import OAI.Geometry.CylinderCovering.Convexity.GeneralCapAreaDefinitions import OAI.Geometry.CylinderCovering.Angular.Taylor namespace OAI open MeasureTheory Set open scoped Interval namespace CylinderCovering.GeneralCapArea lemma continuous_profile : Continuous profile := by unfold profile; fun_prop lemma continuous_idealSpeed : Continuous idealSpeed := by unfold idealSpeed; fun_prop lemma continuous_idealDensity (ε η : ℝ) : Continuous (idealDensity ε η) := by unfold idealDensity profile idealSpeed fun_prop lemma coefficient_polynomial (η q : ℝ) : profile q + η - idealSpeed q / 8 = (η - 1 / 128) + (-1 / 64 : ℝ) * q ^ 2 + (7 / 128 : ℝ) * q ^ 4 := by unfold profile idealSpeed ring lemma coefficient_integral (η : ℝ) : (∫ q in (-1 : ℝ)..1, profile q + η - idealSpeed q / 8) = 2 * η - 1 / 240 := by simp_rw [coefficient_polynomial] rw [intervalIntegral.integral_add, intervalIntegral.integral_add] · simp only [intervalIntegral.integral_const, intervalIntegral.integral_const_mul, integral_pow, smul_eq_mul] norm_num ring · exact (continuous_const.intervalIntegrable _ _) · exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _ · exact (continuous_const.add (continuous_const.mul (continuous_id.pow 2))).intervalIntegrable _ _ · exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _ lemma idealDensity_integral (ε η : ℝ) : (∫ q in (-1 : ℝ)..1, idealDensity ε η q) = 1 / 2 + (2 * η - 1 / 240) * ε ^ 2 := by unfold idealDensity rw [intervalIntegral.integral_add] · rw [intervalIntegral.integral_const_mul, coefficient_integral] simp only [intervalIntegral.integral_const, smul_eq_mul] ring · exact continuous_const.intervalIntegrable _ _ · exact (continuous_const.mul ((continuous_profile.add continuous_const).sub (continuous_idealSpeed.div_const 8))).intervalIntegrable _ _ lemma perturbed_product_remainder {z M A A₀ R E F : ℝ} (hz0 : 0 ≤ z) (hz1 : z ≤ 1) (hM0 : 0 ≤ M) (hM1 : M ≤ 1 / 4) (hA0 : 0 ≤ A) (hA1 : A ≤ 1) (hE : 0 ≤ E) (hR : |R| ≤ 1) (hRE : |R - (1 / 2 + z * M)| ≤ E * z ^ 2) (hAF : |A - A₀| ≤ F * z) : |R ^ 2 / Real.sqrt (1 + z * A) - (1 / 4 + z * (M - A₀ / 8))| ≤ (2 * E + F / 8 + 47 / 128) * z ^ 2 := by have hzM0 : 0 ≤ z * M := mul_nonneg hz0 hM0 have hzM1 : z * M ≤ 1 / 4 := by nlinarith only [mul_le_mul hz1 hM1 hM0 (by norm_num : (0 : ℝ) ≤ 1)] have hbase : |(1 / 2 : ℝ) + z * M| ≤ 1 := by rw [abs_of_nonneg (by linarith)] linarith have hsum : |R + (1 / 2 + z * M)| ≤ 2 := (abs_add_le _ _).trans (by linarith only [hR, hbase]) have hsq : |R ^ 2 - (1 / 2 + z * M) ^ 2| ≤ 2 * E * z ^ 2 := by calc _ = |R - (1 / 2 + z * M)| * |R + (1 / 2 + z * M)| := by rw [← abs_mul]; congr 1; ring _ ≤ (E * z ^ 2) * 2 := mul_le_mul hRE hsum (abs_nonneg _) (by positivity) _ = _ := by ring have hsqrt : 1 ≤ Real.sqrt (1 + z * A) := by exact (Real.le_sqrt (by norm_num) (by positivity)).2 (by nlinarith [mul_nonneg hz0 hA0]) have hradial : |R ^ 2 / Real.sqrt (1 + z * A) - (1 / 2 + z * M) ^ 2 / Real.sqrt (1 + z * A)| ≤ 2 * E * z ^ 2 := by rw [← sub_div, abs_div, abs_of_nonneg (by linarith : 0 ≤ Real.sqrt (1 + z * A))] exact (div_le_self (abs_nonneg _) hsqrt).trans hsq have hspeed : |(1 / 4 + z * (M - A / 8)) - (1 / 4 + z * (M - A₀ / 8))| ≤ F / 8 * z ^ 2 := by calc _ = z / 8 * |A - A₀| := by have hid : (1 / 4 + z * (M - A / 8)) - (1 / 4 + z * (M - A₀ / 8)) = -(z / 8 * (A - A₀)) := by ring rw [hid, abs_neg, abs_mul, abs_of_nonneg (by positivity : 0 ≤ z / 8)] _ ≤ z / 8 * (F * z) := mul_le_mul_of_nonneg_left hAF (by positivity) _ = _ := by ring have htaylor := AngularTaylor.bounded_product_remainder hz0 hz1 hM0 hM1 hA0 hA1 have htri1 := abs_sub_le (R ^ 2 / Real.sqrt (1 + z * A)) ((1 / 2 + z * M) ^ 2 / Real.sqrt (1 + z * A)) (1 / 4 + z * (M - A₀ / 8)) have htri2 := abs_sub_le ((1 / 2 + z * M) ^ 2 / Real.sqrt (1 + z * A)) (1 / 4 + z * (M - A / 8)) (1 / 4 + z * (M - A₀ / 8)) nlinarith only [hradial, hspeed, htaylor, htri1, htri2] lemma profile_bounds {q : ℝ} (hq : |q| ≤ 1) : 0 ≤ profile q ∧ profile q ≤ 1 / 8 := by have hq2 : q ^ 2 ≤ 1 := by nlinarith [(abs_le.mp hq).1, (abs_le.mp hq).2] have hq4 : q ^ 4 ≤ 1 := by nlinarith [sq_nonneg q, mul_self_le_mul_self (sq_nonneg q) hq2] unfold profile constructor · positivity · linarith lemma square_difference {x y E : ℝ} (hE : 0 ≤ E) (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hxy : |x - y| ≤ E) : |x ^ 2 - y ^ 2| ≤ 2 * E := by have hsum : |x + y| ≤ 2 := (abs_add_le _ _).trans (by linarith only [hx, hy]) calc _ = |x - y| * |x + y| := by rw [← abs_mul]; congr 1; ring _ ≤ E * 2 := mul_le_mul hxy hsum (abs_nonneg _) hE _ = _ := by ring theorem capDensity_error {ε η C α β q : ℝ} {R : ℝ → ℝ} (hη0 : 0 ≤ η) (hη1 : η ≤ 1 / 8) (hC : 0 ≤ C) (hε : ε ^ 2 ≤ 1) (hsmall : C * ε ^ 2 ≤ 1 / 16) (hq : |q| ≤ 1) (hα : |α - (1 + q ^ 2) / 4| ≤ C * ε ^ 2) (hβ : |β - q / 2| ≤ C * ε ^ 2) (hR : |R q - (1 / 2 + ε ^ 2 * (profile q + η))| ≤ C * ε ^ 4) : |capDensity ε α β R q - idealDensity ε η q| ≤ (3 * C + 1) * ε ^ 4 := by have hz0 : 0 ≤ ε ^ 2 := sq_nonneg ε have hE : 0 ≤ C * ε ^ 2 := mul_nonneg hC hz0 have hq2 : q ^ 2 ≤ 1 := by nlinarith [(abs_le.mp hq).1, (abs_le.mp hq).2] have ha00 : 0 ≤ (1 + q ^ 2) / 4 := by positivity have ha01 : (1 + q ^ 2) / 4 ≤ 1 / 2 := by linarith have ha0abs : |(1 + q ^ 2) / 4| ≤ 1 / 2 := by rw [abs_of_nonneg ha00] exact ha01 have hb0abs : |q / 2| ≤ 1 / 2 := by rw [abs_div] norm_num linarith have haa : |α| ≤ 9 / 16 := by have h := abs_add_le (α - (1 + q ^ 2) / 4) ((1 + q ^ 2) / 4) rw [sub_add_cancel] at h linarith only [h, hα, ha0abs, hsmall] have hbb : |β| ≤ 9 / 16 := by have h := abs_add_le (β - q / 2) (q / 2) rw [sub_add_cancel] at h linarith only [h, hβ, hb0abs, hsmall] have ha2 : α ^ 2 ≤ 81 / 256 := by nlinarith [(abs_le.mp haa).1, (abs_le.mp haa).2] have hb2 : β ^ 2 ≤ 81 / 256 := by nlinarith [(abs_le.mp hbb).1, (abs_le.mp hbb).2] have hA0 : 0 ≤ α ^ 2 + 2 * β ^ 2 := by positivity have hA1 : α ^ 2 + 2 * β ^ 2 ≤ 1 := by linarith have hM0 : 0 ≤ profile q + η := add_nonneg (profile_bounds hq).1 hη0 have hM1 : profile q + η ≤ 1 / 4 := by linarith [(profile_bounds hq).2] have hbase0 : 0 ≤ (1 / 2 : ℝ) + ε ^ 2 * (profile q + η) := by positivity have hbase1 : (1 / 2 : ℝ) + ε ^ 2 * (profile q + η) ≤ 3 / 4 := by nlinarith only [mul_le_mul hε hM1 hM0 (by norm_num : (0 : ℝ) ≤ 1)] have hCE4 : C * ε ^ 4 ≤ 1 / 16 := by have h := mul_le_mul_of_nonneg_left hε hE nlinarith only [h, hsmall] have hRabs : |R q| ≤ 1 := by have h := abs_add_le (R q - (1 / 2 + ε ^ 2 * (profile q + η))) (1 / 2 + ε ^ 2 * (profile q + η)) rw [sub_add_cancel, abs_of_nonneg hbase0] at h linarith only [h, hR, hbase1, hCE4] have hαsq := square_difference hE (by linarith only [haa] : |α| ≤ 1) (by linarith only [ha0abs] : |(1 + q ^ 2) / 4| ≤ 1) hα have hβsq := square_difference hE (by linarith only [hbb] : |β| ≤ 1) (by linarith only [hb0abs] : |q / 2| ≤ 1) hβ have hspeed : |(α ^ 2 + 2 * β ^ 2) - idealSpeed q| ≤ (6 * C) * ε ^ 2 := by have hid : (α ^ 2 + 2 * β ^ 2) - idealSpeed q = (α ^ 2 - ((1 + q ^ 2) / 4) ^ 2) + 2 * (β ^ 2 - (q / 2) ^ 2) := by unfold idealSpeed ring rw [hid] have h := abs_add_le (α ^ 2 - ((1 + q ^ 2) / 4) ^ 2) (2 * (β ^ 2 - (q / 2) ^ 2)) rw [abs_mul] at h norm_num at h linarith only [h, hαsq, hβsq] have hR' : |R q - (1 / 2 + ε ^ 2 * (profile q + η))| ≤ C * (ε ^ 2) ^ 2 := by convert hR using 1; ring have h := perturbed_product_remainder hz0 hε hM0 hM1 hA0 hA1 hC hRabs hR' hspeed unfold capDensity idealDensity calc _ ≤ (2 * C + (6 * C) / 8 + 47 / 128) * (ε ^ 2) ^ 2 := h _ ≤ (3 * C + 1) * ε ^ 4 := by have hp := mul_nonneg hC (sq_nonneg (ε ^ 2)) nlinarith only [hp, sq_nonneg (ε ^ 2)] lemma integral_error {f : ℝ → ℝ} {a b ε η L : ℝ} (hab : a ≤ b) (hf : IntervalIntegrable f volume a b) (herror : ∀ q ∈ Icc a b, |f q - idealDensity ε η q| ≤ L * ε ^ 4) : |(∫ q in a..b, f q) - (∫ q in a..b, idealDensity ε η q)| ≤ L * ε ^ 4 * (b - a) := by rw [← intervalIntegral.integral_sub hf ((continuous_idealDensity ε η).intervalIntegrable a b)] have h := intervalIntegral.norm_integral_le_of_norm_le_const (a := a) (b := b) (C := L * ε ^ 4) (f := fun q => f q - idealDensity ε η q) ?_ · simpa only [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hab)] using h · intro q hq rw [uIoc_of_le hab] at hq simpa only [Real.norm_eq_abs] using herror q ⟨hq.1.le, hq.2⟩ theorem partition_area_error {N : ℕ} {a : ℕ → ℝ} {f : ℕ → ℝ → ℝ} {ε η L : ℝ} (ha0 : a 0 = -1) (haN : a N = 1) (ha : ∀ j < N, a j ≤ a (j + 1)) (hf : ∀ j < N, IntervalIntegrable (f j) volume (a j) (a (j + 1))) (herror : ∀ j < N, ∀ q ∈ Icc (a j) (a (j + 1)), |f j q - idealDensity ε η q| ≤ L * ε ^ 4) : |(∑ j ∈ Finset.range N, ∫ q in a j..a (j + 1), f j q) - (1 / 2 + (2 * η - 1 / 240) * ε ^ 2)| ≤ 2 * L * ε ^ 4 := by have hideal : (∑ j ∈ Finset.range N, ∫ q in a j..a (j + 1), idealDensity ε η q) = 1 / 2 + (2 * η - 1 / 240) * ε ^ 2 := by rw [intervalIntegral.sum_integral_adjacent_intervals (fun j hj => (continuous_idealDensity ε η).intervalIntegrable (a j) (a (j + 1))), ha0, haN, idealDensity_integral] rw [← hideal, ← Finset.sum_sub_distrib] calc _ ≤ ∑ j ∈ Finset.range N, |(∫ q in a j..a (j + 1), f j q) - (∫ q in a j..a (j + 1), idealDensity ε η q)| := Finset.abs_sum_le_sum_abs _ _ _ ≤ ∑ j ∈ Finset.range N, L * ε ^ 4 * (a (j + 1) - a j) := by apply Finset.sum_le_sum intro j hj exact integral_error (ha j (Finset.mem_range.mp hj)) (hf j (Finset.mem_range.mp hj)) (herror j (Finset.mem_range.mp hj)) _ = 2 * L * ε ^ 4 := by rw [← Finset.mul_sum, Finset.sum_range_sub, haN, ha0] ring lemma partition_monotone {N : ℕ} {a : ℕ → ℝ} (ha : ∀ j < N, a j ≤ a (j + 1)) {i j : ℕ} (hij : i ≤ j) (hj : j ≤ N) : a i ≤ a j := by have h : ∀ k ≥ i, k ≤ N → a i ≤ a k := by intro k hik induction k, hik using Nat.le_induction with | base => intro _; exact le_rfl | succ k hik ih => intro hkN exact (ih (by omega)).trans (ha k (by omega)) exact h j hij hj theorem general_cap_area_expansion {N : ℕ} {a α β : ℕ → ℝ} {R : ℕ → ℝ → ℝ} {ε η C : ℝ} (ha0 : a 0 = -1) (haN : a N = 1) (ha : ∀ j < N, a j ≤ a (j + 1)) (hRcont : ∀ j < N, ContinuousOn (R j) (Icc (a j) (a (j + 1)))) (hη0 : 0 ≤ η) (hη1 : η ≤ 1 / 8) (hC : 0 ≤ C) (hε : ε ^ 2 ≤ 1) (hsmall : C * ε ^ 2 ≤ 1 / 16) (hα : ∀ j < N, ∀ q ∈ Icc (a j) (a (j + 1)), |α j - (1 + q ^ 2) / 4| ≤ C * ε ^ 2) (hβ : ∀ j < N, ∀ q ∈ Icc (a j) (a (j + 1)), |β j - q / 2| ≤ C * ε ^ 2) (hR : ∀ j < N, ∀ q ∈ Icc (a j) (a (j + 1)), |R j q - (1 / 2 + ε ^ 2 * (profile q + η))| ≤ C * ε ^ 4) : |(∑ j ∈ Finset.range N, ∫ q in a j..a (j + 1), capDensity ε (α j) (β j) (R j) q) - (1 / 2 + (2 * η - 1 / 240) * ε ^ 2)| ≤ 2 * (3 * C + 1) * ε ^ 4 := by apply partition_area_error ha0 haN ha · intro j hj apply ContinuousOn.intervalIntegrable_of_Icc (ha j hj) exact ((hRcont j hj).pow 2).div_const _ · intro j hj q hq have haq : |q| ≤ 1 := by have hl : a 0 ≤ a j := partition_monotone ha (Nat.zero_le j) (by omega) have hu : a (j + 1) ≤ a N := partition_monotone ha (by omega) le_rfl rw [ha0] at hl rw [haN] at hu rw [abs_le] exact ⟨hl.trans hq.1, hq.2.trans hu⟩ exact capDensity_error hη0 hη1 hC hε hsmall haq (hα j hj q hq) (hβ j hj q hq) (hR j hj q hq) theorem strict_half_of_area_error {cost ε η L : ℝ} (hε : 0 < ε) (herror : |cost - (1 / 2 + (2 * η - 1 / 240) * ε ^ 2)| ≤ L * ε ^ 4) (hsmall : L * ε ^ 2 < 1 / 240 - 2 * η) : cost < 1 / 2 := by have hgap := mul_lt_mul_of_pos_right hsmall (sq_pos_of_pos hε) have hu := (abs_le.mp herror).2 nlinarith only [hgap, hu] theorem exists_uniform_smallness {η C : ℝ} (hη0 : 0 < η) (hη1 : η < 1 / 480) (hC : 0 ≤ C) : ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, 0 < ε → ε < δ → ε ≤ 1 / 2 ∧ (10 + 4 * C) * ε ≤ η ∧ ε ^ 2 ≤ 1 ∧ C * ε ^ 2 ≤ 1 / 16 ∧ 2 * (3 * C + 1) * ε ^ 2 < 1 / 240 - 2 * η := by let g : ℝ := 1 / 240 - 2 * η have hg : 0 < g := by dsimp [g]; linarith let δ : ℝ := min (1 / 2) (min (η / (10 + 4 * C)) (min (1 / (16 * (C + 1))) (g / (4 * (3 * C + 1))))) have hδ : 0 < δ := by dsimp [δ]; positivity refine ⟨δ, hδ, ?_⟩ intro ε hε0 hεδ have hehalf : ε < 1 / 2 := lt_of_lt_of_le hεδ (min_le_left _ _) have hdiv1 : ε < η / (10 + 4 * C) := lt_of_lt_of_le hεδ ((min_le_right _ _).trans (min_le_left _ _)) have hdiv2 : ε < 1 / (16 * (C + 1)) := lt_of_lt_of_le hεδ ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _))) have hdiv3 : ε < g / (4 * (3 * C + 1)) := lt_of_lt_of_le hεδ ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _))) have hcover : (10 + 4 * C) * ε ≤ η := by have h := (lt_div_iff₀ (by positivity : 0 < 10 + 4 * C)).mp hdiv1 nlinarith only [h] have he1 : ε ≤ 1 := by linarith only [hehalf] have he2 : ε ^ 2 ≤ 1 := by nlinarith only [hε0, he1] have hCE : C * ε ^ 2 ≤ 1 / 16 := by have h := (lt_div_iff₀ (by positivity : 0 < 16 * (C + 1))).mp hdiv2 have hm := mul_le_mul_of_nonneg_left he1 (mul_nonneg hC hε0.le) nlinarith only [h, hm, hε0] have hsaving : 2 * (3 * C + 1) * ε ^ 2 < g := by have h := (lt_div_iff₀ (by positivity : 0 < 4 * (3 * C + 1))).mp hdiv3 have hm := mul_le_mul_of_nonneg_left he1 (show 0 ≤ (3 * C + 1) * ε by positivity) have hp : 0 ≤ (3 * C + 1) * ε := by positivity nlinarith only [h, hm, hp] exact ⟨hehalf.le, hcover, he2, hCE, hsaving⟩ end CylinderCovering.GeneralCapArea end OAI