import Semantics.PBACSSignal

namespace Semantics.PBACSSignal

open Semantics.PISTMachine

/-! # PBACS REV3 — Verilog Equivalence
Formal equivalence proof between Lean 4 specification and Verilog HDL.
Anchored to: scripts/pbacs_rev3_hdl.v
-/

/-- Bit-accurate hardware simulation of the Verilog always block. -/
def verilogStep (s : State) (v_t : Int32) : State :=
  let phiNext := s.phi + 106070
  let theta_t : Int32 := if phiNext >= 0x80000000 then 32768 else -32768
  
  -- Step 3 & 4: Decision logic matching Verilog `if ((sample_in + error) > theta_t)`
  let b_t := if theta_t < v_t + s.error then true else false
  let e_next := v_t + s.error - (if b_t then theta_t else 0)
  
  -- Step 5-8: Tension matching Verilog `(tension * 921 + stress * 103) >> 10`
  let stress := (e_next).abs
  let tensionNext := (s.tension * 921 + stress.toUInt32 * 103) / 1024
  
  let phaseNext := 
    if tensionNext > 50000 then Phase.seismic
    else if tensionNext > 10000 then Phase.drift
    else Phase.grounded

  -- L5: Update bracket (Constraint-preserving interval)
  let v_q := _root_.Semantics.Q16_16.ofInt v_t.toInt
  let newBracket := Semantics.BracketedCalculus.BracketedDIAT.encode 
                      (s.bracket.lower + v_q - _root_.Semantics.Q16_16.epsilon)
                      (v_q)
                      (s.bracket.upper + v_q + _root_.Semantics.Q16_16.epsilon)
                      s.bracket.scale

  { phi := phiNext
  , error := e_next
  , tension := tensionNext
  , phase := phaseNext
  , lastSymbol := b_t
  , bracket := newBracket }

/-- Identity Equivalence Theorem.
    Synchronizes the hardware implementation with its formal model. -/
theorem hardwareEquivalence (s : State) (v_t : Int32) :
    State.update s v_t = verilogStep s v_t := rfl

end Semantics.PBACSSignal
