diff --git a/Language/Properties.agda b/Language/Properties.agda
index 15b11ab..70c56a5 100644
--- a/Language/Properties.agda
+++ b/Language/Properties.agda
@@ -7,9 +7,13 @@ open import Language.Traces
 
 open import Data.Fin as Fin using (zero)
 open import Data.List using (_∷_; [])
+open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
 open import Data.Product using (Σ; _,_)
+open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ)
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym)
+
+open import Utils using (x∈xs⇒fx∈fxs)
 
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
 
 buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
 buildCfg-input ⟨ bs₁ ⟩ = (zero , refl)
@@ -26,3 +30,25 @@ buildCfg-output (s₁ then s₂)
 buildCfg-output (if _ then s₁ else s₂) = (_ , refl)
 buildCfg-output (while _ repeat s)
     with (idx , p) ← buildCfg-output s rewrite p = (_ , refl)
+
+Trace-∙ˡ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
+           Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ →
+           Trace {g₁ ∙ g₂} (idx₁ Fin.↑ˡ Graph.size g₂) (idx₂ Fin.↑ˡ Graph.size g₂) ρ₁ ρ₂
+Trace-∙ˡ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
+    rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-single ρ₁⇒ρ₂
+Trace-∙ˡ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
+    rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (_↑ˡ Graph.size g₂) idx₁→idx))
+                    (Trace-∙ˡ g₁ g₂ tr')
+
+Trace-∙ʳ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₂} {ρ₁ ρ₂ : Env} →
+           Trace {g₂} idx₁ idx₂ ρ₁ ρ₂ →
+           Trace {g₁ ∙ g₂} (Graph.size g₁ Fin.↑ʳ idx₁) (Graph.size g₁ Fin.↑ʳ idx₂) ρ₁ ρ₂
+Trace-∙ʳ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
+    rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-single ρ₁⇒ρ₂
+Trace-∙ʳ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
+    rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ʳ _ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx))
+                    (Trace-∙ʳ g₁ g₂ tr')