import OAI.Analysis.BackwardIntertwiners.ConvolutionBounds namespace OAI namespace BackwardIntertwiners open Filter open scoped Topology noncomputable section def coefficientObstacle (d : ℕ → ℂ) : Set ℕ := {n | 0 < n ∧ d n ≠ 0} def coefficientEnvelope (d : ℕ → ℂ) : ℕ → ℝ := convexEnvelope (coefficientObstacle d) (fun n => -Real.log ‖d n‖) namespace CoefficientEnvelope variable {d : ℕ → ℂ} {G : ℕ → ℝ} variable (hu : ∀ n : ℕ, ∃ k, n < k ∧ d k ≠ 0) variable (hG : DiscreteConvex G) (hG0 : G 0 = 0) variable (hb : ∀ n, ‖d n‖ ≤ Real.exp (-G n)) include hu in lemma unbounded : ∀ n : ℕ, ∃ k, n < k ∧ k ∈ coefficientObstacle d := by intro n obtain ⟨k, hk, hd⟩ := hu n exact ⟨k, hk, by exact ⟨by omega, hd⟩⟩ include hG hG0 hb in lemma minorant : G ∈ convexMinorants (coefficientObstacle d) (fun n => -Real.log ‖d n‖) := by refine ⟨hG0, hG, ?_⟩ intro n hn have hn0 : 0 < ‖d n‖ := norm_pos_iff.mpr hn.2 have hh := Real.log_le_log hn0 (hb n) rw [Real.log_exp] at hh linarith include hu hG hG0 hb in lemma properties : coefficientEnvelope d 0 = 0 ∧ DiscreteConvex (coefficientEnvelope d) ∧ (∀ n, G n ≤ coefficientEnvelope d n) ∧ (∀ n, ‖d n‖ ≤ Real.exp (-coefficientEnvelope d n)) ∧ (∀ n, 0 < coefficientEnvelope d (n+2) - 2*coefficientEnvelope d (n+1) + coefficientEnvelope d n → ‖d (n+1)‖ = Real.exp (-coefficientEnvelope d (n+1))) := by have hm := minorant hG hG0 hb have he : (convexMinorants (coefficientObstacle d) (fun n => -Real.log ‖d n‖)).Nonempty := ⟨G, hm⟩ have hU := unbounded hu refine ⟨ConvexEnvelope.zero hU he, ConvexEnvelope.convex hU he, ConvexEnvelope.le_envelope hU hm, ?_, ?_⟩ · intro n by_cases hn : n = 0 · subst n simpa only [coefficientEnvelope, ConvexEnvelope.zero hU he, hG0] using hb 0 · by_cases hd : d n = 0 · simp only [hd, norm_zero]; exact (Real.exp_pos _).le · have hbelow := ConvexEnvelope.below he (show n ∈ coefficientObstacle d from ⟨by omega, hd⟩) have hlog : Real.log ‖d n‖ ≤ -coefficientEnvelope d n := by change coefficientEnvelope d n ≤ -Real.log ‖d n‖ at hbelow linarith simpa only [Real.exp_log (norm_pos_iff.mpr hd)] using Real.exp_le_exp.mpr hlog · intro n hn obtain ⟨hd, hc⟩ := ConvexEnvelope.contact hU he n hn change coefficientEnvelope d (n+1) = -Real.log ‖d (n+1)‖ at hc rw [hc, neg_neg, Real.exp_log (norm_pos_iff.mpr hd.2)] end CoefficientEnvelope theorem no_both_unbounded {P Q : ℕ → ℝ} {b c : ℤ → ℂ} (hb : EnvelopeBounds P Q b c) (hP : DiscreteConvex P) (hQ : DiscreteConvex Q) (hP0 : P 0 = 0) (hQ0 : Q 0 = 0) {lam δ : ℝ} (hlam : Real.log 3 < lam) (hδ : lam < δ) (hqp : ∀ᶠ n : ℕ in atTop, δ*(n : ℝ)^2 ≤ P n) (hqq : ∀ᶠ n : ℕ in atTop, δ*(n : ℝ)^2 ≤ Q n) (hbu : ∀ n : ℕ, ∃ k, n < k ∧ b (k : ℤ) ≠ 0) (hcu : ∀ n : ℕ, ∃ k, n < k ∧ c (-(k : ℤ)) ≠ 0) (hz : ∀ r : ℕ, 1 ≤ r → (∑' j : ℤ, b j * c (j-r)) = 0) : False := by obtain ⟨hA0, hA, hPA, hcA, hcontactA⟩ := CoefficientEnvelope.properties hcu hP hP0 hb.c_neg obtain ⟨hB0, hB, hQB, hbB, hcontactB⟩ := CoefficientEnvelope.properties hbu hQ hQ0 hb.b_pos let A := coefficientEnvelope (fun n => c (-(n : ℤ))) let B := coefficientEnvelope (fun n => b (n : ℤ)) apply no_correlation_of_envelopes (A := A) (B := B) (b := b) (c := c) ?_ hA hB hA0 hB0 hlam hδ ?_ ?_ hcontactA hcontactB hz · refine ⟨hbB, ?_, ?_, hcA⟩ · intro n exact (hb.b_neg n).trans (Real.exp_le_exp.mpr (hPA n)) · intro n exact (hb.c_pos n).trans (Real.exp_le_exp.mpr (hQB n)) · filter_upwards [hqp] with n hn using hn.trans (hPA n) · filter_upwards [hqq] with n hn using hn.trans (hQB n) def boundedSideTail (B : ℕ → ℝ) (c : ℤ → ℂ) (h : ℤ) (p k : ℕ) : ℝ := Real.exp (B p - B (p+k+1)) * ‖c (h+(k:ℤ)+1)‖ lemma boundedSideTail_nonneg (B : ℕ → ℝ) (c : ℤ → ℂ) (h : ℤ) (p k : ℕ) : 0 ≤ boundedSideTail B c h p k := mul_nonneg (Real.exp_pos _).le (norm_nonneg _) lemma boundedSideTail_mono {B : ℕ → ℝ} (hB : DiscreteConvex B) (c : ℤ → ℂ) (h : ℤ) {p₀ p : ℕ} (hpp : p₀ ≤ p) (k : ℕ) : boundedSideTail B c h p k ≤ boundedSideTail B c h p₀ k := by have hh := hB.increment_mono hpp (k+1) unfold boundedSideTail apply mul_le_mul_of_nonneg_right _ (norm_nonneg _) apply Real.exp_le_exp.mpr simp only [← Nat.add_assoc] at hh linarith lemma boundedSideTail_tendsto {B : ℕ → ℝ} (hB : DiscreteConvex B) (hs : Tendsto (fun n => B (n+1)-B n) atTop atTop) (c : ℤ → ℂ) (h : ℤ) {p₀ : ℕ} (hsum : Summable (boundedSideTail B c h p₀)) : Tendsto (fun p => ∑' k, boundedSideTail B c h p k) atTop (𝓝 0) := by have hpoint (k : ℕ) : Tendsto (fun p => boundedSideTail B c h p k) atTop (𝓝 0) := by have heq : (fun p => boundedSideTail B c h p k) = (fun p => normalizedTailTerm B p k * (Real.exp (-B (k+1)) * ‖c (h+(k:ℤ)+1)‖)) := by funext p unfold boundedSideTail normalizedTailTerm rw [← mul_assoc, ← Real.exp_add] congr 2 ring rw [heq] simpa using (hB.tail_term_tendsto hs k).mul_const (Real.exp (-B (k+1)) * ‖c (h+(k:ℤ)+1)‖) have ht := tendsto_tsum_of_dominated_convergence hsum hpoint (show ∀ᶠ p : ℕ in atTop, ∀ k, ‖boundedSideTail B c h p k‖ ≤ boundedSideTail B c h p₀ k from ?_) · simpa using ht · filter_upwards [eventually_ge_atTop p₀] with p hp k rw [Real.norm_eq_abs, abs_of_nonneg (boundedSideTail_nonneg _ _ _ _ _)] exact boundedSideTail_mono hB c h hp k theorem no_unbounded_against_min {B : ℕ → ℝ} {b c : ℤ → ℂ} (hB : DiscreteConvex B) (hB0 : B 0 = 0) {δ : ℝ} (hδ : 0 < δ) (hquad : ∀ᶠ n : ℕ in atTop, δ*(n:ℝ)^2 ≤ B n) (hb : ∀ n : ℕ, ‖b n‖ ≤ Real.exp (-B n)) (hc : ∀ n, 0 < B (n+2) - 2*B (n+1) + B n → ‖b ((n+1:ℕ):ℤ)‖ = Real.exp (-B (n+1))) {h : ℤ} (hch : c h ≠ 0) (hcmin : ∀ j : ℤ, j < h → c j = 0) {p₀ : ℕ} (hsum : Summable (boundedSideTail B c h p₀)) (hz : ∀ r : ℕ, 1 ≤ r → (∑' j : ℤ, b j * c (j-r)) = 0) : False := by have hchpos := norm_pos_iff.mpr hch have ht := boundedSideTail_tendsto hB (hB.slopes_tendsto hB0 hδ hquad) c h hsum obtain ⟨N, hN⟩ := eventually_atTop.1 (ht (eventually_lt_nhds hchpos)) obtain ⟨n, hn, hcurv⟩ := large_curvature hδ hδ hquad (max N (max p₀ (h.toNat+1))) let p := n+1 have hp₀ : p₀ ≤ p := by dsimp [p]; omega have hpN : N ≤ p := by dsimp [p]; omega have hph : h < (p : ℤ) := by dsimp [p]; omega have hcontact : ‖b (p : ℤ)‖ = Real.exp (-B p) := hc n (by simpa using hcurv) have htailsum : Summable (boundedSideTail B c h p) := by apply hsum.of_norm_bounded intro k rw [Real.norm_eq_abs, abs_of_nonneg (boundedSideTail_nonneg _ _ _ _ _)] exact boundedSideTail_mono hB c h hp₀ k let f := fun i : ℤ => b ((p:ℤ)+i) * c (h+i) have hfneg : ∀ k : ℕ, f (-((k:ℤ)+1)) = 0 := by intro k dsimp [f] rw [hcmin _ (by omega), mul_zero] have hfbound (k : ℕ) : ‖f ((k:ℤ)+1)‖ ≤ Real.exp (-B p) * boundedSideTail B c h p k := by dsimp [f, boundedSideTail] rw [norm_mul, ← mul_assoc, ← Real.exp_add] have he : -B p + (B p - B (p+k+1)) = -B (p+k+1) := by ring rw [he] have hi : (p:ℤ)+((k:ℤ)+1) = ((p+k+1:ℕ):ℤ) := by omega rw [hi] simpa only [add_assoc] using mul_le_mul_of_nonneg_right (hb (p+k+1)) (norm_nonneg (c (h+(k:ℤ)+1))) have hfpos : Summable (fun k : ℕ => ‖f ((k:ℤ)+1)‖) := by apply (htailsum.mul_left (Real.exp (-B p))).of_norm_bounded intro k simpa only [Real.norm_eq_abs, abs_of_nonneg (norm_nonneg _)] using hfbound k have hfn : Summable (fun k : ℕ => ‖f (-((k:ℤ)+1))‖) := by simp only [hfneg, norm_zero] exact summable_zero have hzcenter : (∑' i : ℤ, f i) = 0 := by let r := ((p:ℤ)-h).toNat have hr : 1 ≤ r := by dsimp [r]; omega have hri : (r:ℤ) = (p:ℤ)-h := by dsimp [r]; omega calc (∑' i : ℤ, f i) = ∑' i : ℤ, b ((p:ℤ)+i) * c (((p:ℤ)+i)-(r:ℤ)) := by apply tsum_congr intro i dsimp [f] congr 2 omega _ = ∑' j : ℤ, b j * c (j-r) := by simpa only [Equiv.coe_addLeft] using (Equiv.addLeft (p:ℤ)).tsum_eq (fun j : ℤ => b j * c (j-r)) _ = 0 := hz r hr have hcentral := central_norm_le_of_correlation_zero f hfpos hfn hzcenter simp only [hfneg, norm_zero, tsum_zero, add_zero] at hcentral have hcentral' : Real.exp (-B p) * ‖c h‖ ≤ Real.exp (-B p) * ∑' k, boundedSideTail B c h p k := by calc Real.exp (-B p) * ‖c h‖ = ‖f 0‖ := by simp [f, hcontact] _ ≤ ∑' k : ℕ, ‖f ((k:ℤ)+1)‖ := hcentral _ ≤ ∑' k, Real.exp (-B p) * boundedSideTail B c h p k := hfpos.tsum_le_tsum hfbound (htailsum.mul_left _) _ = _ := tsum_mul_left have hle := (mul_le_mul_iff_right₀ (Real.exp_pos (-B p))).mp hcentral' exact (not_lt_of_ge hle) (hN p hpN) def positiveLog (w : ℤ → ℝ) (n : ℕ) : ℝ := -Real.log (w n) def negativeLog (w : ℤ → ℝ) (n : ℕ) : ℝ := Real.log (w (-(n:ℤ))) def stepRatio (w : ℤ → ℝ) (j : ℤ) : ℝ := w (j+1) / w j lemma stepRatio_summable {w : ℤ → ℝ} (hw : ∀ j, 0 < w j) (hw0 : w 0 = 1) (hP : DiscreteConvex (negativeLog w)) (hQ : DiscreteConvex (positiveLog w)) {δ : ℝ} (hδ : 0 < δ) (hqp : ∀ᶠ n : ℕ in atTop, δ*(n:ℝ)^2 ≤ negativeLog w n) (hqq : ∀ᶠ n : ℕ in atTop, δ*(n:ℝ)^2 ≤ positiveLog w n) : Summable (stepRatio w) := by have hP0 : negativeLog w 0 = 0 := by simp [negativeLog, hw0] have hQ0 : positiveLog w 0 = 0 := by simp [positiveLog, hw0] have hsP := hP.summable_exp_neg_slope hP0 hδ hqp have hsQ := hQ.summable_exp_neg_slope hQ0 hδ hqq apply Summable.of_nat_of_neg_add_one · apply hsQ.congr intro n unfold stepRatio positiveLog have he : -(-Real.log (w (n+1)) - -Real.log (w n)) = Real.log (w (n+1)) - Real.log (w n) := by ring push_cast rw [he, Real.exp_sub, Real.exp_log (hw _), Real.exp_log (hw _)] · apply hsP.congr intro n unfold stepRatio negativeLog have hi : -((n:ℤ)+1)+1 = -(n:ℤ) := by omega rw [hi] push_cast rw [neg_sub, Real.exp_sub, Real.exp_log (hw _), Real.exp_log (hw _)] lemma bounds_of_weight {w : ℤ → ℝ} (hw : ∀ j, 0 < w j) {b c : ℤ → ℂ} (hb : ∀ j, ‖b j‖ ≤ w j) (hc : ∀ j, ‖c j‖ * w j ≤ 1) : EnvelopeBounds (negativeLog w) (positiveLog w) b c := by have hc' (j : ℤ) : ‖c j‖ ≤ (w j)⁻¹ := by simpa only [one_div] using (le_div_iff₀ (hw j)).mpr (hc j) constructor · intro n simpa [positiveLog, Real.exp_log (hw _)] using hb n · intro n simpa [negativeLog, Real.exp_log (hw _)] using hb (-(n:ℤ)) · intro n simpa [positiveLog, Real.exp_neg, Real.exp_log (hw _)] using hc' n · intro n simpa [negativeLog, Real.exp_neg, Real.exp_log (hw _)] using hc' (-(n:ℤ)) lemma boundedSideTail_summable {w : ℤ → ℝ} (hw : ∀ j, 0 < w j) (hdec : Antitone w) (hs : Summable (stepRatio w)) {B : ℕ → ℝ} (hBw : ∀ n : ℕ, Real.exp (-B n) ≤ w n) {c : ℤ → ℂ} (hc : ∀ j, ‖c j‖ * w j ≤ 1) (h : ℤ) {p₀ : ℕ} (hp₀ : h+1 ≤ (p₀:ℤ)) : Summable (boundedSideTail B c h p₀) := by have hsm := hs.comp_injective (i := fun k : ℕ => h+(k:ℤ)+1) (by intro a b hab; dsimp only at hab; omega) apply (hsm.mul_left (Real.exp (B p₀))).of_norm_bounded intro k rw [Real.norm_eq_abs, abs_of_nonneg (boundedSideTail_nonneg _ _ _ _ _)] change boundedSideTail B c h p₀ k ≤ Real.exp (B p₀) * stepRatio w (h+(k:ℤ)+1) have hwle : Real.exp (-B (p₀+k+1)) ≤ w (h+(k:ℤ)+1+1) := by exact (hBw _).trans (hdec (by omega)) have hcw : ‖c (h+(k:ℤ)+1)‖ ≤ 1 / w (h+(k:ℤ)+1) := (le_div_iff₀ (hw _)).mpr (hc _) unfold boundedSideTail stepRatio rw [Real.exp_sub, div_eq_mul_inv, ← Real.exp_neg, mul_assoc] apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le calc Real.exp (-B (p₀+k+1)) * ‖c (h+(k:ℤ)+1)‖ ≤ w (h+(k:ℤ)+1+1) * (1 / w (h+(k:ℤ)+1)) := mul_le_mul hwle hcw (norm_nonneg _) (hw _).le _ = _ := by ring lemma nat_support_unbounded {d : ℤ → ℂ} (hu : ¬∃ m : ℤ, ∀ j : ℤ, d j ≠ 0 → j ≤ m) : ∀ n : ℕ, ∃ k, n < k ∧ d (k:ℤ) ≠ 0 := by push Not at hu intro n obtain ⟨j, hd, hj⟩ := hu n refine ⟨j.toNat, by omega, ?_⟩ have he : (j.toNat:ℤ) = j := by omega simpa only [he] using hd lemma bounded_above_of_min {w : ℤ → ℝ} (hw : ∀ j, 0 < w j) (hdec : Antitone w) (hw0 : w 0 = 1) (hs : Summable (stepRatio w)) (hQ : DiscreteConvex (positiveLog w)) {δ : ℝ} (hδ : 0 < δ) (hqq : ∀ᶠ n : ℕ in atTop, δ*(n:ℝ)^2 ≤ positiveLog w n) {b c : ℤ → ℂ} (hb : ∀ j, ‖b j‖ ≤ w j) (hc : ∀ j, ‖c j‖*w j ≤ 1) {h : ℤ} (hch : c h ≠ 0) (hcmin : ∀ j : ℤ, j < h → c j = 0) (hz : ∀ r : ℕ, 1 ≤ r → (∑' j : ℤ, b j * c (j-r)) = 0) : ∃ m : ℤ, ∀ j : ℤ, b j ≠ 0 → j ≤ m := by by_contra hu have hbu := nat_support_unbounded hu have hQ0 : positiveLog w 0 = 0 := by simp [positiveLog, hw0] obtain ⟨hB0, hB, hQB, hbB, hcontactB⟩ := CoefficientEnvelope.properties hbu hQ hQ0 (bounds_of_weight hw hb hc).b_pos let B := coefficientEnvelope (fun n => b (n:ℤ)) have hquad : ∀ᶠ n : ℕ in atTop, δ*(n:ℝ)^2 ≤ B n := by filter_upwards [hqq] with n hn using hn.trans (hQB n) have hBw (n : ℕ) : Real.exp (-B n) ≤ w n := by have hh := Real.exp_le_exp.mpr (neg_le_neg (hQB n)) simpa only [positiveLog, neg_neg, Real.exp_log (hw _)] using hh exact no_unbounded_against_min hB hB0 hδ hquad hbB hcontactB hch hcmin (boundedSideTail_summable hw hdec hs hBw hc h (p₀ := h.toNat+1) (by omega)) hz end end BackwardIntertwiners end OAI