openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /ClockNames.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.LogarithmName | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.Reparametrization | |
| namespace OAI | |
| /-! Fast names for the internal logarithmic clock and the reciprocal clock | |
| speed. These are operations on rational names, rather than real oracles. -/ | |
| namespace ForcedComputation | |
| open ShearFlows | |
| def shiftedTimeName (b : ℕ → RationalSpaceTime) : ℕ → ℚ := fun n => 1 + (b n).1 | |
| theorem shiftedTimeName_spec {y : SpaceTime} {b : ℕ → RationalSpaceTime} | |
| (hb : IsFastName b y) : IsFastRealName (shiftedTimeName b) (1 + y.1) := by | |
| intro n | |
| have h := (norm_fst_le (y - rationalPoint (b n))).trans (hb n) | |
| simpa only [shiftedTimeName, Rat.cast_add, Rat.cast_one, add_sub_add_left_eq_sub, | |
| Prod.fst_sub, rationalPoint, Real.norm_eq_abs] using h | |
| def clockPointName {y : SpaceTime} (b : ℕ → RationalSpaceTime) | |
| (hy : 0 < 1 + y.1) (hb : IsFastName b y) : ℕ → RationalSpaceTime := fun n => | |
| (logFastName (shiftedTimeName b) hy (shiftedTimeName_spec hb) n, (b n).2) | |
| theorem clockPointName_spec {y : SpaceTime} (b : ℕ → RationalSpaceTime) | |
| (hy : 0 < 1 + y.1) (hb : IsFastName b y) : | |
| IsFastName (clockPointName b hy hb) (logClock y.1, y.2) := by | |
| intro n | |
| apply norm_prod_le_iff.mpr | |
| constructor | |
| · have h := logFastName_spec (shiftedTimeName b) hy (shiftedTimeName_spec hb) n | |
| change |Real.log (1 + y.1) - | |
| (logFastName (shiftedTimeName b) hy (shiftedTimeName_spec hb) n : ℝ)| ≤ errorTolerance n | |
| exact h | |
| · have h := (norm_snd_le (y - rationalPoint (b n))).trans (hb n) | |
| exact h | |
| def reciprocalClockName (b : ℕ → RationalSpaceTime) : ℕ → ℚ := fun n => | |
| (max (1 / 2) (1 + (b (n + 2)).1))⁻¹ | |
| theorem reciprocalClockName_spec {y : SpaceTime} (b : ℕ → RationalSpaceTime) | |
| (hy : (1 / 2 : ℝ) ≤ 1 + y.1) (hb : IsFastName b y) : | |
| IsFastRealName (reciprocalClockName b) ((1 + y.1)⁻¹) := by | |
| intro n | |
| have h : (QBall.mk (1 + (b (n + 2)).1) ((2 : ℚ) ^ (-((n + 2 : ℕ) : ℤ)))).Contains | |
| (1 + y.1) := by | |
| have h := shiftedTimeName_spec hb (n + 2) | |
| simpa only [QBall.Contains, shiftedTimeName, Rat.cast_zpow, Rat.cast_ofNat, | |
| zpow_neg, zpow_natCast, Rat.cast_inv, Rat.cast_pow, errorTolerance] using h | |
| have hy' : ((1 / 2 : ℚ) : ℝ) ≤ 1 + y.1 := by | |
| norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] | |
| exact hy | |
| have hInv := QBall.contains_invLower h (show (0 : ℚ) < 1 / 2 by norm_num) hy' | |
| change |(1 + y.1)⁻¹ - (reciprocalClockName b n : ℝ)| ≤ errorTolerance n | |
| have he : ((((2 : ℚ) ^ (-((n + 2 : ℕ) : ℤ))) / (1 / 2) ^ 2 : ℚ) : ℝ) = | |
| errorTolerance n := by | |
| simp only [Rat.cast_div, Rat.cast_pow, Rat.cast_one, Rat.cast_ofNat, | |
| zpow_neg, zpow_natCast, errorTolerance, pow_add, mul_inv_rev] | |
| norm_num | |
| simpa only [QBall.Contains, QBall.invLower, reciprocalClockName, he] using hInv | |
| end ForcedComputation | |
| end OAI | |