Consolidate per-operator trace lifting into GGraph.Embed + Trace.embed

Each graph-composition operator includes its operands via an index
translation preserving node payloads and edges. Capture that once as
GGraph.Embed (a structure, not a class: for g ; g both inclusions share
the type Embed g (g <~> g), so instance resolution could pick the wrong
copy) with five named witnesses, and replace the five structurally
identical trace-lifting inductions in Properties.lean with a single
generic Trace.embed plus one-line corollaries.

The same witnesses' nodes_eq fields will back the upcoming AST-id/CFG
label bijection, so the per-operator content is stated exactly once.

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

lean/Spa/Language/Graphs.lean

@@ -213,6 +213,65 @@ lemma wrap_outputs (g : GGraph (Option β)) :
     (Option.map h) <$> wrap g = wrap (Option.map h <$> g) := by
   simp [GGraph.wrap, GGraph.map_sequence, GGraph.map_singleton]
 
+/-! ### Embeddings
+
+Each composition operator includes its operands into the result via an index
+translation that preserves node payloads and edges. `Embed` captures exactly
+those two facts, so anything defined from `nodes` and `edges` (traces, node
+labels, …) can be transported along an embedding once, instead of once per
+operator.
+
+`Embed` is deliberately a structure rather than a class: for `g ⤳ g`, both the
+left and the right inclusion inhabit the same type `Embed g (g ⤳ g)`, so
+instance resolution could silently pick the wrong copy. Embeddings into a
+composed graph are non-canonical by design; a named witness says which
+inclusion is meant. -/
+
+/-- An embedding of graph `g` into graph `h`: an index translation that
+    preserves node payloads and edges. -/
+structure Embed (g h : GGraph α) where
+  f : g.Index → h.Index
+  nodes_eq : ∀ i, h.nodes (f i) = g.nodes i
+  edges_mem : ∀ {e : g.Edge}, e ∈ g.edges → (f e.1, f e.2) ∈ h.edges
+
+/-- Embeddings compose. -/
+def Embed.trans {g₁ g₂ g₃ : GGraph α} (e₁ : Embed g₁ g₂) (e₂ : Embed g₂ g₃) :
+    Embed g₁ g₃ where
+  f := e₂.f ∘ e₁.f
+  nodes_eq i := (e₂.nodes_eq (e₁.f i)).trans (e₁.nodes_eq i)
+  edges_mem he := e₂.edges_mem (e₁.edges_mem he)
+
+/-- The left operand's inclusion into a sequenced graph. -/
+def Embed.sequenceLeft (g₁ g₂ : GGraph α) : Embed g₁ (g₁ ⤳ g₂) where
+  f i := i.castAdd g₂.size
+  nodes_eq i := Fin.append_left g₁.nodes g₂.nodes i
+  edges_mem he := List.mem_append_left _ (List.mem_append_left _ (List.mem_map_of_mem _ he))
+
+/-- The right operand's inclusion into a sequenced graph. -/
+def Embed.sequenceRight (g₁ g₂ : GGraph α) : Embed g₂ (g₁ ⤳ g₂) where
+  f i := i.natAdd g₁.size
+  nodes_eq i := Fin.append_right g₁.nodes g₂.nodes i
+  edges_mem he := List.mem_append_left _ (List.mem_append_right _ (List.mem_map_of_mem _ he))
+
+/-- The left operand's inclusion into an overlaid graph. -/
+def Embed.overlayLeft (g₁ g₂ : GGraph α) : Embed g₁ (g₁ ∙ g₂) where
+  f i := i.castAdd g₂.size
+  nodes_eq i := Fin.append_left g₁.nodes g₂.nodes i
+  edges_mem he := List.mem_append_left _ (List.mem_map_of_mem _ he)
+
+/-- The right operand's inclusion into an overlaid graph. -/
+def Embed.overlayRight (g₁ g₂ : GGraph α) : Embed g₂ (g₁ ∙ g₂) where
+  f i := i.natAdd g₁.size
+  nodes_eq i := Fin.append_right g₁.nodes g₂.nodes i
+  edges_mem he := List.mem_append_right _ (List.mem_map_of_mem _ he)
+
+/-- The body's inclusion into a `loop` graph. -/
+def Embed.loop (g : GGraph (Option β)) : Embed g (loop g) where
+  f i := i.natAdd 2
+  nodes_eq i := Fin.append_right (fun _ : Fin 2 => none) g.nodes i
+  edges_mem he := List.mem_append_left _ (List.mem_append_left _
+    (List.mem_append_left _ (List.mem_map_of_mem _ he)))
+
 variable (g : GGraph α)
 
 /-- All the nodes in the graph. -/