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