| import Mathlib | |
| namespace OAI | |
| namespace AngularCubic | |
| lemma endpoint_left (a b : ℝ) : | |
| (1 + a * b) / 2 - a * ((a + b) / 2) = (1 - a ^ 2) / 2 := by ring | |
| lemma endpoint_right (a b : ℝ) : | |
| (1 + a * b) / 2 - b * ((a + b) / 2) = (1 - b ^ 2) / 2 := by ring | |
| lemma square_le_one {x : ℝ} (hx : |x| ≤ 1) : x ^ 2 ≤ 1 := by | |
| have h := abs_le.mp hx | |
| nlinarith [sq_nonneg (1 - x), sq_nonneg (1 + x)] | |
| lemma product_error {x y d : ℝ} (hx : |x| ≤ d) (hy : |y| ≤ 2) : | |
| |x * y| ≤ 2 * d := by | |
| calc | |
| |x * y| = |x| * |y| := abs_mul x y | |
| _ ≤ d * 2 := mul_le_mul hx hy (abs_nonneg y) (le_trans (abs_nonneg x) hx) | |
| _ = 2 * d := by ring | |
| lemma mixed_quartic_bound {a b q r d Dq Dr : ℝ} | |
| (ha : |a - (1 + r ^ 2) / 2| ≤ d) | |
| (hb : |b - (1 + q ^ 2) / 2| ≤ d) | |
| (hq : |q| ≤ 1) (hr : |r| ≤ 1) | |
| (hDq : (q ^ 2 + q ^ 4) / 2 ≤ Dq) | |
| (hDr : (r ^ 2 + r ^ 4) / 2 ≤ Dr) : | |
| a * q ^ 2 + b * r ^ 2 ≤ Dq + Dr + 2 * d := by | |
| have hd : 0 ≤ d := le_trans (abs_nonneg _) ha | |
| have hqa := mul_le_mul_of_nonneg_right (abs_le.mp ha).2 (sq_nonneg q) | |
| have hrb := mul_le_mul_of_nonneg_right (abs_le.mp hb).2 (sq_nonneg r) | |
| have hqd := mul_le_mul_of_nonneg_left (square_le_one hq) hd | |
| have hrd := mul_le_mul_of_nonneg_left (square_le_one hr) hd | |
| nlinarith [sq_nonneg (q ^ 2 - r ^ 2)] | |
| lemma coarse_excess_bound {e d P T : ℝ} | |
| (he : 0 ≤ e) (hehalf : e ≤ 1 / 2) (hd : d ≤ e ^ 2) | |
| (hT : T ≤ 4) (hP : P ≤ e * P + e ^ 2 * T + 4 * e * d) : | |
| P ≤ 12 * e ^ 2 := by | |
| have hTd := mul_le_mul_of_nonneg_left hT (sq_nonneg e) | |
| have hed := mul_le_mul_of_nonneg_left hd (by positivity : 0 ≤ 4 * e) | |
| have hecube : e ^ 3 ≤ e ^ 2 / 2 := by nlinarith [sq_nonneg e] | |
| by_cases hp : P ≤ 0 | |
| · linarith [sq_nonneg e] | |
| · have hp0 : 0 ≤ P := le_of_lt (lt_of_not_ge hp) | |
| have hep := mul_le_mul_of_nonneg_right hehalf hp0 | |
| nlinarith | |
| lemma sharpen_excess_bound {e d P T Q : ℝ} | |
| (he : 0 ≤ e) (hehalf : e ≤ 1 / 2) (hd : d ≤ e ^ 2) | |
| (_hP0 : 0 ≤ P) (hP12 : P ≤ 12 * e ^ 2) | |
| (hT : T ≤ Q + 2 * P + 4 * e) | |
| (hP : P ≤ e * P + e ^ 2 * T + 4 * e * d) : | |
| P ≤ e ^ 2 * Q + 32 * e ^ 3 := by | |
| have h1 := mul_le_mul_of_nonneg_left hT (sq_nonneg e) | |
| have h2 := mul_le_mul_of_nonneg_left hP12 he | |
| have h3 := mul_le_mul_of_nonneg_left hP12 (by positivity : 0 ≤ 2 * e ^ 2) | |
| have h4 := mul_le_mul_of_nonneg_left hd (by positivity : 0 ≤ 4 * e) | |
| have h5 : 24 * e ^ 4 ≤ 12 * e ^ 3 := by | |
| have := mul_nonneg (by linarith : 0 ≤ 1 / 2 - e) (pow_nonneg he 3) | |
| nlinarith | |
| nlinarith | |
| lemma final_excess_bound {e d P Q Dq Dr : ℝ} | |
| (he : 0 ≤ e) (hehalf : e ≤ 1 / 2) (hd : d ≤ e ^ 2) | |
| (hP : P ≤ e ^ 2 * Q + 32 * e ^ 3) | |
| (hQ : Q ≤ Dq + Dr + 2 * d) : | |
| P ≤ e ^ 2 * (Dq + Dr) + 33 * e ^ 3 := by | |
| have h1 := mul_le_mul_of_nonneg_left hQ (sq_nonneg e) | |
| have h2 := mul_le_mul_of_nonneg_left hd (by positivity : 0 ≤ 2 * e ^ 2) | |
| have h3 : 2 * e ^ 4 ≤ e ^ 3 := by | |
| have := mul_nonneg (by linarith : 0 ≤ 1 / 2 - e) (pow_nonneg he 3) | |
| nlinarith | |
| nlinarith | |
| lemma abs_mul_le_one {x y : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1) : | |
| |x * y| ≤ 1 := by | |
| rw [abs_mul] | |
| nlinarith [abs_nonneg x, abs_nonneg y] | |
| lemma three_factor_bound {q a x : ℝ} (hq : |q| ≤ 1) | |
| (ha0 : 0 ≤ a) (ha1 : a ≤ 1) (hx : |x| ≤ 2) : | |
| |q * a * x| ≤ 2 := by | |
| have hqa : |q * a| ≤ 1 := abs_mul_le_one hq (by rwa [abs_of_nonneg ha0]) | |
| simpa using product_error hqa hx | |
| lemma refined_term_bound {e P a b q x y t : ℝ} | |
| (he : 0 ≤ e) (hP : 0 ≤ P) | |
| (ha0 : 0 ≤ a) (ha1 : a ≤ 1) (hb0 : 0 ≤ b) (hb1 : b ≤ 1) | |
| (hq : |q| ≤ 1) (hy : |y| ≤ 2) (ht : t ≤ 1 + P) | |
| (hx : x = q * t - e * b * y) : | |
| q * a * x ≤ a * q ^ 2 + P + 2 * e := by | |
| have haq0 : 0 ≤ a * q ^ 2 := mul_nonneg ha0 (sq_nonneg q) | |
| have haq1 : a * q ^ 2 ≤ 1 := by | |
| have := mul_le_mul_of_nonneg_right ha1 (sq_nonneg q) | |
| nlinarith [square_le_one hq] | |
| have hmain := mul_le_mul_of_nonneg_left ht haq0 | |
| have hcost := mul_le_mul_of_nonneg_right haq1 hP | |
| have habs : |q * a * b * y| ≤ 2 := by | |
| have hqab : |q * a * b| ≤ 1 := | |
| abs_mul_le_one (abs_mul_le_one hq (by rwa [abs_of_nonneg ha0])) | |
| (by rwa [abs_of_nonneg hb0]) | |
| simpa using product_error hqab hy | |
| have herror := mul_le_mul_of_nonneg_left (abs_le.mp habs).1 he | |
| calc | |
| q * a * x = a * q ^ 2 * t - e * (q * a * b * y) := by rw [hx]; ring | |
| _ ≤ a * q ^ 2 + P + 2 * e := by nlinarith | |
| lemma excess_identity {e s t s0 t0 x y q r a b bj bl : ℝ} | |
| (hst : s + t = 2) (hs : s0 = s - e * bj * x) | |
| (ht : t0 = t + e * bl * y) | |
| (hy : y = r * s0 + e * a * x) | |
| (hx : x = q * t0 - e * b * y) : | |
| s0 + t0 - 2 = e * ((bl - q) * y - (bj - r) * x + | |
| r * q * (s0 - t0)) + e ^ 2 * (q * a * x + r * b * y) := by | |
| have hbase : s0 + t0 - 2 = e * (bl * y - bj * x) := by linarith | |
| have hswap : q * y - r * x = r * q * (s0 - t0) + | |
| e * (q * a * x + r * b * y) := by | |
| linear_combination q * hy - r * hx | |
| linear_combination hbase + e * hswap | |
| lemma excess_inequality {e d P s0 t0 x y q r a b bj bl : ℝ} | |
| (he : 0 ≤ e) (hP : |s0 - t0| ≤ P) | |
| (hx : |x| ≤ 2) (hy : |y| ≤ 2) | |
| (hq : |q| ≤ 1) (hr : |r| ≤ 1) | |
| (hbj : |bj - r| ≤ d) (hbl : |bl - q| ≤ d) | |
| (hid : P = e * ((bl - q) * y - (bj - r) * x + | |
| r * q * (s0 - t0)) + e ^ 2 * (q * a * x + r * b * y)) : | |
| P ≤ e * P + e ^ 2 * (q * a * x + r * b * y) + 4 * e * d := by | |
| have h1 := (abs_le.mp (product_error hbl hy)).2 | |
| have h2 := (abs_le.mp (product_error hbj hx)).1 | |
| have h3 : r * q * (s0 - t0) ≤ P := by | |
| have hprod : |r * q| ≤ 1 := abs_mul_le_one hr hq | |
| have hbound : |r * q * (s0 - t0)| ≤ P := by | |
| rw [abs_mul] | |
| have := mul_le_mul hprod hP (abs_nonneg (s0 - t0)) | |
| (by norm_num : (0 : ℝ) ≤ 1) | |
| simpa using this | |
| exact (abs_le.mp hbound).2 | |
| have hsum : (bl - q) * y - (bj - r) * x + r * q * (s0 - t0) ≤ P + 4 * d := by | |
| linarith | |
| have := mul_le_mul_of_nonneg_left hsum he | |
| nlinarith [hid] | |
| theorem radial_restrictions {e d s t s0 t0 x y q r a b bj bl Dj Dl : ℝ} | |
| (he : 0 < e) (hehalf : e ≤ 1 / 2) (hd : d ≤ e ^ 2) | |
| (hst : s + t = 2) | |
| (hs : s0 = s - e * bj * x) (ht : t0 = t + e * bl * y) | |
| (hyEq : y = r * s0 + e * a * x) | |
| (hxEq : x = q * t0 - e * b * y) | |
| (hx : |x| ≤ 2) (hy : |y| ≤ 2) | |
| (hq : |q| ≤ 1) (hr : |r| ≤ 1) | |
| (ha0 : 0 ≤ a) (ha1 : a ≤ 1) (hb0 : 0 ≤ b) (hb1 : b ≤ 1) | |
| (hbj : |bj - r| ≤ d) (hbl : |bl - q| ≤ d) | |
| (ha : |a - (1 + r ^ 2) / 2| ≤ d) | |
| (hb : |b - (1 + q ^ 2) / 2| ≤ d) | |
| (hDj : (r ^ 2 + r ^ 4) / 2 ≤ Dj) | |
| (hDl : (q ^ 2 + q ^ 4) / 2 ≤ Dl) : | |
| s0 ≤ 1 + e ^ 2 * Dj + 100 * e ^ 3 ∨ | |
| t0 ≤ 1 + e ^ 2 * Dl + 100 * e ^ 3 := by | |
| by_contra h | |
| push Not at h | |
| obtain ⟨hsFail, htFail⟩ := h | |
| have he0 : 0 ≤ e := le_of_lt he | |
| have he3 : 0 < e ^ 3 := pow_pos he 3 | |
| have hDj0 : 0 ≤ Dj := by nlinarith only [hDj, sq_nonneg r, sq_nonneg (r ^ 2)] | |
| have hDl0 : 0 ≤ Dl := by nlinarith only [hDl, sq_nonneg q, sq_nonneg (q ^ 2)] | |
| have hs1 : 1 ≤ s0 := by nlinarith only [hsFail, he3, mul_nonneg (sq_nonneg e) hDj0] | |
| have ht1 : 1 ≤ t0 := by nlinarith only [htFail, he3, mul_nonneg (sq_nonneg e) hDl0] | |
| let P := s0 + t0 - 2 | |
| have hP0 : 0 ≤ P := by dsimp [P]; linarith only [hs1, ht1] | |
| have hsP : s0 ≤ 1 + P := by dsimp [P]; linarith only [hs1, ht1] | |
| have htP : t0 ≤ 1 + P := by dsimp [P]; linarith only [hs1, ht1] | |
| have hdiff : |s0 - t0| ≤ P := by | |
| rw [abs_le] | |
| dsimp [P] | |
| constructor <;> linarith only [hs1, ht1] | |
| have hid : P = e * ((bl - q) * y - (bj - r) * x + | |
| r * q * (s0 - t0)) + e ^ 2 * (q * a * x + r * b * y) := | |
| excess_identity hst hs ht hyEq hxEq | |
| have hexcess := excess_inequality he0 hdiff hx hy hq hr hbj hbl hid | |
| have hT : q * a * x + r * b * y ≤ 4 := by | |
| have h1 := (abs_le.mp (three_factor_bound hq ha0 ha1 hx)).2 | |
| have h2 := (abs_le.mp (three_factor_bound hr hb0 hb1 hy)).2 | |
| linarith only [h1, h2] | |
| have hP12 := coarse_excess_bound he0 hehalf hd hT hexcess | |
| have hfirst := refined_term_bound he0 hP0 ha0 ha1 hb0 hb1 hq hy htP hxEq | |
| have hsecond : r * b * y ≤ b * r ^ 2 + P + 2 * e := by | |
| apply refined_term_bound (y := -x) he0 hP0 hb0 hb1 ha0 ha1 hr | |
| (by simpa using hx) hsP | |
| linarith only [hyEq] | |
| have hTsharp : q * a * x + r * b * y ≤ | |
| a * q ^ 2 + b * r ^ 2 + 2 * P + 4 * e := by linarith only [hfirst, hsecond] | |
| have hP32 := sharpen_excess_bound he0 hehalf hd hP0 hP12 hTsharp hexcess | |
| have hQ := mixed_quartic_bound ha hb hq hr hDl hDj | |
| have hP33 := final_excess_bound he0 hehalf hd hP32 hQ | |
| dsimp [P] at hP33 | |
| nlinarith only [hP33, hsFail, htFail, he3] | |
| end AngularCubic | |
| end OAI | |