| import OAI.Analysis.C0Absorption.Grid |
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| namespace OAI |
|
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| open Set Filter Topology |
| open scoped NNReal BigOperators ZeroAtInfty |
| open NormedSpace |
|
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| namespace C0Absorption |
| noncomputable section |
| open Set Filter Topology |
| open scoped NNReal BigOperators ZeroAtInfty |
|
|
| def eta : ℝ := 1/1000 |
| def scheduleH : ℝ := 10000 |
|
|
| theorem eta_pos : 0 < eta := by norm_num [eta] |
| theorem scheduleH_pos : 0 < scheduleH := by norm_num [scheduleH] |
| theorem schedule_speed : Real.pi/scheduleH ≤ eta/2 := by |
| have h := Real.pi_lt_four |
| norm_num [scheduleH,eta] |
| linarith |
|
|
| theorem bandL_pos (j : ℕ) : 0 < (bandL j : ℝ) := by |
| let : Nonempty (BandGrid j) := ⟨fun _ => ⟨0,by constructor; exact neg_nonpos.mpr (Int.natCast_nonneg _); exact Int.natCast_nonneg _⟩⟩ |
| have hc : (0 : ℝ) < Fintype.card (BandGrid j) := by exact_mod_cast Fintype.card_pos |
| simp only [bandL,NNReal.coe_mul,NNReal.coe_ofNat,NNReal.coe_natCast,NNReal.coe_pow] |
| positivity |
|
|
| def rho : ℕ → ℝ |
| | 0 => 1/2 |
| | j+1 => min (Real.exp (-scheduleH)*rho j/2) (eta/(8*bandL (j+1))) |
|
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| theorem rho_pos (j : ℕ) : 0 < rho j := by |
| induction j with |
| | zero => norm_num [rho] |
| | succ j ih => |
| change 0 < min _ _ |
| exact lt_min (by positivity) (div_pos eta_pos (mul_pos (by norm_num) (bandL_pos _))) |
|
|
| theorem rho_step (j : ℕ) : rho (j+1) < Real.exp (-scheduleH)*rho j := by |
| have hp : 0 < Real.exp (-scheduleH)*rho j := mul_pos (Real.exp_pos _) (rho_pos j) |
| exact (min_le_left _ _).trans_lt (by linarith) |
|
|
| theorem rho_half_step (j : ℕ) : rho (j+1) ≤ rho j/2 := by |
| have he : Real.exp (-scheduleH) ≤ 1 := Real.exp_le_one_iff.mpr (by norm_num [scheduleH]) |
| exact (min_le_left _ _).trans (div_le_div_of_nonneg_right |
| (by nlinarith [rho_pos j]) (by norm_num)) |
|
|
| theorem rho_bound (j : ℕ) : rho j ≤ (1/2 : ℝ)^(j+1) := by |
| induction j with |
| | zero => norm_num [rho] |
| | succ j ih => |
| refine (rho_half_step j).trans ?_ |
| rw [pow_succ] |
| linarith |
|
|
| theorem rho_lt_one (j : ℕ) : rho j < 1 := by |
| have hh := rho_bound j |
| have hp : (1/2 : ℝ)^(j+1) < 1 := pow_lt_one₀ (by norm_num) (by norm_num) (by omega) |
| exact hh.trans_lt hp |
|
|
| theorem rho_lip_budget {j : ℕ} (hj : 0 < j) : bandL j*rho j ≤ eta/4 := by |
| obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hj) |
| have hh : rho (k+1) ≤ eta/(8*bandL (k+1)) := min_le_right _ _ |
| have hp := bandL_pos (k+1) |
| have he := (le_div_iff₀ (mul_pos (by norm_num : (0:ℝ)<8) hp)).mp hh |
| linarith [eta_pos] |
|
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| theorem rho_tendsto : Tendsto rho atTop (nhds 0) := by |
| apply squeeze_zero (fun j => (rho_pos j).le) rho_bound |
| exact (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num)).comp |
| (tendsto_add_atTop_nat 1) |
|
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| def ell (j : ℕ) : ℝ := Real.exp (-scheduleH)*rho (j+1) |
|
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| theorem ell_pos (j : ℕ) : 0 < ell j := mul_pos (Real.exp_pos _) (rho_pos _) |
| theorem ell_tendsto : Tendsto ell atTop (nhds 0) := by |
| change Tendsto (fun j => Real.exp (-scheduleH)*rho (j+1)) atTop (nhds 0) |
| convert (tendsto_const_nhds (x := Real.exp (-scheduleH))).mul (rho_tendsto.comp (tendsto_add_atTop_nat 1)) using 1 <;> simp |
|
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| theorem log_rho_step (j : ℕ) : Real.log (rho (j+1)) < Real.log (rho j)-scheduleH := by |
| have hh := Real.strictMonoOn_log (rho_pos (j+1)) |
| (mul_pos (Real.exp_pos _) (rho_pos j)) (rho_step j) |
| rwa [Real.log_mul (Real.exp_pos _).ne' (rho_pos j).ne',Real.log_exp,neg_add_eq_sub] at hh |
|
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| def transitionAngle (j : ℕ) (u : ℝ) : ℝ := |
| Real.pi/(2*scheduleH) * min scheduleH (max 0 (Real.log (rho j)-u)) |
|
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| theorem transitionAngle_nonneg (j : ℕ) (u : ℝ) : 0 ≤ transitionAngle j u := by |
| unfold transitionAngle |
| exact mul_nonneg (div_nonneg Real.pi_pos.le (by norm_num [scheduleH] : 0 ≤ 2*scheduleH)) |
| (le_min scheduleH_pos.le (le_max_left _ _)) |
|
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| theorem transitionAngle_le (j : ℕ) (u : ℝ) : transitionAngle j u ≤ Real.pi/2 := by |
| have hh : min scheduleH (max 0 (Real.log (rho j)-u)) ≤ scheduleH := min_le_left _ _ |
| have hc : 0 ≤ Real.pi/(2*scheduleH) := div_nonneg Real.pi_pos.le (by norm_num [scheduleH]) |
| have hb := mul_le_mul_of_nonneg_left hh hc |
| have he : Real.pi/(2*scheduleH)*scheduleH=Real.pi/2 := by norm_num [scheduleH]; ring |
| exact hb.trans_eq he |
|
|
| theorem transitionAngle_zero (j : ℕ) {u : ℝ} (hu : Real.log (rho j) ≤ u) : |
| transitionAngle j u=0 := by |
| simp only [transitionAngle,max_eq_left (by linarith : Real.log (rho j)-u ≤ 0), |
| min_eq_right scheduleH_pos.le,mul_zero] |
|
|
| theorem transitionAngle_full (j : ℕ) {u : ℝ} (hu : u ≤ Real.log (rho j)-scheduleH) : |
| transitionAngle j u=Real.pi/2 := by |
| unfold transitionAngle |
| rw [min_eq_left (le_max_of_le_right (by linarith : scheduleH ≤ Real.log (rho j)-u))] |
| norm_num [scheduleH] |
| ring |
|
|
| theorem transitionAngle_dichotomy (j : ℕ) (u : ℝ) : |
| transitionAngle (j+1) u=0 ∨ transitionAngle j u=Real.pi/2 := by |
| by_cases hu : Real.log (rho (j+1)) ≤ u |
| · exact Or.inl (transitionAngle_zero (j+1) hu) |
| · exact Or.inr (transitionAngle_full j (by linarith [log_rho_step j])) |
|
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| def transitionInterval (j : ℕ) : Set ℝ := Icc (Real.log (rho j)-scheduleH) (Real.log (rho j)) |
|
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| theorem log_rho_antitone : Antitone (fun j => Real.log (rho j)) := by |
| apply antitone_nat_of_succ_le |
| intro j |
| linarith [log_rho_step j,scheduleH_pos] |
|
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| theorem transitionInterval_disjoint : Pairwise (fun i j => Disjoint (transitionInterval i) (transitionInterval j)) := by |
| suffices ∀ i j, i<j → Disjoint (transitionInterval i) (transitionInterval j) by |
| intro i j hij |
| rcases lt_or_gt_of_ne hij with hh|hh |
| · exact this i j hh |
| · exact (this j i hh).symm |
| intro i j hij |
| have hh := (log_rho_antitone (Nat.succ_le_of_lt hij)).trans_lt (log_rho_step i) |
| apply Set.disjoint_left.mpr |
| intro u hi hj |
| exact (not_le_of_gt hh) (hi.1.trans hj.2) |
|
|
| end |
| end C0Absorption |
|
|
| namespace C0Absorption |
| noncomputable section |
| open Set Filter Topology |
| open scoped NNReal BigOperators ZeroAtInfty |
|
|
| theorem finite_variation_sum_lipschitz {A : Type*} [Fintype A] |
| (f : A → ℝ → ℝ) (C : A → Set ℝ) {K : ℝ≥0} |
| (hdis : Pairwise (fun i j => Disjoint (C i) (C j))) |
| (hf : ∀ i, LipschitzWith K (f i)) |
| (hconst : ∀ i x, x ∉ C i → ∀ᶠ y in nhds x, f i y=f i x) : |
| LipschitzWith K (fun x => ∑ i,f i x) := by |
| classical |
| have hloc (x : ℝ) : ∀ᶠ y in nhds x, |
| dist (∑ i,f i x) (∑ i,f i y) ≤ K*dist x y := by |
| by_cases h : ∃ i, x ∈ C i |
| · obtain ⟨j,hj⟩ := h |
| have hh (i : A) : i ≠ j → ∀ᶠ y in nhds x, f i y=f i x := by |
| intro hij |
| exact hconst i x (fun hi => Set.disjoint_left.mp (hdis hij) hi hj) |
| have he : ∀ᶠ y in nhds x, ∀ i, i≠j → f i y=f i x := |
| Filter.eventually_all.mpr (fun i => by |
| by_cases hij : i=j |
| · exact Filter.Eventually.of_forall (by simp [hij]) |
| · filter_upwards [hh i hij] with y hy using fun _ => hy) |
| filter_upwards [he] with y hy |
| rw [Real.dist_eq,← Finset.sum_sub_distrib, |
| Finset.sum_eq_single j (fun i _ hij => by rw [hy i hij,sub_self]) (by simp)] |
| exact (hf j).dist_le_mul x y |
| · push Not at h |
| filter_upwards [Filter.eventually_all.mpr (fun i => hconst i x (h i))] with y hy |
| simp only [hy,dist_self] |
| positivity |
| have hc : Continuous (fun x => ∑ i,f i x) := continuous_finsetSum _ (fun i _ => (hf i).continuous) |
| have hl := lipschitzOn_convex_of_local (convex_univ : Convex ℝ (univ : Set ℝ)) hc.continuousOn |
| (fun x _ => by simpa only [nhdsWithin_univ] using hloc x) |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| exact hl.dist_le_mul x (mem_univ _) y (mem_univ _) |
|
|
| def angleSpeed : ℝ≥0 := ⟨Real.pi/(2*scheduleH),div_nonneg Real.pi_pos.le (by norm_num [scheduleH])⟩ |
|
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| theorem transitionAngle_lipschitz (j : ℕ) : LipschitzWith angleSpeed (transitionAngle j) := by |
| have hl : LipschitzWith 1 (fun u : ℝ => Real.log (rho j)-u) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro u v |
| simp only [Real.dist_eq,sub_sub_sub_cancel_left,NNReal.coe_one,one_mul,abs_sub_comm,le_refl] |
| have hb := (hl.const_max 0).const_min scheduleH |
| apply LipschitzWith.of_dist_le_mul |
| intro u v |
| change |angleSpeed*(min scheduleH (max 0 (Real.log (rho j)-u)))- |
| angleSpeed*(min scheduleH (max 0 (Real.log (rho j)-v)))| ≤ _ |
| rw [← mul_sub,abs_mul,abs_of_nonneg angleSpeed.coe_nonneg] |
| exact mul_le_mul_of_nonneg_left (by simpa only [NNReal.coe_one,one_mul,Real.dist_eq] using hb.dist_le_mul u v) angleSpeed.coe_nonneg |
|
|
| theorem transitionAngle_antitone (j : ℕ) : Antitone (transitionAngle j) := by |
| intro u v huv |
| apply mul_le_mul_of_nonneg_left _ angleSpeed.coe_nonneg |
| exact min_le_min_left _ (max_le_max_left _ (by linarith)) |
|
|
| theorem transitionAngle_locally_constant (j : ℕ) (u : ℝ) (hu : u ∉ transitionInterval j) : |
| ∀ᶠ v in nhds u, transitionAngle j v=transitionAngle j u := by |
| have h : u < Real.log (rho j)-scheduleH ∨ Real.log (rho j)<u := by |
| change ¬(Real.log (rho j)-scheduleH ≤ u ∧ u ≤ Real.log (rho j)) at hu |
| rcases lt_or_ge u (Real.log (rho j)-scheduleH) with h|h |
| · exact Or.inl h |
| · right |
| by_contra hn |
| exact hu ⟨h,le_of_not_gt hn⟩ |
| rcases h with h|h |
| · filter_upwards [eventually_lt_nhds h] with v hv |
| rw [transitionAngle_full j hv.le,transitionAngle_full j h.le] |
| · filter_upwards [eventually_gt_nhds h] with v hv |
| rw [transitionAngle_zero j hv.le,transitionAngle_zero j h.le] |
|
|
| theorem angle_sum_distance (J : ℕ) (u v : ℝ) : |
| (∑ j : Fin J, |transitionAngle (j+1) u-transitionAngle (j+1) v|) ≤ |
| angleSpeed*dist u v := by |
| have hh := finite_variation_sum_lipschitz |
| (fun j : Fin J => transitionAngle (j+1)) (fun j : Fin J => transitionInterval (j+1)) |
| (fun i j hij => transitionInterval_disjoint (by intro he; exact hij (Fin.ext (by omega)))) |
| (fun j => transitionAngle_lipschitz (j+1)) |
| (fun j => transitionAngle_locally_constant (j+1)) |
| have he (x y : ℝ) (hxy : x≤y) : |
| (∑ j : Fin J, |transitionAngle (j+1) x-transitionAngle (j+1) y|) = |
| dist (∑ j : Fin J,transitionAngle (j+1) x) (∑ j : Fin J,transitionAngle (j+1) y) := by |
| rw [Real.dist_eq,← Finset.sum_sub_distrib,abs_of_nonneg] |
| · apply Finset.sum_congr rfl |
| intro j _ |
| exact abs_of_nonneg (sub_nonneg.mpr (transitionAngle_antitone (j+1) hxy)) |
| · exact Finset.sum_nonneg (fun j _ => sub_nonneg.mpr (transitionAngle_antitone (j+1) hxy)) |
| rcases le_total u v with huv|hvu |
| · rw [he u v huv] |
| exact hh.dist_le_mul u v |
| · have heq : (∑ j : Fin J, |transitionAngle (j+1) u-transitionAngle (j+1) v|)= |
| ∑ j : Fin J, |transitionAngle (j+1) v-transitionAngle (j+1) u| := |
| Finset.sum_congr rfl (fun j _ => abs_sub_comm _ _) |
| rw [heq,he v u hvu] |
| simpa only [dist_comm] using hh.dist_le_mul v u |
|
|
| end |
| end C0Absorption |
|
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| end OAI |
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