Lean migration: Phase 4 (IterProd + FiniteMap lattices)

- Spa.Lattice.IterProd: k-fold product, recursive Lattice instance,
  fixed height k*hA + hB, bot = build of bottoms
- Spa.Lattice.FiniteMap: spine-pinned assoc lists ({l // l.map fst = ks});
  with = the 1100-line Map.agda collapses into positional 'combine'.
  Same lemma inventory (membership, locate, updating, GeneralizedUpdate,
  valuesAt, Provenance-union, le_of_mem_mem) — Nodup is now an explicit
  hypothesis where the Agda Map carried it intrinsically. Fixed height
  |ks|*hB still via transport along the IterProd isomorphism, which no
  longer needs Unique ks (representation is canonical).

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

lean/Spa/Lattice.lean

@@ -103,6 +103,21 @@ theorem BoundedChains.no_longer {α : Type*} [Preorder α] {n : ℕ}
     (h : BoundedChains α n) (c : LTSeries α) : c.length ≠ n + 1 :=
   fun hc => absurd (h c) (by omega)
 
+/-- Re-index a `FixedHeight` along an equality of heights (used where Agda
+just rewrites with arithmetic identities). -/
+def FixedHeight.cast {α : Type*} [Preorder α] {m n : ℕ} (h : m = n)
+    (fh : FixedHeight α m) : FixedHeight α n where
+  bot := fh.bot
+  top := fh.top
+  longestChain := fh.longestChain
+  head_longestChain := fh.head_longestChain
+  last_longestChain := fh.last_longestChain
+  length_longestChain := h ▸ fh.length_longestChain
+  bounded := h ▸ fh.bounded
+
+@[simp] theorem FixedHeight.cast_bot {α : Type*} [Preorder α] {m n : ℕ}
+    (h : m = n) (fh : FixedHeight α m) : (fh.cast h).bot = fh.bot := rfl
+
 /-- Agda: `IsFiniteHeightLattice` / `FiniteHeightLattice` (bundled). -/
 class FiniteHeightLattice (α : Type*) [Lattice α] where
   height : ℕ