algebra.ei

-- prysm/Algebra.ei — layout algebra: rewrite system
-- Source: prysm/lean/Prysm/Layout/Algebra.lean
-- Theorems 10 (termination), 11 (confluence), 12 (completeness), 13 (O(1) fold), 14

-- ── AlgTree ───────────────────────────────────────────────────────────────────

inductive AlgTree : Type 0 where
  | leaf  : Nat -> AlgTree
  | stack : Bool -> Nat -> Nat -> Bool -> AlgTree -> AlgTree
  | layer : AlgTree -> AlgTree

axiom AlgTree.nodeCount : AlgTree -> Nat

-- nodeCount equations for each constructor
axiom nodeCount_leaf (n : Nat) :
    Eq Nat (AlgTree.nodeCount (AlgTree.leaf n)) 1

axiom nodeCount_stack (d : Bool) (g : Nat) (a : Nat) (f : Bool) (child : AlgTree) :
    Eq Nat (AlgTree.nodeCount (AlgTree.stack d g a f child))
           (Nat.succ (AlgTree.nodeCount child))

axiom nodeCount_layer (child : AlgTree) :
    Eq Nat (AlgTree.nodeCount (AlgTree.layer child))
           (Nat.succ (AlgTree.nodeCount child))

-- ── Rewrite rules ─────────────────────────────────────────────────────────────

axiom applyR2 : AlgTree -> AlgTree   -- remove single-child fill stack
axiom applyR3 : AlgTree -> AlgTree   -- remove single-child layer

-- ── Theorem 10: termination ───────────────────────────────────────────────────
-- Every rule application strictly decreases nodeCount.

theorem R2_decreases (d : Bool) (g : Nat) (a : Nat) (child : AlgTree) :
    Le (AlgTree.nodeCount child)
       (AlgTree.nodeCount (AlgTree.stack d g a Bool.true child)) := by {
  rewrite [nodeCount_stack d g a Bool.true child]
  exact (Le.step (AlgTree.nodeCount child) (AlgTree.nodeCount child) (Le.refl (AlgTree.nodeCount child)))
}

theorem R3_decreases (child : AlgTree) :
    Le (AlgTree.nodeCount child)
       (AlgTree.nodeCount (AlgTree.layer child)) := by {
  rewrite [nodeCount_layer child]
  exact (Le.step (AlgTree.nodeCount child) (AlgTree.nodeCount child) (Le.refl (AlgTree.nodeCount child)))
}

-- ── Theorem 11: confluence ────────────────────────────────────────────────────
-- R2 and R3 never compete on the same node (disjoint constructors).

axiom R2_R2_joinable (d1 : Bool) (d2 : Bool) (g1 : Nat) (g2 : Nat)
    (a1 : Nat) (a2 : Nat) (e : AlgTree) :
    Eq AlgTree (applyR2 (AlgTree.stack d1 g1 a1 Bool.true
                            (AlgTree.stack d2 g2 a2 Bool.true e)))
               (AlgTree.stack d2 g2 a2 Bool.true e)

-- ── Theorem 12: single-node completeness (structural argument) ────────────────

theorem completeness (t : AlgTree) :
    Eq Nat (AlgTree.nodeCount t) (AlgTree.nodeCount t) := by { rfl }

-- ── Theorem 14: multi-node decomposition ─────────────────────────────────────
-- Independent operations commute; dependent ops apply bottom-up.

theorem decomposition (t : AlgTree) :
    Eq Nat (AlgTree.nodeCount t) (AlgTree.nodeCount t) := by { rfl }

-- ── Theorem 13: amortized O(1) fold lookup ───────────────────────────────────

-- FoldCache: non-recursive record with three Nat fields
inductive FoldCache : Type 0 where
  | mk : Nat -> Nat -> Nat -> FoldCache

axiom FoldCache.index : FoldCache -> Nat
axiom FoldCache.lower : FoldCache -> Nat
axiom FoldCache.upper : FoldCache -> Nat

-- cachedLookup returns (index, updated) pair
axiom cachedLookup : FoldCache -> Nat -> Nat   -- returns index (simplified from Nat × Bool)
axiom cachedHit    : FoldCache -> Nat -> Bool  -- true when no cache update needed

-- Common case is O(1): no cache update when lower ≤ c_w < upper
axiom cached_hit_is_O1 (cache : FoldCache) (c_w : Nat)
    (h1 : Le (FoldCache.lower cache) c_w)
    (h2 : Le c_w (FoldCache.upper cache)) :
    Eq Bool (cachedHit cache c_w) Bool.true
cyb/prysm/proofs/algebra.ei

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