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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarTransition
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.RecorderRun
namespace OAI
/-! Integer-phase iteration of a periodic planar processor. -/
noncomputable section
namespace ForcedComputation
open ShearFlows
theorem IsPlanarTransition.phase_nat {V : ℝ → Plane → Plane}
{Ψ : ℝ → ℝ → Plane → Plane} (hΨ : IsPlanarTransition V Ψ)
(hV : ∀ y, Function.Periodic (fun t => V t y) 1)
(a : ℝ) (n : ℕ) (t : ℝ) (x : Plane) : Ψ (a + n) t x = Ψ a t x := by
apply hΨ.unique a x (fun s => Ψ (a + n) s x) (hΨ.initial (a + n) x)
intro s
have he : V ((a + n) + s) (Ψ (a + n) s x) = V (a + s) (Ψ (a + n) s x) := by
have h := (hV (Ψ (a + n) s x)).nat_mul n (a + s)
convert h using 1
congr 1
ring
exact (hΨ.ode (a + n) s x).congr_deriv he
theorem IsPlanarTransition.nat_shift {V : ℝ → Plane → Plane}
{Ψ : ℝ → ℝ → Plane → Plane} (hΨ : IsPlanarTransition V Ψ)
(hV : ∀ y, Function.Periodic (fun t => V t y) 1)
(n : ℕ) (s : ℝ) (x : Plane) : Ψ 0 ((n : ℝ) + s) x = Ψ 0 s (Ψ 0 n x) := by
rw [hΨ.compose]
simpa only [zero_add] using hΨ.phase_nat hV 0 n s (Ψ 0 n x)
theorem planar_recorder_steps {Q A : Type*} [DecidableEq Q] [DecidableEq A]
{M : Recorder.Machine Q A} {V : ℝ → Plane → Plane} {Ψ : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition V Ψ) (hV : ∀ y, Function.Periodic (fun t => V t y) 1)
(code : Recorder.Configuration Q A → Plane)
(hone : ∀ {C D}, Recorder.Step M C D → Ψ 0 1 (code C) = code D)
{C D : Recorder.Configuration Q A} {n : ℕ} (h : Recorder.Steps M n C D) :
Ψ 0 n (code C) = code D := by
induction h with
| zero C => simpa only [Nat.cast_zero] using hΨ.initial 0 (code C)
| @next n C D E h hs ih =>
rw [Nat.cast_add, Nat.cast_one, hΨ.nat_shift hV, ih]
exact hone hs
end ForcedComputation
end
end OAI
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