foldMinusOneDecodedStep

formal source

 326/-- The corresponding discrete step on **decoded** Galerkin coefficients.
 327
 328We mimic the VM update on `nuPhi` (via `clampI32`) and then decode a nonnegative magnitude using `decodeMag`. -/
 329noncomputable def foldMinusOneDecodedStep {N : ℕ} (u : GalerkinState N) : GalerkinState N :=
 330  WithLp.toLp 2 (fun i : ((modes N) × Fin 2) =>
 331    let x : ℝ := u i
 332    let m : Int := decodeMag (clampI32 (coeffMag x + (-1)))
 333    let z : Int := (coeffSign x) * m
 334    (z : ℝ))
 335
 336/-- Decoding after a single `FOLD (-1)` step matches the corresponding decoded discrete step. -/
 337lemma decodeCoeff_voxelStep_foldMinusOne_encodeIndex {N : ℕ} (u : GalerkinState N) (i : (modes N) × Fin 2) :
 338    decodeCoeff (voxelStep foldMinusOneProgram (encodeIndex u i)) = (foldMinusOneDecodedStep u) i := by
 339  classical
 340  -- both sides reduce to the same signed-integer expression
 341  simp [decodeCoeff, foldMinusOneDecodedStep, decodeMag, voxelStep_foldMinusOneProgram, encodeIndex, -clampI32]
 342
 343/-- A concrete, proved decay-step simulation instance (after decoding). -/
 344noncomputable def DecodedSimulationHypothesis.foldMinusOne (N : ℕ) : DecodedSimulationHypothesis N :=
 345  { P := foldMinusOneProgram
 346    step := foldMinusOneDecodedStep
 347    comm := by
 348      intro u
 349      classical
 350      -- Prove equality of Galerkin states by pointwise equality of coefficients.
 351      ext i
 352      -- Reduce to the single-voxel lemma `decodeCoeff_voxelStep_foldMinusOne_encodeIndex`.
 353      simp [decodeGalerkin2D, LNALSemantics.independent, encodeGalerkin2D,
 354        decodeCoeff_voxelStep_foldMinusOne_encodeIndex] }
 355
 356end Simulation2D
 357
 358end IndisputableMonolith.ClassicalBridge.Fluids