measure_theory.integral.divergence_theorem ⟷ Mathlib.MeasureTheory.Integral.DivergenceTheorem

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

@@ -508,7 +508,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
         by
         intro a b f
         convert
-          (((volume_preserving_fun_unique (Fin 1) ℝ).symm _).set_integral_preimage_emb
+          (((volume_preserving_fun_unique (Fin 1) ℝ).symm _).setIntegral_preimage_emb
               (MeasurableEquiv.measurableEmbedding _) _ _).symm
         exact ((OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc a b).symm
       simp only [Fin.sum_univ_two, this]

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

@@ -124,7 +124,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
     has_integral_GP_divergence_of_forall_has_deriv_within_at I f f' (s ∩ I.Icc)
       (hs.mono (inter_subset_left _ _)) (fun x hx => Hc _ hx.2) fun x hx =>
       Hd _ ⟨hx.1, fun h => hx.2 ⟨h, hx.1⟩⟩
-  rw [continuousOn_pi] at Hc 
+  rw [continuousOn_pi] at Hc
   refine' (A.unique B).trans (sum_congr rfl fun i hi => _)
   refine' congr_arg₂ Sub.sub _ _
   · have := box.continuous_on_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i))
@@ -176,7 +176,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
   -- Thus it suffices to prove the same about the RHS.
   refine' tendsto_nhds_unique_of_eventuallyEq hI_tendsto _ (eventually_of_forall HJ_eq)
   clear hI_tendsto
-  rw [tendsto_pi_nhds] at hJl hJu 
+  rw [tendsto_pi_nhds] at hJl hJu
   /- We'll need to prove a similar statement about the integrals over the front sides and the
     integrals over the back sides. In order to avoid repeating ourselves, we formulate a lemma. -/
   suffices
@@ -186,7 +186,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
           tendsto (fun k => ∫ x in ((J k).face i).Icc, f (i.insertNth (c k) x) i) at_top
             (𝓝 <| ∫ x in (I.face i).Icc, f (i.insertNth d x) i)
     by
-    rw [box.Icc_eq_pi] at hJ_sub' 
+    rw [box.Icc_eq_pi] at hJ_sub'
     refine' tendsto_finset_sum _ fun i hi => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))
     exacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>
       hJ_sub' k (J k).lower_mem_Icc _ trivial]
@@ -249,7 +249,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
       rw [box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper, ←
         le_div_iff (hvol_pos _)]
       refine'
-        div_le_div_of_le_left εpos.le (hvol_pos _) (prod_le_prod (fun j hj => _) fun j hj => _)
+        div_le_div_of_nonneg_left εpos.le (hvol_pos _) (prod_le_prod (fun j hj => _) fun j hj => _)
       exacts [sub_nonneg.2 (box.lower_le_upper _ _),
         sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
@@ -421,14 +421,14 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
             ∫ x in Icc (e a ∘ i.succAboveEmb) (e b ∘ i.succAboveEmb),
               f (e.symm <| i.insertNth (e a i) x)) :=
       by
-      simp only [← interior_Icc] at Hd 
+      simp only [← interior_Icc] at Hd
       refine'
         integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv e _ _ (fun _ => f)
           (fun _ => F') s hs a b hle (fun i => Hc) (fun x hx i => Hd x hx) _ _ _
       · exact fun x y => (OrderIso.funUnique (Fin 1) ℝ).symm.le_iff_le
       · exact (volume_preserving_fun_unique (Fin 1) ℝ).symm _
       · intro x; rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]
-      · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hle] at Hi 
+      · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hle] at Hi
         exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm
     _ = f b - f a := by
       simp only [Fin.sum_univ_one, e_symm]
@@ -496,7 +496,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
       · exact fun x y => (OrderIso.finTwoArrowIso ℝ).symm.le_iff_le
       · exact (volume_preserving_fin_two_arrow ℝ).symm _
       · exact Fin.forall_fin_two.2 ⟨Hcf, Hcg⟩
-      · rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx 
+      · rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx
         exact Fin.forall_fin_two.2 ⟨Hdf x hx, Hdg x hx⟩
       · intro x; rw [Fin.sum_univ_two]; simp
     _ =
@@ -554,15 +554,15 @@ theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ
   by
   wlog h₁ : a₁ ≤ b₁ generalizing a₁ b₁
   · specialize this b₁ a₁
-    rw [uIcc_comm b₁ a₁, min_comm b₁ a₁, max_comm b₁ a₁] at this 
+    rw [uIcc_comm b₁ a₁, min_comm b₁ a₁, max_comm b₁ a₁] at this
     simp only [intervalIntegral.integral_symm b₁ a₁]
     refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₁))).trans _; abel
   wlog h₂ : a₂ ≤ b₂ generalizing a₂ b₂
   · specialize this b₂ a₂
-    rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this 
+    rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this
     simp only [intervalIntegral.integral_symm b₂ a₂, intervalIntegral.integral_neg]
     refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₂))).trans _; abel
-  simp only [uIcc_of_le h₁, uIcc_of_le h₂, min_eq_left, max_eq_right, h₁, h₂] at Hcf Hcg Hdf Hdg Hi 
+  simp only [uIcc_of_le h₁, uIcc_of_le h₂, min_eq_left, max_eq_right, h₁, h₂] at Hcf Hcg Hdf Hdg Hi
   calc
     ∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) =
         ∫ x in Icc a₁ b₁, ∫ y in Icc a₂ b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) :=

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

@@ -437,14 +437,14 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
 #align measure_theory.integral_eq_of_has_deriv_within_at_off_countable_of_le MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le
 -/
 
-#print MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable /-
+#print MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable /-
 /-- **Fundamental theorem of calculus, part 2**. This version assumes that `f` is continuous on the
 interval and is differentiable off a countable set `s`.
 
 See also `measure_theory.interval_integral.integral_eq_sub_of_has_deriv_right` for a version that
 only assumes right differentiability of `f`.
 -/
-theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}
+theorem integral_eq_of_hasDerivWithinAt_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}
     (hs : s.Countable) (Hc : ContinuousOn f [a, b])
     (Hd : ∀ x ∈ Ioo (min a b) (max a b) \ s, HasDerivAt f (f' x) x)
     (Hi : IntervalIntegrable f' volume a b) : ∫ x in a..b, f' x = f b - f a :=
@@ -455,7 +455,7 @@ theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b
   · simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *
     rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub]
     exact integral_eq_of_has_deriv_within_at_off_countable_of_le f f' hab hs Hc Hd Hi.symm
-#align measure_theory.integral_eq_of_has_deriv_within_at_off_countable MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable
+#align measure_theory.integral_eq_of_has_deriv_within_at_off_countable MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/

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

@@ -428,7 +428,7 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
       · exact fun x y => (OrderIso.funUnique (Fin 1) ℝ).symm.le_iff_le
       · exact (volume_preserving_fun_unique (Fin 1) ℝ).symm _
       · intro x; rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]
-      · rw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi 
+      · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hle] at Hi 
         exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm
     _ = f b - f a := by
       simp only [Fin.sum_univ_one, e_symm]

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

@@ -3,11 +3,11 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathbin.Analysis.BoxIntegral.DivergenceTheorem
-import Mathbin.Analysis.BoxIntegral.Integrability
-import Mathbin.Analysis.Calculus.Deriv.Basic
-import Mathbin.MeasureTheory.Constructions.Prod.Integral
-import Mathbin.MeasureTheory.Integral.IntervalIntegral
+import Analysis.BoxIntegral.DivergenceTheorem
+import Analysis.BoxIntegral.Integrability
+import Analysis.Calculus.Deriv.Basic
+import MeasureTheory.Constructions.Prod.Integral
+import MeasureTheory.Integral.IntervalIntegral
 
 #align_import measure_theory.integral.divergence_theorem from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
 

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module measure_theory.integral.divergence_theorem
-! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.BoxIntegral.DivergenceTheorem
 import Mathbin.Analysis.BoxIntegral.Integrability
@@ -14,6 +9,8 @@ import Mathbin.Analysis.Calculus.Deriv.Basic
 import Mathbin.MeasureTheory.Constructions.Prod.Integral
 import Mathbin.MeasureTheory.Integral.IntervalIntegral
 
+#align_import measure_theory.integral.divergence_theorem from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
+
 /-!
 # Divergence theorem for Bochner integral
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

@@ -260,7 +260,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
 
 variable (a b : ℝⁿ⁺¹)
 
-local notation "face " i => Set.Icc (a ∘ Fin.succAbove i) (b ∘ Fin.succAbove i)
+local notation "face " i => Set.Icc (a ∘ Fin.succAboveEmb i) (b ∘ Fin.succAboveEmb i)
 
 local notation "front_face " i:2000 => Fin.insertNth i (b i)
 
@@ -343,9 +343,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
     (hDF : ∀ x, DF x = ∑ i, f' i x (eL.symm <| e i)) (Hi : IntegrableOn DF (Icc a b)) :
     ∫ x in Icc a b, DF x =
       ∑ i : Fin (n + 1),
-        ((∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
+        ((∫ x in Icc (eL a ∘ i.succAboveEmb) (eL b ∘ i.succAboveEmb),
             f i (eL.symm <| i.insertNth (eL b i) x)) -
-          ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
+          ∫ x in Icc (eL a ∘ i.succAboveEmb) (eL b ∘ i.succAboveEmb),
             f i (eL.symm <| i.insertNth (eL a i) x)) :=
   have he_emb : MeasurableEmbedding eL := eL.toHomeomorph.toMeasurableEquiv.MeasurableEmbedding
   have hIcc : eL ⁻¹' Icc (eL a) (eL b) = Icc a b := by ext1 x;
@@ -359,9 +359,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
       simp only [hIcc, eL.symm_apply_apply]
     _ =
         ∑ i : Fin (n + 1),
-          ((∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
+          ((∫ x in Icc (eL a ∘ i.succAboveEmb) (eL b ∘ i.succAboveEmb),
               f i (eL.symm <| i.insertNth (eL b i) x)) -
-            ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
+            ∫ x in Icc (eL a ∘ i.succAboveEmb) (eL b ∘ i.succAboveEmb),
               f i (eL.symm <| i.insertNth (eL a i) x)) :=
       by
       convert
@@ -419,9 +419,9 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
       simp only [intervalIntegral.integral_of_le hle, set_integral_congr_set_ae Ioc_ae_eq_Icc]
     _ =
         ∑ i : Fin 1,
-          ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
+          ((∫ x in Icc (e a ∘ i.succAboveEmb) (e b ∘ i.succAboveEmb),
               f (e.symm <| i.insertNth (e b i) x)) -
-            ∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
+            ∫ x in Icc (e a ∘ i.succAboveEmb) (e b ∘ i.succAboveEmb),
               f (e.symm <| i.insertNth (e a i) x)) :=
       by
       simp only [← interior_Icc] at Hd 
@@ -488,9 +488,9 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
   calc
     ∫ x in Icc a b, f' x (1, 0) + g' x (0, 1) =
         ∑ i : Fin 2,
-          ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
+          ((∫ x in Icc (e a ∘ i.succAboveEmb) (e b ∘ i.succAboveEmb),
               ![f, g] i (e.symm <| i.insertNth (e b i) x)) -
-            ∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
+            ∫ x in Icc (e a ∘ i.succAboveEmb) (e b ∘ i.succAboveEmb),
               ![f, g] i (e.symm <| i.insertNth (e a i) x)) :=
       by
       refine'

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

@@ -70,16 +70,12 @@ section
 
 variable {n : ℕ}
 
--- mathport name: «exprℝⁿ»
 local notation "ℝⁿ" => Fin n → ℝ
 
--- mathport name: «exprℝⁿ⁺¹»
 local notation "ℝⁿ⁺¹" => Fin (n + 1) → ℝ
 
--- mathport name: «exprEⁿ⁺¹»
 local notation "Eⁿ⁺¹" => Fin (n + 1) → E
 
--- mathport name: «expre »
 local notation "e " i => Pi.single i 1
 
 section
@@ -111,6 +107,7 @@ in several aspects.
 -/
 
 
+#print MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ /-
 /-- An auxiliary lemma for
 `measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable`. This is exactly
 `box_integral.has_integral_GP_divergence_of_forall_has_deriv_within_at` reformulated for the
@@ -140,7 +137,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
     have := (this.integrable_on_compact (box.is_compact_Icc _)).mono_set box.coe_subset_Icc
     exact (this.has_box_integral ⊥ rfl).integral_eq; infer_instance
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₁ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁
+-/
 
+#print MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ /-
 /-- An auxiliary lemma for
 `measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable`. Compared to the previous
 lemma, here we drop the assumption of differentiability on the boundary of the box. -/
@@ -257,18 +256,17 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
       exacts [sub_nonneg.2 (box.lower_le_upper _ _),
         sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
+-/
 
 variable (a b : ℝⁿ⁺¹)
 
--- mathport name: «exprface »
 local notation "face " i => Set.Icc (a ∘ Fin.succAbove i) (b ∘ Fin.succAbove i)
 
--- mathport name: «exprfront_face »
 local notation "front_face " i:2000 => Fin.insertNth i (b i)
 
--- mathport name: «exprback_face »
 local notation "back_face " i:2000 => Fin.insertNth i (a i)
 
+#print MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable /-
 /-- **Divergence theorem** for Bochner integral. If `f : ℝⁿ⁺¹ → Eⁿ⁺¹` is continuous on a rectangular
 box `[a, b] : set ℝⁿ⁺¹`, `a ≤ b`, is differentiable on its interior with derivative
 `f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹` and the divergence `λ x, ∑ i, f' x eᵢ i` is integrable on `[a, b]`,
@@ -310,7 +308,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b) (
     convert
       integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ ⟨a, b, hlt⟩ f f' s hs Hc Hd Hi
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable
+-/
 
+#print MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable' /-
 /-- **Divergence theorem** for a family of functions `f : fin (n + 1) → ℝⁿ⁺¹ → E`. See also
 `measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable'` for a version formulated
 in terms of a vector-valued function `f : ℝⁿ⁺¹ → Eⁿ⁺¹`. -/
@@ -326,9 +326,11 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable' (hle : a ≤ b)
     (fun x => ContinuousLinearMap.pi fun i => f' i x) s hs (continuousOn_pi.2 Hc)
     (fun x hx => hasFDerivAt_pi.2 (Hd x hx)) Hi
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable' MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable'
+-/
 
 end
 
+#print MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv /-
 /-- An auxiliary lemma that is used to specialize the general divergence theorem to spaces that do
 not have the form `fin n → ℝ`. -/
 theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Type _}
@@ -376,6 +378,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
       · rw [← he_vol.integrable_on_comp_preimage he_emb, hIcc]
         simp [← hDF, (· ∘ ·), Hi]
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv
+-/
 
 end
 
@@ -383,18 +386,15 @@ open scoped Interval
 
 open ContinuousLinearMap (smul_right)
 
--- mathport name: «exprℝ¹»
 local notation "ℝ¹" => Fin 1 → ℝ
 
--- mathport name: «exprℝ²»
 local notation "ℝ²" => Fin 2 → ℝ
 
--- mathport name: «exprE¹»
 local notation "E¹" => Fin 1 → E
 
--- mathport name: «exprE²»
 local notation "E²" => Fin 2 → E
 
+#print MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le /-
 /-- **Fundamental theorem of calculus, part 2**. This version assumes that `f` is continuous on the
 interval and is differentiable off a countable set `s`.
 
@@ -438,7 +438,9 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
       have : ∀ c : ℝ, const (Fin 0) c = isEmptyElim := fun c => Subsingleton.elim _ _
       simp [this, volume_pi, measure.pi_of_empty fun _ : Fin 0 => volume]
 #align measure_theory.integral_eq_of_has_deriv_within_at_off_countable_of_le MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le
+-/
 
+#print MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable /-
 /-- **Fundamental theorem of calculus, part 2**. This version assumes that `f` is continuous on the
 interval and is differentiable off a countable set `s`.
 
@@ -457,9 +459,11 @@ theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b
     rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub]
     exact integral_eq_of_has_deriv_within_at_off_countable_of_le f f' hab hs Hc Hd Hi.symm
 #align measure_theory.integral_eq_of_has_deriv_within_at_off_countable MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le /-
 /-- **Divergence theorem** for functions on the plane along rectangles. It is formulated in terms of
 two functions `f g : ℝ × ℝ → E` and an integral over `Icc a b = [a.1, b.1] × [a.2, b.2]`, where
 `a b : ℝ × ℝ`, `a ≤ b`. When thinking of `f` and `g` as the two coordinates of a single function
@@ -521,6 +525,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
         set_integral_congr_set_ae Ioc_ae_eq_Icc]
       abel
 #align measure_theory.integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -528,6 +533,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable /-
 /-- **Divergence theorem** for functions on the plane. It is formulated in terms of two functions
 `f g : ℝ × ℝ → E` and iterated integral `∫ x in a₁..b₁, ∫ y in a₂..b₂, _`, where
 `a₁ a₂ b₁ b₂ : ℝ`. When thinking of `f` and `g` as the two coordinates of a single function
@@ -576,6 +582,7 @@ theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ
             (a₁, a₂) (b₁, b₂) ⟨h₁, h₂⟩ s <;>
         assumption
 #align measure_theory.integral2_divergence_prod_of_has_fderiv_within_at_off_countable MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable
+-/
 
 end MeasureTheory
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -119,10 +119,10 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
     (f : ℝⁿ⁺¹ → Eⁿ⁺¹) (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable)
     (Hc : ContinuousOn f I.Icc) (Hd : ∀ x ∈ I.Icc \ s, HasFDerivWithinAt f (f' x) I.Icc x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) I.Icc) :
-    (∫ x in I.Icc, ∑ i, f' x (e i) i) =
+    ∫ x in I.Icc, ∑ i, f' x (e i) i =
       ∑ i : Fin (n + 1),
-        (∫ x in (I.face i).Icc, f (i.insertNth (I.upper i) x) i) -
-          ∫ x in (I.face i).Icc, f (i.insertNth (I.lower i) x) i :=
+        ((∫ x in (I.face i).Icc, f (i.insertNth (I.upper i) x) i) -
+          ∫ x in (I.face i).Icc, f (i.insertNth (I.lower i) x) i) :=
   by
   simp only [← set_integral_congr_set_ae (box.coe_ae_eq_Icc _)]
   have A := (Hi.mono_set box.coe_subset_Icc).hasBoxIntegral ⊥ rfl
@@ -148,10 +148,10 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
     (f : ℝⁿ⁺¹ → Eⁿ⁺¹) (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable)
     (Hc : ContinuousOn f I.Icc) (Hd : ∀ x ∈ I.Ioo \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) I.Icc) :
-    (∫ x in I.Icc, ∑ i, f' x (e i) i) =
+    ∫ x in I.Icc, ∑ i, f' x (e i) i =
       ∑ i : Fin (n + 1),
-        (∫ x in (I.face i).Icc, f (i.insertNth (I.upper i) x) i) -
-          ∫ x in (I.face i).Icc, f (i.insertNth (I.lower i) x) i :=
+        ((∫ x in (I.face i).Icc, f (i.insertNth (I.upper i) x) i) -
+          ∫ x in (I.face i).Icc, f (i.insertNth (I.lower i) x) i) :=
   by
   /- Choose a monotone sequence `J k` of subboxes that cover the interior of `I` and prove that
     these boxes satisfy the assumptions of the previous lemma. -/
@@ -218,14 +218,14 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
     `{x | x i = c k}` and `{x | x i = d}` over `((J k).face i).Icc` tends to zero as `k → ∞`. Choose
     `ε > 0`. -/
   refine' H.congr_dist (metric.nhds_basis_closed_ball.tendsto_right_iff.2 fun ε εpos => _)
-  have hvol_pos : ∀ J : box (Fin n), 0 < ∏ j, J.upper j - J.lower j := fun J =>
+  have hvol_pos : ∀ J : box (Fin n), 0 < ∏ j, (J.upper j - J.lower j) := fun J =>
     prod_pos fun j hj => sub_pos.2 <| J.lower_lt_upper _
   /- Choose `δ > 0` such that for any `x y ∈ I.Icc` at distance at most `δ`, the distance between
     `f x` and `f y` is at most `ε / volume (I.face i).Icc`, then the distance between the integrals
     is at most `(ε / volume (I.face i).Icc) * volume ((J k).face i).Icc ≤ ε`. -/
   rcases Metric.uniformContinuousOn_iff_le.1
       (I.is_compact_Icc.uniform_continuous_on_of_continuous Hc)
-      (ε / ∏ j, (I.face i).upper j - (I.face i).lower j) (div_pos εpos (hvol_pos (I.face i))) with
+      (ε / ∏ j, ((I.face i).upper j - (I.face i).lower j)) (div_pos εpos (hvol_pos (I.face i))) with
     ⟨δ, δpos, hδ⟩
   refine' (hcd.eventually (Metric.ball_mem_nhds _ δpos)).mono fun k hk => _
   have Hsub : ((J k).face i).Icc ⊆ (I.face i).Icc := box.le_iff_Icc.1 (box.face_mono (hJ_le _) i)
@@ -233,7 +233,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
     integral_sub (Hid.mono_set Hsub) ((Hic _).mono_set Hsub)]
   calc
     ‖∫ x in ((J k).face i).Icc, f (i.insert_nth d x) i - f (i.insert_nth (c k) x) i‖ ≤
-        (ε / ∏ j, (I.face i).upper j - (I.face i).lower j) * (volume ((J k).face i).Icc).toReal :=
+        (ε / ∏ j, ((I.face i).upper j - (I.face i).lower j)) * (volume ((J k).face i).Icc).toReal :=
       by
       refine'
         norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)
@@ -243,7 +243,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
         dist (f (i.insert_nth d x) i) (f (i.insert_nth (c k) x) i) ≤
             dist (f (i.insert_nth d x)) (f (i.insert_nth (c k) x)) :=
           dist_le_pi_dist (f (i.insert_nth d x)) (f (i.insert_nth (c k) x)) i
-        _ ≤ ε / ∏ j, (I.face i).upper j - (I.face i).lower j :=
+        _ ≤ ε / ∏ j, ((I.face i).upper j - (I.face i).lower j) :=
           hδ _ (I.maps_to_insert_nth_face_Icc hd <| Hsub hx) _
             (I.maps_to_insert_nth_face_Icc (hc _) <| Hsub hx) _
       rw [Fin.dist_insertNth_insertNth, dist_self, dist_comm]
@@ -288,9 +288,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b) (
     (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
     (Hd : ∀ x ∈ (Set.pi univ fun i => Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) (Icc a b)) :
-    (∫ x in Icc a b, ∑ i, f' x (e i) i) =
+    ∫ x in Icc a b, ∑ i, f' x (e i) i =
       ∑ i : Fin (n + 1),
-        (∫ x in face i, f ((front_face (i)) x) i) - ∫ x in face i, f ((back_face (i)) x) i :=
+        ((∫ x in face i, f ((front_face (i)) x) i) - ∫ x in face i, f ((back_face (i)) x) i) :=
   by
   rcases em (∃ i, a i = b i) with (⟨i, hi⟩ | hne)
   · -- First we sort out the trivial case `∃ i, a i = b i`.
@@ -319,9 +319,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable' (hle : a ≤ b)
     (hs : s.Countable) (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
     (Hd : ∀ x ∈ (pi Set.univ fun i => Ioo (a i) (b i)) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' i x (e i)) (Icc a b)) :
-    (∫ x in Icc a b, ∑ i, f' i x (e i)) =
+    ∫ x in Icc a b, ∑ i, f' i x (e i) =
       ∑ i : Fin (n + 1),
-        (∫ x in face i, f i ((front_face (i)) x)) - ∫ x in face i, f i ((back_face (i)) x) :=
+        ((∫ x in face i, f i ((front_face (i)) x)) - ∫ x in face i, f i ((back_face (i)) x)) :=
   integral_divergence_of_hasFDerivWithinAt_off_countable a b hle (fun x i => f i x)
     (fun x => ContinuousLinearMap.pi fun i => f' i x) s hs (continuousOn_pi.2 Hc)
     (fun x hx => hasFDerivAt_pi.2 (Hd x hx)) Hi
@@ -339,28 +339,28 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
     (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
     (Hd : ∀ x ∈ interior (Icc a b) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x) (DF : F → E)
     (hDF : ∀ x, DF x = ∑ i, f' i x (eL.symm <| e i)) (Hi : IntegrableOn DF (Icc a b)) :
-    (∫ x in Icc a b, DF x) =
+    ∫ x in Icc a b, DF x =
       ∑ i : Fin (n + 1),
-        (∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
+        ((∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
             f i (eL.symm <| i.insertNth (eL b i) x)) -
           ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
-            f i (eL.symm <| i.insertNth (eL a i) x) :=
+            f i (eL.symm <| i.insertNth (eL a i) x)) :=
   have he_emb : MeasurableEmbedding eL := eL.toHomeomorph.toMeasurableEquiv.MeasurableEmbedding
   have hIcc : eL ⁻¹' Icc (eL a) (eL b) = Icc a b := by ext1 x;
     simp only [Set.mem_preimage, Set.mem_Icc, he_ord]
   have hIcc' : Icc (eL a) (eL b) = eL.symm ⁻¹' Icc a b := by rw [← hIcc, eL.symm_preimage_preimage]
   calc
-    (∫ x in Icc a b, DF x) = ∫ x in Icc a b, ∑ i, f' i x (eL.symm <| e i) := by simp only [hDF]
+    ∫ x in Icc a b, DF x = ∫ x in Icc a b, ∑ i, f' i x (eL.symm <| e i) := by simp only [hDF]
     _ = ∫ x in Icc (eL a) (eL b), ∑ i, f' i (eL.symm x) (eL.symm <| e i) :=
       by
       rw [← he_vol.set_integral_preimage_emb he_emb]
       simp only [hIcc, eL.symm_apply_apply]
     _ =
         ∑ i : Fin (n + 1),
-          (∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
+          ((∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
               f i (eL.symm <| i.insertNth (eL b i) x)) -
             ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
-              f i (eL.symm <| i.insertNth (eL a i) x) :=
+              f i (eL.symm <| i.insertNth (eL a i) x)) :=
       by
       convert
         integral_divergence_of_has_fderiv_within_at_off_countable' (eL a) (eL b)
@@ -408,21 +408,21 @@ differentiability of `f`;
 theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {a b : ℝ} (hle : a ≤ b)
     {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
     (Hd : ∀ x ∈ Ioo a b \ s, HasDerivAt f (f' x) x) (Hi : IntervalIntegrable f' volume a b) :
-    (∫ x in a..b, f' x) = f b - f a :=
+    ∫ x in a..b, f' x = f b - f a :=
   by
   set e : ℝ ≃L[ℝ] ℝ¹ := (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ).symm
   have e_symm : ∀ x, e.symm x = x 0 := fun x => rfl
   set F' : ℝ → ℝ →L[ℝ] E := fun x => smul_right (1 : ℝ →L[ℝ] ℝ) (f' x)
   have hF' : ∀ x y, F' x y = y • f' x := fun x y => rfl
   calc
-    (∫ x in a..b, f' x) = ∫ x in Icc a b, f' x := by
+    ∫ x in a..b, f' x = ∫ x in Icc a b, f' x := by
       simp only [intervalIntegral.integral_of_le hle, set_integral_congr_set_ae Ioc_ae_eq_Icc]
     _ =
         ∑ i : Fin 1,
-          (∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
+          ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
               f (e.symm <| i.insertNth (e b i) x)) -
             ∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
-              f (e.symm <| i.insertNth (e a i) x) :=
+              f (e.symm <| i.insertNth (e a i) x)) :=
       by
       simp only [← interior_Icc] at Hd 
       refine'
@@ -448,7 +448,7 @@ only assumes right differentiability of `f`.
 theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}
     (hs : s.Countable) (Hc : ContinuousOn f [a, b])
     (Hd : ∀ x ∈ Ioo (min a b) (max a b) \ s, HasDerivAt f (f' x) x)
-    (Hi : IntervalIntegrable f' volume a b) : (∫ x in a..b, f' x) = f b - f a :=
+    (Hi : IntervalIntegrable f' volume a b) : ∫ x in a..b, f' x = f b - f a :=
   by
   cases' le_total a b with hab hab
   · simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
@@ -476,18 +476,18 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
     (Hdf : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x)
     (Hdg : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x)
     (Hi : IntegrableOn (fun x => f' x (1, 0) + g' x (0, 1)) (Icc a b)) :
-    (∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
+    ∫ x in Icc a b, f' x (1, 0) + g' x (0, 1) =
       (((∫ x in a.1 ..b.1, g (x, b.2)) - ∫ x in a.1 ..b.1, g (x, a.2)) +
           ∫ y in a.2 ..b.2, f (b.1, y)) -
         ∫ y in a.2 ..b.2, f (a.1, y) :=
   let e : (ℝ × ℝ) ≃L[ℝ] ℝ² := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
   calc
-    (∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
+    ∫ x in Icc a b, f' x (1, 0) + g' x (0, 1) =
         ∑ i : Fin 2,
-          (∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
+          ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
               ![f, g] i (e.symm <| i.insertNth (e b i) x)) -
             ∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
-              ![f, g] i (e.symm <| i.insertNth (e a i) x) :=
+              ![f, g] i (e.symm <| i.insertNth (e a i) x)) :=
       by
       refine'
         integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv e _ _ ![f, g] ![f', g'] s
@@ -503,7 +503,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
           ((∫ x in Icc a.1 b.1, g (x, b.2)) - ∫ x in Icc a.1 b.1, g (x, a.2)) :=
       by
       have :
-        ∀ (a b : ℝ¹) (f : ℝ¹ → E), (∫ x in Icc a b, f x) = ∫ x in Icc (a 0) (b 0), f fun _ => x :=
+        ∀ (a b : ℝ¹) (f : ℝ¹ → E), ∫ x in Icc a b, f x = ∫ x in Icc (a 0) (b 0), f fun _ => x :=
         by
         intro a b f
         convert
@@ -545,7 +545,7 @@ theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ
     (Hdg :
       ∀ x ∈ Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x)
     (Hi : IntegrableOn (fun x => f' x (1, 0) + g' x (0, 1)) ([a₁, b₁] ×ˢ [a₂, b₂])) :
-    (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =
+    ∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) =
       (((∫ x in a₁..b₁, g (x, b₂)) - ∫ x in a₁..b₁, g (x, a₂)) + ∫ y in a₂..b₂, f (b₁, y)) -
         ∫ y in a₂..b₂, f (a₁, y) :=
   by
@@ -561,7 +561,7 @@ theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ
     refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₂))).trans _; abel
   simp only [uIcc_of_le h₁, uIcc_of_le h₂, min_eq_left, max_eq_right, h₁, h₂] at Hcf Hcg Hdf Hdg Hi 
   calc
-    (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =
+    ∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) =
         ∫ x in Icc a₁ b₁, ∫ y in Icc a₂ b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) :=
       by
       simp only [intervalIntegral.integral_of_le, h₁, h₂, set_integral_congr_set_ae Ioc_ae_eq_Icc]

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.integral.divergence_theorem
-! leanprover-community/mathlib commit 3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe
+! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.MeasureTheory.Integral.IntervalIntegral
 /-!
 # Divergence theorem for Bochner integral
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove the Divergence theorem for Bochner integral on a box in
 `ℝⁿ⁺¹ = fin (n + 1) → ℝ`. More precisely, we prove the following theorem.
 

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

@@ -243,7 +243,6 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
         _ ≤ ε / ∏ j, (I.face i).upper j - (I.face i).lower j :=
           hδ _ (I.maps_to_insert_nth_face_Icc hd <| Hsub hx) _
             (I.maps_to_insert_nth_face_Icc (hc _) <| Hsub hx) _
-        
       rw [Fin.dist_insertNth_insertNth, dist_self, dist_comm]
       exact max_le hk.le δpos.lt.le
     _ ≤ ε :=
@@ -254,7 +253,6 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
         div_le_div_of_le_left εpos.le (hvol_pos _) (prod_le_prod (fun j hj => _) fun j hj => _)
       exacts [sub_nonneg.2 (box.lower_le_upper _ _),
         sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]
-    
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
 
 variable (a b : ℝⁿ⁺¹)
@@ -374,7 +372,6 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
           interior_pi_set finite_univ, interior_Icc] using hx.1
       · rw [← he_vol.integrable_on_comp_preimage he_emb, hIcc]
         simp [← hDF, (· ∘ ·), Hi]
-    
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv
 
 end
@@ -437,7 +434,6 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
       simp only [Fin.sum_univ_one, e_symm]
       have : ∀ c : ℝ, const (Fin 0) c = isEmptyElim := fun c => Subsingleton.elim _ _
       simp [this, volume_pi, measure.pi_of_empty fun _ : Fin 0 => volume]
-    
 #align measure_theory.integral_eq_of_has_deriv_within_at_off_countable_of_le MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le
 
 /-- **Fundamental theorem of calculus, part 2**. This version assumes that `f` is continuous on the
@@ -521,7 +517,6 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
       simp only [intervalIntegral.integral_of_le hle.1, intervalIntegral.integral_of_le hle.2,
         set_integral_congr_set_ae Ioc_ae_eq_Icc]
       abel
-    
 #align measure_theory.integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -577,7 +572,6 @@ theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ
           integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le f g f' g'
             (a₁, a₂) (b₁, b₂) ⟨h₁, h₂⟩ s <;>
         assumption
-    
 #align measure_theory.integral2_divergence_prod_of_has_fderiv_within_at_off_countable MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable
 
 end MeasureTheory

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

@@ -141,7 +141,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
 /-- An auxiliary lemma for
 `measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable`. Compared to the previous
 lemma, here we drop the assumption of differentiability on the boundary of the box. -/
-theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Box (Fin (n + 1)))
+theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
     (f : ℝⁿ⁺¹ → Eⁿ⁺¹) (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable)
     (Hc : ContinuousOn f I.Icc) (Hd : ∀ x ∈ I.Ioo \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) I.Icc) :
@@ -255,7 +255,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
       exacts [sub_nonneg.2 (box.lower_le_upper _ _),
         sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]
     
-#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂
+#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
 
 variable (a b : ℝⁿ⁺¹)
 
@@ -283,7 +283,7 @@ We represent both faces `x i = a i` and `x i = b i` as the box
 of `f : ℝⁿ⁺¹ → Eⁿ⁺¹` to these faces are given by `f ∘ back_face i` and `f ∘ front_face i`, where
 `back_face i = fin.insert_nth i (a i)` and `front_face i = fin.insert_nth i (b i)` are embeddings
 `ℝⁿ → ℝⁿ⁺¹` that take `y : ℝⁿ` and insert `a i` (resp., `b i`) as `i`-th coordinate. -/
-theorem integral_divergence_of_has_fderiv_within_at_off_countable (hle : a ≤ b) (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
+theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b) (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
     (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
     (Hd : ∀ x ∈ (Set.pi univ fun i => Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) (Icc a b)) :
@@ -308,12 +308,12 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable (hle : a ≤ b
     have hlt : ∀ i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne ⟨i, hi⟩
     convert
       integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ ⟨a, b, hlt⟩ f f' s hs Hc Hd Hi
-#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable
+#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable
 
 /-- **Divergence theorem** for a family of functions `f : fin (n + 1) → ℝⁿ⁺¹ → E`. See also
 `measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable'` for a version formulated
 in terms of a vector-valued function `f : ℝⁿ⁺¹ → Eⁿ⁺¹`. -/
-theorem integral_divergence_of_has_fderiv_within_at_off_countable' (hle : a ≤ b)
+theorem integral_divergence_of_hasFDerivWithinAt_off_countable' (hle : a ≤ b)
     (f : Fin (n + 1) → ℝⁿ⁺¹ → E) (f' : Fin (n + 1) → ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] E) (s : Set ℝⁿ⁺¹)
     (hs : s.Countable) (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
     (Hd : ∀ x ∈ (pi Set.univ fun i => Ioo (a i) (b i)) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x)
@@ -321,16 +321,16 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable' (hle : a ≤
     (∫ x in Icc a b, ∑ i, f' i x (e i)) =
       ∑ i : Fin (n + 1),
         (∫ x in face i, f i ((front_face (i)) x)) - ∫ x in face i, f i ((back_face (i)) x) :=
-  integral_divergence_of_has_fderiv_within_at_off_countable a b hle (fun x i => f i x)
+  integral_divergence_of_hasFDerivWithinAt_off_countable a b hle (fun x i => f i x)
     (fun x => ContinuousLinearMap.pi fun i => f' i x) s hs (continuousOn_pi.2 Hc)
     (fun x hx => hasFDerivAt_pi.2 (Hd x hx)) Hi
-#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable' MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable'
+#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable' MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable'
 
 end
 
 /-- An auxiliary lemma that is used to specialize the general divergence theorem to spaces that do
 not have the form `fin n → ℝ`. -/
-theorem integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv {F : Type _}
+theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Type _}
     [NormedAddCommGroup F] [NormedSpace ℝ F] [PartialOrder F] [MeasureSpace F] [BorelSpace F]
     (eL : F ≃L[ℝ] ℝⁿ⁺¹) (he_ord : ∀ x y, eL x ≤ eL y ↔ x ≤ y)
     (he_vol : MeasurePreserving eL volume volume) (f : Fin (n + 1) → F → E)
@@ -375,7 +375,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv {F :
       · rw [← he_vol.integrable_on_comp_preimage he_emb, hIcc]
         simp [← hDF, (· ∘ ·), Hi]
     
-#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv
+#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv
 
 end
 
@@ -405,8 +405,8 @@ differentiability of `f`;
 
 * `measure_theory.integral_eq_of_has_deriv_within_at_off_countable` for a version that works both
   for `a ≤ b` and `b ≤ a` at the expense of using unordered intervals instead of `set.Icc`. -/
-theorem integral_eq_of_has_deriv_within_at_off_countable_of_le (f f' : ℝ → E) {a b : ℝ}
-    (hle : a ≤ b) {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
+theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {a b : ℝ} (hle : a ≤ b)
+    {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
     (Hd : ∀ x ∈ Ioo a b \ s, HasDerivAt f (f' x) x) (Hi : IntervalIntegrable f' volume a b) :
     (∫ x in a..b, f' x) = f b - f a :=
   by
@@ -438,7 +438,7 @@ theorem integral_eq_of_has_deriv_within_at_off_countable_of_le (f f' : ℝ → E
       have : ∀ c : ℝ, const (Fin 0) c = isEmptyElim := fun c => Subsingleton.elim _ _
       simp [this, volume_pi, measure.pi_of_empty fun _ : Fin 0 => volume]
     
-#align measure_theory.integral_eq_of_has_deriv_within_at_off_countable_of_le MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable_of_le
+#align measure_theory.integral_eq_of_has_deriv_within_at_off_countable_of_le MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le
 
 /-- **Fundamental theorem of calculus, part 2**. This version assumes that `f` is continuous on the
 interval and is differentiable off a countable set `s`.
@@ -471,7 +471,7 @@ boundary.
 See also `measure_theory.integral2_divergence_prod_of_has_fderiv_within_at_off_countable` for a
 version that does not assume `a ≤ b` and uses iterated interval integral instead of the integral
 over `Icc a b`. -/
-theorem integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le (f g : ℝ × ℝ → E)
+theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f g : ℝ × ℝ → E)
     (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a b : ℝ × ℝ) (hle : a ≤ b) (s : Set (ℝ × ℝ)) (hs : s.Countable)
     (Hcf : ContinuousOn f (Icc a b)) (Hcg : ContinuousOn g (Icc a b))
     (Hdf : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x)
@@ -522,7 +522,7 @@ theorem integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le
         set_integral_congr_set_ae Ioc_ae_eq_Icc]
       abel
     
-#align measure_theory.integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le MeasureTheory.integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le
+#align measure_theory.integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -539,7 +539,7 @@ the normal derivative of `F` along the boundary.
 
 See also `measure_theory.integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le`
 for a version that uses an integral over `Icc a b`, where `a b : ℝ × ℝ`, `a ≤ b`. -/
-theorem integral2_divergence_prod_of_has_fderiv_within_at_off_countable (f g : ℝ × ℝ → E)
+theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ × ℝ → E)
     (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a₁ a₂ b₁ b₂ : ℝ) (s : Set (ℝ × ℝ)) (hs : s.Countable)
     (Hcf : ContinuousOn f ([a₁, b₁] ×ˢ [a₂, b₂])) (Hcg : ContinuousOn g ([a₁, b₁] ×ˢ [a₂, b₂]))
     (Hdf :
@@ -578,7 +578,7 @@ theorem integral2_divergence_prod_of_has_fderiv_within_at_off_countable (f g : 
             (a₁, a₂) (b₁, b₂) ⟨h₁, h₂⟩ s <;>
         assumption
     
-#align measure_theory.integral2_divergence_prod_of_has_fderiv_within_at_off_countable MeasureTheory.integral2_divergence_prod_of_has_fderiv_within_at_off_countable
+#align measure_theory.integral2_divergence_prod_of_has_fderiv_within_at_off_countable MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable
 
 end MeasureTheory
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

@@ -122,7 +122,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
           ∫ x in (I.face i).Icc, f (i.insertNth (I.lower i) x) i :=
   by
   simp only [← set_integral_congr_set_ae (box.coe_ae_eq_Icc _)]
-  have A := (Hi.mono_set box.coe_subset_Icc).has_box_integral ⊥ rfl
+  have A := (Hi.mono_set box.coe_subset_Icc).hasBoxIntegral ⊥ rfl
   have B :=
     has_integral_GP_divergence_of_forall_has_deriv_within_at I f f' (s ∩ I.Icc)
       (hs.mono (inter_subset_left _ _)) (fun x hx => Hc _ hx.2) fun x hx =>

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

@@ -122,7 +122,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
           ∫ x in (I.face i).Icc, f (i.insertNth (I.lower i) x) i :=
   by
   simp only [← set_integral_congr_set_ae (box.coe_ae_eq_Icc _)]
-  have A := (Hi.mono_set box.coe_subset_Icc).hasBoxIntegral ⊥ rfl
+  have A := (Hi.mono_set box.coe_subset_Icc).has_box_integral ⊥ rfl
   have B :=
     has_integral_GP_divergence_of_forall_has_deriv_within_at I f f' (s ∩ I.Icc)
       (hs.mono (inter_subset_left _ _)) (fun x hx => Hc _ hx.2) fun x hx =>

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

@@ -306,8 +306,8 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable (hle : a ≤ b
       rw [this, integral_empty, integral_empty, sub_self]
   · -- In the non-trivial case `∀ i, a i < b i`, we apply a lemma we proved above.
     have hlt : ∀ i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne ⟨i, hi⟩
-    convert integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ ⟨a, b, hlt⟩ f f' s hs Hc
-        Hd Hi
+    convert
+      integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ ⟨a, b, hlt⟩ f f' s hs Hc Hd Hi
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable
 
 /-- **Divergence theorem** for a family of functions `f : fin (n + 1) → ℝⁿ⁺¹ → E`. See also
@@ -361,7 +361,8 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv {F :
             ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
               f i (eL.symm <| i.insertNth (eL a i) x) :=
       by
-      convert integral_divergence_of_has_fderiv_within_at_off_countable' (eL a) (eL b)
+      convert
+        integral_divergence_of_has_fderiv_within_at_off_countable' (eL a) (eL b)
           ((he_ord _ _).2 hle) (fun i x => f i (eL.symm x))
           (fun i x => f' i (eL.symm x) ∘L (eL.symm : ℝⁿ⁺¹ →L[ℝ] F)) (eL.symm ⁻¹' s)
           (hs.preimage eL.symm.injective) _ _ _
@@ -506,7 +507,8 @@ theorem integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le
         ∀ (a b : ℝ¹) (f : ℝ¹ → E), (∫ x in Icc a b, f x) = ∫ x in Icc (a 0) (b 0), f fun _ => x :=
         by
         intro a b f
-        convert(((volume_preserving_fun_unique (Fin 1) ℝ).symm _).set_integral_preimage_emb
+        convert
+          (((volume_preserving_fun_unique (Fin 1) ℝ).symm _).set_integral_preimage_emb
               (MeasurableEquiv.measurableEmbedding _) _ _).symm
         exact ((OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc a b).symm
       simp only [Fin.sum_univ_two, this]

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

@@ -127,7 +127,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
     has_integral_GP_divergence_of_forall_has_deriv_within_at I f f' (s ∩ I.Icc)
       (hs.mono (inter_subset_left _ _)) (fun x hx => Hc _ hx.2) fun x hx =>
       Hd _ ⟨hx.1, fun h => hx.2 ⟨h, hx.1⟩⟩
-  rw [continuousOn_pi] at Hc
+  rw [continuousOn_pi] at Hc 
   refine' (A.unique B).trans (sum_congr rfl fun i hi => _)
   refine' congr_arg₂ Sub.sub _ _
   · have := box.continuous_on_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i))
@@ -169,15 +169,15 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
     tendsto (fun k => ∫ x in (J k).Icc, ∑ i, f' x (e i) i) at_top
       (𝓝 (∫ x in I.Icc, ∑ i, f' x (e i) i)) :=
     by
-    simp only [integrable_on, ← measure.restrict_congr_set (box.Ioo_ae_eq_Icc _)] at Hi⊢
-    rw [← box.Union_Ioo_of_tendsto J.monotone hJl hJu] at Hi⊢
+    simp only [integrable_on, ← measure.restrict_congr_set (box.Ioo_ae_eq_Icc _)] at Hi ⊢
+    rw [← box.Union_Ioo_of_tendsto J.monotone hJl hJu] at Hi ⊢
     exact
       tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo) (box.Ioo.comp J).Monotone
         Hi
   -- Thus it suffices to prove the same about the RHS.
   refine' tendsto_nhds_unique_of_eventuallyEq hI_tendsto _ (eventually_of_forall HJ_eq)
   clear hI_tendsto
-  rw [tendsto_pi_nhds] at hJl hJu
+  rw [tendsto_pi_nhds] at hJl hJu 
   /- We'll need to prove a similar statement about the integrals over the front sides and the
     integrals over the back sides. In order to avoid repeating ourselves, we formulate a lemma. -/
   suffices
@@ -187,9 +187,9 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
           tendsto (fun k => ∫ x in ((J k).face i).Icc, f (i.insertNth (c k) x) i) at_top
             (𝓝 <| ∫ x in (I.face i).Icc, f (i.insertNth d x) i)
     by
-    rw [box.Icc_eq_pi] at hJ_sub'
+    rw [box.Icc_eq_pi] at hJ_sub' 
     refine' tendsto_finset_sum _ fun i hi => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))
-    exacts[fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>
+    exacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>
       hJ_sub' k (J k).lower_mem_Icc _ trivial]
   intro i c d hc hcd
   /- First we prove that the integrals of the restriction of `f` to `{x | x i = d}` over increasing
@@ -207,7 +207,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
     have hIoo : (⋃ k, ((J k).face i).Ioo) = (I.face i).Ioo :=
       box.Union_Ioo_of_tendsto ((box.monotone_face i).comp J.monotone)
         (tendsto_pi_nhds.2 fun _ => hJl _) (tendsto_pi_nhds.2 fun _ => hJu _)
-    simp only [integrable_on, ← measure.restrict_congr_set (box.Ioo_ae_eq_Icc _), ← hIoo] at Hid⊢
+    simp only [integrable_on, ← measure.restrict_congr_set (box.Ioo_ae_eq_Icc _), ← hIoo] at Hid ⊢
     exact
       tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)
         (box.Ioo.monotone.comp ((box.monotone_face i).comp J.monotone)) Hid
@@ -252,7 +252,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
         le_div_iff (hvol_pos _)]
       refine'
         div_le_div_of_le_left εpos.le (hvol_pos _) (prod_le_prod (fun j hj => _) fun j hj => _)
-      exacts[sub_nonneg.2 (box.lower_le_upper _ _),
+      exacts [sub_nonneg.2 (box.lower_le_upper _ _),
         sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]
     
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂
@@ -423,14 +423,14 @@ theorem integral_eq_of_has_deriv_within_at_off_countable_of_le (f f' : ℝ → E
             ∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
               f (e.symm <| i.insertNth (e a i) x) :=
       by
-      simp only [← interior_Icc] at Hd
+      simp only [← interior_Icc] at Hd 
       refine'
         integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv e _ _ (fun _ => f)
           (fun _ => F') s hs a b hle (fun i => Hc) (fun x hx i => Hd x hx) _ _ _
       · exact fun x y => (OrderIso.funUnique (Fin 1) ℝ).symm.le_iff_le
       · exact (volume_preserving_fun_unique (Fin 1) ℝ).symm _
       · intro x; rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]
-      · rw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi
+      · rw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi 
         exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm
     _ = f b - f a := by
       simp only [Fin.sum_univ_one, e_symm]
@@ -495,7 +495,7 @@ theorem integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le
       · exact fun x y => (OrderIso.finTwoArrowIso ℝ).symm.le_iff_le
       · exact (volume_preserving_fin_two_arrow ℝ).symm _
       · exact Fin.forall_fin_two.2 ⟨Hcf, Hcg⟩
-      · rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx
+      · rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx 
         exact Fin.forall_fin_two.2 ⟨Hdf x hx, Hdg x hx⟩
       · intro x; rw [Fin.sum_univ_two]; simp
     _ =
@@ -551,15 +551,15 @@ theorem integral2_divergence_prod_of_has_fderiv_within_at_off_countable (f g : 
   by
   wlog h₁ : a₁ ≤ b₁ generalizing a₁ b₁
   · specialize this b₁ a₁
-    rw [uIcc_comm b₁ a₁, min_comm b₁ a₁, max_comm b₁ a₁] at this
+    rw [uIcc_comm b₁ a₁, min_comm b₁ a₁, max_comm b₁ a₁] at this 
     simp only [intervalIntegral.integral_symm b₁ a₁]
     refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₁))).trans _; abel
   wlog h₂ : a₂ ≤ b₂ generalizing a₂ b₂
   · specialize this b₂ a₂
-    rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this
+    rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this 
     simp only [intervalIntegral.integral_symm b₂ a₂, intervalIntegral.integral_neg]
     refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₂))).trans _; abel
-  simp only [uIcc_of_le h₁, uIcc_of_le h₂, min_eq_left, max_eq_right, h₁, h₂] at Hcf Hcg Hdf Hdg Hi
+  simp only [uIcc_of_le h₁, uIcc_of_le h₂, min_eq_left, max_eq_right, h₁, h₂] at Hcf Hcg Hdf Hdg Hi 
   calc
     (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =
         ∫ x in Icc a₁ b₁, ∫ y in Icc a₂ b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) :=

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

@@ -55,7 +55,7 @@ divergence theorem, Bochner integral
 
 open Set Finset TopologicalSpace Function BoxIntegral MeasureTheory Filter
 
-open BigOperators Classical Topology Interval
+open scoped BigOperators Classical Topology Interval
 
 universe u
 
@@ -378,7 +378,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv {F :
 
 end
 
-open Interval
+open scoped Interval
 
 open ContinuousLinearMap (smul_right)
 

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

@@ -132,12 +132,10 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
   refine' congr_arg₂ Sub.sub _ _
   · have := box.continuous_on_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i))
     have := (this.integrable_on_compact (box.is_compact_Icc _)).mono_set box.coe_subset_Icc
-    exact (this.has_box_integral ⊥ rfl).integral_eq
-    infer_instance
+    exact (this.has_box_integral ⊥ rfl).integral_eq; infer_instance
   · have := box.continuous_on_face_Icc (Hc i) (Set.left_mem_Icc.2 (I.lower_le_upper i))
     have := (this.integrable_on_compact (box.is_compact_Icc _)).mono_set box.coe_subset_Icc
-    exact (this.has_box_integral ⊥ rfl).integral_eq
-    infer_instance
+    exact (this.has_box_integral ⊥ rfl).integral_eq; infer_instance
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₁ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁
 
 /-- An auxiliary lemma for
@@ -347,9 +345,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv {F :
           ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
             f i (eL.symm <| i.insertNth (eL a i) x) :=
   have he_emb : MeasurableEmbedding eL := eL.toHomeomorph.toMeasurableEquiv.MeasurableEmbedding
-  have hIcc : eL ⁻¹' Icc (eL a) (eL b) = Icc a b :=
-    by
-    ext1 x
+  have hIcc : eL ⁻¹' Icc (eL a) (eL b) = Icc a b := by ext1 x;
     simp only [Set.mem_preimage, Set.mem_Icc, he_ord]
   have hIcc' : Icc (eL a) (eL b) = eL.symm ⁻¹' Icc a b := by rw [← hIcc, eL.symm_preimage_preimage]
   calc
@@ -433,8 +429,7 @@ theorem integral_eq_of_has_deriv_within_at_off_countable_of_le (f f' : ℝ → E
           (fun _ => F') s hs a b hle (fun i => Hc) (fun x hx i => Hd x hx) _ _ _
       · exact fun x y => (OrderIso.funUnique (Fin 1) ℝ).symm.le_iff_le
       · exact (volume_preserving_fun_unique (Fin 1) ℝ).symm _
-      · intro x
-        rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]
+      · intro x; rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]
       · rw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi
         exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm
     _ = f b - f a := by
@@ -502,9 +497,7 @@ theorem integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le
       · exact Fin.forall_fin_two.2 ⟨Hcf, Hcg⟩
       · rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx
         exact Fin.forall_fin_two.2 ⟨Hdf x hx, Hdg x hx⟩
-      · intro x
-        rw [Fin.sum_univ_two]
-        simp
+      · intro x; rw [Fin.sum_univ_two]; simp
     _ =
         ((∫ y in Icc a.2 b.2, f (b.1, y)) - ∫ y in Icc a.2 b.2, f (a.1, y)) +
           ((∫ x in Icc a.1 b.1, g (x, b.2)) - ∫ x in Icc a.1 b.1, g (x, a.2)) :=
@@ -560,14 +553,12 @@ theorem integral2_divergence_prod_of_has_fderiv_within_at_off_countable (f g : 
   · specialize this b₁ a₁
     rw [uIcc_comm b₁ a₁, min_comm b₁ a₁, max_comm b₁ a₁] at this
     simp only [intervalIntegral.integral_symm b₁ a₁]
-    refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₁))).trans _
-    abel
+    refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₁))).trans _; abel
   wlog h₂ : a₂ ≤ b₂ generalizing a₂ b₂
   · specialize this b₂ a₂
     rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this
     simp only [intervalIntegral.integral_symm b₂ a₂, intervalIntegral.integral_neg]
-    refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₂))).trans _
-    abel
+    refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h₂))).trans _; abel
   simp only [uIcc_of_le h₁, uIcc_of_le h₂, min_eq_left, max_eq_right, h₁, h₂] at Hcf Hcg Hdf Hdg Hi
   calc
     (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =

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

@@ -4,13 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.integral.divergence_theorem
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
+! 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.BoxIntegral.DivergenceTheorem
 import Mathbin.Analysis.BoxIntegral.Integrability
-import Mathbin.Analysis.Calculus.Deriv
+import Mathbin.Analysis.Calculus.Deriv.Basic
 import Mathbin.MeasureTheory.Constructions.Prod.Integral
 import Mathbin.MeasureTheory.Integral.IntervalIntegral
 

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

@@ -112,9 +112,9 @@ in several aspects.
 `measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable`. This is exactly
 `box_integral.has_integral_GP_divergence_of_forall_has_deriv_within_at` reformulated for the
 Bochner integral. -/
-theorem integral_divergence_of_hasFderivWithinAt_off_countable_aux₁ (I : Box (Fin (n + 1)))
+theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (Fin (n + 1)))
     (f : ℝⁿ⁺¹ → Eⁿ⁺¹) (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable)
-    (Hc : ContinuousOn f I.Icc) (Hd : ∀ x ∈ I.Icc \ s, HasFderivWithinAt f (f' x) I.Icc x)
+    (Hc : ContinuousOn f I.Icc) (Hd : ∀ x ∈ I.Icc \ s, HasFDerivWithinAt f (f' x) I.Icc x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) I.Icc) :
     (∫ x in I.Icc, ∑ i, f' x (e i) i) =
       ∑ i : Fin (n + 1),
@@ -138,14 +138,14 @@ theorem integral_divergence_of_hasFderivWithinAt_off_countable_aux₁ (I : Box (
     have := (this.integrable_on_compact (box.is_compact_Icc _)).mono_set box.coe_subset_Icc
     exact (this.has_box_integral ⊥ rfl).integral_eq
     infer_instance
-#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₁ MeasureTheory.integral_divergence_of_hasFderivWithinAt_off_countable_aux₁
+#align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₁ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁
 
 /-- An auxiliary lemma for
 `measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable`. Compared to the previous
 lemma, here we drop the assumption of differentiability on the boundary of the box. -/
 theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Box (Fin (n + 1)))
     (f : ℝⁿ⁺¹ → Eⁿ⁺¹) (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable)
-    (Hc : ContinuousOn f I.Icc) (Hd : ∀ x ∈ I.Ioo \ s, HasFderivAt f (f' x) x)
+    (Hc : ContinuousOn f I.Icc) (Hd : ∀ x ∈ I.Ioo \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) I.Icc) :
     (∫ x in I.Icc, ∑ i, f' x (e i) i) =
       ∑ i : Fin (n + 1),
@@ -158,8 +158,8 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
   have hJ_sub' : ∀ k, (J k).Icc ⊆ I.Icc := fun k => (hJ_sub k).trans I.Ioo_subset_Icc
   have hJ_le : ∀ k, J k ≤ I := fun k => box.le_iff_Icc.2 (hJ_sub' k)
   have HcJ : ∀ k, ContinuousOn f (J k).Icc := fun k => Hc.mono (hJ_sub' k)
-  have HdJ : ∀ (k), ∀ x ∈ (J k).Icc \ s, HasFderivWithinAt f (f' x) (J k).Icc x := fun k x hx =>
-    (Hd x ⟨hJ_sub k hx.1, hx.2⟩).HasFderivWithinAt
+  have HdJ : ∀ (k), ∀ x ∈ (J k).Icc \ s, HasFDerivWithinAt f (f' x) (J k).Icc x := fun k x hx =>
+    (Hd x ⟨hJ_sub k hx.1, hx.2⟩).HasFDerivWithinAt
   have HiJ : ∀ k, integrable_on (fun x => ∑ i, f' x (e i) i) (J k).Icc := fun k =>
     Hi.mono_set (hJ_sub' k)
   -- Apply the previous lemma to `J k`.
@@ -287,7 +287,7 @@ of `f : ℝⁿ⁺¹ → Eⁿ⁺¹` to these faces are given by `f ∘ back_face
 `ℝⁿ → ℝⁿ⁺¹` that take `y : ℝⁿ` and insert `a i` (resp., `b i`) as `i`-th coordinate. -/
 theorem integral_divergence_of_has_fderiv_within_at_off_countable (hle : a ≤ b) (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
     (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
-    (Hd : ∀ x ∈ (Set.pi univ fun i => Ioo (a i) (b i)) \ s, HasFderivAt f (f' x) x)
+    (Hd : ∀ x ∈ (Set.pi univ fun i => Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) (Icc a b)) :
     (∫ x in Icc a b, ∑ i, f' x (e i) i) =
       ∑ i : Fin (n + 1),
@@ -318,14 +318,14 @@ in terms of a vector-valued function `f : ℝⁿ⁺¹ → Eⁿ⁺¹`. -/
 theorem integral_divergence_of_has_fderiv_within_at_off_countable' (hle : a ≤ b)
     (f : Fin (n + 1) → ℝⁿ⁺¹ → E) (f' : Fin (n + 1) → ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] E) (s : Set ℝⁿ⁺¹)
     (hs : s.Countable) (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
-    (Hd : ∀ x ∈ (pi Set.univ fun i => Ioo (a i) (b i)) \ s, ∀ (i), HasFderivAt (f i) (f' i x) x)
+    (Hd : ∀ x ∈ (pi Set.univ fun i => Ioo (a i) (b i)) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' i x (e i)) (Icc a b)) :
     (∫ x in Icc a b, ∑ i, f' i x (e i)) =
       ∑ i : Fin (n + 1),
         (∫ x in face i, f i ((front_face (i)) x)) - ∫ x in face i, f i ((back_face (i)) x) :=
   integral_divergence_of_has_fderiv_within_at_off_countable a b hle (fun x i => f i x)
     (fun x => ContinuousLinearMap.pi fun i => f' i x) s hs (continuousOn_pi.2 Hc)
-    (fun x hx => hasFderivAt_pi.2 (Hd x hx)) Hi
+    (fun x hx => hasFDerivAt_pi.2 (Hd x hx)) Hi
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable' MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable'
 
 end
@@ -338,7 +338,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv {F :
     (he_vol : MeasurePreserving eL volume volume) (f : Fin (n + 1) → F → E)
     (f' : Fin (n + 1) → F → F →L[ℝ] E) (s : Set F) (hs : s.Countable) (a b : F) (hle : a ≤ b)
     (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
-    (Hd : ∀ x ∈ interior (Icc a b) \ s, ∀ (i), HasFderivAt (f i) (f' i x) x) (DF : F → E)
+    (Hd : ∀ x ∈ interior (Icc a b) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x) (DF : F → E)
     (hDF : ∀ x, DF x = ∑ i, f' i x (eL.symm <| e i)) (Hi : IntegrableOn DF (Icc a b)) :
     (∫ x in Icc a b, DF x) =
       ∑ i : Fin (n + 1),
@@ -478,8 +478,8 @@ over `Icc a b`. -/
 theorem integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le (f g : ℝ × ℝ → E)
     (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a b : ℝ × ℝ) (hle : a ≤ b) (s : Set (ℝ × ℝ)) (hs : s.Countable)
     (Hcf : ContinuousOn f (Icc a b)) (Hcg : ContinuousOn g (Icc a b))
-    (Hdf : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFderivAt f (f' x) x)
-    (Hdg : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFderivAt g (g' x) x)
+    (Hdf : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x)
+    (Hdg : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x)
     (Hi : IntegrableOn (fun x => f' x (1, 0) + g' x (0, 1)) (Icc a b)) :
     (∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
       (((∫ x in a.1 ..b.1, g (x, b.2)) - ∫ x in a.1 ..b.1, g (x, a.2)) +
@@ -548,9 +548,9 @@ theorem integral2_divergence_prod_of_has_fderiv_within_at_off_countable (f g : 
     (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a₁ a₂ b₁ b₂ : ℝ) (s : Set (ℝ × ℝ)) (hs : s.Countable)
     (Hcf : ContinuousOn f ([a₁, b₁] ×ˢ [a₂, b₂])) (Hcg : ContinuousOn g ([a₁, b₁] ×ˢ [a₂, b₂]))
     (Hdf :
-      ∀ x ∈ Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFderivAt f (f' x) x)
+      ∀ x ∈ Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x)
     (Hdg :
-      ∀ x ∈ Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFderivAt g (g' x) x)
+      ∀ x ∈ Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x)
     (Hi : IntegrableOn (fun x => f' x (1, 0) + g' x (0, 1)) ([a₁, b₁] ×ˢ [a₂, b₂])) :
     (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =
       (((∫ x in a₁..b₁, g (x, b₂)) - ∫ x in a₁..b₁, g (x, a₂)) + ∫ y in a₂..b₂, f (b₁, y)) -

mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed

@@ -4,13 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.integral.divergence_theorem
-! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
+! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.BoxIntegral.DivergenceTheorem
 import Mathbin.Analysis.BoxIntegral.Integrability
 import Mathbin.Analysis.Calculus.Deriv
+import Mathbin.MeasureTheory.Constructions.Prod.Integral
 import Mathbin.MeasureTheory.Integral.IntervalIntegral
 
 /-!

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.integral.divergence_theorem
-! leanprover-community/mathlib commit 2196ab363eb097c008d4497125e0dde23fb36db2
+! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.BoxIntegral.DivergenceTheorem
 import Mathbin.Analysis.BoxIntegral.Integrability
+import Mathbin.Analysis.Calculus.Deriv
 import Mathbin.MeasureTheory.Integral.IntervalIntegral
 
 /-!

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -129,11 +129,11 @@ theorem integral_divergence_of_hasFderivWithinAt_off_countable_aux₁ (I : Box (
   refine' (A.unique B).trans (sum_congr rfl fun i hi => _)
   refine' congr_arg₂ Sub.sub _ _
   · have := box.continuous_on_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i))
-    have := (this.integrable_on_compact (box.is_compact_Icc _)).monoSet box.coe_subset_Icc
+    have := (this.integrable_on_compact (box.is_compact_Icc _)).mono_set box.coe_subset_Icc
     exact (this.has_box_integral ⊥ rfl).integral_eq
     infer_instance
   · have := box.continuous_on_face_Icc (Hc i) (Set.left_mem_Icc.2 (I.lower_le_upper i))
-    have := (this.integrable_on_compact (box.is_compact_Icc _)).monoSet box.coe_subset_Icc
+    have := (this.integrable_on_compact (box.is_compact_Icc _)).mono_set box.coe_subset_Icc
     exact (this.has_box_integral ⊥ rfl).integral_eq
     infer_instance
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₁ MeasureTheory.integral_divergence_of_hasFderivWithinAt_off_countable_aux₁
@@ -197,9 +197,9 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
   have hd : d ∈ Icc (I.lower i) (I.upper i) :=
     is_closed_Icc.mem_of_tendsto hcd (eventually_of_forall hc)
   have Hic : ∀ k, integrable_on (fun x => f (i.insert_nth (c k) x) i) (I.face i).Icc := fun k =>
-    (box.continuous_on_face_Icc ((continuous_apply i).comp_continuousOn Hc) (hc k)).integrableOnIcc
+    (box.continuous_on_face_Icc ((continuous_apply i).comp_continuousOn Hc) (hc k)).integrableOn_Icc
   have Hid : integrable_on (fun x => f (i.insert_nth d x) i) (I.face i).Icc :=
-    (box.continuous_on_face_Icc ((continuous_apply i).comp_continuousOn Hc) hd).integrableOnIcc
+    (box.continuous_on_face_Icc ((continuous_apply i).comp_continuousOn Hc) hd).integrableOn_Icc
   have H :
     tendsto (fun k => ∫ x in ((J k).face i).Icc, f (i.insert_nth d x) i) at_top
       (𝓝 <| ∫ x in (I.face i).Icc, f (i.insert_nth d x) i) :=
@@ -227,7 +227,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
   refine' (hcd.eventually (Metric.ball_mem_nhds _ δpos)).mono fun k hk => _
   have Hsub : ((J k).face i).Icc ⊆ (I.face i).Icc := box.le_iff_Icc.1 (box.face_mono (hJ_le _) i)
   rw [mem_closedBall_zero_iff, Real.norm_eq_abs, abs_of_nonneg dist_nonneg, dist_eq_norm, ←
-    integral_sub (Hid.mono_set Hsub) ((Hic _).monoSet Hsub)]
+    integral_sub (Hid.mono_set Hsub) ((Hic _).mono_set Hsub)]
   calc
     ‖∫ x in ((J k).face i).Icc, f (i.insert_nth d x) i - f (i.insert_nth (c k) x) i‖ ≤
         (ε / ∏ j, (I.face i).upper j - (I.face i).lower j) * (volume ((J k).face i).Icc).toReal :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/730c6d4cab72b9d84fcfb9e95e8796e9cd8f40ba

@@ -189,8 +189,8 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
     by
     rw [box.Icc_eq_pi] at hJ_sub'
     refine' tendsto_finset_sum _ fun i hi => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))
-    exacts[fun k => hJ_sub' k (J k).upper_mem_icc _ trivial, fun k =>
-      hJ_sub' k (J k).lower_mem_icc _ trivial]
+    exacts[fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>
+      hJ_sub' k (J k).lower_mem_Icc _ trivial]
   intro i c d hc hcd
   /- First we prove that the integrals of the restriction of `f` to `{x | x i = d}` over increasing
     boxes `((J k).face i).Icc` tend to the desired limit. The proof mostly repeats the one above. -/
@@ -253,7 +253,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ (I : Bo
       refine'
         div_le_div_of_le_left εpos.le (hvol_pos _) (prod_le_prod (fun j hj => _) fun j hj => _)
       exacts[sub_nonneg.2 (box.lower_le_upper _ _),
-        sub_le_sub ((hJ_sub' _ (J _).upper_mem_icc).2 _) ((hJ_sub' _ (J _).lower_mem_icc).1 _)]
+        sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]
     
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂
 

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

@@ -306,8 +306,8 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable (hle : a ≤ b
       rw [this, integral_empty, integral_empty, sub_self]
   · -- In the non-trivial case `∀ i, a i < b i`, we apply a lemma we proved above.
     have hlt : ∀ i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne ⟨i, hi⟩
-    convert
-      integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ ⟨a, b, hlt⟩ f f' s hs Hc Hd Hi
+    convert integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ ⟨a, b, hlt⟩ f f' s hs Hc
+        Hd Hi
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable MeasureTheory.integral_divergence_of_has_fderiv_within_at_off_countable
 
 /-- **Divergence theorem** for a family of functions `f : fin (n + 1) → ℝⁿ⁺¹ → E`. See also
@@ -363,8 +363,7 @@ theorem integral_divergence_of_has_fderiv_within_at_off_countable_of_equiv {F :
             ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
               f i (eL.symm <| i.insertNth (eL a i) x) :=
       by
-      convert
-        integral_divergence_of_has_fderiv_within_at_off_countable' (eL a) (eL b)
+      convert integral_divergence_of_has_fderiv_within_at_off_countable' (eL a) (eL b)
           ((he_ord _ _).2 hle) (fun i x => f i (eL.symm x))
           (fun i x => f' i (eL.symm x) ∘L (eL.symm : ℝⁿ⁺¹ →L[ℝ] F)) (eL.symm ⁻¹' s)
           (hs.preimage eL.symm.injective) _ _ _
@@ -512,8 +511,7 @@ theorem integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le
         ∀ (a b : ℝ¹) (f : ℝ¹ → E), (∫ x in Icc a b, f x) = ∫ x in Icc (a 0) (b 0), f fun _ => x :=
         by
         intro a b f
-        convert
-          (((volume_preserving_fun_unique (Fin 1) ℝ).symm _).set_integral_preimage_emb
+        convert(((volume_preserving_fun_unique (Fin 1) ℝ).symm _).set_integral_preimage_emb
               (MeasurableEquiv.measurableEmbedding _) _ _).symm
         exact ((OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc a b).symm
       simp only [Fin.sum_univ_two, this]

mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module measure_theory.integral.divergence_theorem
-! leanprover-community/mathlib commit 92ca63f0fb391a9ca5f22d2409a6080e786d99f7
+! leanprover-community/mathlib commit 2196ab363eb097c008d4497125e0dde23fb36db2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -458,7 +458,7 @@ theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b
   · simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
     exact integral_eq_of_has_deriv_within_at_off_countable_of_le f f' hab hs Hc Hd Hi
   · simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *
-    rw [intervalIntegral.integral_symm, neg_eq_iff_neg_eq, neg_sub, eq_comm]
+    rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub]
     exact integral_eq_of_has_deriv_within_at_off_countable_of_le f f' hab hs Hc Hd Hi.symm
 #align measure_theory.integral_eq_of_has_deriv_within_at_off_countable MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable
 

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

Done with a global search and replace, and then (to fix the #align lines), replace (#align \S*)setIntegral with $1set_integral.

@@ -118,7 +118,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
       ∑ i : Fin (n + 1),
         ((∫ x in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
           ∫ x in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i) := by
-  simp only [← set_integral_congr_set_ae (Box.coe_ae_eq_Icc _)]
+  simp only [← setIntegral_congr_set_ae (Box.coe_ae_eq_Icc _)]
   have A := (Hi.mono_set Box.coe_subset_Icc).hasBoxIntegral ⊥ rfl
   have B :=
     hasIntegral_GP_divergence_of_forall_hasDerivWithinAt I f f' (s ∩ Box.Icc I)
@@ -170,7 +170,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
       (𝓝 (∫ x in Box.Icc I, ∑ i, f' x (e i) i)) := by
     simp only [IntegrableOn, ← Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi ⊢
     rw [← Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi ⊢
-    exact tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo)
+    exact tendsto_setIntegral_of_monotone (fun k => (J k).measurableSet_Ioo)
       (Box.Ioo.comp J).monotone Hi
   -- Thus it suffices to prove the same about the RHS.
   refine' tendsto_nhds_unique_of_eventuallyEq hI_tendsto _ (eventually_of_forall HJ_eq)
@@ -202,7 +202,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
       Box.iUnion_Ioo_of_tendsto ((Box.monotone_face i).comp J.monotone)
         (tendsto_pi_nhds.2 fun _ => hJl _) (tendsto_pi_nhds.2 fun _ => hJu _)
     simp only [IntegrableOn, ← Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _), ← hIoo] at Hid ⊢
-    exact tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)
+    exact tendsto_setIntegral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)
       (Box.Ioo.monotone.comp ((Box.monotone_face i).comp J.monotone)) Hid
   /- Thus it suffices to show that the distance between the integrals of the restrictions of `f` to
     `{x | x i = c k}` and `{x | x i = d}` over `((J k).face i).Icc` tends to zero as `k → ∞`. Choose
@@ -225,7 +225,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
     ‖∫ x in Box.Icc ((J k).face i), f (i.insertNth d x) i - f (i.insertNth (c k) x) i‖ ≤
         (ε / ∏ j, ((I.face i).upper j - (I.face i).lower j)) *
           (volume (Box.Icc ((J k).face i))).toReal := by
-      refine norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)
+      refine norm_setIntegral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)
         ((J k).face i).measurableSet_Icc fun x hx => ?_
       rw [← dist_eq_norm]
       calc
@@ -277,7 +277,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b)
         ∫ x in face i, f (backFace i x) i) := by
   rcases em (∃ i, a i = b i) with (⟨i, hi⟩ | hne)
   · -- First we sort out the trivial case `∃ i, a i = b i`.
-    rw [volume_pi, ← set_integral_congr_set_ae Measure.univ_pi_Ioc_ae_eq_Icc]
+    rw [volume_pi, ← setIntegral_congr_set_ae Measure.univ_pi_Ioc_ae_eq_Icc]
     have hi' : Ioc (a i) (b i) = ∅ := Ioc_eq_empty hi.not_lt
     have : (pi Set.univ fun j => Ioc (a j) (b j)) = ∅ := univ_pi_eq_empty hi'
     rw [this, integral_empty, sum_eq_zero]
@@ -288,7 +288,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b)
       have : Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] (∅ : Set ℝⁿ) := by
         rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]
         simp [hi]
-      rw [set_integral_congr_set_ae this, set_integral_congr_set_ae this, integral_empty,
+      rw [setIntegral_congr_set_ae this, setIntegral_congr_set_ae this, integral_empty,
         integral_empty, sub_self]
   · -- In the non-trivial case `∀ i, a i < b i`, we apply a lemma we proved above.
     have hlt : ∀ i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne ⟨i, hi⟩
@@ -338,7 +338,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
   calc
     ∫ x in Icc a b, DF x = ∫ x in Icc a b, ∑ i, f' i x (eL.symm <| e i) := by simp only [hDF]
     _ = ∫ x in Icc (eL a) (eL b), ∑ i, f' i (eL.symm x) (eL.symm <| e i) := by
-      rw [← he_vol.set_integral_preimage_emb he_emb]
+      rw [← he_vol.setIntegral_preimage_emb he_emb]
       simp only [hIcc, eL.symm_apply_apply]
     _ = ∑ i : Fin (n + 1),
           ((∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
@@ -389,7 +389,7 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
   have hF' : ∀ x y, F' x y = y • f' x := fun x y => rfl
   calc
     ∫ x in a..b, f' x = ∫ x in Icc a b, f' x := by
-      rw [intervalIntegral.integral_of_le hle, set_integral_congr_set_ae Ioc_ae_eq_Icc]
+      rw [intervalIntegral.integral_of_le hle, setIntegral_congr_set_ae Ioc_ae_eq_Icc]
     _ = ∑ i : Fin 1,
           ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
               f (e.symm <| i.insertNth (e b i) x)) -
@@ -468,7 +468,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
           ((∫ x in Icc a.1 b.1, g (x, b.2)) - ∫ x in Icc a.1 b.1, g (x, a.2)) := by
       have : ∀ (a b : ℝ¹) (f : ℝ¹ → E),
           ∫ x in Icc a b, f x = ∫ x in Icc (a 0) (b 0), f fun _ => x := fun a b f ↦ by
-        convert (((volume_preserving_funUnique (Fin 1) ℝ).symm _).set_integral_preimage_emb
+        convert (((volume_preserving_funUnique (Fin 1) ℝ).symm _).setIntegral_preimage_emb
           (MeasurableEquiv.measurableEmbedding _) f _).symm
         exact ((OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc a b).symm
       simp only [Fin.sum_univ_two, this]
@@ -476,7 +476,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
     _ = (((∫ x in a.1..b.1, g (x, b.2)) - ∫ x in a.1..b.1, g (x, a.2)) +
             ∫ y in a.2..b.2, f (b.1, y)) - ∫ y in a.2..b.2, f (a.1, y) := by
       simp only [intervalIntegral.integral_of_le hle.1, intervalIntegral.integral_of_le hle.2,
-        set_integral_congr_set_ae (Ioc_ae_eq_Icc (α := ℝ) (μ := volume))]
+        setIntegral_congr_set_ae (Ioc_ae_eq_Icc (α := ℝ) (μ := volume))]
       abel
 #align measure_theory.integral_divergence_prod_Icc_of_has_fderiv_within_at_off_countable_of_le MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le
 
@@ -516,8 +516,8 @@ theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ
     (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =
         ∫ x in Icc a₁ b₁, ∫ y in Icc a₂ b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1) := by
       simp only [intervalIntegral.integral_of_le, h₁, h₂,
-        set_integral_congr_set_ae (Ioc_ae_eq_Icc (α := ℝ) (μ := volume))]
-    _ = ∫ x in Icc a₁ b₁ ×ˢ Icc a₂ b₂, f' x (1, 0) + g' x (0, 1) := (set_integral_prod _ Hi).symm
+        setIntegral_congr_set_ae (Ioc_ae_eq_Icc (α := ℝ) (μ := volume))]
+    _ = ∫ x in Icc a₁ b₁ ×ˢ Icc a₂ b₂, f' x (1, 0) + g' x (0, 1) := (setIntegral_prod _ Hi).symm
     _ = (((∫ x in a₁..b₁, g (x, b₂)) - ∫ x in a₁..b₁, g (x, a₂)) + ∫ y in a₂..b₂, f (b₁, y)) -
           ∫ y in a₂..b₂, f (a₁, y) := by
       rw [Icc_prod_Icc] at *

@@ -374,7 +374,7 @@ interval and is differentiable off a countable set `s`.
 
 See also
 
-* `interval_integral.integral_eq_sub_of_has_deriv_right_of_le` for a version that only assumes right
+* `intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le` for a version that only assumes right
 differentiability of `f`;
 
 * `MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable` for a version that works both
@@ -413,7 +413,7 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
 /-- **Fundamental theorem of calculus, part 2**. This version assumes that `f` is continuous on the
 interval and is differentiable off a countable set `s`.
 
-See also `measure_theory.interval_integral.integral_eq_sub_of_has_deriv_right` for a version that
+See also `intervalIntegral.integral_eq_sub_of_hasDeriv_right` for a version that
 only assumes right differentiability of `f`.
 -/
 theorem integral_eq_of_hasDerivWithinAt_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}

The new names and argument orders match the corresponding * lemmas, which I already took care of in a previous PR.

From LeanAPAP

@@ -240,10 +240,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
     _ ≤ ε := by
       rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper,
         ← le_div_iff (hvol_pos _)]
-      refine' div_le_div_of_le_left εpos.le (hvol_pos _)
-        (prod_le_prod (fun j _ => _) fun j _ => _)
-      exacts [sub_nonneg.2 (Box.lower_le_upper _ _),
-        sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]
+      gcongr
+      exacts [hvol_pos _, fun _ _ ↦ sub_nonneg.2 (Box.lower_le_upper _ _),
+        (hJ_sub' _ (J _).upper_mem_Icc).2 _, (hJ_sub' _ (J _).lower_mem_Icc).1 _]
 #align measure_theory.integral_divergence_of_has_fderiv_within_at_off_countable_aux₂ MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
 
 variable (a b : Fin (n + 1) → ℝ)

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -406,7 +406,7 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
       · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hle] at Hi
         exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm
     _ = f b - f a := by
-      simp only [Fin.sum_univ_one, e_symm]
+      simp only [e, Fin.sum_univ_one, e_symm]
       have : ∀ c : ℝ, const (Fin 0) c = isEmptyElim := fun c => Subsingleton.elim _ _
       simp [this, volume_pi, Measure.pi_of_empty fun _ : Fin 0 => volume]
 #align measure_theory.integral_eq_of_has_deriv_within_at_off_countable_of_le MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le

Use Lean 4 syntax fun x ↦ instead, matching the style guide. This is close to exhaustive for doc comments; mathlib has about 460 remaining uses of λ (not all in Lean 3 syntax).

@@ -20,9 +20,10 @@ In this file we prove the Divergence theorem for Bochner integral on a box in
 
 Let `E` be a complete normed space. If `f : ℝⁿ⁺¹ → Eⁿ⁺¹` is
 continuous on a rectangular box `[a, b] : Set ℝⁿ⁺¹`, `a ≤ b`, differentiable on its interior with
-derivative `f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹`, and the divergence `λ x, ∑ i, f' x eᵢ i` is integrable on
-`[a, b]`, where `eᵢ = Pi.single i 1` is the `i`-th basis vector, then its integral is equal to the
-sum of integrals of `f` over the faces of `[a, b]`, taken with appropriate signs. Moreover, the same
+derivative `f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹`, and the divergence `fun x ↦ ∑ i, f' x eᵢ i`
+is integrable on `[a, b]`, where `eᵢ = Pi.single i 1` is the `i`-th basis vector,
+then its integral is equal to the sum of integrals of `f` over the faces of `[a, b]`,
+taken with appropriate signs. Moreover, the same
 is true if the function is not differentiable at countably many points of the interior of `[a, b]`.
 
 Once we prove the general theorem, we deduce corollaries for functions `ℝ → E` and pairs of

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>

@@ -285,8 +285,8 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b)
     rcases eq_or_ne i j with (rfl | hne)
     · simp [hi]
     · rcases Fin.exists_succAbove_eq hne with ⟨i, rfl⟩
-      have : Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] (∅ : Set ℝⁿ)
-      · rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]
+      have : Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] (∅ : Set ℝⁿ) := by
+        rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]
         simp [hi]
       rw [set_integral_congr_set_ae this, set_integral_congr_set_ae this, integral_empty,
         integral_empty, sub_self]

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -8,6 +8,7 @@ import Mathlib.Analysis.BoxIntegral.Integrability
 import Mathlib.Analysis.Calculus.Deriv.Basic
 import Mathlib.MeasureTheory.Constructions.Prod.Integral
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
+import Mathlib.Analysis.Calculus.FDeriv.Equiv
 
 #align_import measure_theory.integral.divergence_theorem from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
 

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

@@ -376,7 +376,7 @@ See also
 * `interval_integral.integral_eq_sub_of_has_deriv_right_of_le` for a version that only assumes right
 differentiability of `f`;
 
-* `MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable` for a version that works both
+* `MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable` for a version that works both
   for `a ≤ b` and `b ≤ a` at the expense of using unordered intervals instead of `Set.Icc`. -/
 theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {a b : ℝ}
     (hle : a ≤ b) {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
@@ -415,7 +415,7 @@ interval and is differentiable off a countable set `s`.
 See also `measure_theory.interval_integral.integral_eq_sub_of_has_deriv_right` for a version that
 only assumes right differentiability of `f`.
 -/
-theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}
+theorem integral_eq_of_hasDerivWithinAt_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}
     (hs : s.Countable) (Hc : ContinuousOn f [[a, b]])
     (Hd : ∀ x ∈ Ioo (min a b) (max a b) \ s, HasDerivAt f (f' x) x)
     (Hi : IntervalIntegrable f' volume a b) : ∫ x in a..b, f' x = f b - f a := by
@@ -425,7 +425,7 @@ theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b
   · simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *
     rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub]
     exact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi.symm
-#align measure_theory.integral_eq_of_has_deriv_within_at_off_countable MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable
+#align measure_theory.integral_eq_of_has_deriv_within_at_off_countable MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable
 
 /-- **Divergence theorem** for functions on the plane along rectangles. It is formulated in terms of
 two functions `f g : ℝ × ℝ → E` and an integral over `Icc a b = [a.1, b.1] × [a.2, b.2]`, where

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -419,7 +419,7 @@ theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b
     (hs : s.Countable) (Hc : ContinuousOn f [[a, b]])
     (Hd : ∀ x ∈ Ioo (min a b) (max a b) \ s, HasDerivAt f (f' x) x)
     (Hi : IntervalIntegrable f' volume a b) : ∫ x in a..b, f' x = f b - f a := by
-  cases' le_total a b with hab hab
+  rcases le_total a b with hab | hab
   · simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
     exact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi
   · simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *

We have in the library the lemma not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter, saying that if a function tends to infinity at a point in an interval [a, b], then its derivative is not interval-integrable on [a, b]. We generalize this result to allow for any set instead of [a, b], and apply it to half-infinite intervals.

In particular, we characterize integrability of x^s on [a, +oo), and deduce that x^s is never integrable on [0, +oo). This makes it possible to remove one assumption in Lemma mellin_comp_rpow on the Mellin transform.

@@ -401,7 +401,7 @@ theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {
       · exact fun x y => (OrderIso.funUnique (Fin 1) ℝ).symm.le_iff_le
       · exact (volume_preserving_funUnique (Fin 1) ℝ).symm _
       · intro x; rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]
-      · rw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi
+      · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hle] at Hi
         exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm
     _ = f b - f a := by
       simp only [Fin.sum_univ_one, e_symm]

@@ -330,7 +330,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
             f i (eL.symm <| i.insertNth (eL b i) x)) -
           ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
             f i (eL.symm <| i.insertNth (eL a i) x)) :=
-  have he_emb : MeasurableEmbedding eL := eL.toHomeomorph.toMeasurableEquiv.measurableEmbedding
+  have he_emb : MeasurableEmbedding eL := eL.toHomeomorph.measurableEmbedding
   have hIcc : eL ⁻¹' Icc (eL a) (eL b) = Icc a b := by
     ext1 x; simp only [Set.mem_preimage, Set.mem_Icc, he_ord]
   have hIcc' : Icc (eL a) (eL b) = eL.symm ⁻¹' Icc a b := by rw [← hIcc, eL.symm_preimage_preimage]

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

This has nice performance benefits.

@@ -316,7 +316,7 @@ end
 
 /-- An auxiliary lemma that is used to specialize the general divergence theorem to spaces that do
 not have the form `Fin n → ℝ`. -/
-theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Type _}
+theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Type*}
     [NormedAddCommGroup F] [NormedSpace ℝ F] [PartialOrder F] [MeasureSpace F] [BorelSpace F]
     (eL : F ≃L[ℝ] ℝⁿ⁺¹) (he_ord : ∀ x y, eL x ≤ eL y ↔ x ≤ y)
     (he_vol : MeasurePreserving eL volume volume) (f : Fin (n + 1) → F → E)

• 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) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module measure_theory.integral.divergence_theorem
-! 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.BoxIntegral.DivergenceTheorem
 import Mathlib.Analysis.BoxIntegral.Integrability
@@ -14,6 +9,8 @@ import Mathlib.Analysis.Calculus.Deriv.Basic
 import Mathlib.MeasureTheory.Constructions.Prod.Integral
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
 
+#align_import measure_theory.integral.divergence_theorem from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
+
 /-!
 # Divergence theorem for Bochner integral
 

@@ -327,7 +327,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
     (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
     (Hd : ∀ x ∈ interior (Icc a b) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x) (DF : F → E)
     (hDF : ∀ x, DF x = ∑ i, f' i x (eL.symm <| e i)) (Hi : IntegrableOn DF (Icc a b)) :
-    (∫ x in Icc a b, DF x) =
+    ∫ x in Icc a b, DF x =
       ∑ i : Fin (n + 1),
         ((∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
             f i (eL.symm <| i.insertNth (eL b i) x)) -
@@ -338,7 +338,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
     ext1 x; simp only [Set.mem_preimage, Set.mem_Icc, he_ord]
   have hIcc' : Icc (eL a) (eL b) = eL.symm ⁻¹' Icc a b := by rw [← hIcc, eL.symm_preimage_preimage]
   calc
-    (∫ x in Icc a b, DF x) = ∫ x in Icc a b, ∑ i, f' i x (eL.symm <| e i) := by simp only [hDF]
+    ∫ x in Icc a b, DF x = ∫ x in Icc a b, ∑ i, f' i x (eL.symm <| e i) := by simp only [hDF]
     _ = ∫ x in Icc (eL a) (eL b), ∑ i, f' i (eL.symm x) (eL.symm <| e i) := by
       rw [← he_vol.set_integral_preimage_emb he_emb]
       simp only [hIcc, eL.symm_apply_apply]
@@ -384,13 +384,13 @@ differentiability of `f`;
 theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {a b : ℝ}
     (hle : a ≤ b) {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
     (Hd : ∀ x ∈ Ioo a b \ s, HasDerivAt f (f' x) x) (Hi : IntervalIntegrable f' volume a b) :
-    (∫ x in a..b, f' x) = f b - f a := by
+    ∫ x in a..b, f' x = f b - f a := by
   set e : ℝ ≃L[ℝ] ℝ¹ := (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ).symm
   have e_symm : ∀ x, e.symm x = x 0 := fun x => rfl
   set F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight (1 : ℝ →L[ℝ] ℝ) (f' x)
   have hF' : ∀ x y, F' x y = y • f' x := fun x y => rfl
   calc
-    (∫ x in a..b, f' x) = ∫ x in Icc a b, f' x := by
+    ∫ x in a..b, f' x = ∫ x in Icc a b, f' x := by
       rw [intervalIntegral.integral_of_le hle, set_integral_congr_set_ae Ioc_ae_eq_Icc]
     _ = ∑ i : Fin 1,
           ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
@@ -421,7 +421,7 @@ only assumes right differentiability of `f`.
 theorem integral_eq_of_has_deriv_within_at_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}
     (hs : s.Countable) (Hc : ContinuousOn f [[a, b]])
     (Hd : ∀ x ∈ Ioo (min a b) (max a b) \ s, HasDerivAt f (f' x) x)
-    (Hi : IntervalIntegrable f' volume a b) : (∫ x in a..b, f' x) = f b - f a := by
+    (Hi : IntervalIntegrable f' volume a b) : ∫ x in a..b, f' x = f b - f a := by
   cases' le_total a b with hab hab
   · simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
     exact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi
@@ -469,7 +469,7 @@ theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f
     _ = ((∫ y in Icc a.2 b.2, f (b.1, y)) - ∫ y in Icc a.2 b.2, f (a.1, y)) +
           ((∫ x in Icc a.1 b.1, g (x, b.2)) - ∫ x in Icc a.1 b.1, g (x, a.2)) := by
       have : ∀ (a b : ℝ¹) (f : ℝ¹ → E),
-          (∫ x in Icc a b, f x) = ∫ x in Icc (a 0) (b 0), f fun _ => x := fun a b f ↦ by
+          ∫ x in Icc a b, f x = ∫ x in Icc (a 0) (b 0), f fun _ => x := fun a b f ↦ by
         convert (((volume_preserving_funUnique (Fin 1) ℝ).symm _).set_integral_preimage_emb
           (MeasurableEquiv.measurableEmbedding _) f _).symm
         exact ((OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc a b).symm

I wrote a script to find lines that contain an odd number of backticks

@@ -39,8 +39,8 @@ Porting note (Yury Kudryashov): I disabled some of these notations because I fai
 work with Lean 4.
 
 * `ℝⁿ`, `ℝⁿ⁺¹`, `Eⁿ⁺¹`: `Fin n → ℝ`, `Fin (n + 1) → ℝ`, `Fin (n + 1) → E`;
-* `face i`: the `i`-th face of the box `[a, b]` as a closed segment in `ℝⁿ`, namely `[a ∘
-  Fin.succAbove i, b ∘ Fin.succAbove i]`;
+* `face i`: the `i`-th face of the box `[a, b]` as a closed segment in `ℝⁿ`, namely
+  `[a ∘ Fin.succAbove i, b ∘ Fin.succAbove i]`;
 * `e i` : `i`-th basis vector `Pi.single i 1`;
 * `frontFace i`, `backFace i`: embeddings `ℝⁿ → ℝⁿ⁺¹` corresponding to the front face
   `{x | x i = b i}` and back face `{x | x i = a i}` of the box `[a, b]`, respectively.

Found with git grep -n "λ [a-zA-Z_ ]*,"

@@ -255,7 +255,7 @@ local notation:max "backFace " i:arg => Fin.insertNth i (a i)
 
 /-- **Divergence theorem** for Bochner integral. If `f : ℝⁿ⁺¹ → Eⁿ⁺¹` is continuous on a rectangular
 box `[a, b] : Set ℝⁿ⁺¹`, `a ≤ b`, is differentiable on its interior with derivative
-`f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹` and the divergence `λ x, ∑ i, f' x eᵢ i` is integrable on `[a, b]`,
+`f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹` and the divergence `fun x ↦ ∑ i, f' x eᵢ i` is integrable on `[a, b]`,
 where `eᵢ = Pi.single i 1` is the `i`-th basis vector, then its integral is equal to the sum of
 integrals of `f` over the faces of `[a, b]`, taken with appropriate signs.
 

These are all doc fixes

@@ -257,7 +257,7 @@ local notation:max "backFace " i:arg => Fin.insertNth i (a i)
 box `[a, b] : Set ℝⁿ⁺¹`, `a ≤ b`, is differentiable on its interior with derivative
 `f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹` and the divergence `λ x, ∑ i, f' x eᵢ i` is integrable on `[a, b]`,
 where `eᵢ = Pi.single i 1` is the `i`-th basis vector, then its integral is equal to the sum of
-integrals of `f` over the faces of `[a, b]`, taken with appropriat signs.
+integrals of `f` over the faces of `[a, b]`, taken with appropriate signs.
 
 Moreover, the same is true if the function is not differentiable at countably many
 points of the interior of `[a, b]`.

@@ -71,15 +71,10 @@ section
 
 variable {n : ℕ}
 
--- local notation "ℝⁿ" => Fin n → ℝ
+local macro:arg t:term:max noWs "ⁿ" : term => `(Fin n → $t)
 
--- set_option quotPrecheck false in
--- local notation:200 "ℝⁿ⁺¹" => Fin (n + 1) → ℝ
+local macro:arg t:term:max noWs "ⁿ⁺¹" : term => `(Fin (n + 1) → $t)
 
--- set_option quotPrecheck false in
--- local notation:200 "Eⁿ⁺¹" => Fin (n + 1) → E
-
--- -- mathport name: «expre »
 local notation "e " i => Pi.single i 1
 
 section
@@ -115,8 +110,8 @@ in several aspects.
 `BoxIntegral.hasIntegral_GP_divergence_of_forall_hasDerivWithinAt` reformulated for the
 Bochner integral. -/
 theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (Fin (n + 1)))
-    (f : (Fin (n + 1) → ℝ) → (Fin (n + 1) → E))
-    (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → E)) (s : Set (Fin (n + 1) → ℝ))
+    (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
+    (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹)
     (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
     (Hd : ∀ x ∈ (Box.Icc I) \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) (Box.Icc I)) :
@@ -147,9 +142,9 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (
 `MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable`. Compared to the previous
 lemma, here we drop the assumption of differentiability on the boundary of the box. -/
 theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
-    (f : (Fin (n + 1) → ℝ) → (Fin (n + 1) → E))
-    (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → E))
-    (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
+    (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
+    (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
+    (s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
     (Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (∑ i, f' · (e i) i) (Box.Icc I)) :
     (∫ x in Box.Icc I, ∑ i, f' x (e i) i) =
@@ -255,8 +250,8 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (
 variable (a b : Fin (n + 1) → ℝ)
 
 local notation "face " i => Set.Icc (a ∘ Fin.succAbove i) (b ∘ Fin.succAbove i)
--- local notation "frontFace " i:2000 => Fin.insertNth i (b i)
--- local notation "backFace " i:2000 => Fin.insertNth i (a i)
+local notation:max "frontFace " i:arg => Fin.insertNth i (b i)
+local notation:max "backFace " i:arg => Fin.insertNth i (a i)
 
 /-- **Divergence theorem** for Bochner integral. If `f : ℝⁿ⁺¹ → Eⁿ⁺¹` is continuous on a rectangular
 box `[a, b] : Set ℝⁿ⁺¹`, `a ≤ b`, is differentiable on its interior with derivative
@@ -274,14 +269,14 @@ of `f : ℝⁿ⁺¹ → Eⁿ⁺¹` to these faces are given by `f ∘ backFace i
 `backFace i = Fin.insertNth i (a i)` and `frontFace i = Fin.insertNth i (b i)` are embeddings
 `ℝⁿ → ℝⁿ⁺¹` that take `y : ℝⁿ` and insert `a i` (resp., `b i`) as `i`-th coordinate. -/
 theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b)
-    (f : (Fin (n + 1) → ℝ) → (Fin (n + 1) → E))
-    (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → E))
-    (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
+    (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
+    (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
+    (s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
     (Hd : ∀ x ∈ (Set.pi univ fun i => Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) (Icc a b)) :
     (∫ x in Icc a b, ∑ i, f' x (e i) i) =
-      ∑ i : Fin (n + 1), ((∫ x in face i, f (i.insertNth (b i) x) i) -
-        ∫ x in face i, f (i.insertNth (a i) x) i) := by
+      ∑ i : Fin (n + 1), ((∫ x in face i, f (frontFace i x) i) -
+        ∫ x in face i, f (backFace i x) i) := by
   rcases em (∃ i, a i = b i) with (⟨i, hi⟩ | hne)
   · -- First we sort out the trivial case `∃ i, a i = b i`.
     rw [volume_pi, ← set_integral_congr_set_ae Measure.univ_pi_Ioc_ae_eq_Icc]
@@ -292,7 +287,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b)
     rcases eq_or_ne i j with (rfl | hne)
     · simp [hi]
     · rcases Fin.exists_succAbove_eq hne with ⟨i, rfl⟩
-      have : Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] (∅ : Set (Fin n → ℝ))
+      have : Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] (∅ : Set ℝⁿ)
       · rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]
         simp [hi]
       rw [set_integral_congr_set_ae this, set_integral_congr_set_ae this, integral_empty,
@@ -307,14 +302,14 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b)
 `MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable'` for a version formulated
 in terms of a vector-valued function `f : ℝⁿ⁺¹ → Eⁿ⁺¹`. -/
 theorem integral_divergence_of_hasFDerivWithinAt_off_countable' (hle : a ≤ b)
-    (f : Fin (n + 1) → (Fin (n + 1) → ℝ) → E)
-    (f' : Fin (n + 1) → (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E) (s : Set (Fin (n + 1) → ℝ))
+    (f : Fin (n + 1) → ℝⁿ⁺¹ → E)
+    (f' : Fin (n + 1) → ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] E) (s : Set ℝⁿ⁺¹)
     (hs : s.Countable) (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
     (Hd : ∀ x ∈ (pi Set.univ fun i => Ioo (a i) (b i)) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x)
     (Hi : IntegrableOn (fun x => ∑ i, f' i x (e i)) (Icc a b)) :
     (∫ x in Icc a b, ∑ i, f' i x (e i)) =
-      ∑ i : Fin (n + 1), ((∫ x in face i, f i (i.insertNth (b i) x)) -
-        ∫ x in face i, f i (i.insertNth (a i) x)) :=
+      ∑ i : Fin (n + 1), ((∫ x in face i, f i (frontFace i x)) -
+        ∫ x in face i, f i (backFace i x)) :=
   integral_divergence_of_hasFDerivWithinAt_off_countable a b hle (fun x i => f i x)
     (fun x => ContinuousLinearMap.pi fun i => f' i x) s hs (continuousOn_pi.2 Hc)
     (fun x hx => hasFDerivAt_pi.2 (Hd x hx)) Hi
@@ -326,7 +321,7 @@ end
 not have the form `Fin n → ℝ`. -/
 theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Type _}
     [NormedAddCommGroup F] [NormedSpace ℝ F] [PartialOrder F] [MeasureSpace F] [BorelSpace F]
-    (eL : F ≃L[ℝ] (Fin (n + 1) → ℝ)) (he_ord : ∀ x y, eL x ≤ eL y ↔ x ≤ y)
+    (eL : F ≃L[ℝ] ℝⁿ⁺¹) (he_ord : ∀ x y, eL x ≤ eL y ↔ x ≤ y)
     (he_vol : MeasurePreserving eL volume volume) (f : Fin (n + 1) → F → E)
     (f' : Fin (n + 1) → F → F →L[ℝ] E) (s : Set F) (hs : s.Countable) (a b : F) (hle : a ≤ b)
     (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
@@ -354,7 +349,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
               f i (eL.symm <| i.insertNth (eL a i) x)) := by
       refine integral_divergence_of_hasFDerivWithinAt_off_countable' (eL a) (eL b)
         ((he_ord _ _).2 hle) (fun i x => f i (eL.symm x))
-        (fun i x => f' i (eL.symm x) ∘L (eL.symm : (Fin (n + 1) → ℝ) →L[ℝ] F)) (eL.symm ⁻¹' s)
+        (fun i x => f' i (eL.symm x) ∘L (eL.symm : ℝⁿ⁺¹ →L[ℝ] F)) (eL.symm ⁻¹' s)
         (hs.preimage eL.symm.injective) ?_ ?_ ?_
       · exact fun i => (Hc i).comp eL.symm.continuousOn hIcc'.subset
       · refine' fun x hx i => (Hd (eL.symm x) ⟨_, hx.2⟩ i).comp x eL.symm.hasFDerivAt
@@ -373,10 +368,8 @@ open scoped Interval
 open ContinuousLinearMap (smulRight)
 
 
-local notation "ℝ¹" => Fin 1 → ℝ
-local notation "ℝ²" => Fin 2 → ℝ
-local notation "E¹" => Fin 1 → E
-local notation "E²" => Fin 2 → E
+local macro:arg t:term:max noWs "¹" : term => `(Fin 1 → $t)
+local macro:arg t:term:max noWs "²" : term => `(Fin 2 → $t)
 
 /-- **Fundamental theorem of calculus, part 2**. This version assumes that `f` is continuous on the
 interval and is differentiable off a countable set `s`.
@@ -535,4 +528,3 @@ theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ
 #align measure_theory.integral2_divergence_prod_of_has_fderiv_within_at_off_countable MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable
 
 end MeasureTheory
-

Co-authored-by: Johan Commelin <johan@commelin.net>