| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| namespace Problem358 |
|
|
| /-- Translation of a finite difference controls its squared interval integral |
| by the squared integral of the original function. -/ |
| theorem forwardDifference_sq_control |
| {f : ℝ → ℝ} {a b h : ℝ} (hh : 0 < h) (habh : h ≤ b - a) |
| (hf : IntervalIntegrable (fun t => (f t) ^ 2) volume a b) |
| (hfmeas : AEStronglyMeasurable f volume) : |
| IntervalIntegrable (fun t => ((f (t + h) - f t) / h) ^ 2) volume a (b - h) ∧ |
| (∫ t in a..b - h, ((f (t + h) - f t) / h) ^ 2) ≤ |
| 4 / h ^ 2 * ∫ t in a..b, (f t) ^ 2 := by |
| have hab : a ≤ b := by linarith |
| have hsmall : a ≤ b - h := by linarith |
| have hleft : a ≤ a + h := by linarith |
| have hright : a + h ≤ b := by linarith |
| have htail : b - h ≤ b := by linarith |
| have hsub {c d : ℝ} (hac : a ≤ c) (hcd : c ≤ d) (hdb : d ≤ b) : |
| IntervalIntegrable (fun t => (f t) ^ 2) volume c d := by |
| apply hf.mono_set |
| rw [uIcc_of_le hcd, uIcc_of_le hab] |
| exact Icc_subset_Icc hac hdb |
| have hshift : IntervalIntegrable (fun t => (f (t + h)) ^ 2) volume a (b - h) := by |
| simpa only [add_sub_cancel_right] using |
| (hsub hleft hright le_rfl).comp_add_right h |
| have hbase := hsub le_rfl hsmall htail |
| have hpoint (t : ℝ) : |
| ((f (t + h) - f t) / h) ^ 2 ≤ |
| (2 / h ^ 2) * ((f (t + h)) ^ 2 + (f t) ^ 2) := by |
| rw [div_pow] |
| apply (div_le_iff₀ (sq_pos_of_pos hh)).mpr |
| field_simp |
| nlinarith [sq_nonneg (f (t + h) + f t)] |
| have hboundInt := (hshift.add hbase).const_mul (2 / h ^ 2) |
| have hmeas : AEStronglyMeasurable |
| (fun t => ((f (t + h) - f t) / h) ^ 2) volume := by |
| have hs := hfmeas.comp_measurePreserving (measurePreserving_add_right volume h) |
| have hd : AEStronglyMeasurable (fun t => (f (t + h) - f t) / h) volume := by |
| simpa only [div_eq_mul_inv, Pi.sub_apply, Function.comp_apply] using |
| (hs.sub hfmeas).mul_const h⁻¹ |
| exact hd.fun_pow 2 |
| have hint : IntervalIntegrable |
| (fun t => ((f (t + h) - f t) / h) ^ 2) volume a (b - h) := by |
| apply hboundInt.mono_fun' hmeas.restrict |
| filter_upwards [] with t |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact hpoint t |
| refine ⟨hint, ?_⟩ |
| have hi := intervalIntegral.integral_mono_on hsmall hint hboundInt |
| (fun t _ => hpoint t) |
| rw [intervalIntegral.integral_const_mul, |
| intervalIntegral.integral_add hshift hbase, |
| intervalIntegral.integral_comp_add_right (fun t => (f t) ^ 2) h] at hi |
| simp only [sub_add_cancel] at hi |
| have hnonneg : 0 ≤ᵐ[volume.restrict (Ioc a b)] (fun t => (f t) ^ 2) := |
| Filter.Eventually.of_forall fun t => sq_nonneg (f t) |
| have h₁ := intervalIntegral.integral_mono_interval hleft hright le_rfl hnonneg hf |
| have h₂ := intervalIntegral.integral_mono_interval le_rfl hsmall htail hnonneg hf |
| have hcoef : 0 ≤ 2 / h ^ 2 := by positivity |
| have hj := mul_le_mul_of_nonneg_left (add_le_add h₁ h₂) hcoef |
| calc |
| _ ≤ (2 / h ^ 2) * ((∫ t in a + h..b, (f t) ^ 2) + |
| ∫ t in a..b - h, (f t) ^ 2) := hi |
| _ ≤ (2 / h ^ 2) * ((∫ t in a..b, (f t) ^ 2) + |
| ∫ t in a..b, (f t) ^ 2) := hj |
| _ = _ := by ring |
| |
| /-- Square-integrability of a forward difference quotient on the shortened interval. -/ |
| theorem intervalIntegrable_forwardDifference_sq |
| {f : ℝ → ℝ} {a b h : ℝ} (hh : 0 < h) (habh : h ≤ b - a) |
| (hf : IntervalIntegrable (fun t => (f t) ^ 2) volume a b) |
| (hfmeas : AEStronglyMeasurable f volume) : |
| IntervalIntegrable (fun t => ((f (t + h) - f t) / h) ^ 2) volume a (b - h) := |
| (forwardDifference_sq_control hh habh hf hfmeas).1 |
| |
| /-- The squared norm of a forward difference is bounded by four times the |
| squared norm of the function, divided by the squared increment. -/ |
| theorem integral_forwardDifference_sq_le |
| {f : ℝ → ℝ} {a b h : ℝ} (hh : 0 < h) (habh : h ≤ b - a) |
| (hf : IntervalIntegrable (fun t => (f t) ^ 2) volume a b) |
| (hfmeas : AEStronglyMeasurable f volume) : |
| (∫ t in a..b - h, ((f (t + h) - f t) / h) ^ 2) ≤ |
| 4 / h ^ 2 * ∫ t in a..b, (f t) ^ 2 := |
| (forwardDifference_sq_control hh habh hf hfmeas).2 |
| |
| end Problem358 |
| |
| end |
| |
| /-! |
| # Pointwise estimates for convex forward difference quotients |
| |
| These are the pointwise ingredients of the coordinate-slice gradient |
| interpolation argument. They do not assert an almost-everywhere derivative |
| existence result or any of the transport theorems. |
| -/ |
| |
| noncomputable section |
| |
| namespace Problem358 |
| |
| /-- Forward difference quotient with step `h`. -/ |
| def forwardQuotient (u : ℝ → ℝ) (h x : ℝ) : ℝ := |
| (u (x + h) - u x) / h |
| |
| /-- Convexity sandwiches a positive-step quotient between the two endpoint |
| derivatives. -/ |
| theorem convex_forwardQuotient_bounds {S : Set ℝ} {u : ℝ → ℝ} |
| {x h p q : ℝ} (hu : ConvexOn ℝ S u) |
| (hx : x ∈ S) (hxh : x + h ∈ S) (hh : 0 < h) |
| (hp : HasDerivAt u p x) (hq : HasDerivAt u q (x + h)) : |
| p ≤ forwardQuotient u h x ∧ forwardQuotient u h x ≤ q := by |
| have hlt : x < x + h := by linarith |
| constructor |
| · simpa [forwardQuotient, slope_def_field] using |
| hu.le_slope_of_hasDerivAt hx hxh hlt hp |
| · simpa [forwardQuotient, slope_def_field] using |
| hu.slope_le_of_hasDerivAt hx hxh hlt hq |
| |
| /-- The nonnegative quotient error is dominated by the derivative increment, |
| which is at most twice the derivative bound. -/ |
| theorem convex_forwardQuotient_error_bounds {S : Set ℝ} {u : ℝ → ℝ} |
| {x h p q L : ℝ} (hu : ConvexOn ℝ S u) |
| (hx : x ∈ S) (hxh : x + h ∈ S) (hh : 0 < h) |
| (hp : HasDerivAt u p x) (hq : HasDerivAt u q (x + h)) |
| (hpL : |p| ≤ L) (hqL : |q| ≤ L) : |
| 0 ≤ forwardQuotient u h x - p ∧ |
| forwardQuotient u h x - p ≤ q - p ∧ q - p ≤ 2 * L := by |
| obtain ⟨hlo, hhi⟩ := convex_forwardQuotient_bounds hu hx hxh hh hp hq |
| obtain ⟨hpLlo, hpLhi⟩ := abs_le.mp hpL |
| obtain ⟨hqLlo, hqLhi⟩ := abs_le.mp hqL |
| exact ⟨by linarith, by linarith, by linarith⟩ |
| |
| /-- Squared quotient error is controlled by the derivative increment. This |
| is the integrand estimate used before translation cancellation on a slice. -/ |
| theorem convex_forwardQuotient_error_sq_le {S : Set ℝ} {u : ℝ → ℝ} |
| {x h p q L : ℝ} (hu : ConvexOn ℝ S u) |
| (hx : x ∈ S) (hxh : x + h ∈ S) (hh : 0 < h) |
| (hp : HasDerivAt u p x) (hq : HasDerivAt u q (x + h)) |
| (hpL : |p| ≤ L) (hqL : |q| ≤ L) : |
| (forwardQuotient u h x - p) ^ 2 ≤ 2 * L * (q - p) := by |
| obtain ⟨he0, heq, hqL'⟩ := |
| convex_forwardQuotient_error_bounds hu hx hxh hh hp hq hpL hqL |
| have hL : 0 ≤ L := (abs_nonneg p).trans hpL |
| have heL : forwardQuotient u h x - p ≤ 2 * L := heq.trans hqL' |
| calc |
| (forwardQuotient u h x - p) ^ 2 ≤ |
| 2 * L * (forwardQuotient u h x - p) := by |
| nlinarith |
| _ ≤ 2 * L * (q - p) := mul_le_mul_of_nonneg_left heq (by positivity) |
| |
| /-- Forward quotients commute with subtraction. -/ |
| theorem forwardQuotient_sub (u v : ℝ → ℝ) (h x : ℝ) : |
| forwardQuotient (fun t => u t - v t) h x = |
| forwardQuotient u h x - forwardQuotient v h x := by |
| unfold forwardQuotient |
| ring |
| |
| /-- Pointwise decomposition bound for the difference of two derivatives. |
| The two last terms are the respective forward quotient errors. -/ |
| theorem derivative_difference_sq_le_quotient_errors |
| (u v : ℝ → ℝ) (h x p q : ℝ) : |
| (p - q) ^ 2 ≤ 3 * |
| ((forwardQuotient (fun t => u t - v t) h x) ^ 2 + |
| (forwardQuotient u h x - p) ^ 2 + |
| (forwardQuotient v h x - q) ^ 2) := by |
| rw [forwardQuotient_sub] |
| nlinarith [sq_nonneg (forwardQuotient u h x - forwardQuotient v h x + |
| (forwardQuotient u h x - p)), |
| sq_nonneg (forwardQuotient u h x - forwardQuotient v h x - |
| (forwardQuotient v h x - q)), |
| sq_nonneg ((forwardQuotient u h x - p) + (forwardQuotient v h x - q))] |
| |
| /-- A difference quotient is bounded by the two endpoint function values. -/ |
| theorem forwardQuotient_sq_le (u : ℝ → ℝ) (h x : ℝ) : |
| (forwardQuotient u h x) ^ 2 ≤ |
| 2 / h ^ 2 * ((u (x + h)) ^ 2 + (u x) ^ 2) := by |
| unfold forwardQuotient |
| have hsq : (u (x + h) - u x) ^ 2 ≤ |
| 2 * ((u (x + h)) ^ 2 + (u x) ^ 2) := by |
| nlinarith [sq_nonneg (u (x + h) + u x)] |
| have := div_le_div_of_nonneg_right hsq (sq_nonneg h) |
| rw [div_pow] |
| calc |
| (u (x + h) - u x) ^ 2 / h ^ 2 ≤ |
| (2 * ((u (x + h)) ^ 2 + (u x) ^ 2)) / h ^ 2 := this |
| _ = _ := by ring |
| |
| /-- Combined pointwise estimate for two convex functions. Integrating the |
| last two derivative increments is the cancellation step of the slice proof. -/ |
| theorem convex_derivative_difference_sq_le {S : Set ℝ} {u v : ℝ → ℝ} |
| {x h p₀ p₁ q₀ q₁ L : ℝ} |
| (hu : ConvexOn ℝ S u) (hv : ConvexOn ℝ S v) |
| (hx : x ∈ S) (hxh : x + h ∈ S) (hh : 0 < h) |
| (hp₀ : HasDerivAt u p₀ x) (hq₀ : HasDerivAt u q₀ (x + h)) |
| (hp₁ : HasDerivAt v p₁ x) (hq₁ : HasDerivAt v q₁ (x + h)) |
| (hp₀L : |p₀| ≤ L) (hq₀L : |q₀| ≤ L) |
| (hp₁L : |p₁| ≤ L) (hq₁L : |q₁| ≤ L) : |
| (p₀ - p₁) ^ 2 ≤ |
| 6 / h ^ 2 * ((u (x + h) - v (x + h)) ^ 2 + (u x - v x) ^ 2) + |
| 6 * L * ((q₀ - p₀) + (q₁ - p₁)) := by |
| have he₀ := convex_forwardQuotient_error_sq_le hu hx hxh hh hp₀ hq₀ hp₀L hq₀L |
| have he₁ := convex_forwardQuotient_error_sq_le hv hx hxh hh hp₁ hq₁ hp₁L hq₁L |
| have hq := forwardQuotient_sq_le (fun t => u t - v t) h x |
| calc |
| (p₀ - p₁) ^ 2 ≤ 3 * |
| ((forwardQuotient (fun t => u t - v t) h x) ^ 2 + |
| (forwardQuotient u h x - p₀) ^ 2 + |
| (forwardQuotient v h x - p₁) ^ 2) := |
| derivative_difference_sq_le_quotient_errors u v h x p₀ p₁ |
| _ ≤ 3 * (2 / h ^ 2 * |
| ((u (x + h) - v (x + h)) ^ 2 + (u x - v x) ^ 2) + |
| 2 * L * (q₀ - p₀) + 2 * L * (q₁ - p₁)) := by |
| exact mul_le_mul_of_nonneg_left (add_le_add (add_le_add hq he₀) he₁) (by norm_num) |
| _ = _ := by ring |
| |
| end Problem358 |
| |
| open MeasureTheory MeasureTheory.Measure |
| |
| namespace Problem358 |
| |
| variable {G : Type*} [AddGroup G] [MeasurableSpace G] [MeasurableAdd G] |
| variable {μ : Measure G} [IsAddRightInvariant μ] |
| |
| /-- A measurable square-integrable function has a square-integrable translate |
| difference. -/ |
| theorem integrable_translation_sub_sq (f : G → ℝ) (v : G) (hfm : Measurable f) |
| (hf : Integrable (fun x => (f x) ^ 2) μ) : |
| Integrable (fun x => (f (x + v) - f x) ^ 2) μ := by |
| have hshift : Integrable (fun x => (f (x + v)) ^ 2) μ := |
| (measurePreserving_add_right μ v).integrable_comp_of_integrable hf |
| have hu := (hshift.const_mul 2).add (hf.const_mul 2) |
| have hm : Measurable (fun x => (f (x + v) - f x) ^ 2) := by fun_prop |
| apply hu.mono' hm.aestronglyMeasurable |
| exact ae_of_all _ fun x => by |
| change ‖(f (x + v) - f x) ^ 2‖ ≤ 2 * (f (x + v)) ^ 2 + 2 * (f x) ^ 2 |
| rw [Real.norm_of_nonneg (sq_nonneg _)] |
| nlinarith [sq_nonneg (f (x + v) + f x)] |
|
|
| /-- Local square-integrability on the common translation domain. -/ |
| theorem integrableOn_translation_sub_sq (K : Set G) (hK : MeasurableSet K) |
| (f : G → ℝ) (v : G) (hfm : Measurable f) |
| (hf : IntegrableOn (fun x => (f x) ^ 2) K μ) : |
| IntegrableOn (fun x => (f (x + v) - f x) ^ 2) |
| (K ∩ (fun x => x + v) ⁻¹' K) μ := by |
| let g := K.indicator f |
| have hg2 : (fun x => (g x) ^ 2) = K.indicator (fun x => (f x) ^ 2) := by |
| funext x |
| by_cases hx : x ∈ K <;> simp [g, hx] |
| have hg : Integrable (fun x => (g x) ^ 2) μ := by |
| rw [hg2] |
| exact hf.integrable_indicator hK |
| have hgi := integrable_translation_sub_sq g v (hfm.indicator hK) hg |
| apply hgi.restrict.congr |
| filter_upwards [ae_restrict_mem (hK.inter (hK.preimage (measurable_id.add_const v)))] with x hx |
| have hx1 : x + v ∈ K := hx.2 |
| simp [g, hx.1, hx1] |
|
|
| /-- Local square-integrability of the divided finite difference. -/ |
| theorem integrableOn_translation_quotient_sq (K : Set G) (hK : MeasurableSet K) |
| (f : G → ℝ) (v : G) (h : ℝ) (hfm : Measurable f) |
| (hf : IntegrableOn (fun x => (f x) ^ 2) K μ) : |
| IntegrableOn (fun x => ((f (x + v) - f x) / h) ^ 2) |
| (K ∩ (fun x => x + v) ⁻¹' K) μ := by |
| simpa only [IntegrableOn, div_pow] using (integrableOn_translation_sub_sq K hK f v hfm hf).div_const (h ^ 2) |
|
|
| /-- Translation does not increase the squared `L²` finite difference by more |
| than four times the original squared norm. No differentiability is required. -/ |
| theorem integral_translation_sub_sq_le (f : G → ℝ) (v : G) |
| (hf : Integrable (fun x => (f x) ^ 2) μ) : |
| (∫ x, (f (x + v) - f x) ^ 2 ∂μ) ≤ 4 * ∫ x, (f x) ^ 2 ∂μ := by |
| have hshift : Integrable (fun x => (f (x + v)) ^ 2) μ := |
| (measurePreserving_add_right μ v).integrable_comp_of_integrable hf |
| have hu := (hshift.const_mul 2).add (hf.const_mul 2) |
| calc |
| (∫ x, (f (x + v) - f x) ^ 2 ∂μ) ≤ |
| ∫ x, 2 * (f (x + v)) ^ 2 + 2 * (f x) ^ 2 ∂μ := by |
| apply integral_mono_of_nonneg (ae_of_all _ fun x => sq_nonneg _) hu |
| exact ae_of_all _ fun x => by |
| change (f (x + v) - f x) ^ 2 ≤ 2 * (f (x + v)) ^ 2 + 2 * (f x) ^ 2 |
| nlinarith [sq_nonneg (f (x + v) + f x)] |
| _ = 4 * ∫ x, (f x) ^ 2 ∂μ := by |
| rw [integral_add (hshift.const_mul 2) (hf.const_mul 2), |
| integral_const_mul, integral_const_mul, |
| integral_add_right_eq_self (fun x => (f x) ^ 2) v] |
| ring |
|
|
| /-- The finite-difference contribution on the set where both endpoints belong |
| to `K` is controlled by the potential's squared norm on `K`. -/ |
| theorem setIntegral_translation_sub_sq_le (K : Set G) (hK : MeasurableSet K) |
| (f : G → ℝ) (v : G) (hf : IntegrableOn (fun x => (f x) ^ 2) K μ) : |
| (∫ x in K ∩ (fun x => x + v) ⁻¹' K, (f (x + v) - f x) ^ 2 ∂μ) ≤ |
| 4 * ∫ x in K, (f x) ^ 2 ∂μ := by |
| let g := K.indicator f |
| have hg2 : (fun x => (g x) ^ 2) = K.indicator (fun x => (f x) ^ 2) := by |
| funext x |
| by_cases hx : x ∈ K <;> simp [g, hx] |
| have hg : Integrable (fun x => (g x) ^ 2) μ := by |
| rw [hg2] |
| exact hf.integrable_indicator hK |
| have hshift : Integrable (fun x => (g (x + v)) ^ 2) μ := |
| (measurePreserving_add_right μ v).integrable_comp_of_integrable hg |
| let S := K ∩ (fun x => x + v) ⁻¹' K |
| have hS : MeasurableSet S := hK.inter (hK.preimage (measurable_id.add_const v)) |
| have hu := (hshift.const_mul 2).add (hg.const_mul 2) |
| calc |
| (∫ x in S, (f (x + v) - f x) ^ 2 ∂μ) ≤ |
| ∫ x in S, 2 * (g (x + v)) ^ 2 + 2 * (g x) ^ 2 ∂μ := by |
| apply integral_mono_of_nonneg (ae_of_all _ fun x => sq_nonneg _) hu.restrict |
| filter_upwards [ae_restrict_mem hS] with x hx |
| have hx0 : x ∈ K := hx.1 |
| have hx1 : x + v ∈ K := hx.2 |
| change (f (x + v) - f x) ^ 2 ≤ 2 * (g (x + v)) ^ 2 + 2 * (g x) ^ 2 |
| simp only [g, Set.indicator_of_mem hx0, Set.indicator_of_mem hx1] |
| nlinarith [sq_nonneg (f (x + v) + f x)] |
| _ ≤ ∫ x, 2 * (g (x + v)) ^ 2 + 2 * (g x) ^ 2 ∂μ := by |
| exact setIntegral_le_integral hu (ae_of_all _ fun x => by positivity) |
| _ = 4 * ∫ x, (g x) ^ 2 ∂μ := by |
| rw [integral_add (hshift.const_mul 2) (hg.const_mul 2), |
| integral_const_mul, integral_const_mul, |
| integral_add_right_eq_self (fun x => (g x) ^ 2) v] |
| ring |
| _ = 4 * ∫ x in K, (f x) ^ 2 ∂μ := by |
| rw [hg2, integral_indicator hK] |
|
|
| /-- The divided finite-difference estimate used in the gradient interpolation |
| argument. It is valid for every real scale (including the harmless zero case). -/ |
| theorem setIntegral_translation_quotient_sq_le (K : Set G) (hK : MeasurableSet K) |
| (f : G → ℝ) (v : G) (h : ℝ) |
| (hf : IntegrableOn (fun x => (f x) ^ 2) K μ) : |
| (∫ x in K ∩ (fun x => x + v) ⁻¹' K, ((f (x + v) - f x) / h) ^ 2 ∂μ) ≤ |
| (4 / h ^ 2) * ∫ x in K, (f x) ^ 2 ∂μ := by |
| simp_rw [div_pow] |
| rw [integral_div] |
| calc |
| (∫ x in K ∩ (fun x => x + v) ⁻¹' K, (f (x + v) - f x) ^ 2 ∂μ) / h ^ 2 ≤ |
| (4 * ∫ x in K, (f x) ^ 2 ∂μ) / h ^ 2 := |
| div_le_div_of_nonneg_right (setIntegral_translation_sub_sq_le K hK f v hf) (sq_nonneg h) |
| _ = (4 / h ^ 2) * ∫ x in K, (f x) ^ 2 ∂μ := by ring |
|
|
| end Problem358 |
|
|
| end |
|
|
| end OAI |
|
|
