File size: 3,429 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.FinitePhaseEvaluation
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.LogarithmName
namespace OAI
/-! Finite rational interval arithmetic for vector-valued derivative formulas.
The stopping test is rational and finite; convergence proves its termination. -/
namespace ForcedComputation
open ShearFlows Filter
open scoped Topology
def realNameBall (a : ℕ → ℚ) (n : ℕ) : QBall :=
⟨a n, (2 : ℚ) ^ (-(n : ℤ))⟩
def vectorNameBall (a : ℕ → RationalVector) (n : ℕ) (j : Fin 3) : QBall :=
realNameBall (fun k => a k j) n
theorem realNameBall_contains {a : ℕ → ℚ} {x : ℝ}
(ha : IsFastRealName a x) (n : ℕ) : (realNameBall a n).Contains x := by
simpa [realNameBall, QBall.Contains, errorTolerance] using ha n
theorem realNameBall_converges {a : ℕ → ℚ} {x : ℝ}
(ha : IsFastRealName a x) : QBall.Converges (realNameBall a) x := by
refine ⟨fastRealName_tendsto ha, ?_⟩
have he : (fun n => ((realNameBall a n).radius : ℝ)) = errorTolerance := by
funext n
simp [realNameBall, errorTolerance]
rw [he]
exact errorTolerance_tendsto
theorem vectorNameBall_contains {a : ℕ → RationalVector} {v : Space}
(ha : IsFastVectorName a v) (n : ℕ) (j : Fin 3) :
(vectorNameBall a n j).Contains (v j) := by
apply realNameBall_contains
intro k
exact (show |v j - (a k j : ℝ)| ≤ ‖v - rationalVector (a k)‖ from
norm_le_pi_norm (v - rationalVector (a k)) j).trans (ha k)
theorem vectorNameBall_converges {a : ℕ → RationalVector} {v : Space}
(ha : IsFastVectorName a v) (j : Fin 3) :
QBall.Converges (fun n => vectorNameBall a n j) (v j) := by
apply realNameBall_converges
intro k
exact (show |v j - (a k j : ℝ)| ≤ ‖v - rationalVector (a k)‖ from
norm_le_pi_norm (v - rationalVector (a k)) j).trans (ha k)
def vectorBallReady (b : ℕ → Fin 3 → QBall) (ε : ℚ) (n : ℕ) : Prop :=
∀ j, (b n j).radius ≤ ε
instance (b : ℕ → Fin 3 → QBall) (ε : ℚ) (n : ℕ) :
Decidable (vectorBallReady b ε n) :=
inferInstanceAs (Decidable (∀ j, (b n j).radius ≤ ε))
theorem vectorBallReady_exists {b : ℕ → Fin 3 → QBall} {v : Space}
(hb : ∀ j, QBall.Converges (fun n => b n j) (v j))
{ε : ℚ} (hε : 0 < ε) : ∃ n, vectorBallReady b ε n := by
have he : (0 : ℝ) < ε := by exact_mod_cast hε
have h : ∀ j, ∀ᶠ n in atTop, ((b n j).radius : ℝ) < ε := fun j =>
(hb j).2.eventually_lt_const he
obtain ⟨n, hn⟩ := (eventually_all.mpr h).exists
exact ⟨n, fun j => by exact_mod_cast (hn j).le⟩
def evaluateVectorBalls (b : ℕ → Fin 3 → QBall) (ε : ℚ)
(h : ∃ n, vectorBallReady b ε n) : RationalVector :=
fun j => (b (Nat.find h) j).center
theorem evaluateVectorBalls_spec {v : Space} (b : ℕ → Fin 3 → QBall)
(hb : ∀ j, QBall.Converges (fun n => b n j) (v j))
(hc : ∀ n j, (b n j).Contains (v j)) (ε : ℚ) (hε : 0 < ε) :
‖v - rationalVector (evaluateVectorBalls b ε (vectorBallReady_exists hb hε))‖ ≤ (ε : ℝ) := by
apply (pi_norm_le_iff_of_nonneg (by exact_mod_cast hε.le)).mpr
intro j
change |v j - ((b (Nat.find (vectorBallReady_exists hb hε)) j).center : ℝ)| ≤ ε
exact (hc _ j).trans (by exact_mod_cast (Nat.find_spec (vectorBallReady_exists hb hε) j))
end ForcedComputation
end OAI
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