| import OAI.Probability.GaussianPropeller.FourAlgebra | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory | |
| open scoped ENNReal | |
| open scoped RealInnerProductSpace | |
| open scoped RealInnerProductSpace | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal RealInnerProductSpace | |
| open Filter | |
| open scoped Topology | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped Topology | |
| open scoped RealInnerProductSpace | |
| open Set Filter | |
| open scoped Topology RealInnerProductSpace | |
| open scoped NNReal | |
| open Set Filter | |
| open scoped Topology RealInnerProductSpace NNReal | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped Topology RealInnerProductSpace | |
| open MeasureTheory Set Filter | |
| open scoped Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped RealInnerProductSpace Topology | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped RealInnerProductSpace Topology ENNReal | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped RealInnerProductSpace Topology ENNReal | |
| open Metric | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped RealInnerProductSpace ENNReal | |
| namespace GaussianPropeller | |
| namespace Scalar | |
| def charge (A u : ℝ) : ℝ := 0.415 * u - (A - 0.15) * u ^ 2 | |
| theorem third_radius_gt {L u : ℝ} (hu : 0 < u) (hL : L ≤ 8 / 9) | |
| (hcharge : (1 - L) * (0.415 / u - 0.734) ≤ 0.116) : 0.23 < u := by | |
| by_contra! h | |
| have hd : (0.415 : ℝ) / 0.23 ≤ 0.415 / u := | |
| div_le_div_of_nonneg_left (by norm_num) hu h | |
| have hm := mul_le_mul_of_nonneg_left (sub_le_sub_right hd 0.734) | |
| (show 0 ≤ 1 - L by linarith) | |
| norm_num at hm | |
| nlinarith | |
| theorem small_minimum_interval {L u t a b M : ℝ} | |
| (ht : 0 ≤ t) (htq : t ≤ 0.066) (hL : L ≤ 8 / 9) | |
| (hua : a ≤ u) (hub : u ≤ b) (ha : 0.066 < a) | |
| (hden : 0 < 0.415 / b - 0.734) (hM : 0 ≤ M) | |
| (hrow : (0.116 : ℝ) ≤ | |
| (1 - (b ^ 2 + 2 * (0.066 : ℝ) ^ 2 + M)) * (0.415 / b - 0.734)) | |
| (hmargin : 8 * (0.066 : ℝ) ^ 2 < (a ^ 2 - (0.066 : ℝ) ^ 2) * M) | |
| (hcharge : (1 - L) * (0.415 / u - 0.734) ≤ 0.116) | |
| (hdet : (u ^ 2 - t ^ 2) * (L - u ^ 2 - 2 * t ^ 2) ≤ 8 * t ^ 2) : False := by | |
| have hup : 0 < u := by linarith | |
| have hb : 0 < b := lt_of_lt_of_le hup hub | |
| have hd := div_le_div_of_nonneg_left (show (0 : ℝ) ≤ 0.415 by norm_num) hup hub | |
| have hm := mul_le_mul_of_nonneg_left (sub_le_sub_right hd 0.734) | |
| (show 0 ≤ 1 - L by linarith) | |
| have hLc : b ^ 2 + 2 * (0.066 : ℝ) ^ 2 + M ≤ L := by | |
| have hh := le_of_mul_le_mul_right ((hm.trans hcharge).trans hrow) hden | |
| linarith | |
| have hub2 : u ^ 2 ≤ b ^ 2 := by nlinarith | |
| have hua2 : a ^ 2 ≤ u ^ 2 := by nlinarith | |
| have ht2 : t ^ 2 ≤ (0.066 : ℝ) ^ 2 := by nlinarith | |
| have hfirst : a ^ 2 - (0.066 : ℝ) ^ 2 ≤ u ^ 2 - t ^ 2 := by linarith | |
| have hfirst0 : 0 ≤ u ^ 2 - t ^ 2 := by nlinarith | |
| have hsecond : M ≤ L - u ^ 2 - 2 * t ^ 2 := by linarith | |
| have hp := mul_le_mul hfirst hsecond hM hfirst0 | |
| nlinarith | |
| theorem minimum_gt {L u t : ℝ} (ht : 0 ≤ t) (hu : 0.23 < u) | |
| (hL : L ≤ 8 / 9) (hL2 : 2 * u ^ 2 ≤ L) | |
| (hcharge : (1 - L) * (0.415 / u - 0.734) ≤ 0.116) | |
| (hdet : (u ^ 2 - t ^ 2) * (L - u ^ 2 - 2 * t ^ 2) ≤ 8 * t ^ 2) : | |
| 0.066 < t := by | |
| by_contra! htq | |
| by_cases hbig : 0.44 ≤ u | |
| · have ht2 : t ^ 2 ≤ (0.066 : ℝ) ^ 2 := by nlinarith | |
| have hu2 : (0.44 : ℝ) ^ 2 ≤ u ^ 2 := by nlinarith | |
| have hfirst : (0.44 : ℝ) ^ 2 - (0.066 : ℝ) ^ 2 ≤ u ^ 2 - t ^ 2 := by linarith | |
| have hsecond : (0.44 : ℝ) ^ 2 - 2 * (0.066 : ℝ) ^ 2 ≤ | |
| L - u ^ 2 - 2 * t ^ 2 := by linarith | |
| have hp := mul_le_mul hfirst hsecond (by norm_num) | |
| (show 0 ≤ u ^ 2 - t ^ 2 by nlinarith) | |
| norm_num at hp | |
| nlinarith | |
| · have hb : u ≤ 0.44 := le_of_lt (lt_of_not_ge hbig) | |
| by_cases h30 : u ≤ 0.30 | |
| · exact small_minimum_interval (a := 0.23) (b := 0.30) (M := 0.72264) | |
| ht htq hL hu.le h30 (by norm_num) (by norm_num) (by norm_num) | |
| (by norm_num) (by norm_num) hcharge hdet | |
| by_cases h35 : u ≤ 0.35 | |
| · exact small_minimum_interval (a := 0.30) (b := 0.35) (M := 0.61198) | |
| ht htq hL (by linarith) h35 (by norm_num) (by norm_num) (by norm_num) | |
| (by norm_num) (by norm_num) hcharge hdet | |
| by_cases h41 : u ≤ 0.41 | |
| · exact small_minimum_interval (a := 0.35) (b := 0.41) (M := 0.40621) | |
| ht htq hL (by linarith) h41 (by norm_num) (by norm_num) (by norm_num) | |
| (by norm_num) (by norm_num) hcharge hdet | |
| · exact small_minimum_interval (a := 0.41) (b := 0.44) (M := 0.24314) | |
| ht htq hL (by linarith) hb (by norm_num) (by norm_num) (by norm_num) | |
| (by norm_num) (by norm_num) hcharge hdet | |
| theorem first_constraint_rational {L c t : ℝ} | |
| (h : (c - t ^ 2) * (L - c - 2 * t ^ 2) ≤ 8 * (3 / Real.pi) ^ 2 * t ^ 2) : | |
| (c - t ^ 2) * (L - c - 2 * t ^ 2) ≤ 8 * t ^ 2 := by | |
| have hratio : (3 : ℝ) / Real.pi ≤ 1 := | |
| (div_le_one Real.pi_pos).mpr Real.pi_gt_three.le | |
| have hratio0 : (0 : ℝ) ≤ 3 / Real.pi := by positivity | |
| have hsq : ((3 : ℝ) / Real.pi) ^ 2 ≤ 1 := by nlinarith | |
| have hh := mul_le_mul_of_nonneg_right hsq (sq_nonneg t) | |
| nlinarith | |
| theorem final_contradiction {c t : ℝ} (ht : 0 < t) (ht12 : t < 0.12) | |
| (hc : 2.6 * t < c) | |
| (hdet : (3 / 2 : ℝ) * (c - t ^ 2) ^ 2 ≤ 10 * (3 / Real.pi) ^ 2 * t ^ 2) : | |
| False := by | |
| have hp : 0 < Real.pi := Real.pi_pos | |
| have hratio : (3 : ℝ) / Real.pi < 150 / 157 := by | |
| rw [div_lt_iff₀ hp] | |
| nlinarith [Real.pi_gt_d2] | |
| have hratio0 : (0 : ℝ) ≤ 3 / Real.pi := by positivity | |
| have hratio2 : ((3 : ℝ) / Real.pi) ^ 2 < (150 / 157 : ℝ) ^ 2 := by nlinarith | |
| have ht2 : 0 < t ^ 2 := sq_pos_of_pos ht | |
| have htop := mul_lt_mul_of_pos_right hratio2 ht2 | |
| have hc' : (2.48 : ℝ) * t < c - t ^ 2 := by nlinarith | |
| have hcsq : ((2.48 : ℝ) * t) ^ 2 < (c - t ^ 2) ^ 2 := by nlinarith | |
| nlinarith | |
| theorem charge_concave {A : ℝ} (hA : 0.15 ≤ A) : | |
| ConcaveOn ℝ Set.univ (charge A) := by | |
| have hlin : ConcaveOn ℝ (Set.univ : Set ℝ) (fun u : ℝ => (0.415 : ℝ) * u) := | |
| (concaveOn_id convex_univ).smul (by norm_num) | |
| have hquad : ConvexOn ℝ (Set.univ : Set ℝ) (fun u : ℝ => (A - 0.15) * u ^ 2) := | |
| (Even.convexOn_pow (by decide : Even (2 : ℕ))).smul (by linarith) | |
| exact hlin.sub hquad | |
| theorem min_charge_le {A a b u : ℝ} (hA : 0.15 ≤ A) | |
| (ha : a ≤ u) (hb : u ≤ b) : min (charge A a) (charge A b) ≤ charge A u := | |
| (charge_concave hA).min_le_of_mem_Icc (Set.mem_univ _) (Set.mem_univ _) ⟨ha, hb⟩ | |
| noncomputable def finalBudget (A k t : ℝ) : ℝ := | |
| charge A (Real.sqrt (2.6 * t)) + k * charge A t | |
| theorem finalBudget_concave {A k : ℝ} (hA : 0.15 ≤ A) (hk : 0 ≤ k) : | |
| ConcaveOn ℝ (Set.Ici 0) (finalBudget A k) := by | |
| have hsqrt : ConcaveOn ℝ (Set.Ici 0) (fun t : ℝ => | |
| (0.415 * Real.sqrt 2.6) * Real.sqrt t) := | |
| Real.strictConcaveOn_sqrt.concaveOn.smul (by positivity) | |
| have hlin : ConvexOn ℝ (Set.Ici 0) (fun t : ℝ => (2.6 * (A - 0.15)) * t) := | |
| (convexOn_id (convex_Ici (0 : ℝ))).smul (by positivity) | |
| have hquad := ((charge_concave hA).subset (Set.subset_univ _) (convex_Ici 0)).smul hk | |
| have heq : Set.EqOn | |
| (fun t : ℝ => (0.415 * Real.sqrt 2.6) * Real.sqrt t - | |
| (2.6 * (A - 0.15)) * t + k * charge A t) | |
| (finalBudget A k) (Set.Ici 0) := by | |
| intro t ht | |
| have ht0 : 0 ≤ t := ht | |
| have hsq := Real.sq_sqrt (show 0 ≤ 2.6 * t by positivity) | |
| dsimp [finalBudget, charge] | |
| rw [hsq, Real.sqrt_mul (by norm_num : (0 : ℝ) ≤ 2.6)] | |
| ring | |
| exact ((hsqrt.sub hlin).add hquad).congr heq | |
| theorem clipped_charge {A s q u δ : ℝ} (hd : 0 < A - 0.15) (hs : 0 < s) | |
| (hq : 0 ≤ q) (hqu : q ≤ u) (hds : (A - 0.15) * s = 0.415) | |
| (hδ0 : 0 ≤ δ) (hδ : charge A u ≤ δ) : | |
| (A - 0.15) * q / (s + q) * (s ^ 2 - u ^ 2) ≤ δ := by | |
| have hden : 0 < s + q := by positivity | |
| have hcoef : 0 ≤ (A - 0.15) * q / (s + q) := by positivity | |
| by_cases hu : s ≤ u | |
| · have hsq : s ^ 2 - u ^ 2 ≤ 0 := by nlinarith | |
| exact (mul_nonpos_of_nonneg_of_nonpos hcoef hsq).trans hδ0 | |
| · have hu' : u ≤ s := le_of_lt (lt_of_not_ge hu) | |
| have hprod : 0 ≤ (A - 0.15) * s * ((s - u) * (u - q)) := | |
| mul_nonneg (by positivity) (mul_nonneg (sub_nonneg.mpr hu') (sub_nonneg.mpr hqu)) | |
| have hdiff : 0 ≤ charge A u - | |
| (A - 0.15) * q / (s + q) * (s ^ 2 - u ^ 2) := by | |
| have heq : (charge A u - (A - 0.15) * q / (s + q) * (s ^ 2 - u ^ 2)) * | |
| (s + q) = (A - 0.15) * s * ((s - u) * (u - q)) := by | |
| dsimp [charge] | |
| rw [← hds] | |
| field_simp | |
| ring | |
| exact nonneg_of_mul_nonneg_left (heq.symm ▸ hprod) hden | |
| linarith | |
| noncomputable def smallRoot (A k : ℝ) : ℝ := | |
| (0.415 - Real.sqrt (0.415 ^ 2 - 4 * (A - 0.15) * (1 - A) / k)) / | |
| (2 * (A - 0.15)) | |
| noncomputable def threshold (A : ℝ) : ℝ := 0.415 / (A - 0.15) | |
| noncomputable def clipCoeff (A : ℝ) : ℝ := (A - 0.15) * 0.42 / (threshold A + 0.42) | |
| theorem smallRoot_five : | |
| 0.066 < smallRoot 0.929 2 ∧ smallRoot 0.929 2 < 0.109 ∧ | |
| 2 * charge 0.929 (smallRoot 0.929 2) = 1 - 0.929 := by | |
| have hsq := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 61607 / 1000000) | |
| have hnn := Real.sqrt_nonneg (61607 / 1000000 : ℝ) | |
| norm_num only [smallRoot, charge] at * | |
| constructor | |
| · nlinarith | |
| constructor | |
| · nlinarith | |
| · nlinarith | |
| theorem smallRoot_six : | |
| 0.066 < smallRoot 0.884 3 ∧ smallRoot 0.884 3 < 0.12 ∧ | |
| 3 * charge 0.884 (smallRoot 0.884 3) = 1 - 0.884 := by | |
| have hsq := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 176099 / 3000000) | |
| have hnn := Real.sqrt_nonneg (176099 / 3000000 : ℝ) | |
| norm_num only [smallRoot, charge] at * | |
| constructor | |
| · nlinarith | |
| constructor | |
| · nlinarith | |
| · nlinarith | |
| structure Regime (A : ℝ) (k : ℕ) : Prop where | |
| A_lower : 0.884 ≤ A | |
| A_upper : A < 1 | |
| k_lower : 2 ≤ k | |
| root_lower : 0.066 < smallRoot A k | |
| root_upper : smallRoot A k < 0.12 | |
| root_eq : (k : ℝ) * charge A (smallRoot A k) = 1 - A | |
| root_cap : 2.6 * smallRoot A k < (threshold A) ^ 2 | |
| root_slope : (A - 0.15) * (0.42 + smallRoot A k) < 0.415 | |
| cap_charge : 1 - A < (k : ℝ) * charge A 0.42 | |
| tail_end_charge : charge A 0.066 ≤ charge A 0.45 | |
| base_charge_pos : 0 < charge A 0.066 | |
| four_margin : 1 - A < clipCoeff A * | |
| (4 * (threshold A) ^ 2 - 1 + ((k : ℝ) - 1) * 0.066 ^ 2) + | |
| ((k : ℝ) - 1) * charge A 0.066 | |
| third_margin : 1 - A < charge A 0.23 + (k : ℝ) * charge A 0.066 | |
| final_margin : 1 - A < finalBudget A k 0.066 | |
| theorem regime_five : Regime 0.929 2 := by | |
| obtain ⟨hl, hu, he⟩ := smallRoot_five | |
| have hsqrt := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 429) | |
| have hsqrt0 := Real.sqrt_nonneg (429 : ℝ) | |
| constructor | |
| · norm_num | |
| · norm_num | |
| · norm_num | |
| · exact hl | |
| · simpa using hu.trans (by norm_num : (0.109 : ℝ) < 0.12) | |
| · exact he | |
| · norm_num [threshold]; nlinarith | |
| · norm_num; nlinarith | |
| · norm_num [charge] | |
| · norm_num [charge] | |
| · norm_num [charge] | |
| · norm_num [clipCoeff, threshold, charge] | |
| · norm_num [charge] | |
| · norm_num [finalBudget, charge] | |
| nlinarith | |
| theorem regime_six : Regime 0.884 3 := by | |
| obtain ⟨hl, hu, he⟩ := smallRoot_six | |
| have hsqrt := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 429) | |
| have hsqrt0 := Real.sqrt_nonneg (429 : ℝ) | |
| constructor | |
| · norm_num | |
| · norm_num | |
| · norm_num | |
| · exact hl | |
| · exact hu | |
| · exact he | |
| · norm_num [threshold]; nlinarith | |
| · norm_num; nlinarith | |
| · norm_num [charge] | |
| · norm_num [charge] | |
| · norm_num [charge] | |
| · norm_num [clipCoeff, threshold, charge] | |
| · norm_num [charge] | |
| · norm_num [finalBudget, charge] | |
| nlinarith | |
| structure RadiusData (m : ℕ) (A : ℝ) where | |
| r : ℕ → ℝ | |
| δ : ℕ → ℝ | |
| pos : ∀ i < m, 0 < r i | |
| ordered : AntitoneOn r (Set.Iio m) | |
| cap : ∀ i < m, r i ≤ 2 / 3 | |
| norm : ∑ i ∈ Finset.range m, r i ^ 2 = 1 | |
| nonneg : ∀ i < m, 0 ≤ δ i | |
| lower : ∀ i < m, charge A (r i) ≤ δ i | |
| budget : ∑ i ∈ Finset.range m, δ i = 1 - A | |
| theorem sum_split (f : ℕ → ℝ) {p m : ℕ} (hp : p ≤ m) : | |
| (∑ i ∈ Finset.range p, f i) + (∑ i ∈ Finset.range (m - p), f (p + i)) = | |
| ∑ i ∈ Finset.range m, f i := by | |
| rw [← Finset.sum_range_add, Nat.add_sub_of_le hp] | |
| theorem sum_prefix_le (f : ℕ → ℝ) {p m : ℕ} (hp : p ≤ m) | |
| (hf : ∀ i < m, 0 ≤ f i) : | |
| (∑ i ∈ Finset.range p, f i) ≤ ∑ i ∈ Finset.range m, f i := by | |
| exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono hp) | |
| (fun i hi _ => hf i (Finset.mem_range.mp hi)) | |
| namespace RadiusData | |
| variable {m : ℕ} {A : ℝ} (D : RadiusData m A) | |
| theorem radius_le {i j : ℕ} (hj : j < m) (hij : i ≤ j) : D.r j ≤ D.r i := | |
| D.ordered (show i ∈ Set.Iio m by exact lt_of_le_of_lt hij hj) hj hij | |
| theorem minimum_le {i : ℕ} (hi : i < m) : D.r (m - 1) ≤ D.r i := | |
| D.radius_le (by omega) (by omega) | |
| theorem prefix_norm {p : ℕ} (hp : p ≤ m) : | |
| (∑ i ∈ Finset.range p, D.r i ^ 2) ≤ 1 := by | |
| simpa [D.norm] using sum_prefix_le (fun i => D.r i ^ 2) hp (fun i _ => sq_nonneg _) | |
| theorem prefix_budget {p : ℕ} (hp : p ≤ m) : | |
| (∑ i ∈ Finset.range p, D.δ i) ≤ 1 - A := by | |
| simpa [D.budget] using sum_prefix_le D.δ hp D.nonneg | |
| theorem first_two_cap (hm : 2 ≤ m) : D.r 0 ^ 2 + D.r 1 ^ 2 ≤ 8 / 9 := by | |
| have h0 := D.cap 0 (by omega) | |
| have h1 := D.cap 1 (by omega) | |
| have hp0 := D.pos 0 (by omega) | |
| have hp1 := D.pos 1 (by omega) | |
| nlinarith | |
| theorem third_twice (hm : 3 ≤ m) : 2 * D.r 2 ^ 2 ≤ D.r 0 ^ 2 + D.r 1 ^ 2 := by | |
| have hp := D.pos 2 (by omega) | |
| have h0 := D.radius_le (by omega : 2 < m) (by omega : 0 ≤ 2) | |
| have h1 := D.radius_le (by omega : 2 < m) (by omega : 1 ≤ 2) | |
| nlinarith | |
| theorem common_charge (hm : 3 ≤ m) (hA : 0.884 ≤ A) : | |
| (1 - (D.r 0 ^ 2 + D.r 1 ^ 2)) * (0.415 / D.r 2 - 0.734) ≤ 0.116 := by | |
| let u := D.r 2 | |
| have hu : 0 < u := D.pos 2 (by omega) | |
| have hterm : ∀ i < m - 2, | |
| D.r (2 + i) ^ 2 * (0.415 / u - (A - 0.15)) ≤ D.δ (2 + i) := by | |
| intro i hi | |
| have hi' : 2 + i < m := by omega | |
| have hri := D.pos (2 + i) hi' | |
| have hle : D.r (2 + i) ≤ u := D.radius_le hi' (by omega) | |
| have hmul : D.r (2 + i) ^ 2 * (0.415 / u) ≤ 0.415 * D.r (2 + i) := by | |
| rw [← mul_div_assoc] | |
| apply (div_le_iff₀ hu).mpr | |
| nlinarith [mul_nonneg (le_of_lt hri) (sub_nonneg.mpr hle)] | |
| have hlower := D.lower (2 + i) hi' | |
| dsimp [charge] at hlower | |
| nlinarith | |
| have hsum := Finset.sum_le_sum (fun i hi => hterm i (Finset.mem_range.mp hi)) | |
| rw [← Finset.sum_mul] at hsum | |
| have hn := sum_split (fun i => D.r i ^ 2) (by omega : 2 ≤ m) | |
| have hb := sum_split D.δ (by omega : 2 ≤ m) | |
| simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add] at hn hb | |
| rw [D.norm] at hn | |
| rw [D.budget] at hb | |
| have hd0 := D.nonneg 0 (by omega) | |
| have hd1 := D.nonneg 1 (by omega) | |
| have hL : 0 ≤ D.r 0 ^ 2 + D.r 1 ^ 2 := by positivity | |
| have hAdj : 0 ≤ (A - 0.884) * (D.r 0 ^ 2 + D.r 1 ^ 2) := by positivity | |
| have hns : (∑ i ∈ Finset.range (m - 2), D.r (2 + i) ^ 2) = | |
| 1 - (D.r 0 ^ 2 + D.r 1 ^ 2) := by linarith | |
| rw [hns] at hsum | |
| dsimp [u] at hsum | |
| nlinarith | |
| theorem tail_coarse_cap (hm : 5 ≤ m) {i : ℕ} (hi : i < m) (hi4 : 4 ≤ i) : | |
| D.r i ≤ 0.45 := by | |
| have hn := D.prefix_norm (by omega : 5 ≤ m) | |
| have hp := D.pos i hi | |
| have h0 := D.radius_le hi (by omega : 0 ≤ i) | |
| have h1 := D.radius_le hi (by omega : 1 ≤ i) | |
| have h2 := D.radius_le hi (by omega : 2 ≤ i) | |
| have h3 := D.radius_le hi (by omega : 3 ≤ i) | |
| have h4 := D.radius_le hi hi4 | |
| simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add] at hn | |
| nlinarith | |
| theorem block_budget {p q : ℕ} (hpq : p + q ≤ m) : | |
| ∑ i ∈ Finset.range q, D.δ (p + i) ≤ 1 - A := by | |
| have h := D.prefix_budget hpq | |
| have hs := sum_split D.δ (by omega : p ≤ p + q) | |
| rw [Nat.add_sub_cancel_left] at hs | |
| have hn : 0 ≤ ∑ i ∈ Finset.range p, D.δ i := | |
| Finset.sum_nonneg (fun i hi => D.nonneg i (by have := Finset.mem_range.mp hi; omega)) | |
| linarith | |
| theorem prefix_block_budget {p q : ℕ} (hpq : p + q ≤ m) : | |
| (∑ i ∈ Finset.range p, D.δ i) + | |
| (∑ i ∈ Finset.range q, D.δ (p + i)) ≤ 1 - A := by | |
| rw [← Finset.sum_range_add] | |
| exact D.prefix_budget hpq | |
| theorem prefix_block_norm {p q : ℕ} (hpq : p + q ≤ m) : | |
| (∑ i ∈ Finset.range p, D.r i ^ 2) + | |
| (∑ i ∈ Finset.range q, D.r (p + i) ^ 2) ≤ 1 := by | |
| rw [← Finset.sum_range_add] | |
| exact D.prefix_norm hpq | |
| theorem minimum_lower (hm : 5 ≤ m) (hA : 0.884 ≤ A) | |
| (hdet : (D.r 2 ^ 2 - D.r (m - 1) ^ 2) * | |
| (D.r 0 ^ 2 + D.r 1 ^ 2 - D.r 2 ^ 2 - 2 * D.r (m - 1) ^ 2) ≤ | |
| 8 * (3 / Real.pi) ^ 2 * D.r (m - 1) ^ 2) : | |
| 0.066 < D.r (m - 1) := by | |
| have hc := D.common_charge (by omega) hA | |
| have hL := D.first_two_cap (by omega) | |
| have hu := third_radius_gt (D.pos 2 (by omega)) hL hc | |
| exact minimum_gt (D.pos (m - 1) (by omega)).le hu hL | |
| (D.third_twice (by omega)) hc (first_constraint_rational hdet) | |
| theorem tail_base_charge {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) | |
| (ht : 0.066 < D.r (m - 1)) {i : ℕ} (hi : i < m) (hi4 : 4 ≤ i) : | |
| charge A 0.066 ≤ D.δ i := by | |
| have h := min_charge_le (by linarith [R.A_lower] : 0.15 ≤ A) | |
| (ht.le.trans (D.minimum_le hi)) (D.tail_coarse_cap hm hi hi4) | |
| rw [min_eq_left R.tail_end_charge] at h | |
| exact h.trans (D.lower i hi) | |
| theorem fourth_cap {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) (hkm : 3 + k ≤ m) | |
| (ht : 0.066 < D.r (m - 1)) : D.r 3 ≤ 0.42 := by | |
| by_contra! h4 | |
| have hk : 2 ≤ k := R.k_lower | |
| have hd : 0 < A - 0.15 := by linarith [R.A_lower] | |
| have hs : 0 < threshold A := div_pos (by norm_num) hd | |
| have hds : (A - 0.15) * threshold A = 0.415 := by | |
| unfold threshold | |
| field_simp | |
| have hcoef : 0 ≤ clipCoeff A := by unfold clipCoeff; positivity | |
| have hfirst : ∀ i < 4, | |
| clipCoeff A * ((threshold A) ^ 2 - D.r i ^ 2) ≤ D.δ i := by | |
| intro i hi | |
| have hi' : i < m := by omega | |
| exact clipped_charge hd hs (by norm_num) | |
| (h4.le.trans (D.radius_le (by omega : 3 < m) (by omega))) | |
| hds (D.nonneg i hi') (D.lower i hi') | |
| have hf := Finset.sum_le_sum (fun i hi => hfirst i (Finset.mem_range.mp hi)) | |
| rw [← Finset.mul_sum, Finset.sum_sub_distrib] at hf | |
| simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_ofNat] at hf | |
| have htailnorm : (k - 1 : ℕ) * (0.066 : ℝ) ^ 2 ≤ | |
| ∑ i ∈ Finset.range (k - 1), D.r (4 + i) ^ 2 := by | |
| have hterm : ∀ i < k - 1, (0.066 : ℝ) ^ 2 ≤ D.r (4 + i) ^ 2 := by | |
| intro i hi | |
| have hi' : 4 + i < m := by omega | |
| have hr := ht.trans_le (D.minimum_le hi') | |
| nlinarith | |
| simpa using Finset.sum_le_sum (fun i hi => hterm i (Finset.mem_range.mp hi)) | |
| have hn := D.prefix_block_norm (by omega : 4 + (k - 1) ≤ m) | |
| have hnm : (∑ i ∈ Finset.range 4, D.r i ^ 2) ≤ | |
| 1 - (k - 1 : ℕ) * (0.066 : ℝ) ^ 2 := by linarith | |
| have hf' := mul_le_mul_of_nonneg_left (sub_le_sub_left hnm (4 * (threshold A) ^ 2)) hcoef | |
| have htbudget : (k - 1 : ℕ) * charge A 0.066 ≤ | |
| ∑ i ∈ Finset.range (k - 1), D.δ (4 + i) := by | |
| have hterm : ∀ i < k - 1, charge A 0.066 ≤ D.δ (4 + i) := by | |
| intro i hi | |
| exact D.tail_base_charge R hm ht (by omega) (by omega) | |
| simpa using Finset.sum_le_sum (fun i hi => hterm i (Finset.mem_range.mp hi)) | |
| have hb := D.prefix_block_budget (by omega : 4 + (k - 1) ≤ m) | |
| have hcast : ((k - 1 : ℕ) : ℝ) = (k : ℝ) - 1 := by | |
| rw [Nat.cast_sub (by omega : 1 ≤ k)] | |
| norm_num | |
| rw [hcast] at hf' htbudget | |
| have hmarg := R.four_margin | |
| nlinarith | |
| theorem tail_cap {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) (hkm : 3 + k ≤ m) | |
| (ht : 0.066 < D.r (m - 1)) {i : ℕ} (hi : i < m) (hi3 : 3 ≤ i) : | |
| D.r i ≤ 0.42 := | |
| (D.radius_le hi hi3).trans (D.fourth_cap R hm hkm ht) | |
| theorem minimum_charge_bound {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) | |
| (hkm : 3 + k ≤ m) (ht : 0.066 < D.r (m - 1)) : | |
| (k : ℝ) * charge A (D.r (m - 1)) ≤ 1 - A := by | |
| have hterm : ∀ i < k, | |
| min (charge A (D.r (m - 1))) (charge A 0.42) ≤ D.δ (3 + i) := by | |
| intro i hi | |
| have hi' : 3 + i < m := by omega | |
| exact (min_charge_le (by linarith [R.A_lower] : 0.15 ≤ A) | |
| (D.minimum_le hi') (D.tail_cap R hm hkm ht hi' (by omega))).trans (D.lower _ hi') | |
| have hs := Finset.sum_le_sum (fun i hi => hterm i (Finset.mem_range.mp hi)) | |
| simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul] at hs | |
| have hb := D.block_budget hkm | |
| by_cases h : charge A (D.r (m - 1)) ≤ charge A 0.42 | |
| · rw [min_eq_left h] at hs | |
| exact hs.trans hb | |
| · rw [min_eq_right (le_of_not_ge h)] at hs | |
| linarith [R.cap_charge] | |
| theorem minimum_le_root {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) | |
| (hkm : 3 + k ≤ m) (ht : 0.066 < D.r (m - 1)) : | |
| D.r (m - 1) ≤ smallRoot A k := by | |
| have hb := D.minimum_charge_bound R hm hkm ht | |
| have hcap := D.tail_cap R hm hkm ht (by omega : m - 1 < m) (by omega : 3 ≤ m - 1) | |
| have hd : 0 ≤ A - 0.15 := by linarith [R.A_lower] | |
| have hk : (0 : ℝ) < k := by exact_mod_cast (lt_of_lt_of_le (by omega : 0 < 2) R.k_lower) | |
| by_contra! h | |
| have hslope : 0 < 0.415 - (A - 0.15) * (D.r (m - 1) + smallRoot A k) := by | |
| have hmul := mul_le_mul_of_nonneg_left hcap hd | |
| linarith [R.root_slope] | |
| have hprod := mul_pos (sub_pos.mpr h) hslope | |
| have hcharge : charge A (smallRoot A k) < charge A (D.r (m - 1)) := by | |
| dsimp [charge] | |
| nlinarith | |
| have hmul := mul_lt_mul_of_pos_left hcharge hk | |
| linarith [R.root_eq] | |
| theorem tail_charge {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) (hkm : 3 + k ≤ m) | |
| (ht : 0.066 < D.r (m - 1)) {i : ℕ} (hi : i < m) (hi3 : 3 ≤ i) : | |
| charge A (D.r (m - 1)) ≤ D.δ i := by | |
| have hbudget := D.minimum_charge_bound R hm hkm ht | |
| have hk : (0 : ℝ) < k := by exact_mod_cast (lt_of_lt_of_le (by omega : 0 < 2) R.k_lower) | |
| have hcharge : charge A (D.r (m - 1)) ≤ charge A 0.42 := by | |
| have h := (lt_of_le_of_lt hbudget R.cap_charge) | |
| nlinarith | |
| have h := min_charge_le (by linarith [R.A_lower] : 0.15 ≤ A) | |
| (D.minimum_le hi) (D.tail_cap R hm hkm ht hi hi3) | |
| rw [min_eq_left hcharge] at h | |
| exact h.trans (D.lower i hi) | |
| theorem third_large {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) (hkm : 3 + k ≤ m) | |
| (ht : 0.066 < D.r (m - 1)) : 2.6 * D.r (m - 1) < D.r 2 ^ 2 := by | |
| let t := D.r (m - 1) | |
| let t₀ := smallRoot A k | |
| have ht0 : 0 < t := by dsimp [t]; linarith | |
| have htroot : t ≤ t₀ := D.minimum_le_root R hm hkm ht | |
| have htrootpos : 0 < t₀ := by dsimp [t₀]; linarith [R.root_lower] | |
| have hd : 0 < A - 0.15 := by linarith [R.A_lower] | |
| have hk : (0 : ℝ) ≤ k := Nat.cast_nonneg k | |
| have htlt : t₀ < 0.12 := R.root_upper | |
| have he_inc : charge A 0.066 ≤ charge A t := by | |
| have hmul := mul_le_mul_of_nonneg_left (show 0.066 + t ≤ 0.42 + t₀ by linarith) hd.le | |
| have hslope : 0 ≤ 0.415 - (A - 0.15) * (t + 0.066) := by | |
| linarith [R.root_slope] | |
| have hprod := mul_nonneg (show 0 ≤ t - 0.066 by dsimp [t]; linarith) hslope | |
| dsimp [charge] | |
| nlinarith | |
| have hfst : 1 - A < charge A 0.23 + (k : ℝ) * charge A t := by | |
| have hm := mul_le_mul_of_nonneg_left he_inc hk | |
| linarith [R.third_margin] | |
| have hspos : 0 < threshold A := div_pos (by norm_num) hd | |
| have hslopeeq : (A - 0.15) * threshold A = 0.415 := by | |
| unfold threshold | |
| field_simp | |
| have hsqrtpos : 0 < Real.sqrt (2.6 * t₀) := Real.sqrt_pos.mpr (by positivity) | |
| have hsqrtsq := Real.sq_sqrt (show 0 ≤ 2.6 * t₀ by positivity) | |
| have hsqrtlt : Real.sqrt (2.6 * t₀) < threshold A := by | |
| have hcap := R.root_cap | |
| dsimp [t₀] at hsqrtsq ⊢ | |
| nlinarith | |
| have hepos : 0 < charge A (Real.sqrt (2.6 * t₀)) := by | |
| have hp := mul_pos hd (sub_pos.mpr hsqrtlt) | |
| have hp' := mul_pos hsqrtpos hp | |
| dsimp [charge] | |
| nlinarith | |
| have hend : 1 - A < finalBudget A k t₀ := by | |
| dsimp [finalBudget] | |
| linarith [R.root_eq] | |
| have hlower := (finalBudget_concave (by linarith [R.A_lower] : 0.15 ≤ A) hk).min_le_of_mem_Icc (by norm_num : (0.066 : ℝ) ∈ Set.Ici 0) | |
| (show t₀ ∈ Set.Ici 0 by exact htrootpos.le) (show t ∈ Set.Icc 0.066 t₀ by exact ⟨ht.le, htroot⟩) | |
| have hsnd : 1 - A < finalBudget A k t := | |
| lt_of_lt_of_le (lt_min R.final_margin hend) hlower | |
| by_contra! hthird | |
| have hthirdpos := D.pos 2 (by omega) | |
| have hthirdlow := third_radius_gt hthirdpos (D.first_two_cap (by omega)) | |
| (D.common_charge (by omega) R.A_lower) | |
| have hsqrt := Real.sq_sqrt (show 0 ≤ 2.6 * t by positivity) | |
| have hthirdhi : D.r 2 ≤ Real.sqrt (2.6 * t) := by | |
| nlinarith [Real.sqrt_nonneg (2.6 * t)] | |
| have hthirdcharge := (min_charge_le (by linarith [R.A_lower] : 0.15 ≤ A) | |
| hthirdlow.le hthirdhi).trans (D.lower 2 (by omega)) | |
| have htailterm : ∀ i < k, charge A t ≤ D.δ (3 + i) := by | |
| intro i hi | |
| exact D.tail_charge R hm hkm ht (by omega) (by omega) | |
| have htail := Finset.sum_le_sum (fun i hi => htailterm i (Finset.mem_range.mp hi)) | |
| simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul] at htail | |
| have hb := D.prefix_block_budget hkm | |
| simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add] at hb | |
| have h0 := D.nonneg 0 (by omega) | |
| have h1 := D.nonneg 1 (by omega) | |
| dsimp [finalBudget] at hsnd | |
| rcases le_total (charge A 0.23) (charge A (Real.sqrt (2.6 * t))) with h | h | |
| · rw [min_eq_left h] at hthirdcharge | |
| linarith | |
| · rw [min_eq_right h] at hthirdcharge | |
| linarith | |
| theorem incompatible {k : ℕ} (R : Regime A k) (hm : 5 ≤ m) (hkm : 3 + k ≤ m) | |
| (hdet₁ : (D.r 2 ^ 2 - D.r (m - 1) ^ 2) * | |
| (D.r 0 ^ 2 + D.r 1 ^ 2 - D.r 2 ^ 2 - 2 * D.r (m - 1) ^ 2) ≤ | |
| 8 * (3 / Real.pi) ^ 2 * D.r (m - 1) ^ 2) | |
| (hdet₂ : (3 / 2 : ℝ) * (D.r 2 ^ 2 - D.r (m - 1) ^ 2) ^ 2 ≤ | |
| 10 * (3 / Real.pi) ^ 2 * D.r (m - 1) ^ 2) : False := by | |
| have ht := D.minimum_lower hm R.A_lower hdet₁ | |
| have htroot := D.minimum_le_root R hm hkm ht | |
| have ht12 := lt_of_le_of_lt htroot R.root_upper | |
| exact final_contradiction (by linarith : 0 < D.r (m - 1)) ht12 | |
| (D.third_large R hm hkm ht) hdet₂ | |
| end RadiusData | |
| theorem scalar_incompatibility {m : ℕ} (hm : 5 ≤ m) (r P : ℕ → ℝ) | |
| (hpos : ∀ i < m, 0 < r i) (hordered : AntitoneOn r (Set.Iio m)) | |
| (hcap : ∀ i < m, r i ≤ 2 / 3) | |
| (hnorm : ∑ i ∈ Finset.range m, r i ^ 2 = 1) | |
| (hprob : ∑ i ∈ Finset.range m, P i = 1) | |
| (hquad : ∀ i < m, (if m = 5 then (0.929 : ℝ) else 0.884) * r i ^ 2 ≤ P i) | |
| (hlin : ∀ i < m, 0.415 * r i + 0.15 * r i ^ 2 ≤ P i) | |
| (hdet₁ : (r 2 ^ 2 - r (m - 1) ^ 2) * | |
| (r 0 ^ 2 + r 1 ^ 2 - r 2 ^ 2 - 2 * r (m - 1) ^ 2) ≤ | |
| 8 * (3 / Real.pi) ^ 2 * r (m - 1) ^ 2) | |
| (hdet₂ : (3 / 2 : ℝ) * (r 2 ^ 2 - r (m - 1) ^ 2) ^ 2 ≤ | |
| 10 * (3 / Real.pi) ^ 2 * r (m - 1) ^ 2) : False := by | |
| let A : ℝ := if m = 5 then 0.929 else 0.884 | |
| let D : RadiusData m A := { | |
| r := r | |
| δ := fun i => P i - A * r i ^ 2 | |
| pos := hpos | |
| ordered := hordered | |
| cap := hcap | |
| norm := hnorm | |
| nonneg := fun i hi => sub_nonneg.mpr (hquad i hi) | |
| lower := by intro i hi; dsimp [charge]; linarith [hlin i hi] | |
| budget := by rw [Finset.sum_sub_distrib, ← Finset.mul_sum, hnorm, hprob, mul_one] } | |
| by_cases hm5 : m = 5 | |
| · have hR : Regime A 2 := by simpa [A, hm5] using regime_five | |
| exact D.incompatible hR hm (by omega) hdet₁ hdet₂ | |
| · have hR : Regime A 3 := by simpa [A, hm5] using regime_six | |
| exact D.incompatible hR hm (by omega) hdet₁ hdet₂ | |
| end Scalar | |
| end GaussianPropeller | |
| end OAI | |