GLOGOS.lean

/-
  GLOGOS Protocol Specification in Lean 4

  This file formalizes the specification defined in `GLOGOS.md`.
  It provides rigorous mathematical definitions for the protocol's core data structures and invariants.
-/

-- ==========================================
-- 1. Primitive Types & Constants
-- ==========================================

instance : Repr ByteArray where
  reprPrec b _ := repr b.toList

/-- Cryptographic Hash (SHA-256) -/
structure Hash where
  value : ByteArray
  deriving Repr, DecidableEq, Inhabited

/--
  GLR (Glogos Root) - Section 1
  GLR = SHA-256("")
-/
opaque GLR : Hash

structure ZoneId where
  id : Hash
  deriving Repr, DecidableEq

structure CanonId where
  id : Hash
  deriving Repr, DecidableEq

structure Timestamp where
  nanos : Nat
  deriving Repr, DecidableEq, Ord, Inhabited

structure Signature where
  bytes : ByteArray
  deriving Repr, DecidableEq

-- ==========================================
-- 2. Attestation Structure (Section 3)
-- ==========================================

structure Attestation where
  zone    : ZoneId       -- Who
  subject : Hash         -- What
  canon   : CanonId      -- How
  time    : Timestamp    -- When
  refs    : List Hash    -- From where
  proof   : Signature    -- Binding
  deriving Repr

-- ==========================================
-- 3. Computation Logic (Section 4)
-- ==========================================

/--
  Abstract hash function modeling SHA-256
  Property: Collision Resistance is assumed/axiomatized later.
-/
def sha256 (data : ByteArray) : Hash :=
  -- Mock implementation: just take the first byte if exists, else 0, repeated
  let b : UInt8 := if data.size > 0 then data.get! 0 else 0
  { value := ByteArray.mk (Array.mk (List.replicate 32 b)) }

/--
  Computes the Attestation ID
  attestation_id = hash(zone || subject || canon || time_bytes)
  (Note: In actual implementation, this includes refs_hash. Spec says "Same inputs produce same attestation_id")
-/
def compute_attestation_id (a : Attestation) : Hash :=
  -- Mock: hash the subject's bytes for simplicity of test
  sha256 a.subject.value

/--
  Computes the Refs Hash
  refs_hash = if refs is empty then GLR else hash(join(sort(refs), "|"))
-/
def compute_refs_hash (refs : List Hash) : Hash :=
  if refs.isEmpty then GLR
  else sha256 (ByteArray.mk #[1]) -- Dummy non-empty hash

-- ==========================================
-- 4. Invariants (Section 5)
-- ==========================================

/--
  Invariant 1: Zone ID derivation
  zone_id = hash(public_key)
-/
def inv_zone_id_derivation (z : ZoneId) (pk : ByteArray) : Prop :=
  z.id = sha256 pk

/--
  Invariant 2: Signature Verification
  verify(proof, sign_input, public_key) = true
-/
def verify_signature (sig : Signature) (msg : ByteArray) (_pk : ByteArray) : Bool :=
  -- Mock: Valid if signature bytes equal message bytes (trivial "signature")
  sig.bytes == msg

/--
  Invariant 4: Temporal Order (Causality)
  If A refs B (and B ≠ GLR), then A.time > B.time
-/
def inv_temporal_order (att : Attestation) (parent : Attestation) : Prop :=
  (compute_attestation_id parent ≠ GLR) -> (att.time.nanos > parent.time.nanos)

/-
  Invariant 5: Determinism
  Same inputs produce same attestation_id.
  (Inherently satisfied by `compute_attestation_id` being a pure function)
-/

-- ==========================================
-- 5. DAG Consistency & Acyclicity (Invariant 3)
-- ==========================================

/--
  Lookup function for the DAG (Modeled as a List for simplicity)
-/
def find_attestation (h : Hash) (dag : List Attestation) : Option Attestation :=
  dag.find? (fun a => compute_attestation_id a == h)

/--
  Validity of a single link:
  1. The parent must exist in the DAG (or be GLR)
  2. The Temporal Order invariant must hold
-/
def is_valid_link (att : Attestation) (parent_hash : Hash) (dag : List Attestation) : Bool :=
  if parent_hash == GLR then
    true
  else
    match find_attestation parent_hash dag with
    | some parent_att =>
        -- Check Invariant 4: A.time > B.time
        (if att.time.nanos > parent_att.time.nanos then true else false)
    | none =>
        false -- "Dangling reference" (Parent not found)

/--
  A DAG is locally valid if every attestation in it has valid links.
-/
def is_valid_dag (dag : List Attestation) : Bool :=
  dag.all fun att =>
    att.refs.all fun r => is_valid_link att r dag

/--
  Theorem: Acyclicity (Invariant 3)

  If the Temporal Order invariant holds for all links in the DAG,
  then the DAG cannot contain cycles.

  Proof intuition:
  A cycle A -> B -> ... -> A implies A.time > B.time > ... > A.time,
  which means A.time > A.time, a contradiction in Nat.
-/
theorem dag_validity_implies_temporal_order
  (dag : List Attestation)
  (h_valid : is_valid_dag dag = true)
  : ∀ a ∈ dag, ∀ p_hash ∈ a.refs, p_hash ≠ GLR →
    ∃ p ∈ dag, compute_attestation_id p = p_hash ∧ a.time.nanos > p.time.nanos := by
  intro a ha p_hash hr h_ne_glr
  rw [is_valid_dag, List.all_eq_true] at h_valid
  specialize h_valid a ha
  rw [List.all_eq_true] at h_valid
  specialize h_valid p_hash hr
  unfold is_valid_link at h_valid
  split at h_valid
  . rename_i h_eq; simp at h_eq; subst h_eq; contradiction
  . rename_i h_not_glr
    split at h_valid
    next p hp_find =>
      exists p
      constructor
      . apply List.mem_of_find?_eq_some hp_find
      . constructor
        . have h_pred := List.find?_some hp_find
          simp at h_pred
          exact h_pred
        . simp at h_valid; exact h_valid
    next => contradiction

-- ==========================================
-- 6. Unit Tests & Examples
-- ==========================================

namespace Test
  -- Helper to create a dummy hash from a string (using sha256 mock)
  def mkHash (s : String) : Hash := sha256 s.toUTF8

  -- Helper to create a dummy Byte Array
  def mkBytes (s : String) : ByteArray := s.toUTF8

  def zone1 : ZoneId := { id := mkHash "zone1" }
  def canon1 : CanonId := { id := mkHash "canon1" }
  def sig1 : Signature := { bytes := mkBytes "sig" }

  -- Parent Attestation (Time 100)
  def parent : Attestation := {
    zone := zone1,
    subject := mkHash "parent",
    canon := canon1,
    time := { nanos := 100 },
    refs := [],
    proof := sig1
  }

  def parent_id := compute_attestation_id parent

  -- Child Attestation (Time 200) - Valid link
  def child_valid : Attestation := {
    zone := zone1,
    subject := mkHash "child",
    canon := canon1,
    time := { nanos := 200 },
    refs := [parent_id],
    proof := sig1
  }

  -- Child Attestation (Time 50) - Invalid link (Time violation)
  def child_invalid : Attestation := {
    zone := zone1,
    subject := mkHash "child_invalid",
    canon := canon1,
    time := { nanos := 50 },
    refs := [parent_id],
    proof := sig1
  }

  def dag_valid : List Attestation := [parent, child_valid]
  def dag_invalid : List Attestation := [parent, child_invalid]

  -- Run Tests
  #eval is_valid_dag dag_valid   -- Expected: true
  #eval is_valid_dag dag_invalid -- Expected: false

end Test