diff --git a/Language/Properties.agda b/Language/Properties.agda
index c6a5af0..9446b42 100644
--- a/Language/Properties.agda
+++ b/Language/Properties.agda
@@ -54,34 +54,34 @@ Trace-∙ʳ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
     Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ʳ _ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx))
                     (Trace-∙ʳ g₁ g₂ tr')
 
-Trace-↦ˡ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
+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)
+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')
+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-↦ˡ tr')
 
-Trace-↦ʳ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₂} {ρ₁ ρ₂ : Env} →
+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)
+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')
+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 (Graph.edges g₁ ↑ˡᵉ Graph.size g₂)
                                         (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx)))
-                    (Trace-↦ʳ g₁ g₂ tr')
+                    (Trace-↦ʳ {g₁} {g₂} tr')
 
-Trace-loop : ∀ (g₁ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
+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 g₁ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂
-Trace-loop g₁ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
-    Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop g₁ tr')
+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-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr')
 
 _++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
        EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →
@@ -100,8 +100,8 @@ _++_ {g₁} {g₂} etr₁ etr₂
                                            (ListMemProp.∈-++⁺ʳ (Graph.size g₁ ↑ʳᵉ Graph.edges g₂)
                                                                (∈-cartesianProduct o∈tr₁ i∈tr₂))
             in
-                (Trace-↦ˡ g₁ g₂ (EndToEndTrace.trace etr₁)) ++⟨ oi∈es ⟩
-                (Trace-↦ʳ g₁ g₂ (EndToEndTrace.trace etr₂))
+                (Trace-↦ˡ {g₁} {g₂} (EndToEndTrace.trace etr₁)) ++⟨ oi∈es ⟩
+                (Trace-↦ʳ {g₁} {g₂} (EndToEndTrace.trace etr₂))
         }
 
 Trace-singleton : ∀ {bss : List BasicStmt} {ρ₁ ρ₂ : Env} →