| import Mathlib | |
| namespace OAI | |
| namespace AngularCartesianValidation | |
| theorem cartesian_failure_displacement | |
| {x y s τ η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1) | |
| (hs0 : 0 ≤ s) (hs1 : s ≤ 1) (hτ : 0 < τ) (hτ1 : τ ≤ 1) | |
| (hη : 0 ≤ η) | |
| (hfirst : 1 / 4 + τ * x * y + | |
| τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4) < s ^ 2) | |
| (hsecond : 1 / 4 - τ * x * y + | |
| τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4) < (1 - s) ^ 2) : | |
| |s - 1 / 2| ≤ 3 * τ ∧ |s - 1 / 2 - τ * x * y| ≤ 10 * τ ^ 2 := by | |
| let u := s - 1 / 2 | |
| have hu0 : -(1 / 2 : ℝ) ≤ u := by dsimp [u]; linarith | |
| have hu1 : u ≤ (1 / 2 : ℝ) := by dsimp [u]; linarith | |
| have hx2 : x ^ 2 ≤ 1 := by | |
| nlinarith only [(abs_le.mp hx).1, (abs_le.mp hx).2] | |
| have hy2 : y ^ 2 ≤ 1 := by | |
| nlinarith only [(abs_le.mp hy).1, (abs_le.mp hy).2] | |
| have hxy : |x * y| ≤ 1 := by | |
| rw [abs_mul] | |
| nlinarith only [mul_le_mul hx hy (abs_nonneg y) (by norm_num : (0 : ℝ) ≤ 1)] | |
| have htxyl : -τ ≤ τ * x * y := by | |
| nlinarith only [mul_le_mul_of_nonneg_left (abs_le.mp hxy).1 hτ.le] | |
| have htxyr : τ * x * y ≤ τ := by | |
| nlinarith only [mul_le_mul_of_nonneg_left (abs_le.mp hxy).2 hτ.le] | |
| have hτsq : τ ^ 2 ≤ τ := by | |
| nlinarith only [mul_nonneg hτ.le (sub_nonneg.mpr hτ1)] | |
| have hA : -(1 / 4 : ℝ) ≤ (y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4 := by | |
| nlinarith only [sq_nonneg y, sq_nonneg (y ^ 2), hx2, hη] | |
| have hB : -(1 / 4 : ℝ) ≤ (x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4 := by | |
| nlinarith only [sq_nonneg x, sq_nonneg (x ^ 2), hy2, hη] | |
| have hfirst' : τ * x * y - τ ^ 2 / 4 < u + u ^ 2 := by | |
| dsimp [u] | |
| nlinarith only [hfirst, mul_le_mul_of_nonneg_left hA (sq_nonneg τ)] | |
| have hsecond' : -τ * x * y - τ ^ 2 / 4 < -u + u ^ 2 := by | |
| dsimp [u] | |
| nlinarith only [hsecond, mul_le_mul_of_nonneg_left hB (sq_nonneg τ)] | |
| have hul : -3 * τ ≤ u := by | |
| by_cases hu : 0 ≤ u | |
| · linarith | |
| · have hmul := mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge hu) | |
| (show 0 ≤ u + 1 / 2 by linarith) | |
| nlinarith only [hmul, hfirst', htxyl, hτsq, hτ.le] | |
| have hur : u ≤ 3 * τ := by | |
| by_cases hu : u ≤ 0 | |
| · linarith | |
| · have hmul := mul_nonpos_of_nonneg_of_nonpos (le_of_not_ge hu) | |
| (show u - 1 / 2 ≤ 0 by linarith) | |
| nlinarith only [hmul, hsecond', htxyr, hτsq, hτ.le] | |
| have hu2 : u ^ 2 ≤ 9 * τ ^ 2 := by | |
| nlinarith only [mul_nonneg (sub_nonneg.mpr hur) | |
| (show 0 ≤ u + 3 * τ by linarith only [hul])] | |
| constructor | |
| · exact abs_le.mpr ⟨by simpa only [neg_mul] using hul, hur⟩ | |
| · change |u - τ * x * y| ≤ 10 * τ ^ 2 | |
| apply abs_le.mpr | |
| constructor | |
| · nlinarith only [hfirst', hu2, sq_nonneg τ] | |
| · nlinarith only [hsecond', hu2, sq_nonneg τ] | |
| theorem cartesian_squared_failures_impossible | |
| {x y s τ η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1) | |
| (hs0 : 0 ≤ s) (hs1 : s ≤ 1) (hτ : 0 < τ) (hτ1 : τ ≤ 1) | |
| (hη : 0 < η) (hsmall : τ ≤ η / 80) | |
| (hfirst : 1 / 4 + τ * x * y + | |
| τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4) < s ^ 2) | |
| (hsecond : 1 / 4 - τ * x * y + | |
| τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4) < (1 - s) ^ 2) : | |
| False := by | |
| obtain ⟨hu, herr⟩ := cartesian_failure_displacement hx hy hs0 hs1 hτ hτ1 hη.le | |
| hfirst hsecond | |
| let u := s - 1 / 2 | |
| change |u| ≤ 3 * τ at hu | |
| change |u - τ * x * y| ≤ 10 * τ ^ 2 at herr | |
| have hxy : |x * y| ≤ 1 := by | |
| rw [abs_mul] | |
| nlinarith only [mul_le_mul hx hy (abs_nonneg y) (by norm_num : (0 : ℝ) ≤ 1)] | |
| have htx : |τ * x * y| ≤ τ := by | |
| calc | |
| |τ * x * y| = τ * |x * y| := by rw [mul_assoc, abs_mul, abs_of_pos hτ] | |
| _ ≤ τ * 1 := mul_le_mul_of_nonneg_left hxy hτ.le | |
| _ = τ := mul_one τ | |
| have hsum : |u + τ * x * y| ≤ 4 * τ := by | |
| exact (abs_add_le _ _).trans (by linarith) | |
| have hsq : |u ^ 2 - (τ * x * y) ^ 2| ≤ 40 * τ ^ 3 := by | |
| calc | |
| |u ^ 2 - (τ * x * y) ^ 2| = |u - τ * x * y| * |u + τ * x * y| := by | |
| rw [← abs_mul] | |
| congr 1 | |
| ring | |
| _ ≤ (10 * τ ^ 2) * (4 * τ) := | |
| mul_le_mul herr hsum (abs_nonneg _) (by positivity) | |
| _ = 40 * τ ^ 3 := by ring | |
| have hsumfail : τ ^ 2 * (x ^ 4 + y ^ 4 + η) < 2 * u ^ 2 := by | |
| dsimp [u] | |
| nlinarith only [hfirst, hsecond] | |
| have hquartic : 2 * x ^ 2 * y ^ 2 ≤ x ^ 4 + y ^ 4 := by | |
| nlinarith only [sq_nonneg (x ^ 2 - y ^ 2)] | |
| have hgap : η * τ ^ 2 < 2 * (u ^ 2 - (τ * x * y) ^ 2) := by | |
| nlinarith only [hsumfail, mul_le_mul_of_nonneg_left hquartic (sq_nonneg τ)] | |
| have hwindow : 80 * τ ^ 3 ≤ η * τ ^ 2 := by | |
| nlinarith only [mul_le_mul_of_nonneg_right hsmall (sq_nonneg τ)] | |
| nlinarith only [hgap, hwindow, (abs_le.mp hsq).2] | |
| theorem cartesian_squared_lower_of_sqrt {Q τ c : ℝ} (hc : 0 ≤ c) | |
| (hroot : 1 / 2 + τ ^ 2 * c < Real.sqrt Q) : | |
| 1 / 4 + τ ^ 2 * c < Q := by | |
| have hbase : 0 < 1 / 2 + τ ^ 2 * c := by positivity | |
| have hQ : 0 ≤ Q := (Real.sqrt_pos.mp (hbase.trans hroot)).le | |
| have hsq := mul_self_lt_mul_self hbase.le hroot | |
| rw [Real.mul_self_sqrt hQ] at hsq | |
| nlinarith only [hsq, sq_nonneg (τ ^ 2 * c)] | |
| theorem cartesian_sqrt_failures_impossible | |
| {x y s τ η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1) | |
| (hs0 : 0 ≤ s) (hs1 : s ≤ 1) (hτ : 0 < τ) (hτ1 : τ ≤ 1) | |
| (hη : 0 < η) (hsmall : τ ≤ η / 80) | |
| (hfirst : 1 / 2 + τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2) < | |
| Real.sqrt (s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4)) | |
| (hsecond : 1 / 2 + τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2) < | |
| Real.sqrt ((1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4)) : | |
| False := by | |
| have hfirst' := cartesian_squared_lower_of_sqrt (by positivity) hfirst | |
| have hsecond' := cartesian_squared_lower_of_sqrt (by positivity) hsecond | |
| apply cartesian_squared_failures_impossible hx hy hs0 hs1 hτ hτ1 hη hsmall | |
| · nlinarith only [hfirst'] | |
| · nlinarith only [hsecond'] | |
| theorem cartesian_edge_one_obstruction (ε x y : ℝ) : | |
| (ε / 2) ^ 2 * (((2 * x) ^ 2 - (2 * y) ^ 2) ^ 2 + 1 / 1000) = | |
| ε ^ 2 * (4 * (x ^ 2 - y ^ 2) ^ 2 + 1 / 4000) := by | |
| ring | |
| end AngularCartesianValidation | |
| end OAI | |