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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 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 81 82 83 84 85 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarProcessorOrbit
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPlanarFlow
namespace OAI
/-! A machine-only planar body for the Euclidean box test. Normalize using
one fixed empty input, then encode every other input at its own point in the
same body. The one-period realization theorems hold for every configuration. -/
noncomputable section
namespace ForcedComputation.BalancedPlanar
open ShearFlows Recorder Recorder.Planar Set
/-- The empty input fixes the spatial normalization of the repeated body. -/
def referenceInput (M : Alternating.Machine) : Alternating.MachineInput := (M, [])
theorem referenceValid {M : Alternating.Machine} (hM : M.WellFormed) :
Alternating.ValidInput (referenceInput M) := by
refine ⟨hM, ?_⟩
intro a ha
simp [referenceInput] at ha
def bodyHamiltonian (M : Alternating.Machine) (hM : M.WellFormed) : FieldExpr :=
normalizedHamiltonian (referenceInput M) (referenceValid hM)
def bodyVelocity (M : Alternating.Machine) (hM : M.WellFormed) : ℝ → Plane → Plane :=
compactVelocity (referenceInput M) (referenceValid hM)
def bodyCode (M : Alternating.Machine) (hM : M.WellFormed)
(C : Configuration (State (freshMachine M)) (Alphabet (freshMachine M))) : Plane :=
normalizedPoint (referenceInput M) (referenceValid hM) C
def inputCode (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : Plane :=
bodyCode I.1 hI.1 (finiteInitializedRecorder (freshInput I) (freshInput_valid hI))
def referenceShift (M : Alternating.Machine) (hM : M.WellFormed) : Fin 2 → ℚ :=
initialShift (freshInput (referenceInput M)) (freshInput_valid (referenceValid hM))
theorem referenceShift_small (M : Alternating.Machine) (hM : M.WellFormed) :
|(referenceShift M hM 0 : ℝ)| ≤ 1 / 512 ∧
|(referenceShift M hM 1 : ℝ)| ≤ 3 / 1024 :=
initialShift_small _ _ (fresh_initial_nonhalting _ (referenceValid hM))
theorem body_step (M : Alternating.Machine) (hM : M.WellFormed)
{Ψ : ℝ → ℝ → Plane → Plane} (hΨ : IsPlanarTransition (planarSlice (bodyHamiltonian M hM)) Ψ)
{C D : Configuration (State (freshMachine M)) (Alphabet (freshMachine M))}
(hs : Step (finiteMachine (freshMachine M) (freshInput_valid (referenceValid hM)).1) C D) :
Ψ 0 1 (bodyCode M hM C) = bodyCode M hM D :=
normalized_planar_step (referenceInput M) (referenceValid hM) hΨ hs
theorem body_step_safe (M : Alternating.Machine) (hM : M.WellFormed)
{Ψ : ℝ → ℝ → Plane → Plane} (hΨ : IsPlanarTransition (planarSlice (bodyHamiltonian M hM)) Ψ)
{C D : Configuration (State (freshMachine M)) (Alphabet (freshMachine M))}
(hs : Step (finiteMachine (freshMachine M) (freshInput_valid (referenceValid hM)).1) C D)
(hD : recorderHalting (freshMachine M) D.control = false) {s : ℝ} (ht : s ∈ Icc (0 : ℝ) 1) :
3 / 16 ≤ Ψ 0 s (bodyCode M hM C) 1 :=
normalized_step_safe (referenceInput M) (referenceValid hM) hΨ hs hD ht
theorem body_step_corridor (M : Alternating.Machine) (hM : M.WellFormed)
{Ψ : ℝ → ℝ → Plane → Plane} (hΨ : IsPlanarTransition (planarSlice (bodyHamiltonian M hM)) Ψ)
{C D : Configuration (State (freshMachine M)) (Alphabet (freshMachine M))}
(hs : Step (finiteMachine (freshMachine M) (freshInput_valid (referenceValid hM)).1) C D)
{s : ℝ} (ht : s ∈ Icc (0 : ℝ) 1) :
Ψ 0 s (bodyCode M hM C) ∈ clockRectangle.carrier :=
normalized_step_corridor (referenceInput M) (referenceValid hM) hΨ hs ht
theorem body_observation (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
{Ψ : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (planarSlice (bodyHamiltonian I.1 hI.1)) Ψ) :
(∃ t : ℝ, 0 ≤ t ∧ Ψ 0 t (inputCode I hI) ∈ observer) ↔ Alternating.Halts I := by
have hδ := (referenceShift_small I.1 hI.1).2
have h := shifted_planar_observation (freshInput I) (freshInput_valid hI)
(referenceShift I.1 hI.1) hδ hΨ
(planarSlice_periodic (normalizedHamiltonian_periodic
(referenceInput I.1) (referenceValid hI.1)))
(body_step I.1 hI.1 hΨ)
(fun {C D} hs _ hD s ht => body_step_safe I.1 hI.1 hΨ hs hD ht)
exact h.trans (freshInput_halts_iff hI)
end ForcedComputation.BalancedPlanar
end
end OAI
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