul_sub_of_deriv_lt /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
@@ -1024,7 +1021,7 @@ theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable
 #align image_sub_lt_mul_sub_of_deriv_lt image_sub_lt_mul_sub_of_deriv_lt
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 #print Convex.image_sub_le_mul_sub_of_deriv_le /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then

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

@@ -84,6 +84,7 @@ open scoped Classical Topology NNReal
 /-! ### One-dimensional fencing inequalities -/
 
 
+#print image_le_of_liminf_slope_right_lt_deriv_boundary' /-
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -129,7 +130,9 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {
     rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
       sub_le_sub_iff_right] at this 
 #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary'
+-/
 
+#print image_le_of_liminf_slope_right_lt_deriv_boundary /-
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -149,7 +152,9 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a
   image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha
     (fun x hx => (hB x).ContinuousAt.ContinuousWithinAt) (fun x hx => (hB x).HasDerivWithinAt) bound
 #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary
+-/
 
+#print image_le_of_liminf_slope_right_le_deriv_boundary /-
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -183,7 +188,9 @@ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b
     continuous_within_at_const.add (continuous_within_at_id.mul continuousWithinAt_const)
   convert continuous_within_at_const.closure_le _ this (Hr x hx) <;> simp
 #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary
+-/
 
+#print image_le_of_deriv_right_lt_deriv_boundary' /-
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -201,7 +208,9 @@ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : 
   image_le_of_liminf_slope_right_lt_deriv_boundary' hf
     (fun x hx r hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound
 #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary'
+-/
 
+#print image_le_of_deriv_right_lt_deriv_boundary /-
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -218,7 +227,9 @@ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : 
   image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha
     (fun x hx => (hB x).ContinuousAt.ContinuousWithinAt) (fun x hx => (hB x).HasDerivWithinAt) bound
 #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary
+-/
 
+#print image_le_of_deriv_right_le_deriv_boundary /-
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -236,6 +247,7 @@ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : 
   image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx r hr =>
     (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr)
 #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary
+-/
 
 /-! ### Vector-valued functions `f : ℝ → E` -/
 
@@ -244,6 +256,7 @@ section
 
 variable {f : ℝ → E} {a b : ℝ}
 
+#print image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -264,7 +277,9 @@ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type _}
   image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB
     hB' bound
 #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary
+-/
 
+#print image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-
 /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -283,7 +298,9 @@ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E}
   image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf
     (fun x hx r hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound
 #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary'
+-/
 
+#print image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-
 /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -302,7 +319,9 @@ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E}
   image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha
     (fun x hx => (hB x).ContinuousAt.ContinuousWithinAt) (fun x hx => (hB x).HasDerivWithinAt) bound
 #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary
+-/
 
+#print image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-
 /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -321,7 +340,9 @@ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E}
   image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB'
     fun x hx r hr => (hf' x hx).liminf_right_slope_norm_le (lt_of_le_of_lt (bound x hx) hr)
 #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary'
+-/
 
+#print image_norm_le_of_norm_deriv_right_le_deriv_boundary /-
 /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
 Let `f` and `B` be continuous functions on `[a, b]` such that
 
@@ -340,7 +361,9 @@ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E}
   image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha
     (fun x hx => (hB x).ContinuousAt.ContinuousWithinAt) (fun x hx => (hB x).HasDerivWithinAt) bound
 #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary
+-/
 
+#print norm_image_sub_le_of_norm_deriv_right_le_segment /-
 /-- A function on `[a, b]` with the norm of the right derivative bounded by `C`
 satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/
 theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ}
@@ -360,7 +383,9 @@ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : 
   convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound
   simp only [g, B]; rw [sub_self, norm_zero, sub_self, MulZeroClass.mul_zero]
 #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment
+-/
 
+#print norm_image_sub_le_of_norm_deriv_le_segment' /-
 /-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
 bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `has_deriv_within_at`
 version. -/
@@ -373,7 +398,9 @@ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ}
       (fun x hx => _) bound
   exact (hf x <| Ico_subset_Icc_self hx).nhdsWithin (Icc_mem_nhdsWithin_Ici hx)
 #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment'
+-/
 
+#print norm_image_sub_le_of_norm_deriv_le_segment /-
 /-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
 bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `deriv_within`
 version. -/
@@ -384,7 +411,9 @@ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : Differentiabl
   refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound
   exact fun x hx => (hf x hx).HasDerivWithinAt
 #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment
+-/
 
+#print norm_image_sub_le_of_norm_deriv_le_segment_01' /-
 /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]`
 bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `has_deriv_within_at`
 version. -/
@@ -394,7 +423,9 @@ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ
   simpa only [sub_zero, mul_one] using
     norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one)
 #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01'
+-/
 
+#print norm_image_sub_le_of_norm_deriv_le_segment_01 /-
 /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]`
 bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `deriv_within` version. -/
 theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ}
@@ -403,14 +434,18 @@ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ}
   simpa only [sub_zero, mul_one] using
     norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one)
 #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01
+-/
 
+#print constant_of_has_deriv_right_zero /-
 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b))
     (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by
   simpa only [MulZeroClass.zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>
     norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv
       (fun y hy => by rw [norm_le_zero_iff]) x hx
 #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero
+-/
 
+#print constant_of_derivWithin_zero /-
 theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b))
     (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a :=
   by
@@ -419,6 +454,7 @@ theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b))
   simpa only [MulZeroClass.zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>
     norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx
 #align constant_of_deriv_within_zero constant_of_derivWithin_zero
+-/
 
 variable {f' g : ℝ → E}
 
@@ -473,6 +509,7 @@ namespace Convex
 
 variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G}
 
+#print Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
 the function is `C`-Lipschitz. Version with `has_fderiv_within`. -/
 theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
@@ -506,7 +543,9 @@ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDeri
   refine' fun t ht => le_of_op_norm_le _ _ _
   exact bound (g t) (segm <| Ico_subset_Icc_self ht)
 #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le
+-/
 
+#print Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
 `s`, then the function is `C`-Lipschitz on `s`. Version with `has_fderiv_within` and
 `lipschitz_on_with`. -/
@@ -518,7 +557,9 @@ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0}
   intro x x_in y y_in
   exact hs.norm_image_sub_le_of_norm_has_fderiv_within_le hf bound y_in x_in
 #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
+-/
 
+#print Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-
 /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function
 differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is
 continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is
@@ -538,7 +579,9 @@ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs
     (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
       (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le
 #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt
+-/
 
+#print Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-
 /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function
 differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is
 continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz
@@ -551,7 +594,9 @@ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ
   (exists_gt _).imp <|
     hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont
 #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt
+-/
 
+#print Convex.norm_image_sub_le_of_norm_fderivWithin_le /-
 /-- The mean value theorem on a convex set: if the derivative of a function within this set is
 bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv_within`. -/
 theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s)
@@ -560,7 +605,9 @@ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f
   hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound xs
     ys
 #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le
+-/
 
+#print Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
 `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv_within` and
 `lipschitz_on_with`. -/
@@ -568,7 +615,9 @@ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : Differenti
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s :=
   hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le
+-/
 
+#print Convex.norm_image_sub_le_of_norm_fderiv_le /-
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`,
 then the function is `C`-Lipschitz. Version with `fderiv`. -/
 theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
@@ -577,7 +626,9 @@ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt
   hs.norm_image_sub_le_of_norm_hasFDerivWithin_le
     (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le
+-/
 
+#print Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
 `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
@@ -585,7 +636,9 @@ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, Dif
   hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
     (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le
+-/
 
+#print Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-
 /-- Variant of the mean value inequality on a convex set, using a bound on the difference between
 the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with
 `has_fderiv_within`. -/
@@ -605,7 +658,9 @@ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDer
     _ = ‖g y - g x‖ := by simp
     _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys
 #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le'
+-/
 
+#print Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-
 /-- Variant of the mean value inequality on a convex set. Version with `fderiv_within`. -/
 theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
@@ -613,7 +668,9 @@ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f
   hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).HasFDerivWithinAt) bound
     xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le'
+-/
 
+#print Convex.norm_image_sub_le_of_norm_fderiv_le' /-
 /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/
 theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
     (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
@@ -621,7 +678,9 @@ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt
   hs.norm_image_sub_le_of_norm_hasFDerivWithin_le'
     (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le'
+-/
 
+#print Convex.is_const_of_fderivWithin_eq_zero /-
 /-- If a function has zero Fréchet derivative at every point of a convex set,
 then it is a constant on this set. -/
 theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s)
@@ -631,13 +690,17 @@ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : Differentiabl
   simpa only [(dist_eq_norm _ _).symm, MulZeroClass.zero_mul, dist_le_zero, eq_comm] using
     hs.norm_image_sub_le_of_norm_fderiv_within_le hf bound hx hy
 #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero
+-/
 
+#print is_const_of_fderiv_eq_zero /-
 theorem is_const_of_fderiv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0)
     (x y : E) : f x = f y :=
   convex_univ.is_const_of_fderivWithin_eq_zero hf.DifferentiableOn
     (fun x _ => by rw [fderivWithin_univ] <;> exact hf' x) trivial trivial
 #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero
+-/
 
+#print Convex.eqOn_of_fderivWithin_eq /-
 /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at
 one point in that set, then they are equal on that set. -/
 theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s)
@@ -650,13 +713,16 @@ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜
   intro z hz
   rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz]
 #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq
+-/
 
+#print eq_of_fderiv_eq /-
 theorem eq_of_fderiv_eq (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g)
     (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g :=
   suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x
   convex_univ.eqOn_of_fderivWithin_eq hf.DifferentiableOn hg.DifferentiableOn uniqueDiffOn_univ
     (fun x hx => by simpa using hf' _) (mem_univ _) hfgx
 #align eq_of_fderiv_eq eq_of_fderiv_eq
+-/
 
 end Convex
 
@@ -664,6 +730,7 @@ namespace Convex
 
 variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜}
 
+#print Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C`, then the function is `C`-Lipschitz. Version with `has_deriv_within`. -/
 theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ}
@@ -672,7 +739,9 @@ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ}
   Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt)
     (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys
 #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
+-/
 
+#print Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
 Version with `has_deriv_within` and `lipschitz_on_with`. -/
@@ -682,7 +751,9 @@ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex 
   Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt)
     (fun x hx => le_trans (by simp) (bound x hx)) hs
 #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le
+-/
 
+#print Convex.norm_image_sub_le_of_norm_derivWithin_le /-
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within
 this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv_within` -/
 theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s)
@@ -691,7 +762,9 @@ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableO
   hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivWithinAt) bound xs
     ys
 #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le
+-/
 
+#print Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
 Version with `deriv_within` and `lipschitz_on_with`. -/
@@ -700,7 +773,9 @@ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ
     LipschitzOnWith C f s :=
   hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le
+-/
 
+#print Convex.norm_image_sub_le_of_norm_deriv_le /-
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/
 theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
@@ -709,7 +784,9 @@ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, Differen
   hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivAt.HasDerivWithinAt)
     bound xs ys
 #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le
+-/
 
+#print Convex.lipschitzOnWith_of_nnnorm_deriv_le /-
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
 Version with `deriv` and `lipschitz_on_with`. -/
@@ -718,7 +795,9 @@ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, Diff
   hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivAt.HasDerivWithinAt)
     bound
 #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le
+-/
 
+#print lipschitzWith_of_nnnorm_deriv_le /-
 /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`,
 then the function is `C`-Lipschitz.  Version with `deriv` and `lipschitz_with`. -/
 theorem lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f)
@@ -726,13 +805,16 @@ theorem lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜
   lipschitz_on_univ.1 <|
     convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x hx => hf x) fun x hx => bound x
 #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le
+-/
 
+#print is_const_of_deriv_eq_zero /-
 /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero,
 then it is a constant function. -/
 theorem is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) :
     f x = f y :=
   is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _
 #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero
+-/
 
 end Convex
 
@@ -749,8 +831,7 @@ variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (I
   (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x)
   (hgd : DifferentiableOn ℝ g (Ioo a b))
 
-include hab hfc hff' hgc hgg'
-
+#print exists_ratio_hasDerivAt_eq_ratio_slope /-
 /-- Cauchy's **Mean Value Theorem**, `has_deriv_at` version. -/
 theorem exists_ratio_hasDerivAt_eq_ratio_slope :
     ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c :=
@@ -765,9 +846,9 @@ theorem exists_ratio_hasDerivAt_eq_ratio_slope :
   rcases exists_hasDerivAt_eq_zero h h' hab hhc hI hhh' with ⟨c, cmem, hc⟩
   exact ⟨c, cmem, sub_eq_zero.1 hc⟩
 #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope
+-/
 
-omit hfc hgc
-
+#print exists_ratio_hasDerivAt_eq_ratio_slope' /-
 /-- Cauchy's **Mean Value Theorem**, extended `has_deriv_at` version. -/
 theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
     (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x)
@@ -796,11 +877,9 @@ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
   rcases exists_hasDerivAt_eq_zero' hab hha hhb hhh' with ⟨c, cmem, hc⟩
   exact ⟨c, cmem, sub_eq_zero.1 hc⟩
 #align exists_ratio_has_deriv_at_eq_ratio_slope' exists_ratio_hasDerivAt_eq_ratio_slope'
+-/
 
-include hfc
-
-omit hgg'
-
+#print exists_hasDerivAt_eq_slope /-
 /-- Lagrange's Mean Value Theorem, `has_deriv_at` version -/
 theorem exists_hasDerivAt_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a) :=
   by
@@ -812,9 +891,9 @@ theorem exists_hasDerivAt_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b
   rw [← hc, mul_div_cancel_left]
   exact ne_of_gt (sub_pos.2 hab)
 #align exists_has_deriv_at_eq_slope exists_hasDerivAt_eq_slope
+-/
 
-omit hff'
-
+#print exists_ratio_deriv_eq_ratio_slope /-
 /-- Cauchy's Mean Value Theorem, `deriv` version. -/
 theorem exists_ratio_deriv_eq_ratio_slope :
     ∃ c ∈ Ioo a b, (g b - g a) * deriv f c = (f b - f a) * deriv g c :=
@@ -823,9 +902,9 @@ theorem exists_ratio_deriv_eq_ratio_slope :
     (deriv g) hgc fun x hx =>
     ((hgd x hx).DifferentiableAt <| IsOpen.mem_nhds isOpen_Ioo hx).HasDerivAt
 #align exists_ratio_deriv_eq_ratio_slope exists_ratio_deriv_eq_ratio_slope
+-/
 
-omit hfc
-
+#print exists_ratio_deriv_eq_ratio_slope' /-
 /-- Cauchy's Mean Value Theorem, extended `deriv` version. -/
 theorem exists_ratio_deriv_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
     (hdf : DifferentiableOn ℝ f <| Ioo a b) (hdg : DifferentiableOn ℝ g <| Ioo a b)
@@ -836,16 +915,20 @@ theorem exists_ratio_deriv_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
     (fun x hx => ((hdf x hx).DifferentiableAt <| Ioo_mem_nhds hx.1 hx.2).HasDerivAt)
     (fun x hx => ((hdg x hx).DifferentiableAt <| Ioo_mem_nhds hx.1 hx.2).HasDerivAt) hfa hga hfb hgb
 #align exists_ratio_deriv_eq_ratio_slope' exists_ratio_deriv_eq_ratio_slope'
+-/
 
+#print exists_deriv_eq_slope /-
 /-- Lagrange's **Mean Value Theorem**, `deriv` version. -/
 theorem exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) :=
   exists_hasDerivAt_eq_slope f (deriv f) hab hfc fun x hx =>
     ((hfd x hx).DifferentiableAt <| IsOpen.mem_nhds isOpen_Ioo hx).HasDerivAt
 #align exists_deriv_eq_slope exists_deriv_eq_slope
+-/
 
 end Interval
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+#print Convex.mul_sub_lt_image_sub_of_lt_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
 `f` grows faster than `C * x` on `D`, i.e., `C * (y - x) < f y - f x` whenever `x, y ∈ D`,
@@ -864,7 +947,9 @@ theorem Convex.mul_sub_lt_image_sub_of_lt_deriv {D : Set ℝ} (hD : Convex ℝ D
   have : C < (f y - f x) / (y - x) := by rw [← ha]; exact hf'_gt _ (hxyD' a_mem)
   exact (lt_div_iff (sub_pos.2 hxy)).1 this
 #align convex.mul_sub_lt_image_sub_of_lt_deriv Convex.mul_sub_lt_image_sub_of_lt_deriv
+-/
 
+#print mul_sub_lt_image_sub_of_lt_deriv /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `C < f'`, then `f` grows faster than
 `C * x`, i.e., `C * (y - x) < f y - f x` whenever `x < y`. -/
 theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
@@ -872,8 +957,10 @@ theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable
   convex_univ.mul_sub_lt_image_sub_of_lt_deriv hf.Continuous.ContinuousOn hf.DifferentiableOn
     (fun x _ => hf'_gt x) x trivial y trivial hxy
 #align mul_sub_lt_image_sub_of_lt_deriv mul_sub_lt_image_sub_of_lt_deriv
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+#print Convex.mul_sub_le_image_sub_of_le_deriv /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
 `f` grows at least as fast as `C * x` on `D`, i.e., `C * (y - x) ≤ f y - f x` whenever `x, y ∈ D`,
@@ -893,7 +980,9 @@ theorem Convex.mul_sub_le_image_sub_of_le_deriv {D : Set ℝ} (hD : Convex ℝ D
   have : C ≤ (f y - f x) / (y - x) := by rw [← ha]; exact hf'_ge _ (hxyD' a_mem)
   exact (le_div_iff (sub_pos.2 hxy')).1 this
 #align convex.mul_sub_le_image_sub_of_le_deriv Convex.mul_sub_le_image_sub_of_le_deriv
+-/
 
+#print mul_sub_le_image_sub_of_le_deriv /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `C ≤ f'`, then `f` grows at least as fast
 as `C * x`, i.e., `C * (y - x) ≤ f y - f x` whenever `x ≤ y`. -/
 theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
@@ -901,8 +990,10 @@ theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable
   convex_univ.mul_sub_le_image_sub_of_le_deriv hf.Continuous.ContinuousOn hf.DifferentiableOn
     (fun x _ => hf'_ge x) x trivial y trivial hxy
 #align mul_sub_le_image_sub_of_le_deriv mul_sub_le_image_sub_of_le_deriv
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+#print Convex.image_sub_lt_mul_sub_of_deriv_lt /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
 `f` grows slower than `C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x, y ∈ D`,
@@ -921,7 +1012,9 @@ theorem Convex.image_sub_lt_mul_sub_of_deriv_lt {D : Set ℝ} (hD : Convex ℝ D
   simpa [-neg_lt_neg_iff] using
     neg_lt_neg (hD.mul_sub_lt_image_sub_of_lt_deriv hf.neg hf'.neg hf'_gt x hx y hy hxy)
 #align convex.image_sub_lt_mul_sub_of_deriv_lt Convex.image_sub_lt_mul_sub_of_deriv_lt
+-/
 
+#print image_sub_lt_mul_sub_of_deriv_lt /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f' < C`, then `f` grows slower than
 `C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x < y`. -/
 theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
@@ -929,8 +1022,10 @@ theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable
   convex_univ.image_sub_lt_mul_sub_of_deriv_lt hf.Continuous.ContinuousOn hf.DifferentiableOn
     (fun x _ => lt_hf' x) x trivial y trivial hxy
 #align image_sub_lt_mul_sub_of_deriv_lt image_sub_lt_mul_sub_of_deriv_lt
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+#print Convex.image_sub_le_mul_sub_of_deriv_le /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then
 `f` grows at most as fast as `C * x` on `D`, i.e., `f y - f x ≤ C * (y - x)` whenever `x, y ∈ D`,
@@ -949,7 +1044,9 @@ theorem Convex.image_sub_le_mul_sub_of_deriv_le {D : Set ℝ} (hD : Convex ℝ D
   simpa [-neg_le_neg_iff] using
     neg_le_neg (hD.mul_sub_le_image_sub_of_le_deriv hf.neg hf'.neg hf'_ge x hx y hy hxy)
 #align convex.image_sub_le_mul_sub_of_deriv_le Convex.image_sub_le_mul_sub_of_deriv_le
+-/
 
+#print image_sub_le_mul_sub_of_deriv_le /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f' ≤ C`, then `f` grows at most as fast
 as `C * x`, i.e., `f y - f x ≤ C * (y - x)` whenever `x ≤ y`. -/
 theorem image_sub_le_mul_sub_of_deriv_le {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
@@ -957,7 +1054,9 @@ theorem image_sub_le_mul_sub_of_deriv_le {f : ℝ → ℝ} (hf : Differentiable
   convex_univ.image_sub_le_mul_sub_of_deriv_le hf.Continuous.ContinuousOn hf.DifferentiableOn
     (fun x _ => le_hf' x) x trivial y trivial hxy
 #align image_sub_le_mul_sub_of_deriv_le image_sub_le_mul_sub_of_deriv_le
+-/
 
+#print Convex.strictMonoOn_of_deriv_pos /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is positive, then
 `f` is a strictly monotone function on `D`.
@@ -971,7 +1070,9 @@ theorem Convex.strictMonoOn_of_deriv_pos {D : Set ℝ} (hD : Convex ℝ D) {f :
     hD.mul_sub_lt_image_sub_of_lt_deriv hf _ hf' x hx y hy
   exact fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).ne').DifferentiableWithinAt
 #align convex.strict_mono_on_of_deriv_pos Convex.strictMonoOn_of_deriv_pos
+-/
 
+#print strictMono_of_deriv_pos /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is positive, then
 `f` is a strictly monotone function.
 Note that we don't require differentiability explicitly as it already implied by the derivative
@@ -983,7 +1084,9 @@ theorem strictMono_of_deriv_pos {f : ℝ → ℝ} (hf' : ∀ x, 0 < deriv f x) :
         (differentiableAt_of_deriv_ne_zero (hf' z).ne').DifferentiableWithinAt.ContinuousWithinAt)
       fun x _ => hf' x
 #align strict_mono_of_deriv_pos strictMono_of_deriv_pos
+-/
 
+#print Convex.monotoneOn_of_deriv_nonneg /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonnegative, then
 `f` is a monotone function on `D`. -/
@@ -993,7 +1096,9 @@ theorem Convex.monotoneOn_of_deriv_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f :
   simpa only [MulZeroClass.zero_mul, sub_nonneg] using
     hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg x hx y hy hxy
 #align convex.monotone_on_of_deriv_nonneg Convex.monotoneOn_of_deriv_nonneg
+-/
 
+#print monotone_of_deriv_nonneg /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonnegative, then
 `f` is a monotone function. -/
 theorem monotone_of_deriv_nonneg {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ x, 0 ≤ deriv f x) :
@@ -1002,7 +1107,9 @@ theorem monotone_of_deriv_nonneg {f : ℝ → ℝ} (hf : Differentiable ℝ f) (
     convex_univ.monotoneOn_of_deriv_nonneg hf.Continuous.ContinuousOn hf.DifferentiableOn fun x _ =>
       hf' x
 #align monotone_of_deriv_nonneg monotone_of_deriv_nonneg
+-/
 
+#print Convex.strictAntiOn_of_deriv_neg /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is negative, then
 `f` is a strictly antitone function on `D`. -/
@@ -1014,7 +1121,9 @@ theorem Convex.strictAntiOn_of_deriv_neg {D : Set ℝ} (hD : Convex ℝ D) {f :
       (fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).Ne).DifferentiableWithinAt) hf' x
       hx y
 #align convex.strict_anti_on_of_deriv_neg Convex.strictAntiOn_of_deriv_neg
+-/
 
+#print strictAnti_of_deriv_neg /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is negative, then
 `f` is a strictly antitone function.
 Note that we don't require differentiability explicitly as it already implied by the derivative
@@ -1026,7 +1135,9 @@ theorem strictAnti_of_deriv_neg {f : ℝ → ℝ} (hf' : ∀ x, deriv f x < 0) :
         (differentiableAt_of_deriv_ne_zero (hf' z).Ne).DifferentiableWithinAt.ContinuousWithinAt)
       fun x _ => hf' x
 #align strict_anti_of_deriv_neg strictAnti_of_deriv_neg
+-/
 
+#print Convex.antitoneOn_of_deriv_nonpos /-
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonpositive, then
 `f` is an antitone function on `D`. -/
@@ -1036,7 +1147,9 @@ theorem Convex.antitoneOn_of_deriv_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f :
   simpa only [MulZeroClass.zero_mul, sub_nonpos] using
     hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy
 #align convex.antitone_on_of_deriv_nonpos Convex.antitoneOn_of_deriv_nonpos
+-/
 
+#print antitone_of_deriv_nonpos /-
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonpositive, then
 `f` is an antitone function. -/
 theorem antitone_of_deriv_nonpos {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ x, deriv f x ≤ 0) :
@@ -1045,7 +1158,9 @@ theorem antitone_of_deriv_nonpos {f : ℝ → ℝ} (hf : Differentiable ℝ f) (
     convex_univ.antitoneOn_of_deriv_nonpos hf.Continuous.ContinuousOn hf.DifferentiableOn fun x _ =>
       hf' x
 #align antitone_of_deriv_nonpos antitone_of_deriv_nonpos
+-/
 
+#print MonotoneOn.convexOn_of_deriv /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
 and `f'` is monotone on the interior, then `f` is convex on `D`. -/
 theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
@@ -1070,7 +1185,9 @@ theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
       rw [← ha, ← hb]
       exact hf'_mono (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb).le)
 #align monotone_on.convex_on_of_deriv MonotoneOn.convexOn_of_deriv
+-/
 
+#print AntitoneOn.concaveOn_of_deriv /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
 and `f'` is antitone on the interior, then `f` is concave on `D`. -/
 theorem AntitoneOn.concaveOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
@@ -1082,7 +1199,9 @@ theorem AntitoneOn.concaveOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
     convert neg_le_neg (h_anti hx hy hxy) <;> convert deriv.neg
   neg_convex_on_iff.mp (this.convex_on_of_deriv hD hf.neg hf'.neg)
 #align antitone_on.concave_on_of_deriv AntitoneOn.concaveOn_of_deriv
+-/
 
+#print StrictMonoOn.exists_slope_lt_deriv_aux /-
 theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
     (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
     ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a :=
@@ -1096,7 +1215,9 @@ theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf
   rw [← ha]
   exact hf'_mono ⟨hxa, hay⟩ ⟨hxa.trans hab, hby⟩ hab
 #align strict_mono_on.exists_slope_lt_deriv_aux StrictMonoOn.exists_slope_lt_deriv_aux
+-/
 
+#print StrictMonoOn.exists_slope_lt_deriv /-
 theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
     (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
     ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a :=
@@ -1133,7 +1254,9 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
         exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le
     linarith
 #align strict_mono_on.exists_slope_lt_deriv StrictMonoOn.exists_slope_lt_deriv
+-/
 
+#print StrictMonoOn.exists_deriv_lt_slope_aux /-
 theorem StrictMonoOn.exists_deriv_lt_slope_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
     (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
     ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) :=
@@ -1147,7 +1270,9 @@ theorem StrictMonoOn.exists_deriv_lt_slope_aux {x y : ℝ} {f : ℝ → ℝ} (hf
   rw [← ha]
   exact hf'_mono ⟨hxb, hba.trans hay⟩ ⟨hxa, hay⟩ hba
 #align strict_mono_on.exists_deriv_lt_slope_aux StrictMonoOn.exists_deriv_lt_slope_aux
+-/
 
+#print StrictMonoOn.exists_deriv_lt_slope /-
 theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
     (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
     ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) :=
@@ -1184,7 +1309,9 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
         exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le
     linarith
 #align strict_mono_on.exists_deriv_lt_slope StrictMonoOn.exists_deriv_lt_slope
+-/
 
+#print StrictMonoOn.strictConvexOn_of_deriv /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
 interior, then `f` is strictly convex on `D`.
 Note that we don't require differentiability, since it is guaranteed at all but at most
@@ -1211,7 +1338,9 @@ theorem StrictMonoOn.strictConvexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {
       apply ha.trans (lt_trans _ hb)
       exact hf' (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb))
 #align strict_mono_on.strict_convex_on_of_deriv StrictMonoOn.strictConvexOn_of_deriv
+-/
 
+#print StrictAntiOn.strictConcaveOn_of_deriv /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f'` is strictly antitone on the
 interior, then `f` is strictly concave on `D`.
 Note that we don't require differentiability, since it is guaranteed at all but at most
@@ -1225,21 +1354,27 @@ theorem StrictAntiOn.strictConcaveOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D)
     convert neg_lt_neg (h_anti hx hy hxy) <;> convert deriv.neg
   neg_strict_convex_on_iff.mp (this.strict_convex_on_of_deriv hD hf.neg)
 #align strict_anti_on.strict_concave_on_of_deriv StrictAntiOn.strictConcaveOn_of_deriv
+-/
 
+#print Monotone.convexOn_univ_of_deriv /-
 /-- If a function `f` is differentiable and `f'` is monotone on `ℝ` then `f` is convex. -/
 theorem Monotone.convexOn_univ_of_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f)
     (hf'_mono : Monotone (deriv f)) : ConvexOn ℝ univ f :=
   (hf'_mono.MonotoneOn _).convexOn_of_deriv convex_univ hf.Continuous.ContinuousOn
     hf.DifferentiableOn
 #align monotone.convex_on_univ_of_deriv Monotone.convexOn_univ_of_deriv
+-/
 
+#print Antitone.concaveOn_univ_of_deriv /-
 /-- If a function `f` is differentiable and `f'` is antitone on `ℝ` then `f` is concave. -/
 theorem Antitone.concaveOn_univ_of_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f)
     (hf'_anti : Antitone (deriv f)) : ConcaveOn ℝ univ f :=
   (hf'_anti.AntitoneOn _).concaveOn_of_deriv convex_univ hf.Continuous.ContinuousOn
     hf.DifferentiableOn
 #align antitone.concave_on_univ_of_deriv Antitone.concaveOn_univ_of_deriv
+-/
 
+#print StrictMono.strictConvexOn_univ_of_deriv /-
 /-- If a function `f` is continuous and `f'` is strictly monotone on `ℝ` then `f` is strictly
 convex. Note that we don't require differentiability, since it is guaranteed at all but at most
 one point by the strict monotonicity of `f'`. -/
@@ -1247,7 +1382,9 @@ theorem StrictMono.strictConvexOn_univ_of_deriv {f : ℝ → ℝ} (hf : Continuo
     (hf'_mono : StrictMono (deriv f)) : StrictConvexOn ℝ univ f :=
   (hf'_mono.StrictMonoOn _).strictConvexOn_of_deriv convex_univ hf.ContinuousOn
 #align strict_mono.strict_convex_on_univ_of_deriv StrictMono.strictConvexOn_univ_of_deriv
+-/
 
+#print StrictAnti.strictConcaveOn_univ_of_deriv /-
 /-- If a function `f` is continuous and `f'` is strictly antitone on `ℝ` then `f` is strictly
 concave. Note that we don't require differentiability, since it is guaranteed at all but at most
 one point by the strict antitonicity of `f'`. -/
@@ -1255,7 +1392,9 @@ theorem StrictAnti.strictConcaveOn_univ_of_deriv {f : ℝ → ℝ} (hf : Continu
     (hf'_anti : StrictAnti (deriv f)) : StrictConcaveOn ℝ univ f :=
   (hf'_anti.StrictAntiOn _).strictConcaveOn_of_deriv convex_univ hf.ContinuousOn
 #align strict_anti.strict_concave_on_univ_of_deriv StrictAnti.strictConcaveOn_univ_of_deriv
+-/
 
+#print convexOn_of_deriv2_nonneg /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
 interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
 theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
@@ -1265,7 +1404,9 @@ theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ →
         rwa [interior_interior]).convexOn_of_deriv
     hD hf hf'
 #align convex_on_of_deriv2_nonneg convexOn_of_deriv2_nonneg
+-/
 
+#print concaveOn_of_deriv2_nonpos /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
 interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
 theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
@@ -1275,7 +1416,9 @@ theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ 
         rwa [interior_interior]).concaveOn_of_deriv
     hD hf hf'
 #align concave_on_of_deriv2_nonpos concaveOn_of_deriv2_nonpos
+-/
 
+#print strictConvexOn_of_deriv2_pos /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on the
 interior, then `f` is strictly convex on `D`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
@@ -1289,7 +1432,9 @@ theorem strictConvexOn_of_deriv2_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
         by rwa [interior_interior]).strictConvexOn_of_deriv
     hD hf
 #align strict_convex_on_of_deriv2_pos strictConvexOn_of_deriv2_pos
+-/
 
+#print strictConcaveOn_of_deriv2_neg /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on the
 interior, then `f` is strictly concave on `D`.
 Note that we don't require twice differentiability explicitly as it already implied by the second
@@ -1303,7 +1448,9 @@ theorem strictConcaveOn_of_deriv2_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
         by rwa [interior_interior]).strictConcaveOn_of_deriv
     hD hf
 #align strict_concave_on_of_deriv2_neg strictConcaveOn_of_deriv2_neg
+-/
 
+#print convexOn_of_deriv2_nonneg' /-
 /-- If a function `f` is twice differentiable on a open convex set `D ⊆ ℝ` and
 `f''` is nonnegative on `D`, then `f` is convex on `D`. -/
 theorem convexOn_of_deriv2_nonneg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
@@ -1312,7 +1459,9 @@ theorem convexOn_of_deriv2_nonneg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ 
   convexOn_of_deriv2_nonneg hD hf'.ContinuousOn (hf'.mono interior_subset)
     (hf''.mono interior_subset) fun x hx => hf''_nonneg x (interior_subset hx)
 #align convex_on_of_deriv2_nonneg' convexOn_of_deriv2_nonneg'
+-/
 
+#print concaveOn_of_deriv2_nonpos' /-
 /-- If a function `f` is twice differentiable on an open convex set `D ⊆ ℝ` and
 `f''` is nonpositive on `D`, then `f` is concave on `D`. -/
 theorem concaveOn_of_deriv2_nonpos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
@@ -1321,7 +1470,9 @@ theorem concaveOn_of_deriv2_nonpos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ 
   concaveOn_of_deriv2_nonpos hD hf'.ContinuousOn (hf'.mono interior_subset)
     (hf''.mono interior_subset) fun x hx => hf''_nonpos x (interior_subset hx)
 #align concave_on_of_deriv2_nonpos' concaveOn_of_deriv2_nonpos'
+-/
 
+#print strictConvexOn_of_deriv2_pos' /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on `D`,
 then `f` is strictly convex on `D`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
@@ -1330,7 +1481,9 @@ theorem strictConvexOn_of_deriv2_pos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
     (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, 0 < ((deriv^[2]) f) x) : StrictConvexOn ℝ D f :=
   strictConvexOn_of_deriv2_pos hD hf fun x hx => hf'' x (interior_subset hx)
 #align strict_convex_on_of_deriv2_pos' strictConvexOn_of_deriv2_pos'
+-/
 
+#print strictConcaveOn_of_deriv2_neg' /-
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on `D`,
 then `f` is strictly concave on `D`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
@@ -1339,7 +1492,9 @@ theorem strictConcaveOn_of_deriv2_neg' {D : Set ℝ} (hD : Convex ℝ D) {f : 
     (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, (deriv^[2]) f x < 0) : StrictConcaveOn ℝ D f :=
   strictConcaveOn_of_deriv2_neg hD hf fun x hx => hf'' x (interior_subset hx)
 #align strict_concave_on_of_deriv2_neg' strictConcaveOn_of_deriv2_neg'
+-/
 
+#print convexOn_univ_of_deriv2_nonneg /-
 /-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonnegative on `ℝ`,
 then `f` is convex on `ℝ`. -/
 theorem convexOn_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
@@ -1348,7 +1503,9 @@ theorem convexOn_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : Differentiable 
   convexOn_of_deriv2_nonneg' convex_univ hf'.DifferentiableOn hf''.DifferentiableOn fun x _ =>
     hf''_nonneg x
 #align convex_on_univ_of_deriv2_nonneg convexOn_univ_of_deriv2_nonneg
+-/
 
+#print concaveOn_univ_of_deriv2_nonpos /-
 /-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonpositive on `ℝ`,
 then `f` is concave on `ℝ`. -/
 theorem concaveOn_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
@@ -1357,7 +1514,9 @@ theorem concaveOn_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : Differentiable
   concaveOn_of_deriv2_nonpos' convex_univ hf'.DifferentiableOn hf''.DifferentiableOn fun x _ =>
     hf''_nonpos x
 #align concave_on_univ_of_deriv2_nonpos concaveOn_univ_of_deriv2_nonpos
+-/
 
+#print strictConvexOn_univ_of_deriv2_pos /-
 /-- If a function `f` is continuous on `ℝ`, and `f''` is strictly positive on `ℝ`,
 then `f` is strictly convex on `ℝ`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
@@ -1366,7 +1525,9 @@ theorem strictConvexOn_univ_of_deriv2_pos {f : ℝ → ℝ} (hf : Continuous f)
     (hf'' : ∀ x, 0 < ((deriv^[2]) f) x) : StrictConvexOn ℝ univ f :=
   strictConvexOn_of_deriv2_pos' convex_univ hf.ContinuousOn fun x _ => hf'' x
 #align strict_convex_on_univ_of_deriv2_pos strictConvexOn_univ_of_deriv2_pos
+-/
 
+#print strictConcaveOn_univ_of_deriv2_neg /-
 /-- If a function `f` is continuous on `ℝ`, and `f''` is strictly negative on `ℝ`,
 then `f` is strictly concave on `ℝ`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
@@ -1375,10 +1536,12 @@ theorem strictConcaveOn_univ_of_deriv2_neg {f : ℝ → ℝ} (hf : Continuous f)
     (hf'' : ∀ x, (deriv^[2]) f x < 0) : StrictConcaveOn ℝ univ f :=
   strictConcaveOn_of_deriv2_neg' convex_univ hf.ContinuousOn fun x _ => hf'' x
 #align strict_concave_on_univ_of_deriv2_neg strictConcaveOn_univ_of_deriv2_neg
+-/
 
 /-! ### Functions `f : E → ℝ` -/
 
 
+#print domain_mvt /-
 /-- Lagrange's Mean Value Theorem, applied to convex domains. -/
 theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
     (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
@@ -1417,6 +1580,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
   use g t; refine' ⟨hseg t <| hIccIoo Ht, _⟩
   simp [g, hMVT']
 #align domain_mvt domain_mvt
+-/
 
 section IsROrC
 
@@ -1434,6 +1598,7 @@ balls over `ℝ` or `ℂ`. For now, we only include the ones that we need.
 variable {𝕜 : Type _} [IsROrC 𝕜] {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type _}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H] {f : G → H} {f' : G → G →L[𝕜] H} {x : G}
 
+#print hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt /-
 /-- Over the reals or the complexes, a continuously differentiable function is strictly
 differentiable. -/
 theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
@@ -1456,7 +1621,9 @@ theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
   refine' (convex_ball _ _).norm_image_sub_le_of_norm_hasFDerivWithin_le' _ hf' h.2 h.1
   exact fun y hy => (hε hy).1.HasFDerivWithinAt
 #align has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
+-/
 
+#print hasStrictDerivAt_of_hasDerivAt_of_continuousAt /-
 /-- Over the reals or the complexes, a continuously differentiable function is strictly
 differentiable. -/
 theorem hasStrictDerivAt_of_hasDerivAt_of_continuousAt {f f' : 𝕜 → G} {x : 𝕜}
@@ -1465,6 +1632,7 @@ theorem hasStrictDerivAt_of_hasDerivAt_of_continuousAt {f f' : 𝕜 → G} {x :
   hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt (hder.mono fun y hy => hy.HasFDerivAt) <|
     (smulRightL 𝕜 𝕜 G 1).Continuous.ContinuousAt.comp hcont
 #align has_strict_deriv_at_of_has_deriv_at_of_continuous_at hasStrictDerivAt_of_hasDerivAt_of_continuousAt
+-/
 
 end IsROrC
 

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

@@ -604,7 +604,6 @@ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDer
     _ = ‖f y - φ y - (f x - φ x)‖ := by abel
     _ = ‖g y - g x‖ := by simp
     _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys
-    
 #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le'
 
 /-- Variant of the mean value inequality on a convex set. Version with `fderiv_within`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -846,7 +846,7 @@ theorem exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b
 
 end Interval
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
 `f` grows faster than `C * x` on `D`, i.e., `C * (y - x) < f y - f x` whenever `x, y ∈ D`,
@@ -874,7 +874,7 @@ theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable
     (fun x _ => hf'_gt x) x trivial y trivial hxy
 #align mul_sub_lt_image_sub_of_lt_deriv mul_sub_lt_image_sub_of_lt_deriv
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
 `f` grows at least as fast as `C * x` on `D`, i.e., `C * (y - x) ≤ f y - f x` whenever `x, y ∈ D`,
@@ -903,7 +903,7 @@ theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable
     (fun x _ => hf'_ge x) x trivial y trivial hxy
 #align mul_sub_le_image_sub_of_le_deriv mul_sub_le_image_sub_of_le_deriv
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
 `f` grows slower than `C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x, y ∈ D`,
@@ -931,7 +931,7 @@ theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable
     (fun x _ => lt_hf' x) x trivial y trivial hxy
 #align image_sub_lt_mul_sub_of_deriv_lt image_sub_lt_mul_sub_of_deriv_lt
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then
 `f` grows at most as fast as `C * x` on `D`, i.e., `f y - f x ≤ C * (y - x)` whenever `x, y ∈ D`,

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, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.calculus.mean_value
-! leanprover-community/mathlib commit 3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe
+! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.Topology.Instances.RealVectorSpace
 /-!
 # The mean value inequality and equalities
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove the following facts:
 
 * `convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s`
@@ -99,8 +102,8 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {
     (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
     (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
   by
-  change Icc a b ⊆ { x | f x ≤ B x }
-  set s := { x | f x ≤ B x } ∩ Icc a b
+  change Icc a b ⊆ {x | f x ≤ B x}
+  set s := {x | f x ≤ B x} ∩ Icc a b
   have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
   have : IsClosed s := by
     simp only [s, inter_comm]
@@ -527,7 +530,7 @@ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs
     (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) :
     ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t :=
   by
-  obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }
+  obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ {y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K}
   exact mem_nhds_within_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))
   rw [inter_comm] at hε 
   refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩
@@ -1101,7 +1104,7 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
   by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
-  · push_neg  at h 
+  · push_neg at h 
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
     obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), (f w - f x) / (w - x) < deriv f a :=
       by
@@ -1152,7 +1155,7 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
   by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h
-  · push_neg  at h 
+  · push_neg at h 
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
     obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), deriv f a < (f w - f x) / (w - x) :=
       by

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

@@ -419,6 +419,7 @@ theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b))
 
 variable {f' g : ℝ → E}
 
+#print eq_of_has_deriv_right_eq /-
 /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`,
   then they are equal everywhere on `[a, b]`. -/
 theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
@@ -431,7 +432,9 @@ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f
       constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by
         simpa only [sub_self] using (derivf y hy).sub (derivg y hy)
 #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq
+-/
 
+#print eq_of_derivWithin_eq /-
 /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere
   on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/
 theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b))
@@ -447,6 +450,7 @@ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b))
     eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuous_on
       gdiff.continuous_on hi
 #align eq_of_deriv_within_eq eq_of_derivWithin_eq
+-/
 
 end
 
@@ -468,9 +472,9 @@ variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
 the function is `C`-Lipschitz. Version with `has_fderiv_within`. -/
-theorem norm_image_sub_le_of_norm_has_fderiv_within_le
-    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
-    (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ :=
+theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
+    (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
+    ‖f y - f x‖ ≤ C * ‖y - x‖ :=
   by
   letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G
   /- By composition with `t ↦ x + t • (y-x)`, we reduce to a statement for functions defined
@@ -498,19 +502,19 @@ theorem norm_image_sub_le_of_norm_has_fderiv_within_le
   apply norm_image_sub_le_of_norm_deriv_le_segment_01' D2
   refine' fun t ht => le_of_op_norm_le _ _ _
   exact bound (g t) (segm <| Ico_subset_Icc_self ht)
-#align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_has_fderiv_within_le
+#align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
 `s`, then the function is `C`-Lipschitz on `s`. Version with `has_fderiv_within` and
 `lipschitz_on_with`. -/
-theorem lipschitzOnWith_of_nnnorm_has_fderiv_within_le {C : ℝ≥0}
+theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0}
     (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C)
     (hs : Convex ℝ s) : LipschitzOnWith C f s :=
   by
   rw [lipschitzOnWith_iff_norm_sub_le]
   intro x x_in y y_in
   exact hs.norm_image_sub_le_of_norm_has_fderiv_within_le hf bound y_in x_in
-#align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_has_fderiv_within_le
+#align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
 
 /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function
 differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is
@@ -528,7 +532,7 @@ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs
   rw [inter_comm] at hε 
   refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩
   exact
-    (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_has_fderiv_within_le
+    (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
       (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le
 #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt
 
@@ -550,8 +554,8 @@ bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv_within`
 theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound
-    xs ys
+  hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound xs
+    ys
 #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
@@ -559,7 +563,7 @@ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f
 `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s :=
-  hs.lipschitzOnWith_of_nnnorm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound
+  hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`,
@@ -567,7 +571,7 @@ then the function is `C`-Lipschitz. Version with `fderiv`. -/
 theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
     (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_fderiv_within_le
+  hs.norm_image_sub_le_of_norm_hasFDerivWithin_le
     (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le
 
@@ -575,16 +579,16 @@ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt
 `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
     (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s :=
-  hs.lipschitzOnWith_of_nnnorm_has_fderiv_within_le
+  hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
     (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le
 
 /-- Variant of the mean value inequality on a convex set, using a bound on the difference between
 the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with
 `has_fderiv_within`. -/
-theorem norm_image_sub_le_of_norm_has_fderiv_within_le'
-    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C)
-    (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ :=
+theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
+    (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
+    ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ :=
   by
   /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem
     applies, `convex.norm_image_sub_le_of_norm_has_fderiv_within_le`. Then, we just need to glue
@@ -596,15 +600,15 @@ theorem norm_image_sub_le_of_norm_has_fderiv_within_le'
     ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp
     _ = ‖f y - φ y - (f x - φ x)‖ := by abel
     _ = ‖g y - g x‖ := by simp
-    _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_has_fderiv_within_le hg bound hs xs ys
+    _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys
     
-#align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_has_fderiv_within_le'
+#align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le'
 
 /-- Variant of the mean value inequality on a convex set. Version with `fderiv_within`. -/
 theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_fderiv_within_le' (fun x hx => (hf x hx).HasFDerivWithinAt) bound
+  hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).HasFDerivWithinAt) bound
     xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le'
 
@@ -612,7 +616,7 @@ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f
 theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
     (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_fderiv_within_le'
+  hs.norm_image_sub_le_of_norm_hasFDerivWithin_le'
     (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le'
 
@@ -660,29 +664,29 @@ variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜}
 
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C`, then the function is `C`-Lipschitz. Version with `has_deriv_within`. -/
-theorem norm_image_sub_le_of_norm_has_deriv_within_le {C : ℝ}
+theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ}
     (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
     (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ :=
-  Convex.norm_image_sub_le_of_norm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt)
+  Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt)
     (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys
-#align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_has_deriv_within_le
+#align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
 
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
 Version with `has_deriv_within` and `lipschitz_on_with`. -/
-theorem lipschitzOnWith_of_nnnorm_has_deriv_within_le {C : ℝ≥0} (hs : Convex ℝ s)
+theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s)
     (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) :
     LipschitzOnWith C f s :=
-  Convex.lipschitzOnWith_of_nnnorm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt)
+  Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).HasFDerivWithinAt)
     (fun x hx => le_trans (by simp) (bound x hx)) hs
-#align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_has_deriv_within_le
+#align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le
 
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within
 this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv_within` -/
 theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_deriv_within_le (fun x hx => (hf x hx).HasDerivWithinAt) bound xs
+  hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivWithinAt) bound xs
     ys
 #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le
 
@@ -692,7 +696,7 @@ Version with `deriv_within` and `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s)
     (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) :
     LipschitzOnWith C f s :=
-  hs.lipschitzOnWith_of_nnnorm_has_deriv_within_le (fun x hx => (hf x hx).HasDerivWithinAt) bound
+  hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le
 
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
@@ -700,8 +704,8 @@ bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/
 theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
     (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_deriv_within_le
-    (fun x hx => (hf x hx).HasDerivAt.HasDerivWithinAt) bound xs ys
+  hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivAt.HasDerivWithinAt)
+    bound xs ys
 #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le
 
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
@@ -709,8 +713,8 @@ bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
 Version with `deriv` and `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
     (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s :=
-  hs.lipschitzOnWith_of_nnnorm_has_deriv_within_le
-    (fun x hx => (hf x hx).HasDerivAt.HasDerivWithinAt) bound
+  hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).HasDerivAt.HasDerivWithinAt)
+    bound
 #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le
 
 /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`,
@@ -1447,7 +1451,7 @@ theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
     exact le_of_lt (hε H').2
   -- apply mean value theorem
   letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G
-  refine' (convex_ball _ _).norm_image_sub_le_of_norm_has_fderiv_within_le' _ hf' h.2 h.1
+  refine' (convex_ball _ _).norm_image_sub_le_of_norm_hasFDerivWithin_le' _ hf' h.2 h.1
   exact fun y hy => (hε hy).1.HasFDerivWithinAt
 #align has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
 

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

@@ -124,7 +124,7 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {
     refine' ⟨z, _, hz⟩
     have := (hfz.trans hzB).le
     rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
-      sub_le_sub_iff_right] at this
+      sub_le_sub_iff_right] at this 
 #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary'
 
 /-- General fencing theorem for continuous functions with an estimate on the derivative.
@@ -425,7 +425,7 @@ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f
     (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b))
     (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y :=
   by
-  simp only [← @sub_eq_zero _ _ (f _)] at hi⊢
+  simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢
   exact
     hi ▸
       constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by
@@ -525,7 +525,7 @@ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs
   by
   obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }
   exact mem_nhds_within_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))
-  rw [inter_comm] at hε
+  rw [inter_comm] at hε 
   refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩
   exact
     (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_has_fderiv_within_le
@@ -639,7 +639,7 @@ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜
     (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) :
     s.EqOn f g := by
   intro y hy
-  suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this
+  suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this 
   refine' hs.is_const_of_fderiv_within_eq_zero (hf.sub hg) _ hx hy
   intro z hz
   rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz]
@@ -802,7 +802,7 @@ theorem exists_hasDerivAt_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b
       fun x hx => hasDerivAt_id x with
     ⟨c, cmem, hc⟩
   use c, cmem
-  simp only [_root_.id, Pi.one_apply, mul_one] at hc
+  simp only [_root_.id, Pi.one_apply, mul_one] at hc 
   rw [← hc, mul_div_cancel_left]
   exact ne_of_gt (sub_pos.2 hab)
 #align exists_has_deriv_at_eq_slope exists_hasDerivAt_eq_slope
@@ -1097,9 +1097,9 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
   by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
-  · push_neg  at h
+  · push_neg  at h 
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
-    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ)(H : a ∈ Ioo x w), (f w - f x) / (w - x) < deriv f a :=
+    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), (f w - f x) / (w - x) < deriv f a :=
       by
       apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
       · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
@@ -1108,7 +1108,7 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
         rw [← hw]
         apply ne_of_lt
         exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
-    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ (b : ℝ)(H : b ∈ Ioo w y), (f y - f w) / (y - w) < deriv f b :=
+    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ (b : ℝ) (H : b ∈ Ioo w y), (f y - f w) / (y - w) < deriv f b :=
       by
       apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
       · refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
@@ -1118,7 +1118,7 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
         apply ne_of_gt
         exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
     refine' ⟨b, ⟨hxw.trans hwb, hby⟩, _⟩
-    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb⊢
+    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
     have : deriv f a * (w - x) < deriv f b * (w - x) :=
       by
       apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _
@@ -1148,9 +1148,9 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
   by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h
-  · push_neg  at h
+  · push_neg  at h 
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
-    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ)(H : a ∈ Ioo x w), deriv f a < (f w - f x) / (w - x) :=
+    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), deriv f a < (f w - f x) / (w - x) :=
       by
       apply StrictMonoOn.exists_deriv_lt_slope_aux _ hxw _ _
       · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
@@ -1159,7 +1159,7 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
         rw [← hw]
         apply ne_of_lt
         exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
-    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ (b : ℝ)(H : b ∈ Ioo w y), deriv f b < (f y - f w) / (y - w) :=
+    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ (b : ℝ) (H : b ∈ Ioo w y), deriv f b < (f y - f w) / (y - w) :=
       by
       apply StrictMonoOn.exists_deriv_lt_slope_aux _ hwy _ _
       · refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
@@ -1169,7 +1169,7 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
         apply ne_of_gt
         exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
     refine' ⟨a, ⟨hxa, haw.trans hwy⟩, _⟩
-    simp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb⊢
+    simp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
     have : deriv f a * (y - w) < deriv f b * (y - w) :=
       by
       apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _
@@ -1388,7 +1388,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
     intro t ht; exact ⟨t, ht, rfl⟩
   have hseg' : Icc 0 1 ⊆ g ⁻¹' s := by
     rw [← image_subset_iff]; unfold image; change ∀ _, _
-    intro z Hz; rw [mem_set_of_eq] at Hz; rcases Hz with ⟨t, Ht, hgt⟩
+    intro z Hz; rw [mem_set_of_eq] at Hz ; rcases Hz with ⟨t, Ht, hgt⟩
     rw [← hgt]; exact hs.segment_subset xs ys (hseg t Ht)
   -- derivative of pullback of f under parametrization
   have hfg :
@@ -1398,7 +1398,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
     have hg : HasDerivAt g (y - x) t :=
       by
       have := ((hasDerivAt_id t).smul_const (y - x)).const_add x
-      rwa [one_smul] at this
+      rwa [one_smul] at this 
     exact (hf (g t) <| hseg' Ht).comp_hasDerivWithinAt _ hg.has_deriv_within_at hseg'
   -- apply 1-variable mean value theorem to pullback
   have hMVT : ∃ t ∈ Ioo (0 : ℝ) 1, (f' (g t) : E → ℝ) (y - x) = (f (g 1) - f (g 0)) / (1 - 0) :=
@@ -1441,7 +1441,7 @@ theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
   refine' ⟨ε, ε0, _⟩
   -- simplify formulas involving the product E × E
   rintro ⟨a, b⟩ h
-  rw [← ball_prod_same, prod_mk_mem_set_prod_eq] at h
+  rw [← ball_prod_same, prod_mk_mem_set_prod_eq] at h 
   -- exploit the choice of ε as the modulus of continuity of f'
   have hf' : ∀ x' ∈ ball x ε, ‖f' x' - f' x‖ ≤ c := by intro x' H'; rw [← dist_eq_norm];
     exact le_of_lt (hε H').2

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

@@ -76,7 +76,7 @@ variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type _} [N
 
 open Metric Set Asymptotics ContinuousLinearMap Filter
 
-open Classical Topology NNReal
+open scoped Classical Topology NNReal
 
 /-! ### One-dimensional fencing inequalities -/
 

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

@@ -355,8 +355,7 @@ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : 
     intro x
     simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))
   convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound
-  simp only [g, B]
-  rw [sub_self, norm_zero, sub_self, MulZeroClass.mul_zero]
+  simp only [g, B]; rw [sub_self, norm_zero, sub_self, MulZeroClass.mul_zero]
 #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment
 
 /-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
@@ -487,13 +486,9 @@ theorem norm_image_sub_le_of_norm_has_fderiv_within_le
     by
     rw [← image_subset_iff, ← segment_eq_image']
     apply hs.segment_subset xs ys
-  have : f x = f (g 0) := by
-    simp only [g]
-    rw [zero_smul, add_zero]
+  have : f x = f (g 0) := by simp only [g]; rw [zero_smul, add_zero]
   rw [this]
-  have : f y = f (g 1) := by
-    simp only [g]
-    rw [one_smul, add_sub_cancel'_right]
+  have : f y = f (g 1) := by simp only [g]; rw [one_smul, add_sub_cancel'_right]
   rw [this]
   have D2 : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t :=
     by
@@ -730,11 +725,7 @@ theorem lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜
 then it is a constant function. -/
 theorem is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) :
     f x = f y :=
-  is_const_of_fderiv_eq_zero hf
-    (fun z => by
-      ext
-      simp [← deriv_fderiv, hf'])
-    _ _
+  is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _
 #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero
 
 end Convex
@@ -759,9 +750,7 @@ theorem exists_ratio_hasDerivAt_eq_ratio_slope :
     ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c :=
   by
   let h x := (g b - g a) * f x - (f b - f a) * g x
-  have hI : h a = h b := by
-    simp only [h]
-    ring
+  have hI : h a = h b := by simp only [h]; ring
   let h' x := (g b - g a) * f' x - (f b - f a) * g' x
   have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx =>
     ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))
@@ -866,9 +855,7 @@ theorem Convex.mul_sub_lt_image_sub_of_lt_deriv {D : Set ℝ} (hD : Convex ℝ D
     subset_sUnion_of_mem ⟨isOpen_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩
   obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x)
   exact exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')
-  have : C < (f y - f x) / (y - x) := by
-    rw [← ha]
-    exact hf'_gt _ (hxyD' a_mem)
+  have : C < (f y - f x) / (y - x) := by rw [← ha]; exact hf'_gt _ (hxyD' a_mem)
   exact (lt_div_iff (sub_pos.2 hxy)).1 this
 #align convex.mul_sub_lt_image_sub_of_lt_deriv Convex.mul_sub_lt_image_sub_of_lt_deriv
 
@@ -891,16 +878,13 @@ theorem Convex.mul_sub_le_image_sub_of_le_deriv {D : Set ℝ} (hD : Convex ℝ D
     ∀ (x) (_ : x ∈ D) (y) (_ : y ∈ D), x ≤ y → C * (y - x) ≤ f y - f x :=
   by
   intro x hx y hy hxy
-  cases' eq_or_lt_of_le hxy with hxy' hxy'
-  · rw [hxy', sub_self, sub_self, MulZeroClass.mul_zero]
+  cases' eq_or_lt_of_le hxy with hxy' hxy'; · rw [hxy', sub_self, sub_self, MulZeroClass.mul_zero]
   have hxyD : Icc x y ⊆ D := hD.ord_connected.out hx hy
   have hxyD' : Ioo x y ⊆ interior D :=
     subset_sUnion_of_mem ⟨isOpen_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩
   obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x)
   exact exists_deriv_eq_slope f hxy' (hf.mono hxyD) (hf'.mono hxyD')
-  have : C ≤ (f y - f x) / (y - x) := by
-    rw [← ha]
-    exact hf'_ge _ (hxyD' a_mem)
+  have : C ≤ (f y - f x) / (y - x) := by rw [← ha]; exact hf'_ge _ (hxyD' a_mem)
   exact (le_div_iff (sub_pos.2 hxy')).1 this
 #align convex.mul_sub_le_image_sub_of_le_deriv Convex.mul_sub_le_image_sub_of_le_deriv
 
@@ -1401,17 +1385,11 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
     by
     rw [segment_eq_image']
     simp only [mem_image, and_imp, add_right_inj]
-    intro t ht
-    exact ⟨t, ht, rfl⟩
+    intro t ht; exact ⟨t, ht, rfl⟩
   have hseg' : Icc 0 1 ⊆ g ⁻¹' s := by
-    rw [← image_subset_iff]
-    unfold image
-    change ∀ _, _
-    intro z Hz
-    rw [mem_set_of_eq] at Hz
-    rcases Hz with ⟨t, Ht, hgt⟩
-    rw [← hgt]
-    exact hs.segment_subset xs ys (hseg t Ht)
+    rw [← image_subset_iff]; unfold image; change ∀ _, _
+    intro z Hz; rw [mem_set_of_eq] at Hz; rcases Hz with ⟨t, Ht, hgt⟩
+    rw [← hgt]; exact hs.segment_subset xs ys (hseg t Ht)
   -- derivative of pullback of f under parametrization
   have hfg :
     ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) ((f' (g t) : E → ℝ) (y - x)) (Icc (0 : ℝ) 1) t :=
@@ -1430,8 +1408,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
     · exact fun t Ht => (hfg t <| hIccIoo Ht).HasDerivAt (Icc_mem_nhds Ht.1 Ht.2)
   -- reinterpret on domain
   rcases hMVT with ⟨t, Ht, hMVT'⟩
-  use g t
-  refine' ⟨hseg t <| hIccIoo Ht, _⟩
+  use g t; refine' ⟨hseg t <| hIccIoo Ht, _⟩
   simp [g, hMVT']
 #align domain_mvt domain_mvt
 
@@ -1466,10 +1443,7 @@ theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
   rintro ⟨a, b⟩ h
   rw [← ball_prod_same, prod_mk_mem_set_prod_eq] at h
   -- exploit the choice of ε as the modulus of continuity of f'
-  have hf' : ∀ x' ∈ ball x ε, ‖f' x' - f' x‖ ≤ c :=
-    by
-    intro x' H'
-    rw [← dist_eq_norm]
+  have hf' : ∀ x' ∈ ball x ε, ‖f' x' - f' x‖ ≤ c := by intro x' H'; rw [← dist_eq_norm];
     exact le_of_lt (hε H').2
   -- apply mean value theorem
   letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G

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

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.calculus.mean_value
-! leanprover-community/mathlib commit 75e7fca56381d056096ce5d05e938f63a6567828
+! leanprover-community/mathlib commit 3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.Analysis.Calculus.Deriv.Slope
 import Mathbin.Analysis.Calculus.LocalExtr
 import Mathbin.Analysis.Convex.Slope
 import Mathbin.Analysis.Convex.Normed

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.calculus.mean_value
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 75e7fca56381d056096ce5d05e938f63a6567828
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -464,7 +464,7 @@ variable {𝕜 G : Type _} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGrou
 
 namespace Convex
 
-variable {f : E → G} {C : ℝ} {s : Set E} {x y : E} {f' : E → E →L[𝕜] G} {φ : E →L[𝕜] G}
+variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G}
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
 the function is `C`-Lipschitz. Version with `has_fderiv_within`. -/
@@ -636,6 +636,26 @@ theorem is_const_of_fderiv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, fd
     (fun x _ => by rw [fderivWithin_univ] <;> exact hf' x) trivial trivial
 #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero
 
+/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at
+one point in that set, then they are equal on that set. -/
+theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s)
+    (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s)
+    (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) :
+    s.EqOn f g := by
+  intro y hy
+  suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this
+  refine' hs.is_const_of_fderiv_within_eq_zero (hf.sub hg) _ hx hy
+  intro z hz
+  rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz]
+#align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq
+
+theorem eq_of_fderiv_eq (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g)
+    (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g :=
+  suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x
+  convex_univ.eqOn_of_fderivWithin_eq hf.DifferentiableOn hg.DifferentiableOn uniqueDiffOn_univ
+    (fun x hx => by simpa using hf' _) (mem_univ _) hfgx
+#align eq_of_fderiv_eq eq_of_fderiv_eq
+
 end Convex
 
 namespace Convex

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

@@ -469,7 +469,7 @@ variable {f : E → G} {C : ℝ} {s : Set E} {x y : E} {f' : E → E →L[𝕜]
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
 the function is `C`-Lipschitz. Version with `has_fderiv_within`. -/
 theorem norm_image_sub_le_of_norm_has_fderiv_within_le
-    (hf : ∀ x ∈ s, HasFderivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
+    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
     (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ :=
   by
   letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G
@@ -497,7 +497,7 @@ theorem norm_image_sub_le_of_norm_has_fderiv_within_le
   have D2 : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t :=
     by
     intro t ht
-    have : HasFderivWithinAt f ((f' (g t)).restrictScalars ℝ) s (g t) := hf (g t) (segm ht)
+    have : HasFDerivWithinAt f ((f' (g t)).restrictScalars ℝ) s (g t) := hf (g t) (segm ht)
     exact this.comp_has_deriv_within_at _ (Dg t).HasDerivWithinAt segm
   apply norm_image_sub_le_of_norm_deriv_le_segment_01' D2
   refine' fun t ht => le_of_op_norm_le _ _ _
@@ -508,7 +508,7 @@ theorem norm_image_sub_le_of_norm_has_fderiv_within_le
 `s`, then the function is `C`-Lipschitz on `s`. Version with `has_fderiv_within` and
 `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_has_fderiv_within_le {C : ℝ≥0}
-    (hf : ∀ x ∈ s, HasFderivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C)
+    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C)
     (hs : Convex ℝ s) : LipschitzOnWith C f s :=
   by
   rw [lipschitzOnWith_iff_norm_sub_le]
@@ -522,19 +522,19 @@ continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖
 `K`-Lipschitz on some neighborhood of `x` within `s`. See also
 `convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at` for a version that claims
 existence of `K` instead of an explicit estimate. -/
-theorem exists_nhdsWithin_lipschitzOnWith_of_hasFderivWithinAt_of_nnnorm_lt (hs : Convex ℝ s)
-    {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFderivWithinAt f (f' y) s y)
+theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s)
+    {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y)
     (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) :
     ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t :=
   by
-  obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFderivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }
+  obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }
   exact mem_nhds_within_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))
   rw [inter_comm] at hε
   refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩
   exact
     (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_has_fderiv_within_le
       (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le
-#align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFderivWithinAt_of_nnnorm_lt
+#align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt
 
 /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function
 differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is
@@ -542,19 +542,19 @@ continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖
 on some neighborhood of `x` within `s`. See also
 `convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt` for a version
 with an explicit estimate on the Lipschitz constant. -/
-theorem exists_nhdsWithin_lipschitzOnWith_of_hasFderivWithinAt (hs : Convex ℝ s) {f : E → G}
-    (hder : ∀ᶠ y in 𝓝[s] x, HasFderivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) :
+theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G}
+    (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) :
     ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t :=
   (exists_gt _).imp <|
-    hs.exists_nhdsWithin_lipschitzOnWith_of_hasFderivWithinAt_of_nnnorm_lt hder hcont
-#align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFderivWithinAt
+    hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont
+#align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt
 
 /-- The mean value theorem on a convex set: if the derivative of a function within this set is
 bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv_within`. -/
 theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_fderiv_within_le (fun x hx => (hf x hx).HasFderivWithinAt) bound
+  hs.norm_image_sub_le_of_norm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound
     xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le
 
@@ -563,7 +563,7 @@ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f
 `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s :=
-  hs.lipschitzOnWith_of_nnnorm_has_fderiv_within_le (fun x hx => (hf x hx).HasFderivWithinAt) bound
+  hs.lipschitzOnWith_of_nnnorm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`,
@@ -572,7 +572,7 @@ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt
     (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x‖ ≤ C * ‖y - x‖ :=
   hs.norm_image_sub_le_of_norm_has_fderiv_within_le
-    (fun x hx => (hf x hx).HasFderivAt.HasFderivWithinAt) bound xs ys
+    (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
@@ -580,21 +580,21 @@ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt
 theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
     (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s :=
   hs.lipschitzOnWith_of_nnnorm_has_fderiv_within_le
-    (fun x hx => (hf x hx).HasFderivAt.HasFderivWithinAt) bound
+    (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le
 
 /-- Variant of the mean value inequality on a convex set, using a bound on the difference between
 the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with
 `has_fderiv_within`. -/
 theorem norm_image_sub_le_of_norm_has_fderiv_within_le'
-    (hf : ∀ x ∈ s, HasFderivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C)
+    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C)
     (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ :=
   by
   /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem
     applies, `convex.norm_image_sub_le_of_norm_has_fderiv_within_le`. Then, we just need to glue
     together the pieces, expressing back `f` in terms of `g`. -/
   let g y := f y - φ y
-  have hg : ∀ x ∈ s, HasFderivWithinAt g (f' x - φ) s x := fun x xs =>
+  have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs =>
     (hf x xs).sub φ.has_fderiv_within_at
   calc
     ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp
@@ -608,7 +608,7 @@ theorem norm_image_sub_le_of_norm_has_fderiv_within_le'
 theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s)
     (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ :=
-  hs.norm_image_sub_le_of_norm_has_fderiv_within_le' (fun x hx => (hf x hx).HasFderivWithinAt) bound
+  hs.norm_image_sub_le_of_norm_has_fderiv_within_le' (fun x hx => (hf x hx).HasFDerivWithinAt) bound
     xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le'
 
@@ -617,7 +617,7 @@ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt
     (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ :=
   hs.norm_image_sub_le_of_norm_has_fderiv_within_le'
-    (fun x hx => (hf x hx).HasFderivAt.HasFderivWithinAt) bound xs ys
+    (fun x hx => (hf x hx).HasFDerivAt.HasFDerivWithinAt) bound xs ys
 #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le'
 
 /-- If a function has zero Fréchet derivative at every point of a convex set,
@@ -647,7 +647,7 @@ bounded by `C`, then the function is `C`-Lipschitz. Version with `has_deriv_with
 theorem norm_image_sub_le_of_norm_has_deriv_within_le {C : ℝ}
     (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
     (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ :=
-  Convex.norm_image_sub_le_of_norm_has_fderiv_within_le (fun x hx => (hf x hx).HasFderivWithinAt)
+  Convex.norm_image_sub_le_of_norm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt)
     (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys
 #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_has_deriv_within_le
 
@@ -657,7 +657,7 @@ Version with `has_deriv_within` and `lipschitz_on_with`. -/
 theorem lipschitzOnWith_of_nnnorm_has_deriv_within_le {C : ℝ≥0} (hs : Convex ℝ s)
     (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) :
     LipschitzOnWith C f s :=
-  Convex.lipschitzOnWith_of_nnnorm_has_fderiv_within_le (fun x hx => (hf x hx).HasFderivWithinAt)
+  Convex.lipschitzOnWith_of_nnnorm_has_fderiv_within_le (fun x hx => (hf x hx).HasFDerivWithinAt)
     (fun x hx => le_trans (by simp) (bound x hx)) hs
 #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_has_deriv_within_le
 
@@ -1370,7 +1370,7 @@ theorem strictConcaveOn_univ_of_deriv2_neg {f : ℝ → ℝ} (hf : Continuous f)
 
 /-- Lagrange's Mean Value Theorem, applied to convex domains. -/
 theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
-    (hf : ∀ x ∈ s, HasFderivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
+    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
     ∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) :=
   by
   have hIccIoo := @Ioo_subset_Icc_self ℝ _ 0 1
@@ -1432,9 +1432,9 @@ variable {𝕜 : Type _} [IsROrC 𝕜] {G : Type _} [NormedAddCommGroup G] [Norm
 
 /-- Over the reals or the complexes, a continuously differentiable function is strictly
 differentiable. -/
-theorem hasStrictFderivAt_of_hasFderivAt_of_continuousAt
-    (hder : ∀ᶠ y in 𝓝 x, HasFderivAt f (f' y) y) (hcont : ContinuousAt f' x) :
-    HasStrictFderivAt f (f' x) x :=
+theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
+    (hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
+    HasStrictFDerivAt f (f' x) x :=
   by
   -- turn little-o definition of strict_fderiv into an epsilon-delta statement
   refine' is_o_iff.mpr fun c hc => metric.eventually_nhds_iff_ball.mpr _
@@ -1453,15 +1453,15 @@ theorem hasStrictFderivAt_of_hasFderivAt_of_continuousAt
   -- apply mean value theorem
   letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G
   refine' (convex_ball _ _).norm_image_sub_le_of_norm_has_fderiv_within_le' _ hf' h.2 h.1
-  exact fun y hy => (hε hy).1.HasFderivWithinAt
-#align has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at hasStrictFderivAt_of_hasFderivAt_of_continuousAt
+  exact fun y hy => (hε hy).1.HasFDerivWithinAt
+#align has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
 
 /-- Over the reals or the complexes, a continuously differentiable function is strictly
 differentiable. -/
 theorem hasStrictDerivAt_of_hasDerivAt_of_continuousAt {f f' : 𝕜 → G} {x : 𝕜}
     (hder : ∀ᶠ y in 𝓝 x, HasDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
     HasStrictDerivAt f (f' x) x :=
-  hasStrictFderivAt_of_hasFderivAt_of_continuousAt (hder.mono fun y hy => hy.HasFderivAt) <|
+  hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt (hder.mono fun y hy => hy.HasFDerivAt) <|
     (smulRightL 𝕜 𝕜 G 1).Continuous.ContinuousAt.comp hcont
 #align has_strict_deriv_at_of_has_deriv_at_of_continuous_at hasStrictDerivAt_of_hasDerivAt_of_continuousAt
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -166,7 +166,7 @@ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b
     by
     intro x hx r hr
     apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
-    · rwa [sub_self, mul_zero, add_zero]
+    · rwa [sub_self, MulZeroClass.mul_zero, add_zero]
     · exact hB.add (continuous_on_const.mul (continuous_id.continuous_on.sub continuousOn_const))
     · intro x hx
       exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r)
@@ -355,7 +355,7 @@ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : 
     simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))
   convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound
   simp only [g, B]
-  rw [sub_self, norm_zero, sub_self, mul_zero]
+  rw [sub_self, norm_zero, sub_self, MulZeroClass.mul_zero]
 #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment
 
 /-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
@@ -403,7 +403,7 @@ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ}
 
 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b))
     (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by
-  simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>
+  simpa only [MulZeroClass.zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>
     norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv
       (fun y hy => by rw [norm_le_zero_iff]) x hx
 #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero
@@ -413,7 +413,7 @@ theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b))
   by
   have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by
     simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx
-  simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>
+  simpa only [MulZeroClass.zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>
     norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx
 #align constant_of_deriv_within_zero constant_of_derivWithin_zero
 
@@ -626,7 +626,7 @@ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : Differentiabl
     (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y :=
   by
   have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero]
-  simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using
+  simpa only [(dist_eq_norm _ _).symm, MulZeroClass.zero_mul, dist_le_zero, eq_comm] using
     hs.norm_image_sub_le_of_norm_fderiv_within_le hf bound hx hy
 #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero
 
@@ -871,7 +871,7 @@ theorem Convex.mul_sub_le_image_sub_of_le_deriv {D : Set ℝ} (hD : Convex ℝ D
   by
   intro x hx y hy hxy
   cases' eq_or_lt_of_le hxy with hxy' hxy'
-  · rw [hxy', sub_self, sub_self, mul_zero]
+  · rw [hxy', sub_self, sub_self, MulZeroClass.mul_zero]
   have hxyD : Icc x y ⊆ D := hD.ord_connected.out hx hy
   have hxyD' : Ioo x y ⊆ interior D :=
     subset_sUnion_of_mem ⟨isOpen_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩
@@ -956,7 +956,8 @@ theorem Convex.strictMonoOn_of_deriv_pos {D : Set ℝ} (hD : Convex ℝ D) {f :
     (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, 0 < deriv f x) : StrictMonoOn f D :=
   by
   rintro x hx y hy
-  simpa only [zero_mul, sub_pos] using hD.mul_sub_lt_image_sub_of_lt_deriv hf _ hf' x hx y hy
+  simpa only [MulZeroClass.zero_mul, sub_pos] using
+    hD.mul_sub_lt_image_sub_of_lt_deriv hf _ hf' x hx y hy
   exact fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).ne').DifferentiableWithinAt
 #align convex.strict_mono_on_of_deriv_pos Convex.strictMonoOn_of_deriv_pos
 
@@ -978,7 +979,7 @@ of the real line. If `f` is differentiable on the interior of `D` and `f'` is no
 theorem Convex.monotoneOn_of_deriv_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
     (hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x) : MonotoneOn f D := fun x hx y hy hxy => by
-  simpa only [zero_mul, sub_nonneg] using
+  simpa only [MulZeroClass.zero_mul, sub_nonneg] using
     hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg x hx y hy hxy
 #align convex.monotone_on_of_deriv_nonneg Convex.monotoneOn_of_deriv_nonneg
 
@@ -997,7 +998,7 @@ of the real line. If `f` is differentiable on the interior of `D` and `f'` is ne
 theorem Convex.strictAntiOn_of_deriv_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, deriv f x < 0) : StrictAntiOn f D :=
   fun x hx y => by
-  simpa only [zero_mul, sub_lt_zero] using
+  simpa only [MulZeroClass.zero_mul, sub_lt_zero] using
     hD.image_sub_lt_mul_sub_of_deriv_lt hf
       (fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).Ne).DifferentiableWithinAt) hf' x
       hx y
@@ -1021,7 +1022,7 @@ of the real line. If `f` is differentiable on the interior of `D` and `f'` is no
 theorem Convex.antitoneOn_of_deriv_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
     (hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0) : AntitoneOn f D := fun x hx y hy hxy => by
-  simpa only [zero_mul, sub_nonpos] using
+  simpa only [MulZeroClass.zero_mul, sub_nonpos] using
     hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy
 #align convex.antitone_on_of_deriv_nonpos Convex.antitoneOn_of_deriv_nonpos
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -829,7 +829,7 @@ theorem exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b
 
 end Interval
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
 `f` grows faster than `C * x` on `D`, i.e., `C * (y - x) < f y - f x` whenever `x, y ∈ D`,
@@ -859,7 +859,7 @@ theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable
     (fun x _ => hf'_gt x) x trivial y trivial hxy
 #align mul_sub_lt_image_sub_of_lt_deriv mul_sub_lt_image_sub_of_lt_deriv
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
 `f` grows at least as fast as `C * x` on `D`, i.e., `C * (y - x) ≤ f y - f x` whenever `x, y ∈ D`,
@@ -891,7 +891,7 @@ theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable
     (fun x _ => hf'_ge x) x trivial y trivial hxy
 #align mul_sub_le_image_sub_of_le_deriv mul_sub_le_image_sub_of_le_deriv
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
 `f` grows slower than `C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x, y ∈ D`,
@@ -919,7 +919,7 @@ theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable
     (fun x _ => lt_hf' x) x trivial y trivial hxy
 #align image_sub_lt_mul_sub_of_deriv_lt image_sub_lt_mul_sub_of_deriv_lt
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x y «expr ∈ » D) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » D) -/
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then
 `f` grows at most as fast as `C * x` on `D`, i.e., `f y - f x ≤ C * (y - x)` whenever `x, y ∈ D`,

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

RCLike is an analytic typeclass, hence should be under Analysis

@@ -9,7 +9,7 @@ import Mathlib.Analysis.Calculus.Deriv.Mul
 import Mathlib.Analysis.Calculus.Deriv.Comp
 import Mathlib.Analysis.Calculus.LocalExtr.Rolle
 import Mathlib.Analysis.Convex.Normed
-import Mathlib.Data.RCLike.Basic
+import Mathlib.Analysis.RCLike.Basic
 
 #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
 

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -9,7 +9,7 @@ import Mathlib.Analysis.Calculus.Deriv.Mul
 import Mathlib.Analysis.Calculus.Deriv.Comp
 import Mathlib.Analysis.Calculus.LocalExtr.Rolle
 import Mathlib.Analysis.Convex.Normed
-import Mathlib.Data.IsROrC.Basic
+import Mathlib.Data.RCLike.Basic
 
 #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
 
@@ -21,7 +21,7 @@ In this file we prove the following facts:
 * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s`
   and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with
   constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the
-  derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`,
+  derivative from a fixed linear map. This lemma and its versions are formulated using `RCLike`,
   so they work both for real and complex derivatives.
 
 * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or
@@ -439,12 +439,12 @@ end
 ### Vector-valued functions `f : E → G`
 
 Theorems in this section work both for real and complex differentiable functions. We use assumptions
-`[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we
+`[RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we
 also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/
 
 section
 
-variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+variable {𝕜 G : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
 
 namespace Convex
 
@@ -1080,7 +1080,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
   simpa [g] using hMVT'.symm
 #align domain_mvt domain_mvt
 
-section IsROrC
+section RCLike
 
 /-!
 ### Vector-valued functions `f : E → F`. Strict differentiability.
@@ -1092,7 +1092,7 @@ make sense and are enough. Many formulations of the mean value inequality could
 balls over `ℝ` or `ℂ`. For now, we only include the ones that we need.
 -/
 
-variable {𝕜 : Type*} [IsROrC 𝕜] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type*}
+variable {𝕜 : Type*} [RCLike 𝕜] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type*}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H] {f : G → H} {f' : G → G →L[𝕜] H} {x : G}
 
 /-- Over the reals or the complexes, a continuously differentiable function is strictly
@@ -1127,4 +1127,4 @@ theorem hasStrictDerivAt_of_hasDerivAt_of_continuousAt {f f' : 𝕜 → G} {x :
     (smulRightL 𝕜 𝕜 G 1).continuous.continuousAt.comp hcont
 #align has_strict_deriv_at_of_has_deriv_at_of_continuous_at hasStrictDerivAt_of_hasDerivAt_of_continuousAt
 
-end IsROrC
+end RCLike

From #9618, #10525 and #10614.

@@ -931,6 +931,9 @@ lemma strictMonoOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f
   strictMonoOn_of_deriv_pos hD hf fun x hx ↦ by
     rw [deriv_eqOn isOpen_interior hf' hx]; exact hf'₀ _ hx
 
+@[deprecated] -- 2024-03-02
+alias StrictMonoOn_of_hasDerivWithinAt_pos := strictMonoOn_of_hasDerivWithinAt_pos
+
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is strictly positive, then
 `f` is a strictly monotone function. -/
 lemma strictMono_of_hasDerivAt_pos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
@@ -1004,12 +1007,18 @@ lemma strictAntiOn_of_hasDerivWithinAt_neg {D : Set ℝ} (hD : Convex ℝ D) {f
   strictAntiOn_of_deriv_neg hD hf fun x hx ↦ by
     rw [deriv_eqOn isOpen_interior hf' hx]; exact hf'₀ _ hx
 
+@[deprecated] -- 2024-03-02
+alias StrictAntiOn_of_hasDerivWithinAt_pos := strictAntiOn_of_hasDerivWithinAt_neg
+
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is strictly positive, then
 `f` is a strictly monotone function. -/
 lemma strictAnti_of_hasDerivAt_neg {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
     (hf' : ∀ x, f' x < 0) : StrictAnti f :=
   strictAnti_of_deriv_neg fun x ↦ by rw [(hf _).deriv]; exact hf' _
 
+@[deprecated] -- 2024-03-02
+alias strictAnti_of_hasDerivAt_pos := strictAnti_of_hasDerivAt_neg
+
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonpositive, then
 `f` is an antitone function on `D`. -/

This PR carves off lemmas about convex functions from Analysis/Calculus/MeanValue.lean and relocates them to a new file. (This is a preparatory step for adding some more lemmas on this topic, since MeanValue.lean is already over-long.)

@@ -8,10 +8,8 @@ import Mathlib.Analysis.Calculus.Deriv.Slope
 import Mathlib.Analysis.Calculus.Deriv.Mul
 import Mathlib.Analysis.Calculus.Deriv.Comp
 import Mathlib.Analysis.Calculus.LocalExtr.Rolle
-import Mathlib.Analysis.Convex.Slope
 import Mathlib.Analysis.Convex.Normed
 import Mathlib.Data.IsROrC.Basic
-import Mathlib.Topology.Instances.RealVectorSpace
 
 #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
 
@@ -1047,354 +1045,8 @@ lemma antitone_of_hasDerivAt_nonpos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt
   antitone_of_deriv_nonpos (fun x ↦ (hf _).differentiableAt) fun x ↦ by
     rw [(hf _).deriv]; exact hf' _
 
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
-and `f'` is monotone on the interior, then `f` is convex on `D`. -/
-theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
-    (hf'_mono : MonotoneOn (deriv f) (interior D)) : ConvexOn ℝ D f :=
-  convexOn_of_slope_mono_adjacent hD
-    (by
-      intro x y z hx hz hxy hyz
-      -- First we prove some trivial inclusions
-      have hxzD : Icc x z ⊆ D := hD.ordConnected.out hx hz
-      have hxyD : Icc x y ⊆ D := (Icc_subset_Icc_right hyz.le).trans hxzD
-      have hxyD' : Ioo x y ⊆ interior D :=
-        subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩
-      have hyzD : Icc y z ⊆ D := (Icc_subset_Icc_left hxy.le).trans hxzD
-      have hyzD' : Ioo y z ⊆ interior D :=
-        subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD⟩
-      -- Then we apply MVT to both `[x, y]` and `[y, z]`
-      obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
-        exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')
-      obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b = (f z - f y) / (z - y) :=
-        exists_deriv_eq_slope f hyz (hf.mono hyzD) (hf'.mono hyzD')
-      rw [← ha, ← hb]
-      exact hf'_mono (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb).le)
-#align monotone_on.convex_on_of_deriv MonotoneOn.convexOn_of_deriv
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
-and `f'` is antitone on the interior, then `f` is concave on `D`. -/
-theorem AntitoneOn.concaveOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
-    (h_anti : AntitoneOn (deriv f) (interior D)) : ConcaveOn ℝ D f :=
-  haveI : MonotoneOn (deriv (-f)) (interior D) := by
-    simpa only [← deriv.neg] using h_anti.neg
-  neg_convexOn_iff.mp (this.convexOn_of_deriv hD hf.neg hf'.neg)
-#align antitone_on.concave_on_of_deriv AntitoneOn.concaveOn_of_deriv
-
-theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
-    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
-    ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a := by
-  have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
-    (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt
-  obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
-    exists_deriv_eq_slope f hxy hf A
-  rcases nonempty_Ioo.2 hay with ⟨b, ⟨hab, hby⟩⟩
-  refine' ⟨b, ⟨hxa.trans hab, hby⟩, _⟩
-  rw [← ha]
-  exact hf'_mono ⟨hxa, hay⟩ ⟨hxa.trans hab, hby⟩ hab
-#align strict_mono_on.exists_slope_lt_deriv_aux StrictMonoOn.exists_slope_lt_deriv_aux
-
-theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
-    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
-    ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a := by
-  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
-  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
-  · push_neg at h
-    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
-    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
-      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
-      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
-      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
-      · intro z hz
-        rw [← hw]
-        apply ne_of_lt
-        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
-    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
-      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
-      · refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
-      · exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
-      · intro z hz
-        rw [← hw]
-        apply ne_of_gt
-        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
-    refine' ⟨b, ⟨hxw.trans hwb, hby⟩, _⟩
-    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
-    have : deriv f a * (w - x) < deriv f b * (w - x) := by
-      apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _
-      · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
-      · rw [← hw]
-        exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le
-    linarith
-#align strict_mono_on.exists_slope_lt_deriv StrictMonoOn.exists_slope_lt_deriv
-
-theorem StrictMonoOn.exists_deriv_lt_slope_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
-    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
-    ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
-  have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
-    (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt
-  obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
-    exists_deriv_eq_slope f hxy hf A
-  rcases nonempty_Ioo.2 hxa with ⟨b, ⟨hxb, hba⟩⟩
-  refine' ⟨b, ⟨hxb, hba.trans hay⟩, _⟩
-  rw [← ha]
-  exact hf'_mono ⟨hxb, hba.trans hay⟩ ⟨hxa, hay⟩ hba
-#align strict_mono_on.exists_deriv_lt_slope_aux StrictMonoOn.exists_deriv_lt_slope_aux
-
-theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
-    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
-    ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
-  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
-  · apply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h
-  · push_neg at h
-    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
-    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, deriv f a < (f w - f x) / (w - x) := by
-      apply StrictMonoOn.exists_deriv_lt_slope_aux _ hxw _ _
-      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
-      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
-      · intro z hz
-        rw [← hw]
-        apply ne_of_lt
-        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
-    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, deriv f b < (f y - f w) / (y - w) := by
-      apply StrictMonoOn.exists_deriv_lt_slope_aux _ hwy _ _
-      · refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
-      · exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
-      · intro z hz
-        rw [← hw]
-        apply ne_of_gt
-        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
-    refine' ⟨a, ⟨hxa, haw.trans hwy⟩, _⟩
-    simp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
-    have : deriv f a * (y - w) < deriv f b * (y - w) := by
-      apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _
-      · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
-      · rw [← hw]
-        exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le
-    linarith
-#align strict_mono_on.exists_deriv_lt_slope StrictMonoOn.exists_deriv_lt_slope
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
-interior, then `f` is strictly convex on `D`.
-Note that we don't require differentiability, since it is guaranteed at all but at most
-one point by the strict monotonicity of `f'`. -/
-theorem StrictMonoOn.strictConvexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : StrictMonoOn (deriv f) (interior D)) : StrictConvexOn ℝ D f :=
-  strictConvexOn_of_slope_strict_mono_adjacent hD fun {x y z} hx hz hxy hyz => by
-    -- First we prove some trivial inclusions
-    have hxzD : Icc x z ⊆ D := hD.ordConnected.out hx hz
-    have hxyD : Icc x y ⊆ D := (Icc_subset_Icc_right hyz.le).trans hxzD
-    have hxyD' : Ioo x y ⊆ interior D :=
-      subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩
-    have hyzD : Icc y z ⊆ D := (Icc_subset_Icc_left hxy.le).trans hxzD
-    have hyzD' : Ioo y z ⊆ interior D :=
-      subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD⟩
-    -- Then we get points `a` and `b` in each interval `[x, y]` and `[y, z]` where the derivatives
-    -- can be compared to the slopes between `x, y` and `y, z` respectively.
-    obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a :=
-      StrictMonoOn.exists_slope_lt_deriv (hf.mono hxyD) hxy (hf'.mono hxyD')
-    obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b < (f z - f y) / (z - y) :=
-      StrictMonoOn.exists_deriv_lt_slope (hf.mono hyzD) hyz (hf'.mono hyzD')
-    apply ha.trans (lt_trans _ hb)
-    exact hf' (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb)
-#align strict_mono_on.strict_convex_on_of_deriv StrictMonoOn.strictConvexOn_of_deriv
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f'` is strictly antitone on the
-interior, then `f` is strictly concave on `D`.
-Note that we don't require differentiability, since it is guaranteed at all but at most
-one point by the strict antitonicity of `f'`. -/
-theorem StrictAntiOn.strictConcaveOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (h_anti : StrictAntiOn (deriv f) (interior D)) :
-    StrictConcaveOn ℝ D f :=
-  have : StrictMonoOn (deriv (-f)) (interior D) := by simpa only [← deriv.neg] using h_anti.neg
-  neg_neg f ▸ (this.strictConvexOn_of_deriv hD hf.neg).neg
-#align strict_anti_on.strict_concave_on_of_deriv StrictAntiOn.strictConcaveOn_of_deriv
-
-/-- If a function `f` is differentiable and `f'` is monotone on `ℝ` then `f` is convex. -/
-theorem Monotone.convexOn_univ_of_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f)
-    (hf'_mono : Monotone (deriv f)) : ConvexOn ℝ univ f :=
-  (hf'_mono.monotoneOn _).convexOn_of_deriv convex_univ hf.continuous.continuousOn
-    hf.differentiableOn
-#align monotone.convex_on_univ_of_deriv Monotone.convexOn_univ_of_deriv
-
-/-- If a function `f` is differentiable and `f'` is antitone on `ℝ` then `f` is concave. -/
-theorem Antitone.concaveOn_univ_of_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f)
-    (hf'_anti : Antitone (deriv f)) : ConcaveOn ℝ univ f :=
-  (hf'_anti.antitoneOn _).concaveOn_of_deriv convex_univ hf.continuous.continuousOn
-    hf.differentiableOn
-#align antitone.concave_on_univ_of_deriv Antitone.concaveOn_univ_of_deriv
-
-/-- If a function `f` is continuous and `f'` is strictly monotone on `ℝ` then `f` is strictly
-convex. Note that we don't require differentiability, since it is guaranteed at all but at most
-one point by the strict monotonicity of `f'`. -/
-theorem StrictMono.strictConvexOn_univ_of_deriv {f : ℝ → ℝ} (hf : Continuous f)
-    (hf'_mono : StrictMono (deriv f)) : StrictConvexOn ℝ univ f :=
-  (hf'_mono.strictMonoOn _).strictConvexOn_of_deriv convex_univ hf.continuousOn
-#align strict_mono.strict_convex_on_univ_of_deriv StrictMono.strictConvexOn_univ_of_deriv
-
-/-- If a function `f` is continuous and `f'` is strictly antitone on `ℝ` then `f` is strictly
-concave. Note that we don't require differentiability, since it is guaranteed at all but at most
-one point by the strict antitonicity of `f'`. -/
-theorem StrictAnti.strictConcaveOn_univ_of_deriv {f : ℝ → ℝ} (hf : Continuous f)
-    (hf'_anti : StrictAnti (deriv f)) : StrictConcaveOn ℝ univ f :=
-  (hf'_anti.strictAntiOn _).strictConcaveOn_of_deriv convex_univ hf.continuousOn
-#align strict_anti.strict_concave_on_univ_of_deriv StrictAnti.strictConcaveOn_univ_of_deriv
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
-interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
-theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
-    (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
-    (hf''_nonneg : ∀ x ∈ interior D, 0 ≤ deriv^[2] f x) : ConvexOn ℝ D f :=
-  (monotoneOn_of_deriv_nonneg hD.interior hf''.continuousOn (by rwa [interior_interior]) <| by
-        rwa [interior_interior]).convexOn_of_deriv
-    hD hf hf'
-#align convex_on_of_deriv2_nonneg convexOn_of_deriv2_nonneg
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
-interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
-theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
-    (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
-    (hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0) : ConcaveOn ℝ D f :=
-  (antitoneOn_of_deriv_nonpos hD.interior hf''.continuousOn (by rwa [interior_interior]) <| by
-        rwa [interior_interior]).concaveOn_of_deriv
-    hD hf hf'
-#align concave_on_of_deriv2_nonpos concaveOn_of_deriv2_nonpos
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
-interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
-lemma convexOn_of_hasDerivWithinAt2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
-    (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x)
-    (hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x) : ConvexOn ℝ D f := by
-  have : (interior D).EqOn (deriv f) f' := deriv_eqOn isOpen_interior hf'
-  refine convexOn_of_deriv2_nonneg hD hf (fun x hx ↦ (hf' _ hx).differentiableWithinAt) ?_ ?_
-  · rw [differentiableOn_congr this]
-    exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt
-  · rintro x hx
-    convert hf''₀ _ hx using 1
-    dsimp
-    rw [deriv_eqOn isOpen_interior (fun y hy ↦ ?_) hx]
-    exact (hf'' _ hy).congr this $ by rw [this hy]
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
-interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
-lemma concaveOn_of_hasDerivWithinAt2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
-    (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x)
-    (hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0) : ConcaveOn ℝ D f := by
-  have : (interior D).EqOn (deriv f) f' := deriv_eqOn isOpen_interior hf'
-  refine concaveOn_of_deriv2_nonpos hD hf (fun x hx ↦ (hf' _ hx).differentiableWithinAt) ?_ ?_
-  · rw [differentiableOn_congr this]
-    exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt
-  · rintro x hx
-    convert hf''₀ _ hx using 1
-    dsimp
-    rw [deriv_eqOn isOpen_interior (fun y hy ↦ ?_) hx]
-    exact (hf'' _ hy).congr this $ by rw [this hy]
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on the
-interior, then `f` is strictly convex on `D`.
-Note that we don't require twice differentiability explicitly as it is already implied by the second
-derivative being strictly positive, except at at most one point. -/
-theorem strictConvexOn_of_deriv2_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, 0 < (deriv^[2] f) x) :
-    StrictConvexOn ℝ D f :=
-  ((strictMonoOn_of_deriv_pos hD.interior fun z hz =>
-          (differentiableAt_of_deriv_ne_zero
-                (hf'' z hz).ne').differentiableWithinAt.continuousWithinAt) <|
-        by rwa [interior_interior]).strictConvexOn_of_deriv
-    hD hf
-#align strict_convex_on_of_deriv2_pos strictConvexOn_of_deriv2_pos
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on the
-interior, then `f` is strictly concave on `D`.
-Note that we don't require twice differentiability explicitly as it already implied by the second
-derivative being strictly negative, except at at most one point. -/
-theorem strictConcaveOn_of_deriv2_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0) :
-    StrictConcaveOn ℝ D f :=
-  ((strictAntiOn_of_deriv_neg hD.interior fun z hz =>
-          (differentiableAt_of_deriv_ne_zero
-                (hf'' z hz).ne).differentiableWithinAt.continuousWithinAt) <|
-        by rwa [interior_interior]).strictConcaveOn_of_deriv
-    hD hf
-#align strict_concave_on_of_deriv2_neg strictConcaveOn_of_deriv2_neg
-
-/-- If a function `f` is twice differentiable on an open convex set `D ⊆ ℝ` and
-`f''` is nonnegative on `D`, then `f` is convex on `D`. -/
-theorem convexOn_of_deriv2_nonneg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D)
-    (hf''_nonneg : ∀ x ∈ D, 0 ≤ (deriv^[2] f) x) : ConvexOn ℝ D f :=
-  convexOn_of_deriv2_nonneg hD hf'.continuousOn (hf'.mono interior_subset)
-    (hf''.mono interior_subset) fun x hx => hf''_nonneg x (interior_subset hx)
-#align convex_on_of_deriv2_nonneg' convexOn_of_deriv2_nonneg'
-
-/-- If a function `f` is twice differentiable on an open convex set `D ⊆ ℝ` and
-`f''` is nonpositive on `D`, then `f` is concave on `D`. -/
-theorem concaveOn_of_deriv2_nonpos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D)
-    (hf''_nonpos : ∀ x ∈ D, deriv^[2] f x ≤ 0) : ConcaveOn ℝ D f :=
-  concaveOn_of_deriv2_nonpos hD hf'.continuousOn (hf'.mono interior_subset)
-    (hf''.mono interior_subset) fun x hx => hf''_nonpos x (interior_subset hx)
-#align concave_on_of_deriv2_nonpos' concaveOn_of_deriv2_nonpos'
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on `D`,
-then `f` is strictly convex on `D`.
-Note that we don't require twice differentiability explicitly as it is already implied by the second
-derivative being strictly positive, except at at most one point. -/
-theorem strictConvexOn_of_deriv2_pos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, 0 < (deriv^[2] f) x) : StrictConvexOn ℝ D f :=
-  strictConvexOn_of_deriv2_pos hD hf fun x hx => hf'' x (interior_subset hx)
-#align strict_convex_on_of_deriv2_pos' strictConvexOn_of_deriv2_pos'
-
-/-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on `D`,
-then `f` is strictly concave on `D`.
-Note that we don't require twice differentiability explicitly as it is already implied by the second
-derivative being strictly negative, except at at most one point. -/
-theorem strictConcaveOn_of_deriv2_neg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, deriv^[2] f x < 0) : StrictConcaveOn ℝ D f :=
-  strictConcaveOn_of_deriv2_neg hD hf fun x hx => hf'' x (interior_subset hx)
-#align strict_concave_on_of_deriv2_neg' strictConcaveOn_of_deriv2_neg'
-
-/-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonnegative on `ℝ`,
-then `f` is convex on `ℝ`. -/
-theorem convexOn_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
-    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonneg : ∀ x, 0 ≤ (deriv^[2] f) x) :
-    ConvexOn ℝ univ f :=
-  convexOn_of_deriv2_nonneg' convex_univ hf'.differentiableOn hf''.differentiableOn fun x _ =>
-    hf''_nonneg x
-#align convex_on_univ_of_deriv2_nonneg convexOn_univ_of_deriv2_nonneg
-
-/-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonpositive on `ℝ`,
-then `f` is concave on `ℝ`. -/
-theorem concaveOn_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
-    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonpos : ∀ x, deriv^[2] f x ≤ 0) :
-    ConcaveOn ℝ univ f :=
-  concaveOn_of_deriv2_nonpos' convex_univ hf'.differentiableOn hf''.differentiableOn fun x _ =>
-    hf''_nonpos x
-#align concave_on_univ_of_deriv2_nonpos concaveOn_univ_of_deriv2_nonpos
-
-/-- If a function `f` is continuous on `ℝ`, and `f''` is strictly positive on `ℝ`,
-then `f` is strictly convex on `ℝ`.
-Note that we don't require twice differentiability explicitly as it is already implied by the second
-derivative being strictly positive, except at at most one point. -/
-theorem strictConvexOn_univ_of_deriv2_pos {f : ℝ → ℝ} (hf : Continuous f)
-    (hf'' : ∀ x, 0 < (deriv^[2] f) x) : StrictConvexOn ℝ univ f :=
-  strictConvexOn_of_deriv2_pos' convex_univ hf.continuousOn fun x _ => hf'' x
-#align strict_convex_on_univ_of_deriv2_pos strictConvexOn_univ_of_deriv2_pos
-
-/-- If a function `f` is continuous on `ℝ`, and `f''` is strictly negative on `ℝ`,
-then `f` is strictly concave on `ℝ`.
-Note that we don't require twice differentiability explicitly as it is already implied by the second
-derivative being strictly negative, except at at most one point. -/
-theorem strictConcaveOn_univ_of_deriv2_neg {f : ℝ → ℝ} (hf : Continuous f)
-    (hf'' : ∀ x, deriv^[2] f x < 0) : StrictConcaveOn ℝ univ f :=
-  strictConcaveOn_of_deriv2_neg' convex_univ hf.continuousOn fun x _ => hf'' x
-#align strict_concave_on_univ_of_deriv2_neg strictConcaveOn_univ_of_deriv2_neg
-
 /-! ### Functions `f : E → ℝ` -/
 
-
 /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
 theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
     (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -1414,7 +1414,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
     · exact fun t Ht => (hfg t <| hsub Ht).hasDerivAt (Icc_mem_nhds Ht.1 Ht.2)
   -- reinterpret on domain
   rcases hMVT with ⟨t, Ht, hMVT'⟩
-  rw [segment_eq_image_lineMap, bex_image_iff]
+  rw [segment_eq_image_lineMap, exists_mem_image]
   refine ⟨t, hsub Ht, ?_⟩
   simpa [g] using hMVT'.symm
 #align domain_mvt domain_mvt

... including Jordan's inequality: 2 / π * x ≤ sin x for 0 ≤ x ≤ π / 2

From LeanAPAP

@@ -927,7 +927,7 @@ theorem strictMono_of_deriv_pos {f : ℝ → ℝ} (hf' : ∀ x, 0 < deriv f x) :
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is strictly positive,
 then `f` is a strictly monotone function on `D`. -/
-lemma StrictMonoOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
+lemma strictMonoOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
     (hf'₀ : ∀ x ∈ interior D, 0 < f' x) : StrictMonoOn f D :=
   strictMonoOn_of_deriv_pos hD hf fun x hx ↦ by
@@ -1000,7 +1000,7 @@ theorem strictAnti_of_deriv_neg {f : ℝ → ℝ} (hf' : ∀ x, deriv f x < 0) :
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is strictly positive,
 then `f` is a strictly monotone function on `D`. -/
-lemma StrictAntiOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
+lemma strictAntiOn_of_hasDerivWithinAt_neg {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
     (hf'₀ : ∀ x ∈ interior D, f' x < 0) : StrictAntiOn f D :=
   strictAntiOn_of_deriv_neg hD hf fun x hx ↦ by
@@ -1008,7 +1008,7 @@ lemma StrictAntiOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f
 
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is strictly positive, then
 `f` is a strictly monotone function. -/
-lemma strictAnti_of_hasDerivAt_pos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
+lemma strictAnti_of_hasDerivAt_neg {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
     (hf' : ∀ x, f' x < 0) : StrictAnti f :=
   strictAnti_of_deriv_neg fun x ↦ by rw [(hf _).deriv]; exact hf' _
 

This covers many instances, but is not exhaustive.

Independently of whether that syntax should be avoided (similar to #10534), I think all these changes are small improvements.

@@ -118,8 +118,8 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {
     have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
       (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
         (Ioi_mem_nhds hrB)
-    obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y
-    exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists
+    obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y :=
+      (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists
     refine' ⟨z, _, hz⟩
     have := (hfz.trans hzB).le
     rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
@@ -494,8 +494,9 @@ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs
     {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y)
     (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) :
     ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by
-  obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }
-  exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))
+  obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0,
+      ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } :=
+    mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))
   rw [inter_comm] at hε
   refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩
   exact
@@ -841,8 +842,8 @@ theorem Convex.mul_sub_le_image_sub_of_le_deriv {D : Set ℝ} (hD : Convex ℝ D
   have hxyD : Icc x y ⊆ D := hD.ordConnected.out hx hy
   have hxyD' : Ioo x y ⊆ interior D :=
     subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩
-  obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x)
-  exact exists_deriv_eq_slope f hxy' (hf.mono hxyD) (hf'.mono hxyD')
+  obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
+    exists_deriv_eq_slope f hxy' (hf.mono hxyD) (hf'.mono hxyD')
   have : C ≤ (f y - f x) / (y - x) := ha ▸ hf'_ge _ (hxyD' a_mem)
   exact (le_div_iff (sub_pos.2 hxy')).1 this
 #align convex.mul_sub_le_image_sub_of_le_deriv Convex.mul_sub_le_image_sub_of_le_deriv
@@ -1063,10 +1064,10 @@ theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
       have hyzD' : Ioo y z ⊆ interior D :=
         subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD⟩
       -- Then we apply MVT to both `[x, y]` and `[y, z]`
-      obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x)
-      exact exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')
-      obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b = (f z - f y) / (z - y)
-      exact exists_deriv_eq_slope f hyz (hf.mono hyzD) (hf'.mono hyzD')
+      obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
+        exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')
+      obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b = (f z - f y) / (z - y) :=
+        exists_deriv_eq_slope f hyz (hf.mono hyzD) (hf'.mono hyzD')
       rw [← ha, ← hb]
       exact hf'_mono (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb).le)
 #align monotone_on.convex_on_of_deriv MonotoneOn.convexOn_of_deriv
@@ -1086,8 +1087,8 @@ theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf
     ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a := by
   have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
     (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt
-  obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x)
-  exact exists_deriv_eq_slope f hxy hf A
+  obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
+    exists_deriv_eq_slope f hxy hf A
   rcases nonempty_Ioo.2 hay with ⟨b, ⟨hab, hby⟩⟩
   refine' ⟨b, ⟨hxa.trans hab, hby⟩, _⟩
   rw [← ha]
@@ -1132,8 +1133,8 @@ theorem StrictMonoOn.exists_deriv_lt_slope_aux {x y : ℝ} {f : ℝ → ℝ} (hf
     ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
   have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
     (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt
-  obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x)
-  exact exists_deriv_eq_slope f hxy hf A
+  obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
+    exists_deriv_eq_slope f hxy hf A
   rcases nonempty_Ioo.2 hxa with ⟨b, ⟨hxb, hba⟩⟩
   refine' ⟨b, ⟨hxb, hba.trans hay⟩, _⟩
   rw [← ha]
@@ -1190,10 +1191,10 @@ theorem StrictMonoOn.strictConvexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {
       subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD⟩
     -- Then we get points `a` and `b` in each interval `[x, y]` and `[y, z]` where the derivatives
     -- can be compared to the slopes between `x, y` and `y, z` respectively.
-    obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a
-    · exact StrictMonoOn.exists_slope_lt_deriv (hf.mono hxyD) hxy (hf'.mono hxyD')
-    obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b < (f z - f y) / (z - y)
-    · exact StrictMonoOn.exists_deriv_lt_slope (hf.mono hyzD) hyz (hf'.mono hyzD')
+    obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a :=
+      StrictMonoOn.exists_slope_lt_deriv (hf.mono hxyD) hxy (hf'.mono hxyD')
+    obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b < (f z - f y) / (z - y) :=
+      StrictMonoOn.exists_deriv_lt_slope (hf.mono hyzD) hyz (hf'.mono hyzD')
     apply ha.trans (lt_trans _ hb)
     exact hf' (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb)
 #align strict_mono_on.strict_convex_on_of_deriv StrictMonoOn.strictConvexOn_of_deriv

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

@@ -102,7 +102,7 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {
   set s := { x | f x ≤ B x } ∩ Icc a b
   have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
   have : IsClosed s := by
-    simp only [inter_comm]
+    simp only [s, inter_comm]
     exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
   apply this.Icc_subset_of_forall_exists_gt ha
   rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
@@ -349,7 +349,7 @@ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : 
     intro x
     simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))
   convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound
-  simp only; rw [sub_self, norm_zero, sub_self, mul_zero]
+  simp only [g, B]; rw [sub_self, norm_zero, sub_self, mul_zero]
 #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment
 
 /-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
@@ -470,7 +470,7 @@ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le
       AffineMap.hasDerivWithinAt_lineMap segm
   have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht =>
     le_of_opNorm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _
-  simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound
+  simpa [g] using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound
 #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
@@ -729,7 +729,7 @@ variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (I
 theorem exists_ratio_hasDerivAt_eq_ratio_slope :
     ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by
   let h x := (g b - g a) * f x - (f b - f a) * g x
-  have hI : h a = h b := by simp only; ring
+  have hI : h a = h b := by simp only [h]; ring
   let h' x := (g b - g a) * f' x - (f b - f a) * g' x
   have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx =>
     ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))
@@ -1415,7 +1415,7 @@ theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ]
   rcases hMVT with ⟨t, Ht, hMVT'⟩
   rw [segment_eq_image_lineMap, bex_image_iff]
   refine ⟨t, hsub Ht, ?_⟩
-  simpa using hMVT'.symm
+  simpa [g] using hMVT'.symm
 #align domain_mvt domain_mvt
 
 section IsROrC

This PR provides variants of deriv lemmas stated in terms of HasDerivAt

From LeanAPAP

@@ -57,8 +57,8 @@ In this file we prove the following facts:
   `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`,
   if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`.
 
-* `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`,
-  `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` :
+* `monotoneOn_of_deriv_nonneg`, `antitoneOn_of_deriv_nonpos`,
+  `strictMono_of_deriv_pos`, `strictAnti_of_deriv_neg` :
   if the derivative of a function is non-negative/non-positive/positive/negative, then
   the function is monotone/antitone/strictly monotone/strictly monotonically
   decreasing.
@@ -904,87 +904,148 @@ of the real line. If `f` is differentiable on the interior of `D` and `f'` is po
 `f` is a strictly monotone function on `D`.
 Note that we don't require differentiability explicitly as it already implied by the derivative
 being strictly positive. -/
-theorem Convex.strictMonoOn_of_deriv_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
+theorem strictMonoOn_of_deriv_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, 0 < deriv f x) : StrictMonoOn f D := by
   intro x hx y hy
   have : DifferentiableOn ℝ f (interior D) := fun z hz =>
     (differentiableAt_of_deriv_ne_zero (hf' z hz).ne').differentiableWithinAt
   simpa only [zero_mul, sub_pos] using
     hD.mul_sub_lt_image_sub_of_lt_deriv hf this hf' x hx y hy
-#align convex.strict_mono_on_of_deriv_pos Convex.strictMonoOn_of_deriv_pos
+#align convex.strict_mono_on_of_deriv_pos strictMonoOn_of_deriv_pos
 
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is positive, then
 `f` is a strictly monotone function.
 Note that we don't require differentiability explicitly as it already implied by the derivative
 being strictly positive. -/
 theorem strictMono_of_deriv_pos {f : ℝ → ℝ} (hf' : ∀ x, 0 < deriv f x) : StrictMono f :=
-  strictMonoOn_univ.1 <| convex_univ.strictMonoOn_of_deriv_pos (fun z _ =>
+  strictMonoOn_univ.1 <| strictMonoOn_of_deriv_pos convex_univ (fun z _ =>
     (differentiableAt_of_deriv_ne_zero (hf' z).ne').differentiableWithinAt.continuousWithinAt)
     fun x _ => hf' x
 #align strict_mono_of_deriv_pos strictMono_of_deriv_pos
 
+/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
+of the real line. If `f` is differentiable on the interior of `D` and `f'` is strictly positive,
+then `f` is a strictly monotone function on `D`. -/
+lemma StrictMonoOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
+    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
+    (hf'₀ : ∀ x ∈ interior D, 0 < f' x) : StrictMonoOn f D :=
+  strictMonoOn_of_deriv_pos hD hf fun x hx ↦ by
+    rw [deriv_eqOn isOpen_interior hf' hx]; exact hf'₀ _ hx
+
+/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is strictly positive, then
+`f` is a strictly monotone function. -/
+lemma strictMono_of_hasDerivAt_pos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
+    (hf' : ∀ x, 0 < f' x) : StrictMono f :=
+  strictMono_of_deriv_pos fun x ↦ by rw [(hf _).deriv]; exact hf' _
+
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonnegative, then
 `f` is a monotone function on `D`. -/
-theorem Convex.monotoneOn_of_deriv_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
+theorem monotoneOn_of_deriv_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
     (hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x) : MonotoneOn f D := fun x hx y hy hxy => by
   simpa only [zero_mul, sub_nonneg] using
     hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg x hx y hy hxy
-#align convex.monotone_on_of_deriv_nonneg Convex.monotoneOn_of_deriv_nonneg
+#align convex.monotone_on_of_deriv_nonneg monotoneOn_of_deriv_nonneg
 
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonnegative, then
 `f` is a monotone function. -/
 theorem monotone_of_deriv_nonneg {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ x, 0 ≤ deriv f x) :
     Monotone f :=
   monotoneOn_univ.1 <|
-    convex_univ.monotoneOn_of_deriv_nonneg hf.continuous.continuousOn hf.differentiableOn fun x _ =>
+    monotoneOn_of_deriv_nonneg convex_univ hf.continuous.continuousOn hf.differentiableOn fun x _ =>
       hf' x
 #align monotone_of_deriv_nonneg monotone_of_deriv_nonneg
 
+/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
+of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonnegative, then
+`f` is a monotone function on `D`. -/
+lemma monotoneOn_of_hasDerivWithinAt_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
+    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
+    (hf'₀ : ∀ x ∈ interior D, 0 ≤ f' x) : MonotoneOn f D :=
+  monotoneOn_of_deriv_nonneg hD hf (fun x hx ↦ (hf' _ hx).differentiableWithinAt) fun x hx ↦ by
+    rw [deriv_eqOn isOpen_interior hf' hx]; exact hf'₀ _ hx
+
+/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonnegative, then
+`f` is a monotone function. -/
+lemma monotone_of_hasDerivAt_nonneg {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
+    (hf' : 0 ≤ f') : Monotone f :=
+  monotone_of_deriv_nonneg (fun x ↦ (hf _).differentiableAt) fun x ↦ by
+    rw [(hf _).deriv]; exact hf' _
+
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is negative, then
 `f` is a strictly antitone function on `D`. -/
-theorem Convex.strictAntiOn_of_deriv_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
+theorem strictAntiOn_of_deriv_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, deriv f x < 0) : StrictAntiOn f D :=
   fun x hx y => by
   simpa only [zero_mul, sub_lt_zero] using
     hD.image_sub_lt_mul_sub_of_deriv_lt hf
       (fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).ne).differentiableWithinAt) hf' x
       hx y
-#align convex.strict_anti_on_of_deriv_neg Convex.strictAntiOn_of_deriv_neg
+#align convex.strict_anti_on_of_deriv_neg strictAntiOn_of_deriv_neg
 
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is negative, then
 `f` is a strictly antitone function.
 Note that we don't require differentiability explicitly as it already implied by the derivative
 being strictly negative. -/
 theorem strictAnti_of_deriv_neg {f : ℝ → ℝ} (hf' : ∀ x, deriv f x < 0) : StrictAnti f :=
-  strictAntiOn_univ.1 <|
-    convex_univ.strictAntiOn_of_deriv_neg
+  strictAntiOn_univ.1 <| strictAntiOn_of_deriv_neg convex_univ
       (fun z _ =>
         (differentiableAt_of_deriv_ne_zero (hf' z).ne).differentiableWithinAt.continuousWithinAt)
       fun x _ => hf' x
 #align strict_anti_of_deriv_neg strictAnti_of_deriv_neg
 
+/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
+of the real line. If `f` is differentiable on the interior of `D` and `f'` is strictly positive,
+then `f` is a strictly monotone function on `D`. -/
+lemma StrictAntiOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
+    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
+    (hf'₀ : ∀ x ∈ interior D, f' x < 0) : StrictAntiOn f D :=
+  strictAntiOn_of_deriv_neg hD hf fun x hx ↦ by
+    rw [deriv_eqOn isOpen_interior hf' hx]; exact hf'₀ _ hx
+
+/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is strictly positive, then
+`f` is a strictly monotone function. -/
+lemma strictAnti_of_hasDerivAt_pos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
+    (hf' : ∀ x, f' x < 0) : StrictAnti f :=
+  strictAnti_of_deriv_neg fun x ↦ by rw [(hf _).deriv]; exact hf' _
+
 /-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
 of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonpositive, then
 `f` is an antitone function on `D`. -/
-theorem Convex.antitoneOn_of_deriv_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
+theorem antitoneOn_of_deriv_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
     (hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0) : AntitoneOn f D := fun x hx y hy hxy => by
   simpa only [zero_mul, sub_nonpos] using
     hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy
-#align convex.antitone_on_of_deriv_nonpos Convex.antitoneOn_of_deriv_nonpos
+#align convex.antitone_on_of_deriv_nonpos antitoneOn_of_deriv_nonpos
 
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonpositive, then
 `f` is an antitone function. -/
 theorem antitone_of_deriv_nonpos {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ x, deriv f x ≤ 0) :
     Antitone f :=
   antitoneOn_univ.1 <|
-    convex_univ.antitoneOn_of_deriv_nonpos hf.continuous.continuousOn hf.differentiableOn fun x _ =>
+    antitoneOn_of_deriv_nonpos convex_univ hf.continuous.continuousOn hf.differentiableOn fun x _ =>
       hf' x
 #align antitone_of_deriv_nonpos antitone_of_deriv_nonpos
 
+/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
+of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonpositive, then
+`f` is an antitone function on `D`. -/
+lemma antitoneOn_of_hasDerivWithinAt_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
+    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
+    (hf'₀ : ∀ x ∈ interior D, f' x ≤ 0) : AntitoneOn f D :=
+  antitoneOn_of_deriv_nonpos hD hf (fun x hx ↦ (hf' _ hx).differentiableWithinAt) fun x hx ↦ by
+    rw [deriv_eqOn isOpen_interior hf' hx]; exact hf'₀ _ hx
+
+/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonpositive, then `f` is an antitone
+function. -/
+lemma antitone_of_hasDerivAt_nonpos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
+    (hf' : f' ≤ 0) : Antitone f :=
+  antitone_of_deriv_nonpos (fun x ↦ (hf _).differentiableAt) fun x ↦ by
+    rw [(hf _).deriv]; exact hf' _
+
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
 and `f'` is monotone on the interior, then `f` is convex on `D`. -/
 theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
@@ -1183,7 +1244,7 @@ interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -
 theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
     (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
     (hf''_nonneg : ∀ x ∈ interior D, 0 ≤ deriv^[2] f x) : ConvexOn ℝ D f :=
-  (hD.interior.monotoneOn_of_deriv_nonneg hf''.continuousOn (by rwa [interior_interior]) <| by
+  (monotoneOn_of_deriv_nonneg hD.interior hf''.continuousOn (by rwa [interior_interior]) <| by
         rwa [interior_interior]).convexOn_of_deriv
     hD hf hf'
 #align convex_on_of_deriv2_nonneg convexOn_of_deriv2_nonneg
@@ -1193,11 +1254,43 @@ interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`.
 theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
     (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
     (hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0) : ConcaveOn ℝ D f :=
-  (hD.interior.antitoneOn_of_deriv_nonpos hf''.continuousOn (by rwa [interior_interior]) <| by
+  (antitoneOn_of_deriv_nonpos hD.interior hf''.continuousOn (by rwa [interior_interior]) <| by
         rwa [interior_interior]).concaveOn_of_deriv
     hD hf hf'
 #align concave_on_of_deriv2_nonpos concaveOn_of_deriv2_nonpos
 
+/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
+interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
+lemma convexOn_of_hasDerivWithinAt2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ}
+    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
+    (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x)
+    (hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x) : ConvexOn ℝ D f := by
+  have : (interior D).EqOn (deriv f) f' := deriv_eqOn isOpen_interior hf'
+  refine convexOn_of_deriv2_nonneg hD hf (fun x hx ↦ (hf' _ hx).differentiableWithinAt) ?_ ?_
+  · rw [differentiableOn_congr this]
+    exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt
+  · rintro x hx
+    convert hf''₀ _ hx using 1
+    dsimp
+    rw [deriv_eqOn isOpen_interior (fun y hy ↦ ?_) hx]
+    exact (hf'' _ hy).congr this $ by rw [this hy]
+
+/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
+interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
+lemma concaveOn_of_hasDerivWithinAt2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ}
+    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
+    (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x)
+    (hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0) : ConcaveOn ℝ D f := by
+  have : (interior D).EqOn (deriv f) f' := deriv_eqOn isOpen_interior hf'
+  refine concaveOn_of_deriv2_nonpos hD hf (fun x hx ↦ (hf' _ hx).differentiableWithinAt) ?_ ?_
+  · rw [differentiableOn_congr this]
+    exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt
+  · rintro x hx
+    convert hf''₀ _ hx using 1
+    dsimp
+    rw [deriv_eqOn isOpen_interior (fun y hy ↦ ?_) hx]
+    exact (hf'' _ hy).congr this $ by rw [this hy]
+
 /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on the
 interior, then `f` is strictly convex on `D`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
@@ -1205,7 +1298,7 @@ derivative being strictly positive, except at at most one point. -/
 theorem strictConvexOn_of_deriv2_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, 0 < (deriv^[2] f) x) :
     StrictConvexOn ℝ D f :=
-  ((hD.interior.strictMonoOn_of_deriv_pos fun z hz =>
+  ((strictMonoOn_of_deriv_pos hD.interior fun z hz =>
           (differentiableAt_of_deriv_ne_zero
                 (hf'' z hz).ne').differentiableWithinAt.continuousWithinAt) <|
         by rwa [interior_interior]).strictConvexOn_of_deriv
@@ -1219,7 +1312,7 @@ derivative being strictly negative, except at at most one point. -/
 theorem strictConcaveOn_of_deriv2_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0) :
     StrictConcaveOn ℝ D f :=
-  ((hD.interior.strictAntiOn_of_deriv_neg fun z hz =>
+  ((strictAntiOn_of_deriv_neg hD.interior fun z hz =>
           (differentiableAt_of_deriv_ne_zero
                 (hf'' z hz).ne).differentiableWithinAt.continuousWithinAt) <|
         by rwa [interior_interior]).strictConcaveOn_of_deriv

Co-authored-by: adomani <adomani@gmail.com>

@@ -469,7 +469,7 @@ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le
     simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _
       AffineMap.hasDerivWithinAt_lineMap segm
   have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht =>
-    le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _
+    le_of_opNorm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _
   simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound
 #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le
 

Subset of #9319

@@ -1118,7 +1118,7 @@ Note that we don't require differentiability, since it is guaranteed at all but
 one point by the strict monotonicity of `f'`. -/
 theorem StrictMonoOn.strictConvexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf : ContinuousOn f D) (hf' : StrictMonoOn (deriv f) (interior D)) : StrictConvexOn ℝ D f :=
-  strictConvexOn_of_slope_strict_mono_adjacent hD <| fun {x y z} hx hz hxy hyz => by
+  strictConvexOn_of_slope_strict_mono_adjacent hD fun {x y z} hx hz hxy hyz => by
     -- First we prove some trivial inclusions
     have hxzD : Icc x z ⊆ D := hD.ordConnected.out hx hz
     have hxyD : Icc x y ⊆ D := (Icc_subset_Icc_right hyz.le).trans hxzD

This PR renames the field Clopens.clopen' -> Clopens.isClopen', and the lemmas

• preimage_closed_of_closed -> ContinuousOn.preimage_isClosed_of_isClosed

as well as: ClopenUpperSet.clopen -> ClopenUpperSet.isClopen connectedComponent_eq_iInter_clopen -> connectedComponent_eq_iInter_isClopen connectedComponent_subset_iInter_clopen -> connectedComponent_subset_iInter_isClopen continuous_boolIndicator_iff_clopen -> continuous_boolIndicator_iff_isClopen continuousOn_boolIndicator_iff_clopen -> continuousOn_boolIndicator_iff_isClopen DiscreteQuotient.ofClopen -> DiscreteQuotient.ofIsClopen disjoint_or_subset_of_clopen -> disjoint_or_subset_of_isClopen exists_clopen_{lower,upper}of_not_le -> exists_isClopen{lower,upper}_of_not_le exists_clopen_of_cofiltered -> exists_isClopen_of_cofiltered exists_clopen_of_totally_separated -> exists_isClopen_of_totally_separated exists_clopen_upper_or_lower_of_ne -> exists_isClopen_upper_or_lower_of_ne IsPreconnected.subset_clopen -> IsPreconnected.subset_isClopen isTotallyDisconnected_of_clopen_set -> isTotallyDisconnected_of_isClopen_set LocallyConstant.ofClopen_fiber_one -> LocallyConstant.ofIsClopen_fiber_one LocallyConstant.ofClopen_fiber_zero -> LocallyConstant.ofIsClopen_fiber_zero LocallyConstant.ofClopen -> LocallyConstant.ofIsClopen preimage_clopen_of_clopen -> preimage_isClopen_of_isClopen TopologicalSpace.Clopens.clopen -> TopologicalSpace.Clopens.isClopen

@@ -103,7 +103,7 @@ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {
   have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
   have : IsClosed s := by
     simp only [inter_comm]
-    exact A.preimage_closed_of_closed isClosed_Icc OrderClosedTopology.isClosed_le'
+    exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
   apply this.Icc_subset_of_forall_exists_gt ha
   rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
   cases' hxB.lt_or_eq with hxB hxB

First calculus prerequisites for Rademacher theorem in #7003.

Add a few lemmas that were available for FDeriv but not for Deriv, weaken assumptions here and there.

@@ -550,6 +550,16 @@ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, Dif
     (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound
 #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le
 
+/-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is
+`C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/
+theorem _root_.lipschitzWith_of_nnnorm_fderiv_le
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G}
+    {C : ℝ≥0} (hf : Differentiable 𝕜 f)
+    (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by
+  let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
+  rw [← lipschitzOn_univ]
+  exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ
+
 /-- Variant of the mean value inequality on a convex set, using a bound on the difference between
 the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with
 `HasFDerivWithinAt`. -/
@@ -595,9 +605,12 @@ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : Differentiabl
     hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy
 #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero
 
-theorem _root_.is_const_of_fderiv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0)
-    (x y : E) : f x = f y :=
-  convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn
+theorem _root_.is_const_of_fderiv_eq_zero
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G}
+    (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0)
+    (x y : E) : f x = f y := by
+  let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
+  exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn
     (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial
 #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero
 
@@ -612,11 +625,14 @@ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜
   rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz]
 #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq
 
-theorem _root_.eq_of_fderiv_eq (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g)
-    (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g :=
+theorem _root_.eq_of_fderiv_eq
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G}
+    (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g)
+    (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by
+  let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x
-  convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ
-    (fun x _ => by simpa using hf' _) (mem_univ _) hfgx
+  exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn
+    uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx
 #align eq_of_fderiv_eq eq_of_fderiv_eq
 
 end Convex

Also rename dimH_image_le_of_locally_lipschitz_on to dimH_image_le_of_locally_lipschitzOn.

@@ -684,7 +684,7 @@ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, Diff
 then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/
 theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f)
     (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f :=
-  lipschitz_on_univ.1 <|
+  lipschitzOn_univ.1 <|
     convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x
 #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le
 

This is a perfect duplicate of HasFDerivWithinAt.mono_of_mem.

Same thing with Deriv instead of FDeriv.

@@ -361,7 +361,7 @@ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ}
   refine'
     norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt)
       (fun x hx => _) bound
-  exact (hf x <| Ico_subset_Icc_self hx).nhdsWithin (Icc_mem_nhdsWithin_Ici hx)
+  exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx)
 #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment'
 
 /-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
@@ -427,9 +427,9 @@ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b))
     (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) :
     ∀ y ∈ Icc a b, f y = g y := by
   have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy =>
-    (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.nhdsWithin (Icc_mem_nhdsWithin_Ici hy)
+    (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)
   have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy =>
-    (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.nhdsWithin (Icc_mem_nhdsWithin_Ici hy)
+    (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)
   exact
     eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn
       gdiff.continuousOn hi

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -71,7 +71,7 @@ In this file we prove the following facts:
 -/
 
 
-variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type _} [NormedAddCommGroup F]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
   [NormedSpace ℝ F]
 
 open Metric Set Asymptotics ContinuousLinearMap Filter
@@ -247,7 +247,7 @@ Let `f` and `B` be continuous functions on `[a, b]` such that
 * we have `f' x < B' x` whenever `‖f x‖ = B x`.
 
 Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/
-theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type _}
+theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*}
     [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b))
     -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x`
     (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r)
@@ -446,7 +446,7 @@ also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/
 
 section
 
-variable {𝕜 G : Type _} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
 
 namespace Convex
 
@@ -1321,7 +1321,7 @@ make sense and are enough. Many formulations of the mean value inequality could
 balls over `ℝ` or `ℂ`. For now, we only include the ones that we need.
 -/
 
-variable {𝕜 : Type _} [IsROrC 𝕜] {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type _}
+variable {𝕜 : Type*} [IsROrC 𝕜] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type*}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H] {f : G → H} {f' : G → G →L[𝕜] H} {x : G}
 
 /-- Over the reals or the complexes, a continuously differentiable function is strictly

Also make f/f' arguments implicit in all versions of Rolle's Theorem.

Fixes #4830

API changes

• exists_Ioo_extr_on_Icc:
  • generalize from functions f : ℝ → ℝ to functions from a conditionally complete linear order to a linear order.
  • make f implicit;
• exists_local_extr_Ioo:
  • rename to exists_isLocalExtr_Ioo;
  • generalize as above;
  • make f implicit;
• exists_isExtrOn_Ioo_of_tendsto, exists_isLocalExtr_Ioo_of_tendsto: new lemmas extracted from the proof of exists_hasDerivAt_eq_zero';
• exists_hasDerivAt_eq_zero, exists_hasDerivAt_eq_zero': make f and f' implicit;
• exists_deriv_eq_zero, exists_deriv_eq_zero': make f implicit.

@@ -5,7 +5,9 @@ Authors: Sébastien Gouëzel, Yury Kudryashov
 -/
 import Mathlib.Analysis.Calculus.Deriv.AffineMap
 import Mathlib.Analysis.Calculus.Deriv.Slope
-import Mathlib.Analysis.Calculus.LocalExtr
+import Mathlib.Analysis.Calculus.Deriv.Mul
+import Mathlib.Analysis.Calculus.Deriv.Comp
+import Mathlib.Analysis.Calculus.LocalExtr.Rolle
 import Mathlib.Analysis.Convex.Slope
 import Mathlib.Analysis.Convex.Normed
 import Mathlib.Data.IsROrC.Basic
@@ -717,7 +719,7 @@ theorem exists_ratio_hasDerivAt_eq_ratio_slope :
     ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))
   have hhc : ContinuousOn h (Icc a b) :=
     (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)
-  rcases exists_hasDerivAt_eq_zero h h' hab hhc hI hhh' with ⟨c, cmem, hc⟩
+  rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩
   exact ⟨c, cmem, sub_eq_zero.1 hc⟩
 #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope
 

• 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, Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.calculus.mean_value
-! leanprover-community/mathlib commit 3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Calculus.Deriv.AffineMap
 import Mathlib.Analysis.Calculus.Deriv.Slope
@@ -16,6 +11,8 @@ import Mathlib.Analysis.Convex.Normed
 import Mathlib.Data.IsROrC.Basic
 import Mathlib.Topology.Instances.RealVectorSpace
 
+#align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
+
 /-!
 # The mean value inequality and equalities
 

@@ -1167,7 +1167,7 @@ theorem StrictAnti.strictConcaveOn_univ_of_deriv {f : ℝ → ℝ} (hf : Continu
 interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
 theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
     (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
-    (hf''_nonneg : ∀ x ∈ interior D, 0 ≤ (deriv^[2]) f x) : ConvexOn ℝ D f :=
+    (hf''_nonneg : ∀ x ∈ interior D, 0 ≤ deriv^[2] f x) : ConvexOn ℝ D f :=
   (hD.interior.monotoneOn_of_deriv_nonneg hf''.continuousOn (by rwa [interior_interior]) <| by
         rwa [interior_interior]).convexOn_of_deriv
     hD hf hf'
@@ -1177,7 +1177,7 @@ theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ →
 interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
 theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
     (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
-    (hf''_nonpos : ∀ x ∈ interior D, (deriv^[2]) f x ≤ 0) : ConcaveOn ℝ D f :=
+    (hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0) : ConcaveOn ℝ D f :=
   (hD.interior.antitoneOn_of_deriv_nonpos hf''.continuousOn (by rwa [interior_interior]) <| by
         rwa [interior_interior]).concaveOn_of_deriv
     hD hf hf'
@@ -1188,7 +1188,7 @@ interior, then `f` is strictly convex on `D`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
 derivative being strictly positive, except at at most one point. -/
 theorem strictConvexOn_of_deriv2_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, 0 < ((deriv^[2]) f) x) :
+    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, 0 < (deriv^[2] f) x) :
     StrictConvexOn ℝ D f :=
   ((hD.interior.strictMonoOn_of_deriv_pos fun z hz =>
           (differentiableAt_of_deriv_ne_zero
@@ -1202,7 +1202,7 @@ interior, then `f` is strictly concave on `D`.
 Note that we don't require twice differentiability explicitly as it already implied by the second
 derivative being strictly negative, except at at most one point. -/
 theorem strictConcaveOn_of_deriv2_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, (deriv^[2]) f x < 0) :
+    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0) :
     StrictConcaveOn ℝ D f :=
   ((hD.interior.strictAntiOn_of_deriv_neg fun z hz =>
           (differentiableAt_of_deriv_ne_zero
@@ -1215,7 +1215,7 @@ theorem strictConcaveOn_of_deriv2_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
 `f''` is nonnegative on `D`, then `f` is convex on `D`. -/
 theorem convexOn_of_deriv2_nonneg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D)
-    (hf''_nonneg : ∀ x ∈ D, 0 ≤ ((deriv^[2]) f) x) : ConvexOn ℝ D f :=
+    (hf''_nonneg : ∀ x ∈ D, 0 ≤ (deriv^[2] f) x) : ConvexOn ℝ D f :=
   convexOn_of_deriv2_nonneg hD hf'.continuousOn (hf'.mono interior_subset)
     (hf''.mono interior_subset) fun x hx => hf''_nonneg x (interior_subset hx)
 #align convex_on_of_deriv2_nonneg' convexOn_of_deriv2_nonneg'
@@ -1224,7 +1224,7 @@ theorem convexOn_of_deriv2_nonneg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ 
 `f''` is nonpositive on `D`, then `f` is concave on `D`. -/
 theorem concaveOn_of_deriv2_nonpos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D)
-    (hf''_nonpos : ∀ x ∈ D, (deriv^[2]) f x ≤ 0) : ConcaveOn ℝ D f :=
+    (hf''_nonpos : ∀ x ∈ D, deriv^[2] f x ≤ 0) : ConcaveOn ℝ D f :=
   concaveOn_of_deriv2_nonpos hD hf'.continuousOn (hf'.mono interior_subset)
     (hf''.mono interior_subset) fun x hx => hf''_nonpos x (interior_subset hx)
 #align concave_on_of_deriv2_nonpos' concaveOn_of_deriv2_nonpos'
@@ -1234,7 +1234,7 @@ then `f` is strictly convex on `D`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
 derivative being strictly positive, except at at most one point. -/
 theorem strictConvexOn_of_deriv2_pos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, 0 < ((deriv^[2]) f) x) : StrictConvexOn ℝ D f :=
+    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, 0 < (deriv^[2] f) x) : StrictConvexOn ℝ D f :=
   strictConvexOn_of_deriv2_pos hD hf fun x hx => hf'' x (interior_subset hx)
 #align strict_convex_on_of_deriv2_pos' strictConvexOn_of_deriv2_pos'
 
@@ -1243,14 +1243,14 @@ then `f` is strictly concave on `D`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
 derivative being strictly negative, except at at most one point. -/
 theorem strictConcaveOn_of_deriv2_neg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
-    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, (deriv^[2]) f x < 0) : StrictConcaveOn ℝ D f :=
+    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, deriv^[2] f x < 0) : StrictConcaveOn ℝ D f :=
   strictConcaveOn_of_deriv2_neg hD hf fun x hx => hf'' x (interior_subset hx)
 #align strict_concave_on_of_deriv2_neg' strictConcaveOn_of_deriv2_neg'
 
 /-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonnegative on `ℝ`,
 then `f` is convex on `ℝ`. -/
 theorem convexOn_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
-    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonneg : ∀ x, 0 ≤ ((deriv^[2]) f) x) :
+    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonneg : ∀ x, 0 ≤ (deriv^[2] f) x) :
     ConvexOn ℝ univ f :=
   convexOn_of_deriv2_nonneg' convex_univ hf'.differentiableOn hf''.differentiableOn fun x _ =>
     hf''_nonneg x
@@ -1259,7 +1259,7 @@ theorem convexOn_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : Differentiable 
 /-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonpositive on `ℝ`,
 then `f` is concave on `ℝ`. -/
 theorem concaveOn_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
-    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonpos : ∀ x, (deriv^[2]) f x ≤ 0) :
+    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonpos : ∀ x, deriv^[2] f x ≤ 0) :
     ConcaveOn ℝ univ f :=
   concaveOn_of_deriv2_nonpos' convex_univ hf'.differentiableOn hf''.differentiableOn fun x _ =>
     hf''_nonpos x
@@ -1270,7 +1270,7 @@ then `f` is strictly convex on `ℝ`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
 derivative being strictly positive, except at at most one point. -/
 theorem strictConvexOn_univ_of_deriv2_pos {f : ℝ → ℝ} (hf : Continuous f)
-    (hf'' : ∀ x, 0 < ((deriv^[2]) f) x) : StrictConvexOn ℝ univ f :=
+    (hf'' : ∀ x, 0 < (deriv^[2] f) x) : StrictConvexOn ℝ univ f :=
   strictConvexOn_of_deriv2_pos' convex_univ hf.continuousOn fun x _ => hf'' x
 #align strict_convex_on_univ_of_deriv2_pos strictConvexOn_univ_of_deriv2_pos
 
@@ -1279,7 +1279,7 @@ then `f` is strictly concave on `ℝ`.
 Note that we don't require twice differentiability explicitly as it is already implied by the second
 derivative being strictly negative, except at at most one point. -/
 theorem strictConcaveOn_univ_of_deriv2_neg {f : ℝ → ℝ} (hf : Continuous f)
-    (hf'' : ∀ x, (deriv^[2]) f x < 0) : StrictConcaveOn ℝ univ f :=
+    (hf'' : ∀ x, deriv^[2] f x < 0) : StrictConcaveOn ℝ univ f :=
   strictConcaveOn_of_deriv2_neg' convex_univ hf.continuousOn fun x _ => hf'' x
 #align strict_concave_on_univ_of_deriv2_neg strictConcaveOn_univ_of_deriv2_neg
 

Changes are of the form

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

@@ -416,7 +416,7 @@ variable {f' g : ℝ → E}
 theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
     (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b))
     (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by
-  simp only [← @sub_eq_zero _ _ (f _)] at hi⊢
+  simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢
   exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by
     simpa only [sub_self] using (derivf y hy).sub (derivg y hy)
 #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq
@@ -1023,7 +1023,7 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
     ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a := by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
-  · push_neg  at h
+  · push_neg at h
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
     obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
       apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
@@ -1042,7 +1042,7 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
         apply ne_of_gt
         exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
     refine' ⟨b, ⟨hxw.trans hwb, hby⟩, _⟩
-    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb⊢
+    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
     have : deriv f a * (w - x) < deriv f b * (w - x) := by
       apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _
       · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
@@ -1069,7 +1069,7 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
     ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
   by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
   · apply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h
-  · push_neg  at h
+  · push_neg at h
     rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
     obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, deriv f a < (f w - f x) / (w - x) := by
       apply StrictMonoOn.exists_deriv_lt_slope_aux _ hxw _ _
@@ -1088,7 +1088,7 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
         apply ne_of_gt
         exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
     refine' ⟨a, ⟨hxa, haw.trans hwy⟩, _⟩
-    simp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb⊢
+    simp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
     have : deriv f a * (y - w) < deriv f b * (y - w) := by
       apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _
       · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)

All of these are doc fixes

@@ -1211,7 +1211,7 @@ theorem strictConcaveOn_of_deriv2_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ
     hD hf
 #align strict_concave_on_of_deriv2_neg strictConcaveOn_of_deriv2_neg
 
-/-- If a function `f` is twice differentiable on a open convex set `D ⊆ ℝ` and
+/-- If a function `f` is twice differentiable on an open convex set `D ⊆ ℝ` and
 `f''` is nonnegative on `D`, then `f` is convex on `D`. -/
 theorem convexOn_of_deriv2_nonneg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
     (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D)

Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>