Add computational reaching-definitions analysis

Introduce a finite-height lattice instance for Bool, then build the
reaching-definitions analysis on top of the forward framework:

* Spa/Lattice/Bool.lean: FiniteHeightLattice Bool (the two-element
  lattice false ≤ true), making FiniteMap A Bool ks a finite-height
  "power set" lattice for free.
* Spa/Analysis/Reaching.lean: DefSet prog = FiniteMap prog.State Bool
  prog.states as the per-variable lattice of definition sites, with a
  StmtEvaluator whose transfer function performs a strong update
  (assignment to k at node s sets k's def-set to {s}).

The analysis computes a least fixed point and produces correct
reaching-definitions sets. Soundness (relating def-sets to actual
execution provenance) is deferred; not yet exposed in Spa.lean.

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

lean/Spa/Analysis/Reaching.lean

@@ -0,0 +1,41 @@
+import Spa.Analysis.Forward
+import Spa.Lattice.Bool
+import Spa.Showable
+
+namespace Spa
+
+open Forward
+
+instance : Showable Bool := ⟨fun b => if b then "true" else "false"⟩
+
+abbrev DefSet (prog : Program) : Type := FiniteMap prog.State Bool prog.states
+
+namespace ReachingAnalysis
+
+variable (prog : Program)
+
+def genSet (s : prog.State) : DefSet prog :=
+  FiniteMap.updating (⊥ : DefSet prog) [s] (fun _ => true)
+
+def eval (s : prog.State) :
+    BasicStmt → VariableValues (DefSet prog) prog → VariableValues (DefSet prog) prog
+  | .assign k _, vs =>
+    FiniteMap.generalizedUpdate id (fun _ _ => genSet prog s) [k] vs
+  | .noop, vs => vs
+
+theorem eval_mono (s : prog.State) (bs : BasicStmt) :
+    Monotone (eval prog s bs) := by
+  cases bs with
+  | assign k e =>
+    exact FiniteMap.generalizedUpdate_monotone monotone_id (fun _ => monotone_const)
+  | noop => exact monotone_id
+
+instance stmtEvaluator : StmtEvaluator (DefSet prog) prog :=
+  ⟨eval prog, eval_mono prog⟩
+
+def output : String :=
+  show' (result (DefSet prog) prog)
+
+end ReachingAnalysis
+
+end Spa