openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /PlanarIteration.lean
| 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 | |