File size: 2,966 Bytes
37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | 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
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