diff --git a/Lattice/Builder.agda b/Lattice/Builder.agda
index 4ba0465..dec59ab 100644
--- a/Lattice/Builder.agda
+++ b/Lattice/Builder.agda
@@ -287,17 +287,49 @@ eqL (plt ∷ plts) (inj₂ v₁ ) (inj₂ v₂) = eqL plts v₁ v₂
 eqL (plt ∷ plts) (inj₁ _) (inj₂ _) = Empty _
 eqL (plt ∷ plts) (inj₂ _) (inj₁ _) = Empty _
 
+eqL-trans : ∀ {a} (L : List (PartialLatticeType a)) {lv₁ lv₂ lv₃ : ListValue L} →  eqL L lv₁ lv₂ → eqL L lv₂ lv₃ → eqL L lv₁ lv₃
+eqL-trans [] {()}
+eqL-trans (plt ∷ plts) {inj₁ v₁} {inj₁ v₂} {inj₁ v₃} v₁≈v₂ v₂≈v₃ = PartialLatticeType.≈-trans plt v₁≈v₂ v₂≈v₃
+eqL-trans (plt ∷ plts) {inj₂ lv₁} {inj₂ lv₂} {inj₂ lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqL-trans plts lv₁≈lv₂ lv₂≈lv₃
+
+eqL-sym : ∀ {a} (L : List (PartialLatticeType a)) {lv₁ lv₂ : ListValue L} →  eqL L lv₁ lv₂ → eqL L lv₂ lv₁
+eqL-sym [] {()}
+eqL-sym (plt ∷ plts) {inj₁ v₁} {inj₁ v₂} v₁≈v₂ = PartialLatticeType.≈-sym plt v₁≈v₂
+eqL-sym (plt ∷ plts) {inj₂ lv₁} {inj₂ lv₂} lv₁≈lv₂ = eqL-sym plts lv₁≈lv₂
+
 eqL-refl : ∀ {a} (L : List (PartialLatticeType a)) (lv : ListValue L) →  eqL L lv lv
 eqL-refl [] ()
 eqL-refl (plt ∷ plts) (inj₁ v) = IsEquivalence.≈-refl (PartialLatticeType.≈-equiv plt)
 eqL-refl (plt ∷ plts) (inj₂ v) = eqL-refl plts v
 
+instance
+    eqL-Equivalence : ∀ {a} {L : List (PartialLatticeType a)} → IsEquivalence (ListValue L) (eqL L)
+    eqL-Equivalence {L = L} = record
+        { ≈-trans = eqL-trans L
+        ; ≈-sym = eqL-sym L
+        ; ≈-refl = λ {lv} → eqL-refl L lv
+        }
+
 eqLv : ∀ {a} (L : Layer a) →  LayerValue L → LayerValue L → Set a
 eqLv L = eqL (toList L)
 
+eqLv-trans : ∀ {a} (L : Layer a) →  {lv₁ lv₂ lv₃ : LayerValue L} → eqLv L lv₁ lv₂ → eqLv L lv₂ lv₃ → eqLv L lv₁ lv₃
+eqLv-trans L {lv₁} {lv₂} {lv₃} = eqL-trans (toList L) {lv₁} {lv₂} {lv₃}
+
+eqLv-sym : ∀ {a} (L : Layer a) →  {lv₁ lv₂ : LayerValue L} → eqLv L lv₁ lv₂ → eqLv L lv₂ lv₁
+eqLv-sym L {lv₁} {lv₂} = eqL-sym (toList L) {lv₁} {lv₂}
+
 eqLv-refl : ∀ {a} (L : Layer a) →  (lv : LayerValue L) → eqLv L lv lv
 eqLv-refl L = eqL-refl (toList L)
 
+instance
+    eqLv-Equivalence : ∀ {a} {L : Layer a} → IsEquivalence (LayerValue L) (eqLv L)
+    eqLv-Equivalence {L = L} = record
+        { ≈-trans = λ {lv₁} {lv₂} {lv₃} → eqLv-trans L {lv₁} {lv₂} {lv₃}
+        ; ≈-sym = λ {lv₁} {lv₂} → eqLv-sym L {lv₁} {lv₂}
+        ; ≈-refl = λ {lv} → eqLv-refl L lv
+        }
+
 eqPath' : ∀ {a} (Ls : Layers a) → Path' Ls → Path' Ls → Set a
 eqPath' (single l) lv₁ lv₂ = eqLv l lv₁ lv₂
 eqPath' (add-via-least l ls) (inj₁ lv₁) (inj₁ lv₂) = eqLv l lv₁ lv₂
@@ -309,6 +341,20 @@ eqPath' (add-via-greatest l ls) (inj₂ p₁) (inj₂ p₂) = eqPath' ls p₁ p
 eqPath' (add-via-greatest l ls) (inj₁ lv₁) (inj₂ p₂) = Empty _
 eqPath' (add-via-greatest l ls) (inj₂ p₁) (inj₁ lv₂) = Empty _
 
+eqPath'-trans : ∀ {a} (Ls : Layers a) →  {p₁ p₂ p₃ : Path' Ls} →  eqPath' Ls p₁ p₂ → eqPath' Ls p₂ p₃ → eqPath' Ls p₁ p₃
+eqPath'-trans (single l) {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqLv-trans l {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃
+eqPath'-trans (add-via-least l ls) {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqLv-trans l {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃
+eqPath'-trans (add-via-least l ls) {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} p₁≈p₂ p₂≈p₃ = eqPath'-trans ls p₁≈p₂ p₂≈p₃
+eqPath'-trans (add-via-greatest l ls) {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqLv-trans l {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃
+eqPath'-trans (add-via-greatest l ls) {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} p₁≈p₂ p₂≈p₃ = eqPath'-trans ls p₁≈p₂ p₂≈p₃
+
+eqPath'-sym : ∀ {a} (Ls : Layers a) →  {p₁ p₂ : Path' Ls} →  eqPath' Ls p₁ p₂ → eqPath' Ls p₂ p₁
+eqPath'-sym (single l) {lv₁} {lv₂} lv₁≈lv₂ = eqLv-sym l {lv₁} {lv₂} lv₁≈lv₂
+eqPath'-sym (add-via-least l ls) {inj₁ lv₁} {inj₁ lv₂}lv₁≈lv₂ = eqLv-sym l {lv₁} {lv₂} lv₁≈lv₂
+eqPath'-sym (add-via-least l ls) {inj₂ p₁} {inj₂ p₂} p₁≈p₂ = eqPath'-sym ls p₁≈p₂
+eqPath'-sym (add-via-greatest l ls) {inj₁ lv₁} {inj₁ lv₂}lv₁≈lv₂ = eqLv-sym l {lv₁} {lv₂} lv₁≈lv₂
+eqPath'-sym (add-via-greatest l ls) {inj₂ p₁} {inj₂ p₂} p₁≈p₂ = eqPath'-sym ls p₁≈p₂
+
 eqPath'-refl : ∀ {a} (Ls : Layers a) →  (p : Path' Ls) →  eqPath' Ls p p
 eqPath'-refl (single l) lv = eqLv-refl l lv
 eqPath'-refl (add-via-least l ls) (inj₁ lv) = eqLv-refl l lv
@@ -316,6 +362,14 @@ eqPath'-refl (add-via-least l ls) (inj₂ p) = eqPath'-refl ls p
 eqPath'-refl (add-via-greatest l ls) (inj₁ lv) = eqLv-refl l lv
 eqPath'-refl (add-via-greatest l ls) (inj₂ p) = eqPath'-refl ls p
 
+instance
+    eqPath'-Equivalence : ∀ {a} {Ls : Layers a} → IsEquivalence (Path' Ls) (eqPath' Ls)
+    eqPath'-Equivalence {Ls = Ls} = record
+        { ≈-trans = eqPath'-trans Ls
+        ; ≈-sym = eqPath'-sym Ls
+        ; ≈-refl = λ {x} → eqPath'-refl Ls x
+        }
+
 -- Now that we can compare paths for "equality", we can state and
 -- prove theorems such as commutativity and idempotence.