analysis.calculus.parametric_integral ⟷ Mathlib.Analysis.Calculus.ParametricIntegral

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

@@ -63,7 +63,7 @@ open TopologicalSpace MeasureTheory Filter Metric
 
 open scoped Topology Filter
 
-variable {α : Type _} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type _} [IsROrC 𝕜] {E : Type _}
+variable {α : Type _} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type _} [RCLike 𝕜] {E : Type _}
   [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type _}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H]
 

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

@@ -99,7 +99,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       simp only [norm_sub_rev (F x₀ _)]
       refine' h_lipsch.mono fun a ha => (ha x x_in).trans _
       rw [mul_comm ε]
-      rw [mem_ball, dist_eq_norm] at x_in 
+      rw [mem_ball, dist_eq_norm] at x_in
       exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)
     exact
       integrable_of_norm_sub_le (hF_meas x x_in) hF_int
@@ -161,7 +161,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       by
       ext x
       rw [norm_smul_of_nonneg (nneg _)]
-    rwa [hasFDerivAt_iff_tendsto, this] at ha 
+    rwa [hasFDerivAt_iff_tendsto, this] at ha
 #align has_fderiv_at_integral_of_dominated_loc_of_lip' hasFDerivAt_integral_of_dominated_loc_of_lip'
 -/
 
@@ -243,14 +243,14 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
       h_diff with
     hF'_int key
   replace hF'_int : integrable F' μ
-  · rw [← integrable_norm_iff hm] at hF'_int 
+  · rw [← integrable_norm_iff hm] at hF'_int
     simpa only [L, (· ∘ ·), integrable_norm_iff, hF'_meas, one_mul, norm_one,
       ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_restrict_scalarsL',
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using
       hF'_int
   refine' ⟨hF'_int, _⟩
   simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff ⊢
-  rwa [ContinuousLinearMap.integral_comp_comm _ hF'_int] at key 
+  rwa [ContinuousLinearMap.integral_comp_comm _ hF'_int] at key
   all_goals infer_instance
 #align has_deriv_at_integral_of_dominated_loc_of_lip hasDerivAt_integral_of_dominated_loc_of_lip
 -/

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

@@ -146,7 +146,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := (add_le_add _ _)
       _ ≤ b a + ‖F' a‖ := _
     exact mul_le_mul_of_nonneg_left ha_bound (nneg _)
-    apply mul_le_mul_of_nonneg_left ((F' a).le_op_norm _) (nneg _)
+    apply mul_le_mul_of_nonneg_left ((F' a).le_opNorm _) (nneg _)
     by_cases h : ‖x - x₀‖ = 0
     · simpa [h] using add_nonneg (b_nonneg a) (norm_nonneg (F' a))
     · field_simp [h]

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

@@ -3,8 +3,8 @@ Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
 -/
-import Mathbin.Analysis.Calculus.MeanValue
-import Mathbin.MeasureTheory.Integral.SetIntegral
+import Analysis.Calculus.MeanValue
+import MeasureTheory.Integral.SetIntegral
 
 #align_import analysis.calculus.parametric_integral from "leanprover-community/mathlib"@"c20927220ef87bb4962ba08bf6da2ce3cf50a6dd"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
-
-! This file was ported from Lean 3 source module analysis.calculus.parametric_integral
-! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Calculus.MeanValue
 import Mathbin.MeasureTheory.Integral.SetIntegral
 
+#align_import analysis.calculus.parametric_integral from "leanprover-community/mathlib"@"c20927220ef87bb4962ba08bf6da2ce3cf50a6dd"
+
 /-!
 # Derivatives of integrals depending on parameters
 

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

@@ -70,6 +70,7 @@ variable {α : Type _} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type _} [I
   [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type _}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H]
 
+#print hasFDerivAt_integral_of_dominated_loc_of_lip' /-
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming `F x₀` is
 integrable, `‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖` for `x` in a ball around `x₀` for ae `a` with
 integrable Lipschitz bound `bound` (with a ball radius independent of `a`), and `F x` is
@@ -165,7 +166,9 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       rw [norm_smul_of_nonneg (nneg _)]
     rwa [hasFDerivAt_iff_tendsto, this] at ha 
 #align has_fderiv_at_integral_of_dominated_loc_of_lip' hasFDerivAt_integral_of_dominated_loc_of_lip'
+-/
 
+#print hasFDerivAt_integral_of_dominated_loc_of_lip /-
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
 `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
 (with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
@@ -186,7 +189,9 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F : H → α → E} {F' :
   replace bound_integrable := bound_integrable.norm
   apply hasFDerivAt_integral_of_dominated_loc_of_lip' δ_pos <;> assumption
 #align has_fderiv_at_integral_of_dominated_loc_of_lip hasFDerivAt_integral_of_dominated_loc_of_lip
+-/
 
+#print hasFDerivAt_integral_of_dominated_of_fderiv_le /-
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
 `F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
 derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
@@ -217,7 +222,9 @@ theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F : H → α → E} {F'
     (hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this
         bound_integrable diff_x₀).2
 #align has_fderiv_at_integral_of_dominated_of_fderiv_le hasFDerivAt_integral_of_dominated_of_fderiv_le
+-/
 
+#print hasDerivAt_integral_of_dominated_loc_of_lip /-
 /-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : 𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
 assuming `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
 (with ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is
@@ -249,7 +256,9 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
   rwa [ContinuousLinearMap.integral_comp_comm _ hF'_int] at key 
   all_goals infer_instance
 #align has_deriv_at_integral_of_dominated_loc_of_lip hasDerivAt_integral_of_dominated_loc_of_lip
+-/
 
+#print hasDerivAt_integral_of_dominated_loc_of_deriv_le /-
 /-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : ℝ`, assuming
 `F x₀` is integrable, `x ↦ F x a` is differentiable on an interval around `x₀` for ae `a`
 (with interval radius independent of `a`) with derivative uniformly bounded by an integrable
@@ -277,4 +286,5 @@ theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le {F : 𝕜 → α → E}
     hasDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this bound_integrable
       diff_x₀
 #align has_deriv_at_integral_of_dominated_loc_of_deriv_le hasDerivAt_integral_of_dominated_loc_of_deriv_le
+-/
 

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

@@ -117,7 +117,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
   have h_ball : ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos
   have :
     ∀ᶠ x in 𝓝 x₀,
-      ‖x - x₀‖⁻¹ * ‖((∫ a, F x a ∂μ) - ∫ a, F x₀ a ∂μ) - (∫ a, F' a ∂μ) (x - x₀)‖ =
+      ‖x - x₀‖⁻¹ * ‖∫ a, F x a ∂μ - ∫ a, F x₀ a ∂μ - (∫ a, F' a ∂μ) (x - x₀)‖ =
         ‖∫ a, ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ‖ :=
     by
     apply mem_of_superset (ball_mem_nhds _ ε_pos)
@@ -127,7 +127,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
     exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
       hF'_int.apply_continuous_linear_map _]
   rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←
-    show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp]
+    show ∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ = 0 by simp]
   apply tendsto_integral_filter_of_dominated_convergence
   · filter_upwards [h_ball] with _ x_in
     apply ae_strongly_measurable.const_smul

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

@@ -147,7 +147,6 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
         rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _
       _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := (add_le_add _ _)
       _ ≤ b a + ‖F' a‖ := _
-      
     exact mul_le_mul_of_nonneg_left ha_bound (nneg _)
     apply mul_le_mul_of_nonneg_left ((F' a).le_op_norm _) (nneg _)
     by_cases h : ‖x - x₀‖ = 0

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
 
 ! This file was ported from Lean 3 source module analysis.calculus.parametric_integral
-! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
+! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.MeasureTheory.Integral.SetIntegral
 /-!
 # Derivatives of integrals depending on parameters
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A parametric integral is a function with shape `f = λ x : H, ∫ a : α, F x a ∂μ` for some
 `F : H → α → E`, where `H` and `E` are normed spaces and `α` is a measured space with measure `μ`.
 

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

@@ -126,7 +126,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
   rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←
     show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp]
   apply tendsto_integral_filter_of_dominated_convergence
-  · filter_upwards [h_ball]with _ x_in
+  · filter_upwards [h_ball] with _ x_in
     apply ae_strongly_measurable.const_smul
     exact ((hF_meas _ x_in).sub (hF_meas _ x₀_in)).sub (hF'_meas.apply_continuous_linear_map _)
   · apply mem_of_superset h_ball

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

@@ -207,7 +207,7 @@ theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F : H → α → E} {F'
     apply (h_diff.and h_bound).mono
     rintro a ⟨ha_deriv, ha_bound⟩
     refine'
-      (convex_ball _ _).lipschitzOnWith_of_nnnorm_has_fderiv_within_le
+      (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
         (fun x x_in => (ha_deriv x x_in).HasFDerivWithinAt) fun x x_in => _
     rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]
     exact (ha_bound x x_in).trans (le_abs_self _)
@@ -267,7 +267,7 @@ theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le {F : 𝕜 → α → E}
     apply (h_diff.and h_bound).mono
     rintro a ⟨ha_deriv, ha_bound⟩
     refine'
-      (convex_ball _ _).lipschitzOnWith_of_nnnorm_has_deriv_within_le
+      (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
         (fun x x_in => (ha_deriv x x_in).HasDerivWithinAt) fun x x_in => _
     rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]
     exact (ha_bound x x_in).trans (le_abs_self _)

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

@@ -98,7 +98,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       simp only [norm_sub_rev (F x₀ _)]
       refine' h_lipsch.mono fun a ha => (ha x x_in).trans _
       rw [mul_comm ε]
-      rw [mem_ball, dist_eq_norm] at x_in
+      rw [mem_ball, dist_eq_norm] at x_in 
       exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)
     exact
       integrable_of_norm_sub_le (hF_meas x x_in) hF_int
@@ -121,7 +121,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
     intro x x_in
     rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub,
       ← ContinuousLinearMap.integral_apply hF'_int]
-    exacts[hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
+    exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
       hF'_int.apply_continuous_linear_map _]
   rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←
     show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp]
@@ -161,7 +161,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       by
       ext x
       rw [norm_smul_of_nonneg (nneg _)]
-    rwa [hasFDerivAt_iff_tendsto, this] at ha
+    rwa [hasFDerivAt_iff_tendsto, this] at ha 
 #align has_fderiv_at_integral_of_dominated_loc_of_lip' hasFDerivAt_integral_of_dominated_loc_of_lip'
 
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
@@ -237,14 +237,14 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
       h_diff with
     hF'_int key
   replace hF'_int : integrable F' μ
-  · rw [← integrable_norm_iff hm] at hF'_int
+  · rw [← integrable_norm_iff hm] at hF'_int 
     simpa only [L, (· ∘ ·), integrable_norm_iff, hF'_meas, one_mul, norm_one,
       ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_restrict_scalarsL',
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using
       hF'_int
   refine' ⟨hF'_int, _⟩
-  simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff⊢
-  rwa [ContinuousLinearMap.integral_comp_comm _ hF'_int] at key
+  simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff ⊢
+  rwa [ContinuousLinearMap.integral_comp_comm _ hF'_int] at key 
   all_goals infer_instance
 #align has_deriv_at_integral_of_dominated_loc_of_lip hasDerivAt_integral_of_dominated_loc_of_lip
 

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

@@ -61,7 +61,7 @@ noncomputable section
 
 open TopologicalSpace MeasureTheory Filter Metric
 
-open Topology Filter
+open scoped Topology Filter
 
 variable {α : Type _} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type _} [IsROrC 𝕜] {E : Type _}
   [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type _}

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

@@ -74,8 +74,8 @@ ae-measurable for `x` in the same ball. See `has_fderiv_at_integral_of_dominated
 slightly less general but usually more useful version. -/
 theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
     {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε)
-    (hF_meas : ∀ x ∈ ball x₀ ε, AeStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
-    (hF'_meas : AeStronglyMeasurable F' μ)
+    (hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
+    (hF'_meas : AEStronglyMeasurable F' μ)
     (h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
     (bound_integrable : Integrable (bound : α → ℝ) μ)
     (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀) :
@@ -169,8 +169,8 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
 (with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
 for `x` in a possibly smaller neighborhood of `x₀`. -/
 theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
-    {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AeStronglyMeasurable (F x) μ)
-    (hF_int : Integrable (F x₀) μ) (hF'_meas : AeStronglyMeasurable F' μ)
+    {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
+    (hF_int : Integrable (F x₀) μ) (hF'_meas : AEStronglyMeasurable F' μ)
     (h_lip : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (fun x => F x a) (ball x₀ ε))
     (bound_integrable : Integrable (bound : α → ℝ) μ)
     (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀) :
@@ -191,8 +191,8 @@ derivative norm uniformly bounded by an integrable function (the ball radius is
 and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
 theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F : H → α → E} {F' : H → α → H →L[𝕜] E}
     {x₀ : H} {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε)
-    (hF_meas : ∀ᶠ x in 𝓝 x₀, AeStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
-    (hF'_meas : AeStronglyMeasurable (F' x₀) μ)
+    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
+    (hF'_meas : AEStronglyMeasurable (F' x₀) μ)
     (h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a)
     (bound_integrable : Integrable (bound : α → ℝ) μ)
     (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x a) (F' x a) x) :
@@ -221,8 +221,8 @@ assuming `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball ar
 (with ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is
 ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
 theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F' : α → E} {x₀ : 𝕜} {ε : ℝ}
-    (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AeStronglyMeasurable (F x) μ)
-    (hF_int : Integrable (F x₀) μ) (hF'_meas : AeStronglyMeasurable F' μ) {bound : α → ℝ}
+    (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
+    (hF_int : Integrable (F x₀) μ) (hF'_meas : AEStronglyMeasurable F' μ) {bound : α → ℝ}
     (h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (fun x => F x a) (ball x₀ ε))
     (bound_integrable : Integrable (bound : α → ℝ) μ)
     (h_diff : ∀ᵐ a ∂μ, HasDerivAt (fun x => F x a) (F' a) x₀) :
@@ -253,8 +253,8 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
 (with interval radius independent of `a`) with derivative uniformly bounded by an integrable
 function, and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
 theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le {F : 𝕜 → α → E} {F' : 𝕜 → α → E} {x₀ : 𝕜}
-    {ε : ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AeStronglyMeasurable (F x) μ)
-    (hF_int : Integrable (F x₀) μ) (hF'_meas : AeStronglyMeasurable (F' x₀) μ) {bound : α → ℝ}
+    {ε : ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
+    (hF_int : Integrable (F x₀) μ) (hF'_meas : AEStronglyMeasurable (F' x₀) μ) {bound : α → ℝ}
     (h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
     (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasDerivAt (fun x => F x a) (F' x a) x) :
     Integrable (F' x₀) μ ∧ HasDerivAt (fun n => ∫ a, F n a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ :=

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

@@ -72,14 +72,14 @@ integrable, `‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖` for `x` in a b
 integrable Lipschitz bound `bound` (with a ball radius independent of `a`), and `F x` is
 ae-measurable for `x` in the same ball. See `has_fderiv_at_integral_of_dominated_loc_of_lip` for a
 slightly less general but usually more useful version. -/
-theorem hasFderivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
+theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
     {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε)
     (hF_meas : ∀ x ∈ ball x₀ ε, AeStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
     (hF'_meas : AeStronglyMeasurable F' μ)
     (h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
     (bound_integrable : Integrable (bound : α → ℝ) μ)
-    (h_diff : ∀ᵐ a ∂μ, HasFderivAt (fun x => F x a) (F' a) x₀) :
-    Integrable F' μ ∧ HasFderivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
+    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀) :
+    Integrable F' μ ∧ HasFDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
   by
   have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
   have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x => inv_nonneg.mpr (norm_nonneg _)
@@ -123,7 +123,7 @@ theorem hasFderivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       ← ContinuousLinearMap.integral_apply hF'_int]
     exacts[hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
       hF'_int.apply_continuous_linear_map _]
-  rw [hasFderivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←
+  rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←
     show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp]
   apply tendsto_integral_filter_of_dominated_convergence
   · filter_upwards [h_ball]with _ x_in
@@ -161,20 +161,20 @@ theorem hasFderivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       by
       ext x
       rw [norm_smul_of_nonneg (nneg _)]
-    rwa [hasFderivAt_iff_tendsto, this] at ha
-#align has_fderiv_at_integral_of_dominated_loc_of_lip' hasFderivAt_integral_of_dominated_loc_of_lip'
+    rwa [hasFDerivAt_iff_tendsto, this] at ha
+#align has_fderiv_at_integral_of_dominated_loc_of_lip' hasFDerivAt_integral_of_dominated_loc_of_lip'
 
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
 `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
 (with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
 for `x` in a possibly smaller neighborhood of `x₀`. -/
-theorem hasFderivAt_integral_of_dominated_loc_of_lip {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
+theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
     {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AeStronglyMeasurable (F x) μ)
     (hF_int : Integrable (F x₀) μ) (hF'_meas : AeStronglyMeasurable F' μ)
     (h_lip : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (fun x => F x a) (ball x₀ ε))
     (bound_integrable : Integrable (bound : α → ℝ) μ)
-    (h_diff : ∀ᵐ a ∂μ, HasFderivAt (fun x => F x a) (F' a) x₀) :
-    Integrable F' μ ∧ HasFderivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
+    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀) :
+    Integrable F' μ ∧ HasFDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
   by
   obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ x ∈ ball x₀ δ, ae_strongly_measurable (F x) μ ∧ x ∈ ball x₀ ε
   exact eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x₀ ε_pos))
@@ -182,25 +182,25 @@ theorem hasFderivAt_integral_of_dominated_loc_of_lip {F : H → α → E} {F' :
   replace h_lip : ∀ᵐ a : α ∂μ, ∀ x ∈ ball x₀ δ, ‖F x a - F x₀ a‖ ≤ |bound a| * ‖x - x₀‖
   exact h_lip.mono fun a lip x hx => lip.norm_sub_le (hδε x hx) (mem_ball_self ε_pos)
   replace bound_integrable := bound_integrable.norm
-  apply hasFderivAt_integral_of_dominated_loc_of_lip' δ_pos <;> assumption
-#align has_fderiv_at_integral_of_dominated_loc_of_lip hasFderivAt_integral_of_dominated_loc_of_lip
+  apply hasFDerivAt_integral_of_dominated_loc_of_lip' δ_pos <;> assumption
+#align has_fderiv_at_integral_of_dominated_loc_of_lip hasFDerivAt_integral_of_dominated_loc_of_lip
 
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
 `F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
 derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
 and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
-theorem hasFderivAt_integral_of_dominated_of_fderiv_le {F : H → α → E} {F' : H → α → H →L[𝕜] E}
+theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F : H → α → E} {F' : H → α → H →L[𝕜] E}
     {x₀ : H} {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε)
     (hF_meas : ∀ᶠ x in 𝓝 x₀, AeStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
     (hF'_meas : AeStronglyMeasurable (F' x₀) μ)
     (h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a)
     (bound_integrable : Integrable (bound : α → ℝ) μ)
-    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasFderivAt (fun x => F x a) (F' x a) x) :
-    HasFderivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ :=
+    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x a) (F' x a) x) :
+    HasFDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ :=
   by
   letI : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H
   have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
-  have diff_x₀ : ∀ᵐ a ∂μ, HasFderivAt (fun x => F x a) (F' x₀ a) x₀ :=
+  have diff_x₀ : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ :=
     h_diff.mono fun a ha => ha x₀ x₀_in
   have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) :=
     by
@@ -208,13 +208,13 @@ theorem hasFderivAt_integral_of_dominated_of_fderiv_le {F : H → α → E} {F'
     rintro a ⟨ha_deriv, ha_bound⟩
     refine'
       (convex_ball _ _).lipschitzOnWith_of_nnnorm_has_fderiv_within_le
-        (fun x x_in => (ha_deriv x x_in).HasFderivWithinAt) fun x x_in => _
+        (fun x x_in => (ha_deriv x x_in).HasFDerivWithinAt) fun x x_in => _
     rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]
     exact (ha_bound x x_in).trans (le_abs_self _)
   exact
-    (hasFderivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this
+    (hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this
         bound_integrable diff_x₀).2
-#align has_fderiv_at_integral_of_dominated_of_fderiv_le hasFderivAt_integral_of_dominated_of_fderiv_le
+#align has_fderiv_at_integral_of_dominated_of_fderiv_le hasFDerivAt_integral_of_dominated_of_fderiv_le
 
 /-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : 𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
 assuming `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
@@ -229,11 +229,11 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
     Integrable F' μ ∧ HasDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
   by
   set L : E →L[𝕜] 𝕜 →L[𝕜] E := ContinuousLinearMap.smulRightL 𝕜 𝕜 E 1
-  replace h_diff : ∀ᵐ a ∂μ, HasFderivAt (fun x => F x a) (L (F' a)) x₀ :=
-    h_diff.mono fun x hx => hx.HasFderivAt
+  replace h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀ :=
+    h_diff.mono fun x hx => hx.HasFDerivAt
   have hm : ae_strongly_measurable (L ∘ F') μ := L.continuous.comp_ae_strongly_measurable hF'_meas
   cases'
-    hasFderivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable
+    hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable
       h_diff with
     hF'_int key
   replace hF'_int : integrable F' μ
@@ -243,7 +243,7 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using
       hF'_int
   refine' ⟨hF'_int, _⟩
-  simp_rw [hasDerivAt_iff_hasFderivAt] at h_diff⊢
+  simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff⊢
   rwa [ContinuousLinearMap.integral_comp_comm _ hF'_int] at key
   all_goals infer_instance
 #align has_deriv_at_integral_of_dominated_loc_of_lip hasDerivAt_integral_of_dominated_loc_of_lip

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

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
 
 ! This file was ported from Lean 3 source module analysis.calculus.parametric_integral
-! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
+! 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.Calculus.MeanValue
 import Mathbin.MeasureTheory.Integral.SetIntegral
 
 /-!
@@ -145,7 +146,7 @@ theorem hasFderivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       _ ≤ b a + ‖F' a‖ := _
       
     exact mul_le_mul_of_nonneg_left ha_bound (nneg _)
-    apply mul_le_mul_of_nonneg_left ((F' a).le_opNorm _) (nneg _)
+    apply mul_le_mul_of_nonneg_left ((F' a).le_op_norm _) (nneg _)
     by_cases h : ‖x - x₀‖ = 0
     · simpa [h] using add_nonneg (b_nonneg a) (norm_nonneg (F' a))
     · field_simp [h]

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

@@ -4,12 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
 
 ! This file was ported from Lean 3 source module analysis.calculus.parametric_integral
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Integral.SetIntegral
-import Mathbin.Analysis.Calculus.MeanValue
 
 /-!
 # Derivatives of integrals depending on parameters

mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba

@@ -146,7 +146,7 @@ theorem hasFderivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       _ ≤ b a + ‖F' a‖ := _
       
     exact mul_le_mul_of_nonneg_left ha_bound (nneg _)
-    apply mul_le_mul_of_nonneg_left ((F' a).le_op_norm _) (nneg _)
+    apply mul_le_mul_of_nonneg_left ((F' a).le_opNorm _) (nneg _)
     by_cases h : ‖x - x₀‖ = 0
     · simpa [h] using add_nonneg (b_nonneg a) (norm_nonneg (F' a))
     · field_simp [h]

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

@@ -139,10 +139,10 @@ theorem hasFderivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ =
           ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • F' a (x - x₀)‖ :=
         by rw [smul_sub]
-      _ ≤ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := norm_sub_le _ _
+      _ ≤ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := (norm_sub_le _ _)
       _ = ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖F' a (x - x₀)‖ := by
         rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _
-      _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := add_le_add _ _
+      _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := (add_le_add _ _)
       _ ≤ b a + ‖F' a‖ := _
       
     exact mul_le_mul_of_nonneg_left ha_bound (nneg _)

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

A PR accompanying #12339.

Zulip discussion

@@ -129,7 +129,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜]
     exact ((hF_meas _ x_in).sub (hF_meas _ x₀_in)).sub (hF'_meas.apply_continuousLinearMap _)
   · refine mem_of_superset h_ball fun x hx ↦ ?_
     apply (h_diff.and h_lipsch).mono
-    rintro a ⟨-, ha_bound⟩
+    on_goal 1 => rintro a ⟨-, ha_bound⟩
     show ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ ≤ b a + ‖F' a‖
     replace ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ := ha_bound x hx
     calc

This is a PR that's a temporary measure to improve performance of congr!/convert, and the implementation may change in a future PR with a new version of congr!.

Introduces two typeclasses that are meant to quickly evaluate in common cases of Subsingleton and IsEmpty. Makes congr! use these typeclasses rather than Subsingleton.

Local Subsingleton/IsEmpty instances are included as Fast instances. To get congr!/convert to reason about subsingleton types, you can add such instances to the local context. Or, you can apply Subsingleton.elim yourself.

Zulip discussion

@@ -106,8 +106,8 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜]
   /- Discard the trivial case where `E` is not complete, as all integrals vanish. -/
   by_cases hE : CompleteSpace E; swap
   · rcases subsingleton_or_nontrivial H with hH|hH
-    · convert hasFDerivAt_of_subsingleton _ _
-      exact hH
+    · have : Subsingleton (H →L[𝕜] E) := inferInstance
+      convert hasFDerivAt_of_subsingleton _ x₀
     · have : ¬(CompleteSpace (H →L[𝕜] E)) := by
         simpa [SeparatingDual.completeSpace_continuousLinearMap_iff] using hE
       simp only [integral, hE, ↓reduceDite, this]

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -136,7 +136,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜]
       ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ =
           ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • F' a (x - x₀)‖ :=
         by rw [smul_sub]
-      _ ≤ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := (norm_sub_le _ _)
+      _ ≤ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := norm_sub_le _ _
       _ = ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖F' a (x - x₀)‖ := by
         rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _
       _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := by

When computing the integral of a function taking values in a noncomplete space, we use the junk value 0. This means that several theorems about integrals hold without completeness assumptions for trivial reasons. We use this to drop several completeness assumptions here and there in mathlib. This involves one nontrivial mathematical fact, that E →L[𝕜] F is complete iff F is complete, for which we add the missing direction (from left to right) in this PR.

@@ -6,6 +6,7 @@ Authors: Patrick Massot
 import Mathlib.Analysis.Calculus.MeanValue
 import Mathlib.MeasureTheory.Integral.DominatedConvergence
 import Mathlib.MeasureTheory.Integral.SetIntegral
+import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual
 
 #align_import analysis.calculus.parametric_integral from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
 
@@ -61,7 +62,7 @@ open TopologicalSpace MeasureTheory Filter Metric
 open scoped Topology Filter
 
 variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type*} [RCLike 𝕜] {E : Type*}
-  [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type*}
+  [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type*}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H]
 
 variable {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ}
@@ -102,6 +103,15 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜]
       exact ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)
     b_int.mono' hF'_meas this
   refine ⟨hF'_int, ?_⟩
+  /- Discard the trivial case where `E` is not complete, as all integrals vanish. -/
+  by_cases hE : CompleteSpace E; swap
+  · rcases subsingleton_or_nontrivial H with hH|hH
+    · convert hasFDerivAt_of_subsingleton _ _
+      exact hH
+    · have : ¬(CompleteSpace (H →L[𝕜] E)) := by
+        simpa [SeparatingDual.completeSpace_continuousLinearMap_iff] using hE
+      simp only [integral, hE, ↓reduceDite, this]
+      exact hasFDerivAt_const 0 x₀
   have h_ball : ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos
   have : ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖((∫ a, F x a ∂μ) - ∫ a, F x₀ a ∂μ) - (∫ a, F' a ∂μ) (x - x₀)‖ =
       ‖∫ a, ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ‖ := by
@@ -268,6 +278,9 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using
       hF'_int
   refine ⟨hF'_int, ?_⟩
+  by_cases hE : CompleteSpace E; swap
+  · simp [integral, hE]
+    exact hasDerivAt_const x₀ 0
   simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff ⊢
   simpa only [(· ∘ ·), ContinuousLinearMap.integral_comp_comm _ hF'_int] using key
 #align has_deriv_at_integral_of_dominated_loc_of_lip hasDerivAt_integral_of_dominated_loc_of_lip

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

@@ -60,7 +60,7 @@ open TopologicalSpace MeasureTheory Filter Metric
 
 open scoped Topology Filter
 
-variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type*} [IsROrC 𝕜] {E : Type*}
+variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type*} [RCLike 𝕜] {E : Type*}
   [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type*}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H]
 

Suggested by @loefflerd. Only code motion (and cosmetic adaptions, such as minimising import and open statements).

Pre-requisite for #11108 and (morally) #11110.

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
 -/
 import Mathlib.Analysis.Calculus.MeanValue
-import Mathlib.MeasureTheory.Integral.IntervalIntegral
+import Mathlib.MeasureTheory.Integral.DominatedConvergence
 import Mathlib.MeasureTheory.Integral.SetIntegral
 
 #align_import analysis.calculus.parametric_integral from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"

Based on code from the sphere eversion project.

Co-authored-by: David Loeffler <d.loeffler.01@cantab.net>

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
 -/
 import Mathlib.Analysis.Calculus.MeanValue
+import Mathlib.MeasureTheory.Integral.IntervalIntegral
 import Mathlib.MeasureTheory.Integral.SetIntegral
 
 #align_import analysis.calculus.parametric_integral from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
@@ -37,7 +38,7 @@ variable.
   A subtle point is that the "near x₀" in the last condition has to be uniform in `a`. This is
   controlled by a positive number `ε`.
 
-* `hasFDerivAt_integral_of_dominated_of_fderiv_le`: this version assume `fun x ↦ F x a` has
+* `hasFDerivAt_integral_of_dominated_of_fderiv_le`: this version assumes `fun x ↦ F x a` has
    derivative `F' x a` for `x` near `x₀` and `F' x` is bounded by an integrable function independent
    from `x` near `x₀`.
 
@@ -165,6 +166,31 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F' : α → H →L[𝕜] E
   apply hasFDerivAt_integral_of_dominated_loc_of_lip' δ_pos <;> assumption
 #align has_fderiv_at_integral_of_dominated_loc_of_lip hasFDerivAt_integral_of_dominated_loc_of_lip
 
+/-- Differentiation under integral of `x ↦ ∫ x in a..b, F x t` at a given point `x₀ ∈ (a,b)`,
+assuming `F x₀` is integrable on `(a,b)`, that `x ↦ F x t` is Lipschitz on a ball around `x₀`
+for almost every `t` (with a ball radius independent of `t`) with integrable Lipschitz bound,
+and `F x` is a.e.-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
+theorem hasFDerivAt_integral_of_dominated_loc_of_lip_interval [NormedSpace ℝ H] {μ : Measure ℝ}
+    {F : H → ℝ → E} {F' : ℝ → H →L[ℝ] E} {a b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε)
+    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) <| μ.restrict (Ι a b))
+    (hF_int : IntervalIntegrable (F x₀) μ a b)
+    (hF'_meas : AEStronglyMeasurable F' <| μ.restrict (Ι a b))
+    (h_lip : ∀ᵐ t ∂μ.restrict (Ι a b),
+      LipschitzOnWith (Real.nnabs <| bound t) (F · t) (ball x₀ ε))
+    (bound_integrable : IntervalIntegrable bound μ a b)
+    (h_diff : ∀ᵐ t ∂μ.restrict (Ι a b), HasFDerivAt (F · t) (F' t) x₀) :
+    IntervalIntegrable F' μ a b ∧
+      HasFDerivAt (fun x ↦ ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' t ∂μ) x₀ := by
+  simp_rw [AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff, eventually_and] at hF_meas hF'_meas
+  rw [ae_restrict_uIoc_iff] at h_lip h_diff
+  have H₁ :=
+    hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas.1 hF_int.1 hF'_meas.1 h_lip.1
+      bound_integrable.1 h_diff.1
+  have H₂ :=
+    hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas.2 hF_int.2 hF'_meas.2 h_lip.2
+      bound_integrable.2 h_diff.2
+  exact ⟨⟨H₁.1, H₂.1⟩, H₁.2.sub H₂.2⟩
+
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
 `F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
 derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
@@ -178,7 +204,7 @@ theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F' : H → α → H →L
     HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ := by
   letI : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H
   have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
-  have diff_x₀ : ∀ᵐ a ∂μ, HasFDerivAt (fun x ↦ F x a) (F' x₀ a) x₀ :=
+  have diff_x₀ : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' x₀ a) x₀ :=
     h_diff.mono fun a ha ↦ ha x₀ x₀_in
   have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (F · a) (ball x₀ ε) := by
     apply (h_diff.and h_bound).mono
@@ -191,6 +217,27 @@ theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F' : H → α → H →L
     bound_integrable diff_x₀).2
 #align has_fderiv_at_integral_of_dominated_of_fderiv_le hasFDerivAt_integral_of_dominated_of_fderiv_le
 
+/-- Differentiation under integral of `x ↦ ∫ x in a..b, F x a` at a given point `x₀`, assuming
+`F x₀` is integrable on `(a,b)`, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
+derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
+and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
+theorem hasFDerivAt_integral_of_dominated_of_fderiv_le'' [NormedSpace ℝ H] {μ : Measure ℝ}
+    {F : H → ℝ → E} {F' : H → ℝ → H →L[ℝ] E} {a b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε)
+    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) <| μ.restrict (Ι a b))
+    (hF_int : IntervalIntegrable (F x₀) μ a b)
+    (hF'_meas : AEStronglyMeasurable (F' x₀) <| μ.restrict (Ι a b))
+    (h_bound : ∀ᵐ t ∂μ.restrict (Ι a b), ∀ x ∈ ball x₀ ε, ‖F' x t‖ ≤ bound t)
+    (bound_integrable : IntervalIntegrable bound μ a b)
+    (h_diff : ∀ᵐ t ∂μ.restrict (Ι a b), ∀ x ∈ ball x₀ ε, HasFDerivAt (F · t) (F' x t) x) :
+    HasFDerivAt (fun x ↦ ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' x₀ t ∂μ) x₀ := by
+  rw [ae_restrict_uIoc_iff] at h_diff h_bound
+  simp_rw [AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff, eventually_and] at hF_meas hF'_meas
+  exact
+    (hasFDerivAt_integral_of_dominated_of_fderiv_le ε_pos hF_meas.1 hF_int.1 hF'_meas.1 h_bound.1
+          bound_integrable.1 h_diff.1).sub
+      (hasFDerivAt_integral_of_dominated_of_fderiv_le ε_pos hF_meas.2 hF_int.2 hF'_meas.2 h_bound.2
+        bound_integrable.2 h_diff.2)
+
 section
 
 variable {F : 𝕜 → α → E} {x₀ : 𝕜}

This covers many instances, but is not exhaustive.

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

@@ -156,8 +156,8 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F' : α → H →L[𝕜] E
     (bound_integrable : Integrable (bound : α → ℝ) μ)
     (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a) x₀) :
     Integrable F' μ ∧ HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
-  obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ x ∈ ball x₀ δ, AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε
-  exact eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x₀ ε_pos))
+  obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ x ∈ ball x₀ δ, AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε :=
+    eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x₀ ε_pos))
   choose hδ_meas hδε using hδ
   replace h_lip : ∀ᵐ a : α ∂μ, ∀ x ∈ ball x₀ δ, ‖F x a - F x₀ a‖ ≤ |bound a| * ‖x - x₀‖ :=
     h_lip.mono fun a lip x hx ↦ lip.norm_sub_le (hδε x hx) (mem_ball_self ε_pos)

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

@@ -216,7 +216,7 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0
     hF'_int key
   replace hF'_int : Integrable F' μ := by
     rw [← integrable_norm_iff hm] at hF'_int
-    simpa only [(· ∘ ·), integrable_norm_iff, hF'_meas, one_mul, norm_one,
+    simpa only [L, (· ∘ ·), integrable_norm_iff, hF'_meas, one_mul, norm_one,
       ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_restrict_scalarsL',
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using
       hF'_int

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).

@@ -11,7 +11,7 @@ import Mathlib.MeasureTheory.Integral.SetIntegral
 /-!
 # Derivatives of integrals depending on parameters
 
-A parametric integral is a function with shape `f = λ x : H, ∫ a : α, F x a ∂μ` for some
+A parametric integral is a function with shape `f = fun x : H ↦ ∫ a : α, F x a ∂μ` for some
 `F : H → α → E`, where `H` and `E` are normed spaces and `α` is a measured space with measure `μ`.
 
 We already know from `continuous_of_dominated` in `Mathlib/MeasureTheory/Integral/Bochner.lean` how
@@ -31,8 +31,8 @@ variable.
   - `F x` is ae-measurable for x near `x₀`,
   - `F x₀` is integrable,
   - `fun x ↦ F x a` has derivative `F' a : H →L[ℝ] E` at `x₀` which is ae-measurable,
-  - `λ x, F x a` is locally Lipschitz near `x₀` for almost every `a`, with a Lipschitz bound which
-    is integrable with respect to `a`.
+  - `fun x ↦ F x a` is locally Lipschitz near `x₀` for almost every `a`,
+    with a Lipschitz bound which is integrable with respect to `a`.
 
   A subtle point is that the "near x₀" in the last condition has to be uniform in `a`. This is
   controlled by a positive number `ε`.

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>

@@ -82,9 +82,9 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜]
   set b : α → ℝ := fun a ↦ |bound a|
   have b_int : Integrable b μ := bound_integrable.norm
   have b_nonneg : ∀ a, 0 ≤ b a := fun a ↦ abs_nonneg _
-  replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
-  exact h_lipsch.mono fun a ha x hx ↦
-    (ha x hx).trans <| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _)
+  replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ :=
+    h_lipsch.mono fun a ha x hx ↦
+      (ha x hx).trans <| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _)
   have hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ := fun x x_in ↦ by
     have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by
       simp only [norm_sub_rev (F x₀ _)]
@@ -159,8 +159,8 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F' : α → H →L[𝕜] E
   obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ x ∈ ball x₀ δ, AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε
   exact eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x₀ ε_pos))
   choose hδ_meas hδε using hδ
-  replace h_lip : ∀ᵐ a : α ∂μ, ∀ x ∈ ball x₀ δ, ‖F x a - F x₀ a‖ ≤ |bound a| * ‖x - x₀‖
-  exact h_lip.mono fun a lip x hx ↦ lip.norm_sub_le (hδε x hx) (mem_ball_self ε_pos)
+  replace h_lip : ∀ᵐ a : α ∂μ, ∀ x ∈ ball x₀ δ, ‖F x a - F x₀ a‖ ≤ |bound a| * ‖x - x₀‖ :=
+    h_lip.mono fun a lip x hx ↦ lip.norm_sub_le (hδε x hx) (mem_ball_self ε_pos)
   replace bound_integrable := bound_integrable.norm
   apply hasFDerivAt_integral_of_dominated_loc_of_lip' δ_pos <;> assumption
 #align has_fderiv_at_integral_of_dominated_loc_of_lip hasFDerivAt_integral_of_dominated_loc_of_lip
@@ -214,8 +214,8 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0
     hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable
       h_diff with
     hF'_int key
-  replace hF'_int : Integrable F' μ
-  · rw [← integrable_norm_iff hm] at hF'_int
+  replace hF'_int : Integrable F' μ := by
+    rw [← integrable_norm_iff hm] at hF'_int
     simpa only [(· ∘ ·), integrable_norm_iff, hF'_meas, one_mul, norm_one,
       ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_restrict_scalarsL',
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using

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

@@ -129,7 +129,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜]
       _ = ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖F' a (x - x₀)‖ := by
         rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _
       _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := by
-        gcongr; exact (F' a).le_op_norm _
+        gcongr; exact (F' a).le_opNorm _
       _ ≤ b a + ‖F' a‖ := ?_
     simp only [← div_eq_inv_mul]
     apply_rules [add_le_add, div_le_of_nonneg_of_le_mul] <;> first | rfl | positivity

• collect some very common variables
• use refine and \mapsto instead of refine' and => (both are preferred now)

@@ -63,31 +63,32 @@ variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type*} [IsR
   [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type*}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H]
 
+variable {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ}
+
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming `F x₀` is
 integrable, `‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖` for `x` in a ball around `x₀` for ae `a` with
 integrable Lipschitz bound `bound` (with a ball radius independent of `a`), and `F x` is
 ae-measurable for `x` in the same ball. See `hasFDerivAt_integral_of_dominated_loc_of_lip` for a
 slightly less general but usually more useful version. -/
-theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
-    {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε)
+theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε)
     (hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
     (hF'_meas : AEStronglyMeasurable F' μ)
     (h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
     (bound_integrable : Integrable (bound : α → ℝ) μ)
-    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀) :
-    Integrable F' μ ∧ HasFDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
+    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a) x₀) :
+    Integrable F' μ ∧ HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
   have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
-  have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x => inv_nonneg.mpr (norm_nonneg _)
-  set b : α → ℝ := fun a => |bound a|
+  have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x ↦ inv_nonneg.mpr (norm_nonneg _)
+  set b : α → ℝ := fun a ↦ |bound a|
   have b_int : Integrable b μ := bound_integrable.norm
-  have b_nonneg : ∀ a, 0 ≤ b a := fun a => abs_nonneg _
+  have b_nonneg : ∀ a, 0 ≤ b a := fun a ↦ abs_nonneg _
   replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
-  exact h_lipsch.mono fun a ha x hx =>
+  exact h_lipsch.mono fun a ha x hx ↦
     (ha x hx).trans <| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _)
   have hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ := fun x x_in ↦ by
     have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by
       simp only [norm_sub_rev (F x₀ _)]
-      refine' h_lipsch.mono fun a ha => (ha x x_in).trans _
+      refine h_lipsch.mono fun a ha ↦ (ha x x_in).trans ?_
       rw [mul_comm ε]
       rw [mem_ball, dist_eq_norm] at x_in
       exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)
@@ -97,9 +98,9 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
     have : ∀ᵐ a ∂μ, ‖F' a‖ ≤ b a := by
       apply (h_diff.and h_lipsch).mono
       rintro a ⟨ha_diff, ha_lip⟩
-      refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)
+      exact ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)
     b_int.mono' hF'_meas this
-  refine' ⟨hF'_int, _⟩
+  refine ⟨hF'_int, ?_⟩
   have h_ball : ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos
   have : ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖((∫ a, F x a ∂μ) - ∫ a, F x₀ a ∂μ) - (∫ a, F' a ∂μ) (x - x₀)‖ =
       ‖∫ a, ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ‖ := by
@@ -135,9 +136,9 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
   · exact b_int.add hF'_int.norm
   · apply h_diff.mono
     intro a ha
-    suffices Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0) by simpa
+    suffices Tendsto (fun x ↦ ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0) by simpa
     rw [tendsto_zero_iff_norm_tendsto_zero]
-    have : (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - F' a (x - x₀)‖) = fun x =>
+    have : (fun x ↦ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - F' a (x - x₀)‖) = fun x ↦
         ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ := by
       ext x
       rw [norm_smul_of_nonneg (nneg _)]
@@ -148,18 +149,18 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
 `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
 (with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
 for `x` in a possibly smaller neighborhood of `x₀`. -/
-theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
-    {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
+theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F' : α → H →L[𝕜] E}
+    (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
     (hF_int : Integrable (F x₀) μ) (hF'_meas : AEStronglyMeasurable F' μ)
-    (h_lip : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (fun x => F x a) (ball x₀ ε))
+    (h_lip : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
     (bound_integrable : Integrable (bound : α → ℝ) μ)
-    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀) :
-    Integrable F' μ ∧ HasFDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
+    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a) x₀) :
+    Integrable F' μ ∧ HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
   obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ x ∈ ball x₀ δ, AEStronglyMeasurable (F x) μ ∧ x ∈ ball x₀ ε
   exact eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x₀ ε_pos))
   choose hδ_meas hδε using hδ
   replace h_lip : ∀ᵐ a : α ∂μ, ∀ x ∈ ball x₀ δ, ‖F x a - F x₀ a‖ ≤ |bound a| * ‖x - x₀‖
-  exact h_lip.mono fun a lip x hx => lip.norm_sub_le (hδε x hx) (mem_ball_self ε_pos)
+  exact h_lip.mono fun a lip x hx ↦ lip.norm_sub_le (hδε x hx) (mem_ball_self ε_pos)
   replace bound_integrable := bound_integrable.norm
   apply hasFDerivAt_integral_of_dominated_loc_of_lip' δ_pos <;> assumption
 #align has_fderiv_at_integral_of_dominated_loc_of_lip hasFDerivAt_integral_of_dominated_loc_of_lip
@@ -168,44 +169,46 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip {F : H → α → E} {F' :
 `F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
 derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
 and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
-theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F : H → α → E} {F' : H → α → H →L[𝕜] E}
-    {x₀ : H} {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε)
+theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F' : H → α → H →L[𝕜] E} (ε_pos : 0 < ε)
     (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
     (hF'_meas : AEStronglyMeasurable (F' x₀) μ)
     (h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a)
     (bound_integrable : Integrable (bound : α → ℝ) μ)
-    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x a) (F' x a) x) :
-    HasFDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ := by
+    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (F · a) (F' x a) x) :
+    HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ := by
   letI : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H
   have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
-  have diff_x₀ : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀ :=
-    h_diff.mono fun a ha => ha x₀ x₀_in
-  have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε) := by
+  have diff_x₀ : ∀ᵐ a ∂μ, HasFDerivAt (fun x ↦ F x a) (F' x₀ a) x₀ :=
+    h_diff.mono fun a ha ↦ ha x₀ x₀_in
+  have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (F · a) (ball x₀ ε) := by
     apply (h_diff.and h_bound).mono
     rintro a ⟨ha_deriv, ha_bound⟩
-    refine'
-      (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
-        (fun x x_in => (ha_deriv x x_in).hasFDerivWithinAt) fun x x_in => _
+    refine (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
+      (fun x x_in ↦ (ha_deriv x x_in).hasFDerivWithinAt) fun x x_in ↦ ?_
     rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]
     exact (ha_bound x x_in).trans (le_abs_self _)
   exact (hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this
     bound_integrable diff_x₀).2
 #align has_fderiv_at_integral_of_dominated_of_fderiv_le hasFDerivAt_integral_of_dominated_of_fderiv_le
 
+section
+
+variable {F : 𝕜 → α → E} {x₀ : 𝕜}
+
 /-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : 𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
 assuming `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
 (with ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is
 ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
-theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F' : α → E} {x₀ : 𝕜} {ε : ℝ}
-    (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
-    (hF_int : Integrable (F x₀) μ) (hF'_meas : AEStronglyMeasurable F' μ) {bound : α → ℝ}
-    (h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (fun x => F x a) (ball x₀ ε))
+theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0 < ε)
+    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
+    (hF'_meas : AEStronglyMeasurable F' μ)
+    (h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
     (bound_integrable : Integrable (bound : α → ℝ) μ)
-    (h_diff : ∀ᵐ a ∂μ, HasDerivAt (fun x => F x a) (F' a) x₀) :
-    Integrable F' μ ∧ HasDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
+    (h_diff : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' a) x₀) :
+    Integrable F' μ ∧ HasDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
   set L : E →L[𝕜] 𝕜 →L[𝕜] E := ContinuousLinearMap.smulRightL 𝕜 𝕜 E 1
-  replace h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀ :=
-    h_diff.mono fun x hx => hx.hasFDerivAt
+  replace h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (L (F' a)) x₀ :=
+    h_diff.mono fun x hx ↦ hx.hasFDerivAt
   have hm : AEStronglyMeasurable (L ∘ F') μ := L.continuous.comp_aestronglyMeasurable hF'_meas
   cases'
     hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable
@@ -217,7 +220,7 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
       ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_restrict_scalarsL',
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using
       hF'_int
-  refine' ⟨hF'_int, _⟩
+  refine ⟨hF'_int, ?_⟩
   simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff ⊢
   simpa only [(· ∘ ·), ContinuousLinearMap.integral_comp_comm _ hF'_int] using key
 #align has_deriv_at_integral_of_dominated_loc_of_lip hasDerivAt_integral_of_dominated_loc_of_lip
@@ -226,23 +229,25 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
 `F x₀` is integrable, `x ↦ F x a` is differentiable on an interval around `x₀` for ae `a`
 (with interval radius independent of `a`) with derivative uniformly bounded by an integrable
 function, and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
-theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le {F : 𝕜 → α → E} {F' : 𝕜 → α → E} {x₀ : 𝕜}
-    {ε : ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
-    (hF_int : Integrable (F x₀) μ) (hF'_meas : AEStronglyMeasurable (F' x₀) μ) {bound : α → ℝ}
+theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le (ε_pos : 0 < ε)
+    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
+    {F' : 𝕜 → α → E} (hF'_meas : AEStronglyMeasurable (F' x₀) μ)
     (h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
-    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasDerivAt (fun x => F x a) (F' x a) x) :
-    Integrable (F' x₀) μ ∧ HasDerivAt (fun n => ∫ a, F n a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ := by
+    (h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasDerivAt (F · a) (F' x a) x) :
+    Integrable (F' x₀) μ ∧ HasDerivAt (fun n ↦ ∫ a, F n a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ := by
   have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
-  have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ :=
-    h_diff.mono fun a ha => ha x₀ x₀_in
-  have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x : 𝕜 => F x a) (ball x₀ ε) := by
+  have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' x₀ a) x₀ :=
+    h_diff.mono fun a ha ↦ ha x₀ x₀_in
+  have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x : 𝕜 ↦ F x a) (ball x₀ ε) := by
     apply (h_diff.and h_bound).mono
     rintro a ⟨ha_deriv, ha_bound⟩
-    refine' (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
-      (fun x x_in => (ha_deriv x x_in).hasDerivWithinAt) fun x x_in => _
+    refine (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
+      (fun x x_in ↦ (ha_deriv x x_in).hasDerivWithinAt) fun x x_in ↦ ?_
     rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]
     exact (ha_bound x x_in).trans (le_abs_self _)
   exact
     hasDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this bound_integrable
       diff_x₀
 #align has_deriv_at_integral_of_dominated_loc_of_deriv_le hasDerivAt_integral_of_dominated_loc_of_deriv_le
+
+end

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

This has nice performance benefits.

@@ -59,8 +59,8 @@ open TopologicalSpace MeasureTheory Filter Metric
 
 open scoped Topology Filter
 
-variable {α : Type _} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type _} [IsROrC 𝕜] {E : Type _}
-  [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type _}
+variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {𝕜 : Type*} [IsROrC 𝕜] {E : Type*}
+  [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type*}
   [NormedAddCommGroup H] [NormedSpace 𝕜 H]
 
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming `F x₀` is

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot
-
-! This file was ported from Lean 3 source module analysis.calculus.parametric_integral
-! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Calculus.MeanValue
 import Mathlib.MeasureTheory.Integral.SetIntegral
 
+#align_import analysis.calculus.parametric_integral from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
+
 /-!
 # Derivatives of integrals depending on parameters
 

@@ -17,8 +17,8 @@ import Mathlib.MeasureTheory.Integral.SetIntegral
 A parametric integral is a function with shape `f = λ x : H, ∫ a : α, F x a ∂μ` for some
 `F : H → α → E`, where `H` and `E` are normed spaces and `α` is a measured space with measure `μ`.
 
-We already know from `continuous_of_dominated` in `measure_theory.integral.bochner` how to
-guarantee that `f` is continuous using the dominated convergence theorem. In this file,
+We already know from `continuous_of_dominated` in `Mathlib/MeasureTheory/Integral/Bochner.lean` how
+to guarantee that `f` is continuous using the dominated convergence theorem. In this file,
 we want to express the derivative of `f` as the integral of the derivative of `F` with respect
 to `x`.
 
@@ -92,7 +92,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
       simp only [norm_sub_rev (F x₀ _)]
       refine' h_lipsch.mono fun a ha => (ha x x_in).trans _
       rw [mul_comm ε]
-      rw [mem_ball, dist_eq_norm] at x_in 
+      rw [mem_ball, dist_eq_norm] at x_in
       exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)
     exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int
       (bound_integrable.norm.const_mul ε) this
@@ -144,7 +144,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' :
         ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ := by
       ext x
       rw [norm_smul_of_nonneg (nneg _)]
-    rwa [hasFDerivAt_iff_tendsto, this] at ha 
+    rwa [hasFDerivAt_iff_tendsto, this] at ha
 #align has_fderiv_at_integral_of_dominated_loc_of_lip' hasFDerivAt_integral_of_dominated_loc_of_lip'
 
 /-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
@@ -215,7 +215,7 @@ theorem hasDerivAt_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F'
       h_diff with
     hF'_int key
   replace hF'_int : Integrable F' μ
-  · rw [← integrable_norm_iff hm] at hF'_int 
+  · rw [← integrable_norm_iff hm] at hF'_int
     simpa only [(· ∘ ·), integrable_norm_iff, hF'_meas, one_mul, norm_one,
       ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_restrict_scalarsL',
       ContinuousLinearMap.norm_restrictScalars, ContinuousLinearMap.norm_smulRightL_apply] using
@@ -249,4 +249,3 @@ theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le {F : 𝕜 → α → E}
     hasDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this bound_integrable
       diff_x₀
 #align has_deriv_at_integral_of_dominated_loc_of_deriv_le hasDerivAt_integral_of_dominated_loc_of_deriv_le
-

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

@@ -33,14 +33,14 @@ variable.
 * `hasFDerivAt_integral_of_dominated_loc_of_lip`: this version assumes that
   - `F x` is ae-measurable for x near `x₀`,
   - `F x₀` is integrable,
-  - `λ x, F x a` has derivative `F' a : H →L[ℝ] E` at `x₀` which is ae-measurable,
+  - `fun x ↦ F x a` has derivative `F' a : H →L[ℝ] E` at `x₀` which is ae-measurable,
   - `λ x, F x a` is locally Lipschitz near `x₀` for almost every `a`, with a Lipschitz bound which
     is integrable with respect to `a`.
 
   A subtle point is that the "near x₀" in the last condition has to be uniform in `a`. This is
   controlled by a positive number `ε`.
 
-* `hasFDerivAt_integral_of_dominated_of_fderiv_le`: this version assume `λ x, F x a` has
+* `hasFDerivAt_integral_of_dominated_of_fderiv_le`: this version assume `fun x ↦ F x a` has
    derivative `F' x a` for `x` near `x₀` and `F' x` is bounded by an integrable function independent
    from `x` near `x₀`.