analysis.calculus.fderiv_symmetric ⟷ Mathlib.Analysis.Calculus.FDeriv.Symmetric

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

@@ -166,7 +166,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           refine' ⟨_, xt_mem t ⟨ht.1, ht.2.le⟩⟩
           rw [add_assoc, add_mem_ball_iff_norm]
           exact I.trans_lt hδ
-        simpa only [mem_set_of_eq, add_assoc x, add_sub_cancel'] using sδ H
+        simpa only [mem_set_of_eq, add_assoc x, add_sub_cancel_left] using sδ H
       _ ≤ ε * (‖h • v‖ + ‖h • w‖) * ‖h • w‖ :=
         by
         apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
@@ -224,17 +224,17 @@ theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) •
   have h2v : x + (2 : ℝ) • v ∈ interior s :=
     by
     convert s_conv.add_smul_sub_mem_interior xs h4v A using 1
-    simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]
+    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
     norm_num
   have h2w : x + (2 : ℝ) • w ∈ interior s :=
     by
     convert s_conv.add_smul_sub_mem_interior xs h4w A using 1
-    simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]
+    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
     norm_num
   have hvw : x + (v + w) ∈ interior s :=
     by
     convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1
-    simp only [smul_smul, one_div, add_sub_cancel', add_right_inj, smul_add, smul_sub]
+    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj, smul_add, smul_sub]
     norm_num
     abel
   have h2vw : x + (2 • v + w) ∈ interior s :=
@@ -246,7 +246,7 @@ theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) •
   have hvww : x + (v + w) + w ∈ interior s :=
     by
     convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1
-    simp only [one_div, add_sub_cancel', inv_smul_smul₀, add_sub_add_right_eq_sub, Ne.def,
+    simp only [one_div, add_sub_cancel_left, inv_smul_smul₀, add_sub_add_right_eq_sub, Ne.def,
       not_false_iff, bit0_eq_zero, one_ne_zero]
     rw [two_smul]
     abel

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

@@ -81,7 +81,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   apply is_o.trans_is_O (is_o_iff.2 fun ε εpos => _) (is_O_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
   -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
   -- good up to `δ`.
-  rw [HasFDerivWithinAt, HasFDerivAtFilter, is_o_iff] at hx 
+  rw [HasFDerivWithinAt, HasFDerivAtFilter, is_o_iff] at hx
   rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
   have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ :=
     by
@@ -102,7 +102,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
     rw [← smul_smul]
     apply s_conv.interior.add_smul_mem this _ ht
-    rw [add_assoc] at hw 
+    rw [add_assoc] at hw
     rw [add_assoc, ← smul_add]
     exact s_conv.add_smul_mem_interior xs hw ⟨hpos, h_lt_1.le⟩
   -- define a function `g` on `[0,1]` (identified with `[v, v + w]`) such that `g 1 - g 0` is the
@@ -322,7 +322,7 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
     intro m
     apply nhdsWithin_le_nhds
     apply A m
-    rw [mem_interior_iff_mem_nhds] at hy 
+    rw [mem_interior_iff_mem_nhds] at hy
     exact interior_mem_nhds.2 hy
   -- we choose `t m > 0` such that `x + 4 (z + (t m) m)` belongs to the interior of `s`, for any
   -- vector `m`.
@@ -335,7 +335,7 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
       s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts 0) (ts m)
     simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, add_right_inj,
       ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', add_zero,
-      ContinuousLinearMap.zero_apply, smul_zero, ContinuousLinearMap.map_zero] at this 
+      ContinuousLinearMap.zero_apply, smul_zero, ContinuousLinearMap.map_zero] at this
     exact smul_right_injective F (tpos m).ne' this
   -- applying `second_derivative_within_at_symmetric_of_mem_interior` to the vectors `z + (t v) v`
   -- and `z + (t w) w`, we deduce that `f'' v w = f'' w v`. Cross terms involving `z` can be
@@ -343,10 +343,10 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
   have : f'' (z + t v • v) (z + t w • w) = f'' (z + t w • w) (z + t v • v) :=
     s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts w) (ts v)
   simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, smul_add, smul_smul,
-    ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this 
-  rw [← sub_eq_zero] at this 
-  abel at this 
-  simp only [one_zsmul, neg_smul, sub_eq_zero, mul_comm, ← sub_eq_add_neg] at this 
+    ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this
+  rw [← sub_eq_zero] at this
+  abel at this
+  simp only [one_zsmul, neg_smul, sub_eq_zero, mul_comm, ← sub_eq_add_neg] at this
   apply smul_right_injective F _ this
   simp [(tpos v).ne', (tpos w).ne']
 #align convex.second_derivative_within_at_symmetric Convex.second_derivative_within_at_symmetric

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

@@ -157,7 +157,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           ContinuousLinearMap.add_apply, Pi.smul_apply, smul_sub, smul_add, smul_smul, ← sub_sub,
           ContinuousLinearMap.coe_smul', Pi.sub_apply, ContinuousLinearMap.map_smul, this]
       _ ≤ ‖f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)‖ * ‖h • w‖ :=
-        (ContinuousLinearMap.le_op_norm _ _)
+        (ContinuousLinearMap.le_opNorm _ _)
       _ ≤ ε * ‖h • v + (t * h) • w‖ * ‖h • w‖ :=
         by
         apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)

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

@@ -185,7 +185,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   · congr 1
     dsimp only [g]
     simp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, MulZeroClass.zero_mul, sub_zero,
-      zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow']
+      zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow]
     abel
   ·
     simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,

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

@@ -3,7 +3,7 @@ Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Analysis.Calculus.MeanValue
+import Analysis.Calculus.MeanValue
 
 #align_import analysis.calculus.fderiv_symmetric from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"
 

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

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.calculus.fderiv_symmetric
-! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Calculus.MeanValue
 
+#align_import analysis.calculus.fderiv_symmetric from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"
+
 /-!
 # Symmetry of the second derivative
 

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

@@ -63,8 +63,7 @@ variable {E F : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCom
   {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
   (hx : HasFDerivWithinAt f' f'' (interior s) x)
 
-include s_conv xs hx hf
-
+#print Convex.taylor_approx_two_segment /-
 /-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
 differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one can
 Taylor-expand to order two the function `f` on the segment `[x + h v, x + h (v + w)]`, giving a
@@ -195,7 +194,9 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,
       mul_assoc, pow_bit0_abs, norm_nonneg, abs_pow]
 #align convex.taylor_approx_two_segment Convex.taylor_approx_two_segment
+-/
 
+#print Convex.isLittleO_alternate_sum_square /-
 /-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
 along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
 In a setting where `f` is not guaranteed to be continuous at `f`, we can still
@@ -261,7 +262,9 @@ theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) •
     ContinuousLinearMap.map_smul]
   abel
 #align convex.is_o_alternate_sum_square Convex.isLittleO_alternate_sum_square
+-/
 
+#print Convex.second_derivative_within_at_symmetric_of_mem_interior /-
 /-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
 differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one
 has `f'' v w = f'' w v`. Superseded by `convex.second_derivative_within_at_symmetric`, which
@@ -292,9 +295,9 @@ theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
       field_simp [LT.lt.ne' hpos, SMul.smul]
   simpa only [sub_eq_zero] using is_o_const_const_iff.1 B
 #align convex.second_derivative_within_at_symmetric_of_mem_interior Convex.second_derivative_within_at_symmetric_of_mem_interior
+-/
 
-omit s_conv xs hx hf
-
+#print Convex.second_derivative_within_at_symmetric /-
 /-- If a function is differentiable inside a convex set with nonempty interior, and has a second
 derivative at a point of this convex set, then this second derivative is symmetric. -/
 theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
@@ -350,7 +353,9 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
   apply smul_right_injective F _ this
   simp [(tpos v).ne', (tpos w).ne']
 #align convex.second_derivative_within_at_symmetric Convex.second_derivative_within_at_symmetric
+-/
 
+#print second_derivative_symmetric_of_eventually /-
 /-- If a function is differentiable around `x`, and has two derivatives at `x`, then the second
 derivative is symmetric. -/
 theorem second_derivative_symmetric_of_eventually {f : E → F} {f' : E → E →L[ℝ] F}
@@ -364,11 +369,14 @@ theorem second_derivative_symmetric_of_eventually {f : E → F} {f' : E → E 
     Convex.second_derivative_within_at_symmetric (convex_ball x ε) A
       (fun y hy => hε (interior_subset hy)) (Metric.mem_ball_self εpos) hx.has_fderiv_within_at v w
 #align second_derivative_symmetric_of_eventually second_derivative_symmetric_of_eventually
+-/
 
+#print second_derivative_symmetric /-
 /-- If a function is differentiable, and has two derivatives at `x`, then the second
 derivative is symmetric. -/
 theorem second_derivative_symmetric {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
     (hf : ∀ y, HasFDerivAt f (f' y) y) (hx : HasFDerivAt f' f'' x) (v w : E) : f'' v w = f'' w v :=
   second_derivative_symmetric_of_eventually (Filter.eventually_of_forall hf) hx v w
 #align second_derivative_symmetric second_derivative_symmetric
+-/
 

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

@@ -152,7 +152,6 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           (add_le_add le_rfl
             (mul_le_mul_of_nonneg_right ht.2.le (mul_nonneg hpos.le (norm_nonneg _))))
         _ = h * (‖v‖ + ‖w‖) := by ring
-        
     calc
       ‖g' t‖ = ‖(f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)) (h • w)‖ :=
         by
@@ -182,7 +181,6 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
         exact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le
       _ = ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
         simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le]; ring
-      
   -- conclude using the mean value inequality
   have I : ‖g 1 - g 0‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
     simpa only [mul_one, sub_zero] using

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.fderiv_symmetric
-! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
+! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Analysis.Calculus.MeanValue
 /-!
 # Symmetry of the second derivative
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We show that, over the reals, the second derivative is symmetric.
 
 The most precise result is `convex.second_derivative_within_at_symmetric`. It asserts that,
@@ -93,7 +96,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
     mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
       ⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
-  filter_upwards [E1, E2, self_mem_nhdsWithin]with h hδ h_lt_1 hpos
+  filter_upwards [E1, E2, self_mem_nhdsWithin] with h hδ h_lt_1 hpos
   -- we consider `h` small enough that all points under consideration belong to this ball,
   -- and also with `0 < h < 1`.
   replace hpos : 0 < h := hpos
@@ -271,7 +274,8 @@ theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
   by
   have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 :=
     by
-    convert(s_conv.is_o_alternate_sum_square hf xs hx h4v h4w).sub
+    convert
+      (s_conv.is_o_alternate_sum_square hf xs hx h4v h4w).sub
         (s_conv.is_o_alternate_sum_square hf xs hx h4w h4v)
     ext h
     simp only [add_comm, smul_add, smul_sub]
@@ -286,7 +290,7 @@ theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
       rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
       congr 1
       field_simp [LT.lt.ne' hpos]
-    · filter_upwards [self_mem_nhdsWithin]with _ hpos
+    · filter_upwards [self_mem_nhdsWithin] with _ hpos
       field_simp [LT.lt.ne' hpos, SMul.smul]
   simpa only [sub_eq_zero] using is_o_const_const_iff.1 B
 #align convex.second_derivative_within_at_symmetric_of_mem_interior Convex.second_derivative_within_at_symmetric_of_mem_interior
@@ -343,7 +347,7 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
   simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, smul_add, smul_smul,
     ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this 
   rw [← sub_eq_zero] at this 
-  abel  at this 
+  abel at this 
   simp only [one_zsmul, neg_smul, sub_eq_zero, mul_comm, ← sub_eq_add_neg] at this 
   apply smul_right_injective F _ this
   simp [(tpos v).ne', (tpos w).ne']

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

@@ -82,7 +82,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   apply is_o.trans_is_O (is_o_iff.2 fun ε εpos => _) (is_O_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
   -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
   -- good up to `δ`.
-  rw [HasFDerivWithinAt, HasFDerivAtFilter, is_o_iff] at hx
+  rw [HasFDerivWithinAt, HasFDerivAtFilter, is_o_iff] at hx 
   rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
   have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ :=
     by
@@ -103,7 +103,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
     rw [← smul_smul]
     apply s_conv.interior.add_smul_mem this _ ht
-    rw [add_assoc] at hw
+    rw [add_assoc] at hw 
     rw [add_assoc, ← smul_add]
     exact s_conv.add_smul_mem_interior xs hw ⟨hpos, h_lt_1.le⟩
   -- define a function `g` on `[0,1]` (identified with `[v, v + w]`) such that `g 1 - g 0` is the
@@ -320,7 +320,7 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
     intro m
     apply nhdsWithin_le_nhds
     apply A m
-    rw [mem_interior_iff_mem_nhds] at hy
+    rw [mem_interior_iff_mem_nhds] at hy 
     exact interior_mem_nhds.2 hy
   -- we choose `t m > 0` such that `x + 4 (z + (t m) m)` belongs to the interior of `s`, for any
   -- vector `m`.
@@ -333,7 +333,7 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
       s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts 0) (ts m)
     simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, add_right_inj,
       ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', add_zero,
-      ContinuousLinearMap.zero_apply, smul_zero, ContinuousLinearMap.map_zero] at this
+      ContinuousLinearMap.zero_apply, smul_zero, ContinuousLinearMap.map_zero] at this 
     exact smul_right_injective F (tpos m).ne' this
   -- applying `second_derivative_within_at_symmetric_of_mem_interior` to the vectors `z + (t v) v`
   -- and `z + (t w) w`, we deduce that `f'' v w = f'' w v`. Cross terms involving `z` can be
@@ -341,10 +341,10 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
   have : f'' (z + t v • v) (z + t w • w) = f'' (z + t w • w) (z + t v • v) :=
     s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts w) (ts v)
   simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, smul_add, smul_smul,
-    ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this
-  rw [← sub_eq_zero] at this
-  abel  at this
-  simp only [one_zsmul, neg_smul, sub_eq_zero, mul_comm, ← sub_eq_add_neg] at this
+    ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this 
+  rw [← sub_eq_zero] at this 
+  abel  at this 
+  simp only [one_zsmul, neg_smul, sub_eq_zero, mul_comm, ← sub_eq_add_neg] at this 
   apply smul_right_injective F _ this
   simp [(tpos v).ne', (tpos w).ne']
 #align convex.second_derivative_within_at_symmetric Convex.second_derivative_within_at_symmetric

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

@@ -53,7 +53,7 @@ rectangle are contained in `s` by convexity. The general case follows by lineari
 
 open Asymptotics Set
 
-open Topology
+open scoped Topology
 
 variable {E F : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
   [NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) {f : E → F} {f' : E → E →L[ℝ] F}

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

@@ -177,10 +177,8 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
         refine' add_le_add le_rfl _
         simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]
         exact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le
-      _ = ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 :=
-        by
-        simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le]
-        ring
+      _ = ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
+        simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le]; ring
       
   -- conclude using the mean value inequality
   have I : ‖g 1 - g 0‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by

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

@@ -57,8 +57,8 @@ open Topology
 
 variable {E F : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
   [NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) {f : E → F} {f' : E → E →L[ℝ] F}
-  {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFderivAt f (f' x) x) {x : E} (xs : x ∈ s)
-  (hx : HasFderivWithinAt f' f'' (interior s) x)
+  {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
+  (hx : HasFDerivWithinAt f' f'' (interior s) x)
 
 include s_conv xs hx hf
 
@@ -82,7 +82,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   apply is_o.trans_is_O (is_o_iff.2 fun ε εpos => _) (is_O_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
   -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
   -- good up to `δ`.
-  rw [HasFderivWithinAt, HasFderivAtFilter, is_o_iff] at hx
+  rw [HasFDerivWithinAt, HasFDerivAtFilter, is_o_iff] at hx
   rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
   have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ :=
     by
@@ -299,8 +299,8 @@ omit s_conv xs hx hf
 derivative at a point of this convex set, then this second derivative is symmetric. -/
 theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
     (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
-    (hf : ∀ x ∈ interior s, HasFderivAt f (f' x) x) {x : E} (xs : x ∈ s)
-    (hx : HasFderivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v :=
+    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
+    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v :=
   by
   /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
     then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
@@ -354,7 +354,7 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
 /-- If a function is differentiable around `x`, and has two derivatives at `x`, then the second
 derivative is symmetric. -/
 theorem second_derivative_symmetric_of_eventually {f : E → F} {f' : E → E →L[ℝ] F}
-    {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ᶠ y in 𝓝 x, HasFderivAt f (f' y) y) (hx : HasFderivAt f' f'' x)
+    {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hx : HasFDerivAt f' f'' x)
     (v w : E) : f'' v w = f'' w v :=
   by
   rcases Metric.mem_nhds_iff.1 hf with ⟨ε, εpos, hε⟩
@@ -368,7 +368,7 @@ theorem second_derivative_symmetric_of_eventually {f : E → F} {f' : E → E 
 /-- If a function is differentiable, and has two derivatives at `x`, then the second
 derivative is symmetric. -/
 theorem second_derivative_symmetric {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
-    (hf : ∀ y, HasFderivAt f (f' y) y) (hx : HasFderivAt f' f'' x) (v w : E) : f'' v w = f'' w v :=
+    (hf : ∀ y, HasFDerivAt f (f' y) y) (hx : HasFDerivAt f' f'' x) (v w : E) : f'' v w = f'' w v :=
   second_derivative_symmetric_of_eventually (Filter.eventually_of_forall hf) hx v w
 #align second_derivative_symmetric second_derivative_symmetric
 

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

@@ -4,13 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.fderiv_symmetric
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.Analysis.Calculus.Deriv
 import Mathbin.Analysis.Calculus.MeanValue
-import Mathbin.Analysis.Convex.Topology
 
 /-!
 # Symmetry of the second derivative

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

@@ -203,7 +203,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
 along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
 In a setting where `f` is not guaranteed to be continuous at `f`, we can still
 get this if we use a quadrilateral based at `h v + h w`. -/
-theorem Convex.isOCat_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
+theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
     (h4w : x + (4 : ℝ) • w ∈ interior s) :
     (fun h : ℝ =>
         f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) -
@@ -263,7 +263,7 @@ theorem Convex.isOCat_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v
     ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul',
     ContinuousLinearMap.map_smul]
   abel
-#align convex.is_o_alternate_sum_square Convex.isOCat_alternate_sum_square
+#align convex.is_o_alternate_sum_square Convex.isLittleO_alternate_sum_square
 
 /-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
 differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -275,8 +275,7 @@ theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
   by
   have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 :=
     by
-    convert
-      (s_conv.is_o_alternate_sum_square hf xs hx h4v h4w).sub
+    convert(s_conv.is_o_alternate_sum_square hf xs hx h4v h4w).sub
         (s_conv.is_o_alternate_sum_square hf xs hx h4w h4v)
     ext h
     simp only [add_comm, smul_add, smul_sub]

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

@@ -91,7 +91,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
       (continuous_id.mul continuous_const).ContinuousWithinAt
     apply (tendsto_order.1 this).2 δ
-    simpa only [zero_mul] using δpos
+    simpa only [MulZeroClass.zero_mul] using δpos
   have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
     mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
       ⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
@@ -191,8 +191,8 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   convert I using 1
   · congr 1
     dsimp only [g]
-    simp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero, zero_smul, Ne.def,
-      not_false_iff, bit0_eq_zero, zero_pow']
+    simp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, MulZeroClass.zero_mul, sub_zero,
+      zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow']
     abel
   ·
     simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,

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

@@ -148,8 +148,8 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           simp only [norm_smul, Real.norm_eq_abs, hpos.le, abs_of_nonneg, abs_mul, ht.left,
             mul_assoc]
         _ ≤ h * ‖v‖ + 1 * (h * ‖w‖) :=
-          add_le_add le_rfl
-            (mul_le_mul_of_nonneg_right ht.2.le (mul_nonneg hpos.le (norm_nonneg _)))
+          (add_le_add le_rfl
+            (mul_le_mul_of_nonneg_right ht.2.le (mul_nonneg hpos.le (norm_nonneg _))))
         _ = h * (‖v‖ + ‖w‖) := by ring
         
     calc
@@ -161,7 +161,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           ContinuousLinearMap.add_apply, Pi.smul_apply, smul_sub, smul_add, smul_smul, ← sub_sub,
           ContinuousLinearMap.coe_smul', Pi.sub_apply, ContinuousLinearMap.map_smul, this]
       _ ≤ ‖f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)‖ * ‖h • w‖ :=
-        ContinuousLinearMap.le_op_norm _ _
+        (ContinuousLinearMap.le_op_norm _ _)
       _ ≤ ε * ‖h • v + (t * h) • w‖ * ‖h • w‖ :=
         by
         apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)

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

@@ -166,7 +166,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   convert I using 1
   · congr 1
     simp only [g, Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero,
-      zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow]
+      zero_smul, Ne, not_false_iff, bit0_eq_zero, zero_pow]
     abel
   · simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,
       hpos.le, mul_assoc, pow_bit0_abs, norm_nonneg, abs_pow]

Lemma names around cancellation of multiplication and division are a mess.

This PR renames a handful of them according to the following table (each big row contains the multiplicative statement, then the three rows contain the GroupWithZero lemma name, the Group lemma, the AddGroup lemma name).

| Statement | New name | Old name | |

@@ -150,7 +150,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           refine' ⟨_, xt_mem t ⟨ht.1, ht.2.le⟩⟩
           rw [add_assoc, add_mem_ball_iff_norm]
           exact I.trans_lt hδ
-        simpa only [mem_setOf_eq, add_assoc x, add_sub_cancel'] using sδ H
+        simpa only [mem_setOf_eq, add_assoc x, add_sub_cancel_left] using sδ H
       _ ≤ ε * (‖h • v‖ + ‖h • w‖) * ‖h • w‖ := by
         gcongr
         apply (norm_add_le _ _).trans
@@ -196,15 +196,15 @@ theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) •
     abel
   have h2v : x + (2 : ℝ) • v ∈ interior s := by
     convert s_conv.add_smul_sub_mem_interior xs h4v A using 1
-    simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]
+    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
     norm_num
   have h2w : x + (2 : ℝ) • w ∈ interior s := by
     convert s_conv.add_smul_sub_mem_interior xs h4w A using 1
-    simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]
+    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
     norm_num
   have hvw : x + (v + w) ∈ interior s := by
     convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1
-    simp only [smul_smul, one_div, add_sub_cancel', add_right_inj, smul_add, smul_sub]
+    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj, smul_add, smul_sub]
     norm_num
     abel
   have h2vw : x + (2 • v + w) ∈ interior s := by
@@ -214,7 +214,8 @@ theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) •
     abel
   have hvww : x + (v + w) + w ∈ interior s := by
     convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1
-    rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel', inv_smul_smul₀ two_ne_zero, two_smul]
+    rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel_left, inv_smul_smul₀ two_ne_zero,
+      two_smul]
     abel
   have TA1 := s_conv.taylor_approx_two_segment hf xs hx h2vw h2vww
   have TA2 := s_conv.taylor_approx_two_segment hf xs hx hvw hvww

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

@@ -165,7 +165,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
       norm_image_sub_le_of_norm_deriv_le_segment' g_deriv g'_bound 1 (right_mem_Icc.2 zero_le_one)
   convert I using 1
   · congr 1
-    simp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero,
+    simp only [g, Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero,
       zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow]
     abel
   · simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,

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

This follows on from #6964.

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

@@ -119,8 +119,8 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     · apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]
     · apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]
     · suffices H : HasDerivWithinAt (fun u => ((u * h) ^ 2 / 2) • f'' w w)
-          ((((2 : ℕ) : ℝ) * (t * h) ^ (2 - 1) * (1 * h) / 2) • f'' w w) (Icc 0 1) t
-      · convert H using 2
+          ((((2 : ℕ) : ℝ) * (t * h) ^ (2 - 1) * (1 * h) / 2) • f'' w w) (Icc 0 1) t by
+        convert H using 2
         ring
       apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_id',
         HasDerivAt.pow, HasDerivAt.mul_const]

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

@@ -143,7 +143,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           ContinuousLinearMap.add_apply, Pi.smul_apply, smul_sub, smul_add, smul_smul, ← sub_sub,
           ContinuousLinearMap.coe_smul', Pi.sub_apply, ContinuousLinearMap.map_smul, this]
       _ ≤ ‖f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)‖ * ‖h • w‖ :=
-        (ContinuousLinearMap.le_op_norm _ _)
+        (ContinuousLinearMap.le_opNorm _ _)
       _ ≤ ε * ‖h • v + (t * h) • w‖ * ‖h • w‖ := by
         apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
         have H : x + h • v + (t * h) • w ∈ Metric.ball x δ ∩ interior s := by

This involves moving lemmas from Algebra.GroupPower.Ring to Algebra.GroupWithZero.Basic and changing some 0 < n assumptions to n ≠ 0.

From LeanAPAP

@@ -166,7 +166,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   convert I using 1
   · congr 1
     simp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero,
-      zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow']
+      zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow]
     abel
   · simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,
       hpos.le, mul_assoc, pow_bit0_abs, norm_nonneg, abs_pow]

This way we can easily change the definition so that it works for topological vector spaces without generalizing any of the theorems right away.

@@ -76,7 +76,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     (isLittleO_iff.2 fun ε εpos => _) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
   -- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
   -- good up to `δ`.
-  rw [HasFDerivWithinAt, HasFDerivAtFilter, isLittleO_iff] at hx
+  rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff] at hx
   rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
   have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ := by
     have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=

We clean up heart beat bumps after #6474.

@@ -58,7 +58,6 @@ variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddComm
   {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
   (hx : HasFDerivWithinAt f' f'' (interior s) x)
 
-set_option maxHeartbeats 300000 in
 /-- Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
 differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one can
 Taylor-expand to order two the function `f` on the segment `[x + h v, x + h (v + w)]`, giving a

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -83,7 +83,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
       (continuous_id.mul continuous_const).continuousWithinAt
     apply (tendsto_order.1 this).2 δ
-    simpa only [MulZeroClass.zero_mul] using δpos
+    simpa only [zero_mul] using δpos
   have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
     mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
       ⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
@@ -166,7 +166,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
       norm_image_sub_le_of_norm_deriv_le_segment' g_deriv g'_bound 1 (right_mem_Icc.2 zero_le_one)
   convert I using 1
   · congr 1
-    simp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, MulZeroClass.zero_mul, sub_zero,
+    simp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero,
       zero_smul, Ne.def, not_false_iff, bit0_eq_zero, zero_pow']
     abel
   · simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,

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

This has nice performance benefits.

@@ -53,7 +53,7 @@ open Asymptotics Set
 
 open scoped Topology
 
-variable {E F : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
   [NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) {f : E → F} {f' : E → E →L[ℝ] F}
   {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
   (hx : HasFDerivWithinAt f' f'' (interior s) x)

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.

@@ -3,6 +3,7 @@ Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
+import Mathlib.Analysis.Calculus.Deriv.Pow
 import Mathlib.Analysis.Calculus.MeanValue
 
 #align_import analysis.calculus.fderiv_symmetric from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.calculus.fderiv_symmetric
-! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Calculus.MeanValue
 
+#align_import analysis.calculus.fderiv_symmetric from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
+
 /-!
 # Symmetry of the second derivative
 

Following on from #4702, another hundred sample uses of the gcongr tactic.

@@ -136,9 +136,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
         _ = h * ‖v‖ + t * (h * ‖w‖) := by
           simp only [norm_smul, Real.norm_eq_abs, hpos.le, abs_of_nonneg, abs_mul, ht.left,
             mul_assoc]
-        _ ≤ h * ‖v‖ + 1 * (h * ‖w‖) :=
-          (add_le_add le_rfl
-            (mul_le_mul_of_nonneg_right ht.2.le (mul_nonneg hpos.le (norm_nonneg _))))
+        _ ≤ h * ‖v‖ + 1 * (h * ‖w‖) := by gcongr; exact ht.2.le
         _ = h * (‖v‖ + ‖w‖) := by ring
     calc
       ‖g' t‖ = ‖(f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)) (h • w)‖ := by
@@ -157,10 +155,9 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
           exact I.trans_lt hδ
         simpa only [mem_setOf_eq, add_assoc x, add_sub_cancel'] using sδ H
       _ ≤ ε * (‖h • v‖ + ‖h • w‖) * ‖h • w‖ := by
-        apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
-        apply mul_le_mul_of_nonneg_left _ εpos.le
+        gcongr
         apply (norm_add_le _ _).trans
-        refine' add_le_add le_rfl _
+        gcongr
         simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]
         exact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le
       _ = ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by