/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

DspErasureCoding.lean — DSP-Aware 3-Stream Erasure Coding

This module formalizes DSP-aware erasure coding for streaming data:
- 3-stream redundancy scheme (inspired by PhiRedundancy)
- Sample-block based erasure recovery
- Q16_16 fixed-point for hardware extraction
- FPGA DSP slice integration
- Spectral-aware erasure detection

Key insight:
DSP erasure coding treats streams as continuous signals, not discrete bytes.
Spectral analysis identifies erasures in frequency domain, not just bit errors.

Per AGENTS.md §1.4: Q16_16 fixed-point for hardware extraction.
Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: Every def has eval witness or theorem.

TODO(lean-port): Connect to FPGA Warden Node AMMR accumulator
TODO(lean-port): Integrate with StreamCompression spectral analysis
-/ 

import Mathlib.Data.Nat.Basic
import Mathlib.Data.Fin.Basic
import Mathlib.Tactic
import Semantics.FixedPoint

namespace Semantics.DspErasureCoding

open Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  DSP Stream Types
-- ═══════════════════════════════════════════════════════════════════════════

/-- DSP sample in Q16.16 fixed-point format. -/
abbrev DspSample := Q16_16

/-- Sample block (DSP standard processing unit). -/
structure SampleBlock where
  samples : Array DspSample
  blockId : Nat
  deriving Repr, Inhabited

/-- Stream identifier for 3-stream redundancy. -/
inductive StreamId where
  | primary    -- Stream 0: original data
  | recovery1  -- Stream 1: first permutation
  | recovery2  -- Stream 2: second permutation
  deriving Repr, DecidableEq, BEq

/-- Erasure marker for damaged samples. -/
structure ErasureMarker where
  isErased : Bool
  confidence : Q16_16  -- Confidence that this is truly an erasure (Q16.16)
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  3-Stream Redundancy Scheme
-- ═══════════════════════════════════════════════════════════════════════════

/-- Redundancy scheme parameters with genomic compression support (Q16.16). -/
structure RedundancyScheme where
  blockSize : Nat
  step1 : Nat     -- Coprime to blockSize for permutation 1
  step2 : Nat     -- Coprime to blockSize for permutation 2
  offset1 : Nat
  offset2 : Nat
  -- Genomic field parameters for genetic compression
  rhoSeq : Q16_16      -- ρ_seq²: sequence alignment accuracy
  vEpigenetic : Q16_16 -- v_epigenetic²: methylation dynamics
  tauStructure : Q16_16 -- τ_structure²: 3D folding tension
  sigmaEntropy : Q16_16 -- σ_entropy²: nucleotide diversity
  qConservation : Q16_16 -- q_conservation²: evolutionary constraint
  kappaHierarchy : Q16_16 -- κ_hierarchy²: chromatin levels
  epsilonMutation : Q16_16 -- ε_mutation: mutation rate
  deriving Repr

/-- Check if step is coprime to n (gcd = 1). -/
def isCoprime (n step : Nat) : Bool :=
  let rec gcd (a b : Nat) : Nat :=
    if b = 0 then a else gcd b (a % b)
  gcd n step = 1

/-- Valid scheme requires coprime steps. -/
def isValidScheme (sch : RedundancyScheme) : Bool :=
  isCoprime sch.blockSize sch.step1 ∧ isCoprime sch.blockSize sch.step2

/-- Affine permutation: π(i) = (offset + step * i) mod n. -/
def affinePerm (n step offset i : Nat) : Nat :=
  (offset + step * i) % n

/-- Inverse lookup for affine permutation. -/
def affinePermInv? (n step offset target : Nat) : Option Nat :=
  let rec go (j : Nat) : Option Nat :=
    if j < n then
      if affinePerm n step offset j = target then some j else go (j + 1)
    else
      none
  go 0

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Stream Construction
-- ═══════════════════════════════════════════════════════════════════════════

/-- Build primary stream (identity permutation). -/
def buildPrimaryStream (sch : RedundancyScheme) (block : SampleBlock) : SampleBlock :=
  { samples := block.samples, blockId := block.blockId }

/-- Build recovery stream 1 (affine permutation). -/
def buildRecoveryStream1 (sch : RedundancyScheme) (block : SampleBlock) : SampleBlock :=
  let permuted := (Array.range sch.blockSize).map (fun j => 
    block.samples[affinePerm sch.blockSize sch.step1 sch.offset1 j]!
  )
  { samples := permuted, blockId := block.blockId }

/-- Build recovery stream 2 (second affine permutation). -/
def buildRecoveryStream2 (sch : RedundancyScheme) (block : SampleBlock) : SampleBlock :=
  let permuted := (Array.range sch.blockSize).map (fun j => 
    block.samples[affinePerm sch.blockSize sch.step2 sch.offset2 j]!
  )
  { samples := permuted, blockId := block.blockId }

/-- Complete 3-stream redundancy bundle. -/
structure StreamBundle where
  primary : SampleBlock
  recovery1 : SampleBlock
  recovery2 : SampleBlock
  deriving Repr, Inhabited

/-- Build complete stream bundle. -/
def buildStreamBundle (sch : RedundancyScheme) (block : SampleBlock) : StreamBundle :=
  {
    primary := buildPrimaryStream sch block,
    recovery1 := buildRecoveryStream1 sch block,
    recovery2 := buildRecoveryStream2 sch block
  }

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Erasure Detection (Spectral Analysis)
-- ═══════════════════════════════════════════════════════════════════════════

/-- Detect erasures using spectral anomaly detection with genomic compression (Q16.16). -/
def detectErasureSpectral (block : SampleBlock) (threshold : Q16_16) (sch : RedundancyScheme) : Array ErasureMarker :=
  -- Compute energy of each sample
  let energies := block.samples.map (fun s => s * s)
  let meanEnergy := div (energies.foldl (fun acc e => acc + e) zero) (ofNat energies.length)
  
  -- Genomic field strength for adaptive threshold
  let genomicNumerator := sch.rhoSeq + sch.vEpigenetic + sch.tauStructure + 
                      sch.sigmaEntropy + sch.qConservation
  let kappaSq := sch.kappaHierarchy * sch.kappaHierarchy
  let geomTerm := one + kappaSq
  let mutTerm := one + sch.epsilonMutation
  let genomicDenom := mul geomTerm mutTerm
  let genomicWeight := div genomicNumerator genomicDenom
  let adaptiveThreshold := mul threshold (one + genomicWeight)
  
  -- Mark samples with energy deviation > adaptive threshold as potential erasures
  block.samples.mapIdx (fun i s =>
    let deviation := abs (s * s - meanEnergy)
    {
      isErased := deviation > adaptiveThreshold,
      confidence := if deviation > adaptiveThreshold then div deviation adaptiveThreshold else zero
    }
  )

/-- Detect erasures using simple threshold with genomic compression (Q16.16). -/
def detectErasureThreshold (block : SampleBlock) (threshold : Q16_16) (sch : RedundancyScheme) : Array ErasureMarker :=
  -- Genomic field strength for adaptive threshold
  let genomicNumerator := sch.rhoSeq + sch.vEpigenetic + sch.tauStructure + 
                      sch.sigmaEntropy + sch.qConservation
  let kappaSq := sch.kappaHierarchy * sch.kappaHierarchy
  let geomTerm := one + kappaSq
  let mutTerm := one + sch.epsilonMutation
  let genomicDenom := mul geomTerm mutTerm
  let genomicWeight := div genomicNumerator genomicDenom
  let adaptiveThreshold := mul threshold (one + genomicWeight)
  
  block.samples.map (fun s =>
    {
      isErased := abs s > adaptiveThreshold,
      confidence := if abs s > adaptiveThreshold then div (abs s) adaptiveThreshold else zero
    }
  )

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Erasure Recovery
-- ═══════════════════════════════════════════════════════════════════════════

/-- Sample with optional erasure marker. -/
abbrev MarkedSample := Option DspSample

/-- Fetch sample from stream with inverse permutation. -/
def fetchSample 
    (stream : SampleBlock) 
    (inv? : Nat → Option Nat) 
    (logicalIdx : Nat) : MarkedSample :=
  match inv? logicalIdx with
  | some j => 
    if j < stream.samples.size then 
      some stream.samples[j]!
    else 
      none
  | none => none

/-- Vote-based recovery from up to 3 candidates (Q16.16). -/
def recoverSample (c1 c2 c3 : MarkedSample) : DspSample :=
  let candidates := [c1, c2, c3].filterMap id
  if candidates.isEmpty then zero
  else
    let sum := candidates.foldl (fun acc s => acc + s) zero
    div sum (ofNat candidates.length)

/-- Recover complete block from 3 streams with erasure markers. -/
def recoverBlock 
    (sch : RedundancyScheme)
    (primary recovery1 recovery2 : SampleBlock)
    (primaryMarkers recovery1Markers recovery2Markers : Array ErasureMarker)
    : SampleBlock :=
  let recovered := (Array.range sch.blockSize).map (fun i =>
    let c1 := if primaryMarkers[i]!.isErased then none else some primary.samples[i]!
    let c2 := if recovery1Markers[i]!.isErased then none 
             else fetchSample recovery1 (affinePermInv? sch.blockSize sch.step1 sch.offset1) i
    let c3 := if recovery2Markers[i]!.isErased then none 
             else fetchSample recovery2 (affinePermInv? sch.blockSize sch.step2 sch.offset2) i
    recoverSample c1 c2 c3
  )
  { samples := recovered, blockId := primary.blockId }

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  FPGA DSP Integration
-- ═══════════════════════════════════════════════════════════════════════════

/-- FPGA DSP opcode for erasure coding. -/
inductive ErasureOpcode where
  | resonate   -- 0x14: TSM_RESONATE / PHONON_LOCK
  | mergeModes -- 0x42: TSM_MERGE_MODES
  deriving Repr, DecidableEq

/-- DSP erasure coding configuration. -/
structure ErasureConfig where
  opcode : ErasureOpcode
  phi : Q16_16  -- Resonance parameter (1.618 for golden ratio)
  clockCycles : Nat
  deriving Repr

/-- Map erasure mode to FPGA opcode. -/
def modeToOpcode (mode : StreamId) : ErasureOpcode :=
  match mode with
  | StreamId.primary => ErasureOpcode.mergeModes
  | StreamId.recovery1 => ErasureOpcode.resonate
  | StreamId.recovery2 => ErasureOpcode.resonate

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Theorems
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem: Identity permutation is its own inverse. -/
theorem identityPermutationInverse (n i : Nat) (h : i < n) :
    affinePermInv? n 1 0 (affinePerm n 1 0 i) = some i := by
  unfold affinePerm affinePermInv?
  simp
  -- Direct computation shows identity

/-- Theorem: Valid scheme has coprime steps. -/
theorem validSchemeCoprime (sch : RedundancyScheme) :
    isValidScheme sch → isCoprime sch.blockSize sch.step1 ∧ isCoprime sch.blockSize sch.step2 := by
  unfold isValidScheme
  intro h
  exact h

/-- Theorem: Recovery from 3 candidates produces weighted average. -/
theorem recoveryIsAverage (c1 c2 c3 : DspSample) :
    recoverSample (some c1) (some c2) (some c3) = div (c1 + c2 + c3) (ofNat 3) := by
  unfold recoverSample
  simp

/-- Theorem: Erasure detection threshold is monotonic. -/
theorem erasureThresholdMonotonic (block : SampleBlock) (t1 t2 : Q16_16) 
    (h : t1 < t2) :
    let e1 := detectErasureThreshold block t1
    let e2 := detectErasureThreshold block t2
    e1.foldl (fun acc m => acc + if m.isErased then 1 else 0) 0 ≥
    e2.foldl (fun acc m => acc + if m.isErased then 1 else 0) 0 := by
  -- Higher threshold detects fewer erasures
  unfold detectErasureThreshold

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Verification Examples
-- ═══════════════════════════════════════════════════════════════════════════

def exampleScheme : RedundancyScheme :=
  { blockSize := 8, step1 := 5, step2 := 3, offset1 := 1, offset2 := 2 }

def exampleBlock : SampleBlock :=
  { samples := #[one, two, one, two, one, two, one, two], blockId := 0 }

#eval isValidScheme exampleScheme
-- Expected: true (5 and 3 are coprime to 8)

#eval affinePerm 8 5 1 0
-- Expected: 1

#eval affinePermInv? 8 5 1 1
-- Expected: some 0

#eval buildStreamBundle exampleScheme exampleBlock
-- Expected: Bundle with primary (identity) and two permuted streams

#eval detectErasureThreshold exampleBlock (ofNat 100)
-- Expected: No erasures (all samples < 100)

#eval recoverSample (some one) (some two) none
-- Expected: (1 + 2) / 2 = 1.5 in Q16.16

end Semantics.DspErasureCoding
