diff --git a/Language/Properties.agda b/Language/Properties.agda
index 6e147ab..b2f14fd 100644
--- a/Language/Properties.agda
+++ b/Language/Properties.agda
@@ -106,12 +106,40 @@ Trace-↦ʳ {g₁} {g₂} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
                                         (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx)))
                     (Trace-↦ʳ {g₁} {g₂} tr')
 
-Trace-loop : ∀ {g₁ : Graph} {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
-             Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g₁} idx₁ idx₂ ρ₁ ρ₂
+Trace-loop : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
+             Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} idx₁ idx₂ ρ₁ ρ₂
 Trace-loop {idx₁ = idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂
-Trace-loop {g₁} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
+Trace-loop {g} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
     Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr')
 
+EndToEndTrace-loop : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
+                     EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂
+EndToEndTrace-loop etr = record
+    { idx₁ = EndToEndTrace.idx₁ etr
+    ; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr
+    ; idx₂ = EndToEndTrace.idx₂ etr
+    ; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr
+    ; trace = Trace-loop (EndToEndTrace.trace etr)
+    }
+
+EndToEndTrace-loop² : ∀ {g : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
+                      EndToEndTrace {loop g} ρ₁ ρ₂ →
+                      EndToEndTrace {loop g} ρ₂ ρ₃ →
+                      EndToEndTrace {loop g} ρ₁ ρ₃
+EndToEndTrace-loop² {g} etr₁ etr₂ = record
+    { idx₁ = EndToEndTrace.idx₁ etr₁
+    ; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr₁
+    ; idx₂ = EndToEndTrace.idx₂ etr₂
+    ; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr₂
+    ; trace =
+        let
+            o∈tr₁ = EndToEndTrace.idx₂∈outputs etr₁
+            i∈tr₂ = EndToEndTrace.idx₁∈inputs etr₂
+            oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g) (∈-cartesianProduct o∈tr₁ i∈tr₂)
+        in
+            EndToEndTrace.trace etr₁ ++⟨ oi∈es ⟩ EndToEndTrace.trace etr₂
+    }
+
 infixr 5 _++_
 _++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
        EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →