| import Mathlib | |
| namespace OAI | |
| namespace CylinderCovering.GeneralCap | |
| theorem quartic_bounds {q : ℝ} (hq : |q| ≤ 1) : | |
| 0 ≤ (q ^ 2 + q ^ 4) / 16 ∧ (q ^ 2 + q ^ 4) / 16 ≤ 1 / 8 := by | |
| have hq2 : q ^ 2 ≤ 1 := by nlinarith only [(abs_le.mp hq).1, (abs_le.mp hq).2] | |
| have hq4 : q ^ 4 ≤ 1 := by nlinarith [sq_nonneg (q ^ 2 - 1)] | |
| constructor <;> nlinarith [sq_nonneg q, sq_nonneg (q ^ 2)] | |
| theorem first_crossing {N : ℕ} (hN : 0 < N) (f : ℕ → ℝ) (z : ℝ) | |
| (h0 : f 0 ≤ z) (hN' : z ≤ f N) : | |
| ∃ j : ℕ, j < N ∧ f j ≤ z ∧ z ≤ f (j + 1) := by | |
| have hex : ∃ i : ℕ, 0 < i ∧ i ≤ N ∧ z ≤ f i := ⟨N, hN, le_rfl, hN'⟩ | |
| let i := Nat.find hex | |
| have hi : 0 < i ∧ i ≤ N ∧ z ≤ f i := Nat.find_spec hex | |
| have him : i - 1 + 1 = i := by omega | |
| refine ⟨i - 1, by omega, ?_, by simpa [him] using hi.2.2⟩ | |
| by_cases h : i = 1 | |
| · simpa [h] using h0 | |
| · have hsmall : i - 1 < Nat.find hex := by dsimp [i] at *; omega | |
| have hmin := Nat.find_min hex hsmall | |
| by_contra! hz | |
| exact hmin ⟨by omega, by omega, le_of_lt hz⟩ | |
| theorem weighted_errors {ε E x y e f : ℝ} (he : 0 ≤ ε) (hE : 0 ≤ E) | |
| (hxy : |x| + |y| ≤ 1) (he' : |e| ≤ E) (hf : |f| ≤ E) : | |
| -ε * x * e + ε * y * f ≤ ε * E := by | |
| have h1 : -ε * x * e ≤ ε * |x| * E := by | |
| calc | |
| -ε * x * e ≤ |-ε * x * e| := le_abs_self _ | |
| _ = ε * |x| * |e| := by simp [abs_mul, abs_of_nonneg he] | |
| _ ≤ ε * |x| * E := mul_le_mul_of_nonneg_left he' (by positivity) | |
| have h2 : ε * y * f ≤ ε * |y| * E := by | |
| calc | |
| ε * y * f ≤ |ε * y * f| := le_abs_self _ | |
| _ = ε * |y| * |f| := by simp [abs_mul, abs_of_nonneg he] | |
| _ ≤ ε * |y| * E := mul_le_mul_of_nonneg_left hf (by positivity) | |
| have h3 := mul_le_mul_of_nonneg_right hxy (show 0 ≤ ε * E by positivity) | |
| nlinarith | |
| theorem rough_angle {ε x y q a r : ℝ} (he : 0 ≤ ε) | |
| (hx : |x| ≤ 1) (hq : |q| ≤ 1) (ha : |a| ≤ 1) | |
| (hr0 : 1 / 2 ≤ r) (hr1 : r ≤ 1 / 2 + ε) | |
| (hy : y = r * q + ε * x * a) : |y - q / 2| ≤ 2 * ε := by | |
| have heq : y - q / 2 = (r - 1 / 2) * q + ε * x * a := by rw [hy]; ring | |
| have h1 : |(r - 1 / 2) * q| ≤ ε := by | |
| rw [abs_mul, abs_of_nonneg (by linarith : 0 ≤ r - 1 / 2)] | |
| calc | |
| (r - 1 / 2) * |q| ≤ (r - 1 / 2) * 1 := | |
| mul_le_mul_of_nonneg_left hq (by linarith) | |
| _ ≤ ε := by linarith | |
| have h2 : |ε * x * a| ≤ ε := by | |
| rw [abs_mul, abs_mul, abs_of_nonneg he] | |
| calc | |
| ε * |x| * |a| ≤ ε * 1 * 1 := | |
| mul_le_mul (mul_le_mul_of_nonneg_left hx he) ha (abs_nonneg _) (by positivity) | |
| _ = ε := by ring | |
| rw [heq] | |
| exact (abs_add_le _ _).trans (by linarith) | |
| theorem rough_square {ε x p : ℝ} (he : 0 ≤ ε) (hx : |x| ≤ 1) (hp : |p| ≤ 1) | |
| (h : |x - p / 2| ≤ 2 * ε) : |x ^ 2 - p ^ 2 / 4| ≤ 3 * ε := by | |
| have hsum : |x + p / 2| ≤ 3 / 2 := by | |
| have h' := abs_add_le x (p / 2) | |
| rw [abs_div] at h' | |
| norm_num at h' | |
| linarith | |
| calc | |
| |x ^ 2 - p ^ 2 / 4| = |x - p / 2| * |x + p / 2| := by | |
| rw [← abs_mul]; congr 1; ring | |
| _ ≤ (2 * ε) * (3 / 2) := mul_le_mul h hsum (abs_nonneg _) (by positivity) | |
| _ = 3 * ε := by ring | |
| theorem cross_term_bound {ε E x y p q a b : ℝ} | |
| (he : 0 ≤ ε) (hE : 0 ≤ E) (hp : |p| ≤ 1) (hq : |q| ≤ 1) | |
| (ha0 : 0 ≤ a) (ha1 : a ≤ 1) (hb0 : 0 ≤ b) (hb1 : b ≤ 1) | |
| (hxp : |x ^ 2 - p ^ 2 / 4| ≤ 3 * ε) | |
| (hyq : |y ^ 2 - q ^ 2 / 4| ≤ 3 * ε) | |
| (ha : |a - (1 + q ^ 2) / 4| ≤ E) | |
| (hb : |b - (1 + p ^ 2) / 4| ≤ E) : | |
| x ^ 2 * a + y ^ 2 * b ≤ | |
| (p ^ 2 + p ^ 4) / 16 + (q ^ 2 + q ^ 4) / 16 + 6 * ε + E / 2 := by | |
| have hp2 : p ^ 2 ≤ 1 := by nlinarith only [(abs_le.mp hp).1, (abs_le.mp hp).2] | |
| have hq2 : q ^ 2 ≤ 1 := by nlinarith only [(abs_le.mp hq).1, (abs_le.mp hq).2] | |
| have hxa : x ^ 2 * a ≤ p ^ 2 / 4 * ((1 + q ^ 2) / 4) + 3 * ε + E / 4 := by | |
| have h1 := mul_le_mul_of_nonneg_right (abs_le.mp hxp).2 ha0 | |
| have h2 := mul_le_mul_of_nonneg_left ha1 he | |
| have h3 := mul_le_mul_of_nonneg_left (abs_le.mp ha).2 (sq_nonneg p) | |
| have h4 := mul_le_mul_of_nonneg_right hp2 hE | |
| nlinarith | |
| have hyb : y ^ 2 * b ≤ q ^ 2 / 4 * ((1 + p ^ 2) / 4) + 3 * ε + E / 4 := by | |
| have h1 := mul_le_mul_of_nonneg_right (abs_le.mp hyq).2 hb0 | |
| have h2 := mul_le_mul_of_nonneg_left hb1 he | |
| have h3 := mul_le_mul_of_nonneg_left (abs_le.mp hb).2 (sq_nonneg q) | |
| have h4 := mul_le_mul_of_nonneg_right hq2 hE | |
| nlinarith | |
| nlinarith [sq_nonneg (p ^ 2 - q ^ 2)] | |
| theorem radial_identity {ε x y q p a b β γ r s : ℝ} | |
| (hrs : r + s = 1 - ε * x * β + ε * y * γ) | |
| (hy : y = r * q + ε * x * a) (hx : x = s * p - ε * y * b) : | |
| (r - 1 / 2) * (1 - ε * x * q) + (s - 1 / 2) * (1 + ε * y * p) = | |
| ε ^ 2 * (x ^ 2 * a + y ^ 2 * b) - ε * x * (β - q / 2) + ε * y * (γ - p / 2) := by | |
| have hq : r * q = y - ε * x * a := by linarith | |
| have hp : s * p = x + ε * y * b := by linarith | |
| calc | |
| _ = r + s - 1 - ε * x * (r * q) + ε * x * q / 2 + | |
| ε * y * (s * p) - ε * y * p / 2 := by ring | |
| _ = _ := by rw [hrs, hq, hp]; ring | |
| theorem radial_lower_est {ε M r f : ℝ} (heLt : ε < 1) | |
| (hM : 0 ≤ M) (hr : 1 / 2 + ε ^ 2 * M < r) (hf : 1 - ε ≤ f) : | |
| ε ^ 2 * M * (1 - ε) < (r - 1 / 2) * f := by | |
| calc | |
| _ ≤ ε ^ 2 * M * f := mul_le_mul_of_nonneg_left hf (by positivity) | |
| _ < _ := mul_lt_mul_of_pos_right (by linarith only [hr]) (by linarith only [heLt, hf]) | |
| theorem radial_failure {ε K x y q p a b β γ M N r s : ℝ} | |
| (he0 : 0 < ε) (heLt : ε < 1) (hK : 0 ≤ K) | |
| (hxy : |x| + |y| ≤ 1) (hq : |q| ≤ 1) (hp : |p| ≤ 1) | |
| (hM0 : 0 ≤ M) (hN0 : 0 ≤ N) | |
| (heβ : |β - q / 2| ≤ K * ε ^ 2) (heγ : |γ - p / 2| ≤ K * ε ^ 2) | |
| (hrs : r + s = 1 - ε * x * β + ε * y * γ) | |
| (hy : y = r * q + ε * x * a) (hx : x = s * p - ε * y * b) | |
| (hr : 1 / 2 + ε ^ 2 * M < r) (hs : 1 / 2 + ε ^ 2 * N < s) : | |
| (M + N) * (1 - ε) < (x ^ 2 * a + y ^ 2 * b) + K * ε := by | |
| have he := he0.le | |
| have hx1 : |x| ≤ 1 := by linarith only [hxy, abs_nonneg y] | |
| have hy1 : |y| ≤ 1 := by linarith only [hxy, abs_nonneg x] | |
| have habsq : |x * q| ≤ 1 := by | |
| rw [abs_mul] | |
| exact (mul_le_mul hx1 hq (abs_nonneg _) (by norm_num)).trans_eq (by norm_num) | |
| have hbpsq : |y * p| ≤ 1 := by | |
| rw [abs_mul] | |
| exact (mul_le_mul hy1 hp (abs_nonneg _) (by norm_num)).trans_eq (by norm_num) | |
| have hf1 : 1 - ε ≤ 1 - ε * x * q := by | |
| nlinarith only [mul_le_mul_of_nonneg_left (abs_le.mp habsq).2 he] | |
| have hf2 : 1 - ε ≤ 1 + ε * y * p := by | |
| nlinarith only [mul_le_mul_of_nonneg_left (abs_le.mp hbpsq).1 he] | |
| have hstrict1 := radial_lower_est heLt hM0 hr hf1 | |
| have hstrict2 := radial_lower_est heLt hN0 hs hf2 | |
| have hid := radial_identity hrs hy hx | |
| have herr := weighted_errors he (show 0 ≤ K * ε ^ 2 by positivity) hxy heβ heγ | |
| have hscaled : ε ^ 2 * ((M + N) * (1 - ε)) < | |
| ε ^ 2 * ((x ^ 2 * a + y ^ 2 * b) + K * ε) := by | |
| nlinarith only [hstrict1, hstrict2, hid, herr] | |
| exact (mul_lt_mul_iff_right₀ (sq_pos_of_pos he0)).mp hscaled | |
| theorem radial_exclusion {ε K η x y q p a b β γ r s R Q : ℝ} | |
| (he0 : 0 < ε) (he1 : ε ≤ 1 / 2) (hK : 0 ≤ K) | |
| (hη0 : 0 < η) (hη1 : η ≤ 1 / 480) | |
| (hsmall : (10 + 4 * K) * ε ≤ η) | |
| (hxy : |x| + |y| ≤ 1) (hq : |q| ≤ 1) (hp : |p| ≤ 1) | |
| (heβ : |β - q / 2| ≤ K * ε ^ 2) (heγ : |γ - p / 2| ≤ K * ε ^ 2) | |
| (hea : |a - (1 + q ^ 2) / 4| ≤ K * ε ^ 2) | |
| (heb : |b - (1 + p ^ 2) / 4| ≤ K * ε ^ 2) | |
| (hR : 1 / 2 + ε ^ 2 * ((q ^ 2 + q ^ 4) / 16 + η) - K * ε ^ 4 ≤ R) | |
| (hQ : 1 / 2 + ε ^ 2 * ((p ^ 2 + p ^ 4) / 16 + η) - K * ε ^ 4 ≤ Q) | |
| (hrs : r + s = 1 - ε * x * β + ε * y * γ) | |
| (hy : y = r * q + ε * x * a) (hx : x = s * p - ε * y * b) : | |
| r ≤ R ∨ s ≤ Q := by | |
| by_contra! hbad | |
| have he : 0 ≤ ε := he0.le | |
| have heLt : ε < 1 := by linarith only [he1] | |
| have hKE : K * ε ≤ η / 4 := by nlinarith only [hsmall, he0] | |
| have hKE0 : 0 ≤ K * ε := mul_nonneg hK he | |
| have hEη : K * ε ^ 2 ≤ η / 2 := by | |
| have h1 := mul_le_mul_of_nonneg_right hKE he | |
| have h2 := mul_le_mul_of_nonneg_left he1 hη0.le | |
| nlinarith only [h1, h2, hη0] | |
| have hE1 : K * ε ^ 2 ≤ 1 / 4 := by linarith only [hEη, hη1] | |
| have hEε : K * ε ^ 2 ≤ K * ε := by | |
| have h := mul_le_mul_of_nonneg_left he1 hKE0 | |
| nlinarith only [h, hKE0] | |
| have hE0 : 0 ≤ K * ε ^ 2 := by positivity | |
| have hp2 : p ^ 2 ≤ 1 := by nlinarith only [(abs_le.mp hp).1, (abs_le.mp hp).2] | |
| have hq2 : q ^ 2 ≤ 1 := by nlinarith only [(abs_le.mp hq).1, (abs_le.mp hq).2] | |
| have ha0 : 0 ≤ a := by nlinarith only [hE1, (abs_le.mp hea).1, sq_nonneg q] | |
| have hb0 : 0 ≤ b := by nlinarith only [hE1, (abs_le.mp heb).1, sq_nonneg p] | |
| have ha1 : a ≤ 1 := by nlinarith only [hE1, hq2, (abs_le.mp hea).2] | |
| have hb1 : b ≤ 1 := by nlinarith only [hE1, hp2, (abs_le.mp heb).2] | |
| have hβ : |β| ≤ 1 := by | |
| apply abs_le.mpr | |
| constructor <;> nlinarith only [hE1, (abs_le.mp heβ).1, (abs_le.mp heβ).2, | |
| (abs_le.mp hq).1, (abs_le.mp hq).2] | |
| have hγ : |γ| ≤ 1 := by | |
| apply abs_le.mpr | |
| constructor <;> nlinarith only [hE1, (abs_le.mp heγ).1, (abs_le.mp heγ).2, | |
| (abs_le.mp hp).1, (abs_le.mp hp).2] | |
| let M := (q ^ 2 + q ^ 4) / 16 + η - K * ε ^ 2 | |
| let N := (p ^ 2 + p ^ 4) / 16 + η - K * ε ^ 2 | |
| have hM0 : 0 ≤ M := by dsimp [M]; linarith only [hEη, hη0, (quartic_bounds hq).1] | |
| have hN0 : 0 ≤ N := by dsimp [N]; linarith only [hEη, hη0, (quartic_bounds hp).1] | |
| have hMN : M + N ≤ 3 := by | |
| dsimp [M, N] | |
| linarith only [(quartic_bounds hq).2, (quartic_bounds hp).2, hη1, hE0] | |
| have hr : 1 / 2 + ε ^ 2 * M < r := by | |
| dsimp [M] | |
| nlinarith only [hR, hbad.1] | |
| have hs : 1 / 2 + ε ^ 2 * N < s := by | |
| dsimp [N] | |
| nlinarith only [hQ, hbad.2] | |
| have hr0 : 1 / 2 < r := by linarith only [hr, mul_nonneg (sq_nonneg ε) hM0] | |
| have hs0 : 1 / 2 < s := by linarith only [hs, mul_nonneg (sq_nonneg ε) hN0] | |
| have hsum : r + s ≤ 1 + ε := by | |
| have err := weighted_errors he (by norm_num : (0 : ℝ) ≤ 1) hxy hβ hγ | |
| rw [hrs] | |
| linarith only [err] | |
| have hr1 : r ≤ 1 / 2 + ε := by linarith only [hsum, hs0] | |
| have hs1 : s ≤ 1 / 2 + ε := by linarith only [hsum, hr0] | |
| have hx1 : |x| ≤ 1 := by linarith only [hxy, abs_nonneg y] | |
| have hy1 : |y| ≤ 1 := by linarith only [hxy, abs_nonneg x] | |
| have hqrough := rough_angle he hx1 hq (by rwa [abs_of_nonneg ha0]) hr0.le hr1 hy | |
| have hprough : |x - p / 2| ≤ 2 * ε := by | |
| apply rough_angle he hy1 hp (a := -b) (by simpa [abs_of_nonneg hb0] using hb1) hs0.le hs1 | |
| linear_combination hx | |
| have hcross := cross_term_bound he hE0 hp hq ha0 ha1 hb0 hb1 | |
| (rough_square he hx1 hp hprough) (rough_square he hy1 hq hqrough) hea heb | |
| have hfail := radial_failure he0 heLt hK hxy hq hp hM0 hN0 heβ heγ hrs hy hx hr hs | |
| have hMNε := mul_le_mul_of_nonneg_right hMN he | |
| dsimp [M, N] at hfail hMNε | |
| nlinarith only [hfail, hcross, hMNε, hEε, hsmall, hη0, hKE0, he0] | |
| theorem node_bounds {N : ℕ} (node : ℕ → ℝ) | |
| (h0 : node 0 = -1) (hN : node N = 1) | |
| (hstep : ∀ j < N, node j < node (j + 1)) : | |
| ∀ i ≤ N, |node i| ≤ 1 := by | |
| have hmono : MonotoneOn node (Set.Iic N) := | |
| (strictMonoOn_Iic_of_lt_succ hstep).monotoneOn | |
| intro i hi | |
| have hl := hmono (show 0 ∈ Set.Iic N from Nat.zero_le N) hi (Nat.zero_le i) | |
| have hu := hmono hi (show N ∈ Set.Iic N by simp) hi | |
| rw [h0] at hl | |
| rw [hN] at hu | |
| exact abs_le.mpr ⟨hl, hu⟩ | |
| theorem angular_selection {N : ℕ} (hN : 0 < N) | |
| (node α β boundary : ℕ → ℝ) | |
| (hnode0 : node 0 = -1) (hnodeN : node N = 1) | |
| (hboundary0 : boundary 0 = 0) (hboundaryN : boundary N = 0) | |
| (hinc : ∀ j < N, node j < node (j + 1)) | |
| (hleft : ∀ j < N, α j - node j * β j = boundary j) | |
| (hright : ∀ j < N, α j - node (j + 1) * β j = boundary (j + 1)) | |
| {r z ℓ : ℝ} (hz : |z| ≤ r) : | |
| ∃ j : ℕ, j < N ∧ ∃ q : ℝ, q ∈ Set.Icc (node j) (node (j + 1)) ∧ | |
| 0 ≤ r - ℓ * β j ∧ z = (r - ℓ * β j) * q + ℓ * α j := by | |
| let f : ℕ → ℝ := fun i => node i * r + ℓ * boundary i | |
| have hf0 : f 0 = -r := by simp [f, hnode0, hboundary0] | |
| have hfN : f N = r := by simp [f, hnodeN, hboundaryN] | |
| obtain ⟨j, hj, hlo, hhi⟩ := first_crossing hN f z | |
| (by rw [hf0]; exact (abs_le.mp hz).1) | |
| (by rw [hfN]; exact (abs_le.mp hz).2) | |
| have he0 : f j = node j * (r - ℓ * β j) + ℓ * α j := by | |
| dsimp [f] | |
| rw [← hleft j hj] | |
| ring | |
| have he1 : f (j + 1) = node (j + 1) * (r - ℓ * β j) + ℓ * α j := by | |
| dsimp [f] | |
| rw [← hright j hj] | |
| ring | |
| rw [he0] at hlo | |
| rw [he1] at hhi | |
| have hlo' : node j * (r - ℓ * β j) ≤ z - ℓ * α j := by linarith only [hlo] | |
| have hhi' : z - ℓ * α j ≤ node (j + 1) * (r - ℓ * β j) := by linarith only [hhi] | |
| have hnonneg : 0 ≤ r - ℓ * β j := by | |
| have hmul : 0 ≤ (node (j + 1) - node j) * (r - ℓ * β j) := by | |
| nlinarith only [hlo', hhi'] | |
| exact nonneg_of_mul_nonneg_right hmul (sub_pos.mpr (hinc j hj)) | |
| by_cases hzero : r - ℓ * β j = 0 | |
| · refine ⟨j, hj, node j, ⟨le_rfl, (hinc j hj).le⟩, hnonneg, ?_⟩ | |
| rw [hzero] | |
| rw [hzero] at hlo' hhi' | |
| linarith only [hlo', hhi'] | |
| · have hpos : 0 < r - ℓ * β j := lt_of_le_of_ne hnonneg (Ne.symm hzero) | |
| refine ⟨j, hj, (z - ℓ * α j) / (r - ℓ * β j), | |
| ⟨(le_div_iff₀ hpos).mpr hlo', (div_le_iff₀ hpos).mpr hhi'⟩, hnonneg, ?_⟩ | |
| field_simp | |
| ring | |
| theorem coordinate_coverage {N : ℕ} (hN : 0 < N) | |
| (node α β boundary : ℕ → ℝ) (R : ℕ → ℝ → ℝ) | |
| (hnode0 : node 0 = -1) (hnodeN : node N = 1) | |
| (hnode : ∀ i ≤ N, |node i| ≤ 1) | |
| (hboundary0 : boundary 0 = 0) (hboundaryN : boundary N = 0) | |
| (hinc : ∀ j < N, node j < node (j + 1)) | |
| (hleft : ∀ j < N, α j - node j * β j = boundary j) | |
| (hright : ∀ j < N, α j - node (j + 1) * β j = boundary (j + 1)) | |
| {ε K η : ℝ} (he0 : 0 < ε) (he1 : ε ≤ 1 / 2) (hK : 0 ≤ K) | |
| (hη0 : 0 < η) (hη1 : η ≤ 1 / 480) (hsmall : (10 + 4 * K) * ε ≤ η) | |
| (hcoeff : ∀ j < N, ∀ q ∈ Set.Icc (node j) (node (j + 1)), | |
| |α j - (1 + q ^ 2) / 4| ≤ K * ε ^ 2 ∧ |β j - q / 2| ≤ K * ε ^ 2) | |
| (hcap : ∀ j < N, ∀ q ∈ Set.Icc (node j) (node (j + 1)), | |
| 1 / 2 + ε ^ 2 * ((q ^ 2 + q ^ 4) / 16 + η) - K * ε ^ 4 ≤ R j q) | |
| {x y t : ℝ} (hx : |x| ≤ 1 - t) (hy : |y| ≤ t) : | |
| (∃ j : ℕ, j < N ∧ ∃ q : ℝ, q ∈ Set.Icc (node j) (node (j + 1)) ∧ | |
| 0 ≤ t - ε * x * β j ∧ t - ε * x * β j ≤ R j q ∧ | |
| y = (t - ε * x * β j) * q + ε * x * α j) ∨ | |
| (∃ j : ℕ, j < N ∧ ∃ p : ℝ, p ∈ Set.Icc (node j) (node (j + 1)) ∧ | |
| 0 ≤ 1 - t + ε * y * β j ∧ 1 - t + ε * y * β j ≤ R j p ∧ | |
| x = (1 - t + ε * y * β j) * p - ε * y * α j) := by | |
| obtain ⟨j, hj, q, hqI, hr, hy'⟩ := angular_selection hN node α β boundary | |
| hnode0 hnodeN hboundary0 hboundaryN hinc hleft hright (ℓ := ε * x) hy | |
| obtain ⟨k, hk, p, hpI, hs, hx'⟩ := angular_selection hN node α β boundary | |
| hnode0 hnodeN hboundary0 hboundaryN hinc hleft hright (ℓ := -(ε * y)) hx | |
| have hs' : 0 ≤ 1 - t + ε * y * β k := by | |
| simpa only [neg_mul, sub_neg_eq_add] using hs | |
| have hx'' : x = (1 - t + ε * y * β k) * p - ε * y * α k := by | |
| linear_combination hx' | |
| have hq : |q| ≤ 1 := by | |
| apply abs_le.mpr | |
| exact ⟨(abs_le.mp (hnode j hj.le)).1.trans hqI.1, | |
| hqI.2.trans (abs_le.mp (hnode (j + 1) hj)).2⟩ | |
| have hp : |p| ≤ 1 := by | |
| apply abs_le.mpr | |
| exact ⟨(abs_le.mp (hnode k hk.le)).1.trans hpI.1, | |
| hpI.2.trans (abs_le.mp (hnode (k + 1) hk)).2⟩ | |
| have hcq := hcoeff j hj q hqI | |
| have hcp := hcoeff k hk p hpI | |
| have hcuts := radial_exclusion he0 he1 hK hη0 hη1 hsmall | |
| (show |x| + |y| ≤ 1 by linarith only [hx, hy]) hq hp hcq.2 hcp.2 hcq.1 hcp.1 | |
| (hcap j hj q hqI) (hcap k hk p hpI) | |
| (show (t - ε * x * β j) + (1 - t + ε * y * β k) = | |
| 1 - ε * x * β j + ε * y * β k by ring) hy' hx'' | |
| exact hcuts.elim | |
| (fun h => Or.inl ⟨j, hj, q, hqI, hr, h, hy'⟩) | |
| (fun h => Or.inr ⟨k, hk, p, hpI, hs', h, hx''⟩) | |
| theorem half_lt_cap {ε K η q R : ℝ} | |
| (he0 : 0 < ε) (he1 : ε ≤ 1 / 2) (_hK : 0 ≤ K) (hη : 0 < η) | |
| (hsmall : (10 + 4 * K) * ε ≤ η) | |
| (hR : 1 / 2 + ε ^ 2 * ((q ^ 2 + q ^ 4) / 16 + η) - K * ε ^ 4 ≤ R) : | |
| 1 / 2 < R := by | |
| have hKE : K * ε ≤ η / 4 := by nlinarith only [hsmall, he0] | |
| have hEη : K * ε ^ 2 ≤ η / 2 := by | |
| have h1 := mul_le_mul_of_nonneg_right hKE he0.le | |
| have h2 := mul_le_mul_of_nonneg_left he1 hη.le | |
| nlinarith only [h1, h2, hη] | |
| have hd : 0 ≤ (q ^ 2 + q ^ 4) / 16 := by positivity | |
| have hm : 0 < (q ^ 2 + q ^ 4) / 16 + η - K * ε ^ 2 := by | |
| linarith only [hd, hEη, hη] | |
| have hprod := mul_pos (sq_pos_of_pos he0) hm | |
| nlinarith only [hR, hprod] | |
| theorem exists_positive_threshold {K η : ℝ} (hK : 0 ≤ K) (hη : 0 < η) : | |
| ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, 0 < ε → ε ≤ δ → | |
| ε ≤ 1 / 2 ∧ (10 + 4 * K) * ε ≤ η := by | |
| have hd : 0 < 10 + 4 * K := by linarith only [hK] | |
| refine ⟨min (1 / 2) (η / (10 + 4 * K)), lt_min (by norm_num) (div_pos hη hd), ?_⟩ | |
| intro ε _ hε | |
| refine ⟨hε.trans (min_le_left _ _), ?_⟩ | |
| have hdiv := hε.trans (min_le_right _ _) | |
| have hmul := (le_div_iff₀ hd).mp hdiv | |
| nlinarith only [hmul] | |
| end CylinderCovering.GeneralCap | |
| end OAI | |