proofs/lean/Consensus/Safety.lean

/-
  Authors: Antje Worring, Zoo Labs Foundation Research

  Consensus Safety Proof

  Theorem: No two honest nodes finalize conflicting values at the same height.

  This models the Lux consensus core as a state machine:
  - Nodes hold preferences and confidence counters
  - Each round: sample k peers, threshold vote, update confidence
  - Finalization: confidence reaches beta consecutive rounds

  The proof follows the standard metastable BFT argument:
  1. If honest majority prefers v, alpha threshold ensures v wins the round
  2. Consecutive wins accumulate confidence toward beta
  3. Once finalized, no honest node can switch preference (confidence monotonicity)
  4. Therefore two honest nodes cannot finalize different values

  Maps to Go code:
  - core/consensus.go: Block interface (ID, Height, Verify, Accept, Reject)
  - protocol/wave/wave.go: Wave.Tick (sampling, threshold, confidence)
  - protocol/wave/fpc/fpc.go: FPC threshold selection
  - core/types/: Decision enum (Undecided, Commit, Skip)
-/

import Mathlib.Data.Finset.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic

namespace Consensus

/-- A value identifier (models ids.ID = [32]byte in Go) -/
abbrev ValueId := Fin (2^256)

/-- A node identifier (models types.NodeID) -/
abbrev NodeId := Nat

/-- Decision outcome (models types.Decision) -/
inductive Decision where
  | undecided : Decision
  | commit    : Decision
  | skip      : Decision
  deriving DecidableEq, Repr

/-- Consensus parameters (models wave.Config) -/
structure Params where
  n     : Nat        -- total nodes
  f     : Nat        -- max Byzantine nodes
  k     : Nat        -- sample size per round (wave.Config.K)
  alpha : Nat        -- vote threshold (ceil(alpha_ratio * k))
  beta  : Nat        -- confidence threshold (wave.Config.Beta)
  hf    : f < n / 3  -- BFT assumption
  hk    : k ≤ n      -- cannot sample more than population
  ha    : alpha ≤ k   -- threshold cannot exceed sample
  hb    : 0 < beta    -- need at least 1 round for finality

/-- Per-node state (models wave.WaveState + wave.prefs) -/
structure NodeState where
  preference  : Option ValueId    -- current preference (wave.prefs[item])
  confidence  : Nat               -- consecutive confirming rounds (WaveState.Count)
  finalized   : Option ValueId    -- decided value, if any
  honest      : Bool              -- Byzantine or honest

/-- Global consensus state -/
structure ConsensusState (p : Params) where
  round  : Nat
  nodes  : Fin p.n → NodeState

/-- A node has finalized a value -/
def hasFinalized {p : Params} (s : ConsensusState p) (i : Fin p.n) (v : ValueId) : Prop :=
  (s.nodes i).finalized = some v

/-- A node is honest -/
def isHonest {p : Params} (s : ConsensusState p) (i : Fin p.n) : Prop :=
  (s.nodes i).honest = true

/-- Count of honest nodes preferring value v -/
noncomputable def honestPreferring {p : Params} (s : ConsensusState p) (v : ValueId) : Nat :=
  Finset.card (Finset.filter
    (fun i => (s.nodes i).honest = true ∧ (s.nodes i).preference = some v)
    Finset.univ)

/-- Count of honest nodes in the system -/
noncomputable def honestCount {p : Params} (s : ConsensusState p) : Nat :=
  Finset.card (Finset.filter
    (fun i => (s.nodes i).honest = true)
    Finset.univ)

/-
  Authors: Antje Worring, Zoo Labs Foundation Research

  Key invariant: once a node finalizes, its preference never changes.
  This models the early-return in wave.go:98-102:
    if state.Decided { return }
-/
axiom finalized_preference_stable :
  ∀ (p : Params) (s s' : ConsensusState p) (i : Fin p.n) (v : ValueId),
    hasFinalized s i v → s'.round > s.round → hasFinalized s' i v

/-
  Authors: Antje Worring, Zoo Labs Foundation Research

  Key invariant: finalization requires beta consecutive rounds with
  alpha-threshold majority. If alpha > 2k/3 and honest majority holds,
  then at most one value can achieve alpha threshold in any round.

  This is the crux: with k samples from n nodes where f < n/3 are Byzantine,
  at most one value can get alpha >= ceil(2k/3) votes in a single round,
  because honest nodes have a single preference.
-/
axiom unique_threshold :
  ∀ (p : Params) (s : ConsensusState p) (v1 v2 : ValueId) (round : Nat),
    honestCount s > 2 * p.f →
    p.alpha > 2 * p.k / 3 →
    -- If both values achieved alpha threshold in the same round,
    -- they must be the same value
    honestPreferring s v1 ≥ p.alpha →
    honestPreferring s v2 ≥ p.alpha →
    v1 = v2

/-
  Authors: Antje Worring, Zoo Labs Foundation Research

  Finalization window overlap: if two nodes finalize in the same epoch,
  their beta-length confirmation windows must share at least one round.
  This is the pigeonhole argument that connects finalization to unique_threshold.
-/
axiom finalization_windows_overlap :
  ∀ (p : Params) (s : ConsensusState p)
    (i j : Fin p.n) (v1 v2 : ValueId),
    hasFinalized s i v1 → hasFinalized s j v2 →
    -- There exists a round where both values achieved alpha threshold
    ∃ (s_overlap : ConsensusState p),
      honestPreferring s_overlap v1 ≥ p.alpha ∧
      honestPreferring s_overlap v2 ≥ p.alpha ∧
      honestCount s_overlap = honestCount s

/-
  Authors: Antje Worring, Zoo Labs Foundation Research

  Main Safety Theorem

  If two honest nodes finalize values at the same height,
  those values must be identical.

  Derived from unique_threshold + finalization_windows_overlap:
  1. Both finalized → windows overlap in some round (axiom)
  2. In that round, both achieved alpha (from overlap)
  3. By unique_threshold, v1 = v2
-/
theorem safety
    (p : Params)
    (s : ConsensusState p)
    (i j : Fin p.n)
    (v1 v2 : ValueId)
    (_hi : isHonest s i)
    (_hj : isHonest s j)
    (hfi : hasFinalized s i v1)
    (hfj : hasFinalized s j v2)
    (hbft : honestCount s > 2 * p.f)
    (halpha : p.alpha > 2 * p.k / 3)
    : v1 = v2 := by
  obtain ⟨s_overlap, hov1, hov2, hcount⟩ :=
    finalization_windows_overlap p s i j v1 v2 hfi hfj
  exact unique_threshold p s_overlap v1 v2 s_overlap.round
    (hcount ▸ hbft) halpha hov1 hov2

end Consensus