Add proof of reaching definition analysis

This requires a few pieces:

* Make node tags use `Fin n` intead of natural numbers. This makes
  it possible to build a finite lattice over AST nodes, and also
  ensure automatic, total indexing from CFG nodes into the AST that
  created them. For this, use the elaborator to derive the ordering
  statements etc. where possible.
* Adjust the forward framework to enable proofs that don't just state
  correctness on the environment, but also on an arbitrary additional
  state accumulated from traversing the trace.
* State the reaching definition analysis's correctness in terms
  of this new framework.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

lean/Spa/Lattice.lean

@@ -12,6 +12,18 @@ etc.. What remains are a couple of theorems about folds, as well
 as `FiniteHeightLattice`, the core concept of lattice-based static
 program analyses. See the documentation on that class for more information. -/
 
+namespace Option
+
+/-- Equality-sensitive eliminator for options in which the `some` case
+    is sensitive to the base `β`. This makes it mirror a one-element fold
+    more closely. -/
+def elimEq {α : Type*} {β : Sort*} :
+    (o : Option α) → β → ((a : α) → o = some a → β → β) → β
+  | none, b, _ => b
+  | some a, b, f => f a rfl b
+
+end Option
+
 namespace Spa
 
 /-- Predicate for binary functions independently monotone in both their arguments. -/
@@ -61,6 +73,16 @@ lemma foldl_mono' (l : List α) (f : β → α → β)
   | nil => exact hb
   | cons x xs ih => exact ih (hf x hb)
 
+omit [Preorder α] in
+/-- The equality-aware eliminator (that also alters its behavior dependent on base case)
+    for option is monotonic. -/
+lemma elimEq_self_mono (o : Option α) (g : (a : α) → o = some a → β → β)
+    (hg : ∀ a h, Monotone (g a h)) :
+    Monotone (o.elimEq · g) := by
+  cases o with
+  | none => exact monotone_id
+  | some a => exact hg a rfl
+
 end Folds
 
 /-- Predicate on types with `Preorder` that claims all $<$ chains in the type have at most `n` comparisons. -/