Loogle!

Result

Found 6 declarations mentioning ContinuousWithinAt and intervalIntegral.

• intervalIntegral.continuousWithinAt_primitive 📋 Mathlib.MeasureTheory.Integral.DominatedConvergence
  {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b₀ b₁ b₂ : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} (hb₀ : μ {b₀} = 0) (h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)) : ContinuousWithinAt (fun b => ∫ (x : ℝ) in a..b, f x ∂μ) (Set.Icc b₁ b₂) b₀
• intervalIntegral.continuousWithinAt_of_dominated_interval 📋 Mathlib.MeasureTheory.Integral.DominatedConvergence
  {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} {X : Type u_3} [TopologicalSpace X] [FirstCountableTopology X] {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ} {s : Set X} (hF_meas : ∀ᶠ (x : X) in nhdsWithin x₀ s, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Set.uIoc a b))) (h_bound : ∀ᶠ (x : X) in nhdsWithin x₀ s, ∀ᵐ (t : ℝ) ∂μ, t ∈ Set.uIoc a b → ‖F x t‖ ≤ bound t) (bound_integrable : IntervalIntegrable bound μ a b) (h_cont : ∀ᵐ (t : ℝ) ∂μ, t ∈ Set.uIoc a b → ContinuousWithinAt (fun x => F x t) s x₀) : ContinuousWithinAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) s x₀
• intervalIntegral.derivWithin_integral_right 📋 Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
  {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) {s t : Set ℝ} [intervalIntegral.FTCFilter b (nhdsWithin b s) (nhdsWithin b t)] (hmeas : StronglyMeasurableAtFilter f (nhdsWithin b t) MeasureTheory.volume) (hb : ContinuousWithinAt f t b) (hs : UniqueDiffWithinAt ℝ s b := by uniqueDiffWithinAt_Ici_Iic_univ) : derivWithin (fun u => ∫ (x : ℝ) in a..u, f x) s b = f b
• intervalIntegral.derivWithin_integral_left 📋 Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
  {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) {s t : Set ℝ} [intervalIntegral.FTCFilter a (nhdsWithin a s) (nhdsWithin a t)] (hmeas : StronglyMeasurableAtFilter f (nhdsWithin a t) MeasureTheory.volume) (ha : ContinuousWithinAt f t a) (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ) : derivWithin (fun u => ∫ (x : ℝ) in u..b, f x) s a = -f a
• intervalIntegral.integral_hasDerivWithinAt_right 📋 Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
  {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) {s t : Set ℝ} [intervalIntegral.FTCFilter b (nhdsWithin b s) (nhdsWithin b t)] (hmeas : StronglyMeasurableAtFilter f (nhdsWithin b t) MeasureTheory.volume) (hb : ContinuousWithinAt f t b) : HasDerivWithinAt (fun u => ∫ (x : ℝ) in a..u, f x) (f b) s b
• intervalIntegral.integral_hasDerivWithinAt_left 📋 Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
  {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) {s t : Set ℝ} [intervalIntegral.FTCFilter a (nhdsWithin a s) (nhdsWithin a t)] (hmeas : StronglyMeasurableAtFilter f (nhdsWithin a t) MeasureTheory.volume) (ha : ContinuousWithinAt f t a) : HasDerivWithinAt (fun u => ∫ (x : ℝ) in u..b, f x) (-f a) s a

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision c3b95a7