openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /BalancedPlanarProgram.lean
| 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 | |