diff --git a/Language/Properties.agda b/Language/Properties.agda
index b2f14fd..65641d3 100644
--- a/Language/Properties.agda
+++ b/Language/Properties.agda
@@ -140,6 +140,17 @@ EndToEndTrace-loop² {g} etr₁ etr₂ = record
             EndToEndTrace.trace etr₁ ++⟨ oi∈es ⟩ EndToEndTrace.trace etr₂
     }
 
+Trace-optional : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
+                 Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {optional g} (2 Fin.↑ʳ idx₁) (2 Fin.↑ʳ idx₂) ρ₁ ρ₂
+Trace-optional = {!!}
+
+EndToEndTrace-optional : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
+                         EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {optional g} ρ₁ ρ₂
+EndToEndTrace-optional = {!!}
+
+EndToEndTrace-optional-ε : ∀ {g : Graph} {ρ : Env} → EndToEndTrace {optional g} ρ ρ
+EndToEndTrace-optional-ε = {!!}
+
 infixr 5 _++_
 _++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
        EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →
@@ -189,3 +200,6 @@ buildCfg-sufficient (⇒ˢ-if-false ρ₁ ρ₂ _ s₁ s₂ _ ρ₁,s₂⇒ρ₂
     Trace-singleton[] ρ₁ ++
     (EndToEndTrace-∙ʳ {buildCfg s₁} (buildCfg-sufficient ρ₁,s₂⇒ρ₂)) ++
     Trace-singleton[] ρ₂
+buildCfg-sufficient (⇒ˢ-while-true ρ₁ ρ₂ ρ₃ _ _ s _ _ ρ₁,s⇒ρ₂ ρ₂,ws⇒ρ₃) = {!!}
+buildCfg-sufficient (⇒ˢ-while-false ρ _ s _) =
+    EndToEndTrace-optional-ε {loop (buildCfg s)} {ρ}