Reorganize proofs to make 'Program' accessible to files in Language/

lean/Spa/Language/Properties.lean

@@ -23,13 +23,6 @@ namespace Spa
 
 open Graph
 
-lemma Fin.castAdd_ne_natAdd {n m : ℕ} (i : Fin n) (j : Fin m) :
-    Fin.castAdd m i ≠ Fin.natAdd n j := by
-  intro h
-  have := congrArg Fin.val h
-  simp only [Fin.coe_castAdd, Fin.coe_natAdd] at this
-  omega
-
 section Embeddings
 
 variable {g₁ g₂ : Graph} {ρ₁ ρ₂ : Env}
@@ -246,49 +239,17 @@ noncomputable def Stmt.cfg_sufficient {s : Stmt} {ρ₁ ρ₂ : Env}
   | whileFalse ρ e s _ =>
     exact EndToEndTrace.loop_empty
 
-/-- The input / entry node generated by `Graph.wrap`. -/
-def Graph.wrapInput (g : Graph) : (Graph.wrap g).Index :=
-  (0 : Fin 1).castAdd ((g ⤳ Graph.singleton none).size)
+namespace Program
 
-/-- The output / exit node generated by `Graph.wrap`. -/
-def Graph.wrapOutput (g : Graph) : (Graph.wrap g).Index :=
-  Fin.natAdd 1 ((Fin.natAdd g.size (0 : Fin 1)))
+noncomputable def trace (p : Program) {ρ : Env} (h : EvalStmt [] p.rootStmt ρ) :
+    Trace p.cfg p.initialState p.finalState [] ρ := by
+  obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := EndToEndTrace.wrap (Stmt.cfg_sufficient h)
+  rw [Graph.wrap_inputs, List.mem_singleton] at h₁
+  rw [Graph.wrap_outputs, List.mem_singleton] at h₂
+  subst h₁; subst h₂
+  exact tr
 
-/-- The `Graph.wrapInput` is, indeed, the graph's only input after `Graph.wrap`. -/
-lemma Graph.wrap_inputs (g : Graph) :
-    (Graph.wrap g).inputs = [g.wrapInput] := rfl
+end Program
 
-/-- The `Graph.wrapInput` is, indeed, the graph's only output after `Graph.wrap`. -/
-lemma Graph.wrap_outputs (g : Graph) :
-    (Graph.wrap g).outputs = [g.wrapOutput] := rfl
-
-/-- When sequencing (proven here with `Graph.singleton` on the left), no edges
-    exist from the right-hand graph back to the left. -/
-private lemma not_mem_edges_castAdd_sequence {g₂ : Graph} (i : Fin 1)
-    (idx : (Graph.singleton none ⤳ g₂).Index) :
-    ((idx, i.castAdd g₂.size) : (Graph.singleton none ⤳ g₂).Edge)
-      ∉ (Graph.singleton none ⤳ g₂).edges := by
-  intro h
-  rcases List.mem_append.mp h with h' | h'
-  · rcases List.mem_append.mp h' with h'' | h''
-    · -- lifted edges of `singleton []`: there are none
-      simp [Graph.singleton, List.finCastAddProd] at h''
-    · -- lifted edges of g₂: targets are natAdd
-      obtain ⟨e, _, heq⟩ := List.mem_map.mp h''
-      exact Fin.castAdd_ne_natAdd i e.2 (congrArg Prod.snd heq).symm
-  · -- product edges: targets are natAdd'd inputs of g₂
-    obtain ⟨-, hb⟩ := List.mem_product.mp h'
-    obtain ⟨j, -, heq⟩ := List.mem_map.mp hb
-    exact Fin.castAdd_ne_natAdd i j heq.symm
-
-/-- The input node of a graph after `Graph.wrap` has no predecessors. -/
-lemma Graph.wrap_predecessors_eq_nil (g : Graph) (idx : (Graph.wrap g).Index)
-    (h : idx ∈ (Graph.wrap g).inputs) :
-    (Graph.wrap g).predecessors idx = [] := by
-  rw [Graph.wrap_inputs, List.mem_singleton] at h
-  subst h
-  rw [GGraph.predecessors, List.filter_eq_nil_iff]
-  intro idx' _
-  simpa using not_mem_edges_castAdd_sequence (g₂ := g ⤳ Graph.singleton none) 0 idx'
 
 end Spa