Lean migration: Phase 5 (language, CFGs, traces, Program)

- Spa.Language.Base: Expr/BasicStmt/Stmt + HasVar relations; StringSet
  lifts to Finset String
- Spa.Language.Semantics: Value/Env/Env.Mem, big-step relations,
  LatticeInterpretation (respects-≈ field drops out with =)
- Spa.Language.Graphs: Graph with nodes : Fin size → List BasicStmt
  (Vec lookup lemmas lift to Fin.append_left/right), comp/link/loop/
  skipto/singleton/wrap/buildCfg, predecessors via List.finRange
- Spa.Language.Traces: Trace + EndToEndTrace (Prop-valued)
- Spa.Language.Properties: trace embeddings, loop lemmas,
  buildCfg_sufficient; the 80-line Fin-disjointness block reduces to
  castAdd_ne_natAdd + mathlib list lemmas
- Spa.Language: Program (vars via Finset.sort — toList is noncomputable)

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

lean/Spa/Language/Traces.lean

@@ -0,0 +1,38 @@
+/-
+Port of `Language/Traces.agda`.
+
+Correspondence:
+  Trace          ↦ Trace (a `Prop`-valued inductive; only used in proofs)
+  _++⟨_⟩_        ↦ Trace.concat
+  EndToEndTrace  ↦ EndToEndTrace (a `Prop`-valued structure, like `∃`; its
+                   fields are accessed by destructuring inside proofs)
+-/
+import Spa.Language.Semantics
+import Spa.Language.Graphs
+
+namespace Spa
+
+/-- Agda: `Trace`. -/
+inductive Trace (g : Graph) : g.Index → g.Index → Env → Env → Prop
+  | single {ρ₁ ρ₂ : Env} {idx : g.Index} :
+      EvalBasicStmts ρ₁ (g.nodes idx) ρ₂ → Trace g idx idx ρ₁ ρ₂
+  | edge {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : g.Index} :
+      EvalBasicStmts ρ₁ (g.nodes idx₁) ρ₂ → (idx₁, idx₂) ∈ g.edges →
+      Trace g idx₂ idx₃ ρ₂ ρ₃ → Trace g idx₁ idx₃ ρ₁ ρ₃
+
+/-- Agda: `_++⟨_⟩_`. -/
+theorem Trace.concat {g : Graph} {idx₁ idx₂ idx₃ idx₄ : g.Index}
+    {ρ₁ ρ₂ ρ₃ : Env} (tr₁ : Trace g idx₁ idx₂ ρ₁ ρ₂)
+    (he : (idx₂, idx₃) ∈ g.edges) (tr₂ : Trace g idx₃ idx₄ ρ₂ ρ₃) :
+    Trace g idx₁ idx₄ ρ₁ ρ₃ := by
+  induction tr₁ with
+  | single hbs => exact Trace.edge hbs he tr₂
+  | edge hbs he' _ ih => exact Trace.edge hbs he' (ih he tr₂)
+
+/-- Agda: `EndToEndTrace` (an existential package, destructured in proofs). -/
+inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Prop
+  | intro (idx₁ : g.Index) (idx₁_mem : idx₁ ∈ g.inputs)
+      (idx₂ : g.Index) (idx₂_mem : idx₂ ∈ g.outputs)
+      (trace : Trace g idx₁ idx₂ ρ₁ ρ₂) : EndToEndTrace g ρ₁ ρ₂
+
+end Spa