diff --git a/collect/src/main/kotlin/com/certora/collect/AbstractTreapMap.kt b/collect/src/main/kotlin/com/certora/collect/AbstractTreapMap.kt
index b0dcf0b..2cbf207 100644
--- a/collect/src/main/kotlin/com/certora/collect/AbstractTreapMap.kt
+++ b/collect/src/main/kotlin/com/certora/collect/AbstractTreapMap.kt
@@ -8,11 +8,8 @@ import kotlinx.collections.immutable.ImmutableSet
     Base class for TreapMap implementations.  Provides the Map operations; derived classes deal with type-specific
     behavior such as hash collisions.  See [Treap] for an overview of all of this.
  */
-internal sealed class AbstractTreapMap<@Treapable K, V, @Treapable S : AbstractTreapMap<K, V, S>>(
-    left: S?,
-    right: S?
-) : TreapMap<K, V>, Treap<K, S>(left, right) {
-
+internal sealed class AbstractTreapMap<@Treapable K, V, @Treapable S : AbstractTreapMap<K, V, S>>
+: TreapMap<K, V>, Treap<K, S>() {
     /**
         Derived classes override to create an apropriate node containing the given entry.
      */
@@ -139,13 +136,7 @@ internal sealed class AbstractTreapMap<@Treapable K, V, @Treapable S : AbstractT
             override fun iterator() = entrySequence().iterator()
         }
 
-    override val keys: ImmutableSet<K>
-        get() = object: AbstractSet<K>(), ImmutableSet<K> {
-            override val size get() = this@AbstractTreapMap.size
-            override fun isEmpty() = this@AbstractTreapMap.isEmpty()
-            override operator fun contains(element: K) = containsKey(element)
-            override operator fun iterator() = entrySequence().map { it.key }.iterator()
-        }
+    abstract override val keys: TreapSet<K>
 
     override val values: ImmutableCollection<V>
         get() = object: AbstractCollection<V>(), ImmutableCollection<V> {
diff --git a/collect/src/main/kotlin/com/certora/collect/AbstractTreapSet.kt b/collect/src/main/kotlin/com/certora/collect/AbstractTreapSet.kt
index 4900c4e..2895c15 100644
--- a/collect/src/main/kotlin/com/certora/collect/AbstractTreapSet.kt
+++ b/collect/src/main/kotlin/com/certora/collect/AbstractTreapSet.kt
@@ -6,10 +6,7 @@ import com.certora.forkjoin.*
     Base class for TreapSet implementations.  Provides the Set operations; derived classes deal with type-specific
     behavior such as hash collisions.  See `Treap` for an overview of all of this.
  */
-internal sealed class AbstractTreapSet<@Treapable E, S : AbstractTreapSet<E, S>>(
-    left: S?,
-    right: S?
-) : TreapSet<E>, Treap<E, S>(left, right) {
+internal sealed class AbstractTreapSet<@Treapable E, S : AbstractTreapSet<E, S>> : TreapSet<E>, Treap<E, S>() {
     /**
         Derived classes override to create an apropriate node containing the given element.
      */
diff --git a/collect/src/main/kotlin/com/certora/collect/EmptyTreapMap.kt b/collect/src/main/kotlin/com/certora/collect/EmptyTreapMap.kt
index a99cc4a..d877184 100644
--- a/collect/src/main/kotlin/com/certora/collect/EmptyTreapMap.kt
+++ b/collect/src/main/kotlin/com/certora/collect/EmptyTreapMap.kt
@@ -66,7 +66,7 @@ internal class EmptyTreapMap<@Treapable K, V> private constructor() : TreapMap<K
         m.asSequence().map { MapEntry(it.key, null to it.value) }
 
     override val entries: ImmutableSet<Map.Entry<K, V>> get() = persistentSetOf<Map.Entry<K, V>>()
-    override val keys: ImmutableSet<K> get() = persistentSetOf<K>()
+    override val keys: TreapSet<K> get() = treapSetOf<K>()
     override val values: ImmutableCollection<V> get() = persistentSetOf<V>()
 
     @Suppress("Treapability", "UNCHECKED_CAST")
diff --git a/collect/src/main/kotlin/com/certora/collect/HashTreapMap.kt b/collect/src/main/kotlin/com/certora/collect/HashTreapMap.kt
index cea3982..37a9052 100644
--- a/collect/src/main/kotlin/com/certora/collect/HashTreapMap.kt
+++ b/collect/src/main/kotlin/com/certora/collect/HashTreapMap.kt
@@ -15,9 +15,9 @@ internal class HashTreapMap<@Treapable K, V>(
     override val key: K,
     override val value: V,
     override val next: KeyValuePairList.More<K, V>? = null,
-    left: HashTreapMap<K, V>? = null,
-    right: HashTreapMap<K, V>? = null
-) : AbstractTreapMap<K, V, HashTreapMap<K, V>>(left, right), TreapKey.Hashed<K>, KeyValuePairList<K, V> {
+    override val left: HashTreapMap<K, V>? = null,
+    override val right: HashTreapMap<K, V>? = null
+) : AbstractTreapMap<K, V, HashTreapMap<K, V>>(), TreapKey.Hashed<K>, KeyValuePairList<K, V> {
 
     override fun hashCode() = computeHashCode()
 
@@ -350,12 +350,30 @@ internal class HashTreapMap<@Treapable K, V>(
         forEachPair { (k, v) -> action(MapEntry(k, v)) }
         right?.forEachEntry(action)
     }
+
+    override val keys get() = KeySet(this)
+
+    class KeySet<@Treapable K>(
+        private val map: HashTreapMap<K, *>
+    ) : AbstractHashTreapSet<K>() {
+        override fun hashCode() = super.hashCode()
+
+        override val element get() = map.key
+        override val left get() = map.left?.keys
+        override val right get() = map.right?.keys
+        override val next get() = map.next?.let { More(it) }
+
+        class More<K>(val mapMore: KeyValuePairList.More<K, *>) : ElementList<K>, java.io.Serializable {
+            override val element get() = mapMore.key
+            override val next get() = mapMore.next?.let { More(it) }
+        }
+    }
 }
 
 internal interface KeyValuePairList<K, V> {
     abstract val key: K
     abstract val value: V
-    abstract val next: More<K, V>?
+    abstract val next: KeyValuePairList<K, V>?
     operator fun component1() = key
     operator fun component2() = value
 
diff --git a/collect/src/main/kotlin/com/certora/collect/HashTreapSet.kt b/collect/src/main/kotlin/com/certora/collect/HashTreapSet.kt
index 987e503..45a1c08 100644
--- a/collect/src/main/kotlin/com/certora/collect/HashTreapSet.kt
+++ b/collect/src/main/kotlin/com/certora/collect/HashTreapSet.kt
@@ -8,12 +8,8 @@ package com.certora.collect
     This is just a simple linked list, so operations on it are either O(N) or O(N^2), but collisions are assumed to be
     rare enough that these lists will be very small - usually just one element.
  */
-internal class HashTreapSet<@Treapable E>(
-    override val element: E,
-    override val next: ElementList.More<E>? = null,
-    left: HashTreapSet<E>? = null,
-    right: HashTreapSet<E>? = null
-) : AbstractTreapSet<E, HashTreapSet<E>>(left, right), TreapKey.Hashed<E>, ElementList<E> {
+internal abstract class AbstractHashTreapSet<@Treapable E>
+    : AbstractTreapSet<E, AbstractHashTreapSet<E>>(), TreapKey.Hashed<E>, ElementList<E> {
 
     override fun hashCode(): Int = computeHashCode()
 
@@ -45,7 +41,7 @@ internal class HashTreapSet<@Treapable E>(
         return count
     }
 
-    override fun copyWith(left: HashTreapSet<E>?, right: HashTreapSet<E>?): HashTreapSet<E> =
+    override fun copyWith(left: AbstractHashTreapSet<E>?, right: AbstractHashTreapSet<E>?) =
         HashTreapSet(element, next, left, right)
 
     fun withElement(element: E) = when {
@@ -53,7 +49,7 @@ internal class HashTreapSet<@Treapable E>(
         else -> HashTreapSet(this.element, ElementList.More(element, this.next), this.left, this.right)
     }
 
-    override fun shallowEquals(that: HashTreapSet<E>): Boolean {
+    override fun shallowEquals(that: AbstractHashTreapSet<E>): Boolean {
         forEachNodeElement {
             if (!that.shallowContains(it)) {
                 return false
@@ -84,7 +80,7 @@ internal class HashTreapSet<@Treapable E>(
         return false
     }
 
-    override fun shallowContainsAll(elements: HashTreapSet<E>): Boolean {
+    override fun shallowContainsAll(elements: AbstractHashTreapSet<E>): Boolean {
         elements.forEachNodeElement {
             if (!this.shallowContains(it)) {
                 return false
@@ -93,7 +89,7 @@ internal class HashTreapSet<@Treapable E>(
         return true
     }
 
-    override fun shallowContainsAny(elements: HashTreapSet<E>): Boolean {
+    override fun shallowContainsAny(elements: AbstractHashTreapSet<E>): Boolean {
         elements.forEachNodeElement {
             if (this.shallowContains(it)) {
                 return true
@@ -102,13 +98,13 @@ internal class HashTreapSet<@Treapable E>(
         return false
     }
 
-    override fun shallowAdd(that: HashTreapSet<E>): HashTreapSet<E> {
+    override fun shallowAdd(that: AbstractHashTreapSet<E>): AbstractHashTreapSet<E> {
         // add is only called with a single element
         check (that.next == null) { "add with multiple elements?" }
         return this.withElement(that.element)
     }
 
-    override fun shallowUnion(that: HashTreapSet<E>): HashTreapSet<E> {
+    override fun shallowUnion(that: AbstractHashTreapSet<E>): AbstractHashTreapSet<E> {
         var result = this
         that.forEachNodeElement {
             result = result.withElement(it)
@@ -116,7 +112,7 @@ internal class HashTreapSet<@Treapable E>(
         return result
     }
 
-    override fun shallowDifference(that: HashTreapSet<E>): HashTreapSet<E>? {
+    override fun shallowDifference(that: AbstractHashTreapSet<E>): AbstractHashTreapSet<E>? {
         // Fast path for the most common case
         if (this.next == null) {
             if (that.shallowContains(this.element)) {
@@ -126,7 +122,7 @@ internal class HashTreapSet<@Treapable E>(
             }
         }
 
-        var result: HashTreapSet<E>? = null
+        var result: AbstractHashTreapSet<E>? = null
         var changed = false
         this.forEachNodeElement {
             if (!that.shallowContains(it)) {
@@ -143,7 +139,7 @@ internal class HashTreapSet<@Treapable E>(
         }
     }
 
-    override fun shallowIntersect(that: HashTreapSet<E>): HashTreapSet<E>? {
+    override fun shallowIntersect(that: AbstractHashTreapSet<E>): AbstractHashTreapSet<E>? {
         // Fast path for the most common case
         if (this.next == null) {
             if (that.shallowContains(this.element)) {
@@ -153,7 +149,7 @@ internal class HashTreapSet<@Treapable E>(
             }
         }
 
-        var result: HashTreapSet<E>? = null
+        var result: AbstractHashTreapSet<E>? = null
         var changed = false
         this.forEachNodeElement {
             if (that.shallowContains(it)) {
@@ -170,7 +166,7 @@ internal class HashTreapSet<@Treapable E>(
         }
     }
 
-    override fun shallowRemove(element: E): HashTreapSet<E>? {
+    override fun shallowRemove(element: E): AbstractHashTreapSet<E>? {
         // Fast path for the most common case
         if (this.next == null) {
             if (this.element == element) {
@@ -180,7 +176,7 @@ internal class HashTreapSet<@Treapable E>(
             }
         }
 
-        var result: HashTreapSet<E>? = null
+        var result: AbstractHashTreapSet<E>? = null
         var changed = false
         this.forEachNodeElement {
             if (it != element) {
@@ -197,8 +193,8 @@ internal class HashTreapSet<@Treapable E>(
         }
     }
 
-    override fun shallowRemoveAll(predicate: (E) -> Boolean): HashTreapSet<E>? {
-        var result: HashTreapSet<E>? = null
+    override fun shallowRemoveAll(predicate: (E) -> Boolean): AbstractHashTreapSet<E>? {
+        var result: AbstractHashTreapSet<E>? = null
         var removed = false
         this.forEachNodeElement {
             if (predicate(it)) {
@@ -253,10 +249,17 @@ internal class HashTreapSet<@Treapable E>(
 
 internal interface ElementList<E> {
     val element: E
-    val next: More<E>?
+    val next: ElementList<E>?
 
     class More<E>(
         override val element: E,
-        override val next: More<E>?
+        override val next: ElementList<E>?
     ) : ElementList<E>, java.io.Serializable
 }
+
+internal class HashTreapSet<@Treapable E>(
+    override val element: E,
+    override val next: ElementList<E>? = null,
+    override val left: AbstractHashTreapSet<E>? = null,
+    override val right: AbstractHashTreapSet<E>? = null
+) : AbstractHashTreapSet<E>()
diff --git a/collect/src/main/kotlin/com/certora/collect/SortedTreapMap.kt b/collect/src/main/kotlin/com/certora/collect/SortedTreapMap.kt
index 0c3617a..73362fc 100644
--- a/collect/src/main/kotlin/com/certora/collect/SortedTreapMap.kt
+++ b/collect/src/main/kotlin/com/certora/collect/SortedTreapMap.kt
@@ -9,11 +9,9 @@ import kotlinx.collections.immutable.PersistentMap
 internal class SortedTreapMap<@Treapable K, V>(
     val key: K,
     val value: V,
-    left: SortedTreapMap<K, V>? = null,
-    right: SortedTreapMap<K, V>? = null
-) : AbstractTreapMap<K, V, SortedTreapMap<K, V>>(left, right), TreapKey.Sorted<K> {
-
-    init { check(key is Comparable<*>?) { "SortedTreapMap keys must be Comparable" } }
+    override val left: SortedTreapMap<K, V>? = null,
+    override val right: SortedTreapMap<K, V>? = null
+) : AbstractTreapMap<K, V, SortedTreapMap<K, V>>(), TreapKey.Sorted<K> {
 
     override fun hashCode() = computeHashCode()
 
@@ -163,4 +161,16 @@ internal class SortedTreapMap<@Treapable K, V>(
         action(this.asEntry())
         right?.forEachEntry(action)
     }
+
+    override val keys get() = KeySet(this)
+
+    class KeySet<@Treapable K>(
+        private val map: SortedTreapMap<K, *>
+    ) : AbstractSortedTreapSet<K>() {
+        override fun hashCode() = super.hashCode()
+
+        override val treapKey get() = map.key
+        override val left get() = map.left?.keys
+        override val right get() = map.right?.keys
+    }
 }
diff --git a/collect/src/main/kotlin/com/certora/collect/SortedTreapSet.kt b/collect/src/main/kotlin/com/certora/collect/SortedTreapSet.kt
index 5704f0a..518f049 100644
--- a/collect/src/main/kotlin/com/certora/collect/SortedTreapSet.kt
+++ b/collect/src/main/kotlin/com/certora/collect/SortedTreapSet.kt
@@ -6,13 +6,8 @@ import kotlinx.collections.immutable.PersistentSet
     A TreapSet specific to Comparable elements.  Iterates in the order defined by the objects.  We store one element per
     Treap node, with the element itself as the Treap key.
  */
-internal class SortedTreapSet<@Treapable E>(
-    override val treapKey: E,
-    left: SortedTreapSet<E>? = null,
-    right: SortedTreapSet<E>? = null
-) : AbstractTreapSet<E, SortedTreapSet<E>>(left, right), TreapKey.Sorted<E> {
-
-    init { check(treapKey is Comparable<*>?) { "SortedTreapSet elements must be Comparable" } }
+internal abstract class AbstractSortedTreapSet<@Treapable E>
+    : AbstractTreapSet<E, AbstractSortedTreapSet<E>>(), TreapKey.Sorted<E> {
 
     override fun hashCode(): Int = computeHashCode()
 
@@ -31,29 +26,36 @@ internal class SortedTreapSet<@Treapable E>(
     override val self get() = this
     override fun iterator(): Iterator<E> = this.asTreapSequence().map { it.treapKey }.iterator()
 
-    override fun shallowEquals(that: SortedTreapSet<E>): Boolean = this.compareKeyTo(that) == 0
+    override fun shallowEquals(that: AbstractSortedTreapSet<E>): Boolean = this.compareKeyTo(that) == 0
     override val shallowSize: Int get() = 1
 
-    override fun copyWith(left: SortedTreapSet<E>?, right: SortedTreapSet<E>?): SortedTreapSet<E> = SortedTreapSet(treapKey, left, right)
+    override fun copyWith(left: AbstractSortedTreapSet<E>?, right: AbstractSortedTreapSet<E>?) =
+        SortedTreapSet(treapKey, left, right)
 
     // Since these are only called for Treap nodes with the same key, and each of our nodes stores a single element,
     // these are trivial.
     override fun shallowContains(element: E) = true
-    override fun shallowContainsAll(elements: SortedTreapSet<E>) = true
-    override fun shallowContainsAny(elements: SortedTreapSet<E>) = true
+    override fun shallowContainsAll(elements: AbstractSortedTreapSet<E>) = true
+    override fun shallowContainsAny(elements: AbstractSortedTreapSet<E>) = true
     override fun shallowFindEqual(element: E) = treapKey.takeIf { it == element }
-    override fun shallowAdd(that: SortedTreapSet<E>) = this
-    override fun shallowUnion(that: SortedTreapSet<E>) = this
-    override fun shallowDifference(that: SortedTreapSet<E>) = null
-    override fun shallowIntersect(that: SortedTreapSet<E>) = this
-    override fun shallowRemove(element: E): SortedTreapSet<E>? = null
-    override fun shallowRemoveAll(predicate: (E) -> Boolean): SortedTreapSet<E>? = this.takeIf { !predicate(treapKey) }
-    override fun shallowComputeHashCode(): Int = treapKey.hashCode()
-    override fun shallowGetSingleElement(): E = treapKey
-    override fun arbitraryOrNull(): E? = treapKey
-    override fun shallowForEach(action: (element: E) -> Unit): Unit { action(treapKey) }
-    override fun <R : Any> shallowMapReduce(map: (E) -> R, reduce: (R, R) -> R): R = map(treapKey)
+    override fun shallowAdd(that: AbstractSortedTreapSet<E>) = this
+    override fun shallowUnion(that: AbstractSortedTreapSet<E>) = this
+    override fun shallowDifference(that: AbstractSortedTreapSet<E>) = null
+    override fun shallowIntersect(that: AbstractSortedTreapSet<E>) = this
+    override fun shallowRemove(element: E) = null
+    override fun shallowRemoveAll(predicate: (E) -> Boolean) = this.takeIf { !predicate(treapKey) }
+    override fun shallowComputeHashCode() = treapKey.hashCode()
+    override fun shallowGetSingleElement() = treapKey
+    override fun arbitraryOrNull() = treapKey
+    override fun shallowForEach(action: (element: E) -> Unit) { action(treapKey) }
+    override fun <R : Any> shallowMapReduce(map: (E) -> R, reduce: (R, R) -> R) = map(treapKey)
 
     override fun containsAny(predicate: (E) -> Boolean): Boolean =
         predicate(treapKey) || left?.containsAny(predicate) == true || right?.containsAny(predicate) == true
 }
+
+internal class SortedTreapSet<@Treapable E>(
+    override val treapKey: E,
+    override val left: AbstractSortedTreapSet<E>? = null,
+    override val right: AbstractSortedTreapSet<E>? = null
+) : AbstractSortedTreapSet<E>()
diff --git a/collect/src/main/kotlin/com/certora/collect/Treap.kt b/collect/src/main/kotlin/com/certora/collect/Treap.kt
index 7a456ea..f10b171 100644
--- a/collect/src/main/kotlin/com/certora/collect/Treap.kt
+++ b/collect/src/main/kotlin/com/certora/collect/Treap.kt
@@ -65,11 +65,9 @@ package com.certora.collect
 
     - Treaps impose special requirements on keys if they are serialized.  See [Treapable].
 */
-internal abstract class Treap<@Treapable T, S : Treap<T, S>>(
-    @JvmField val left: S?,
-    @JvmField val right: S?
-) : TreapKey<T>, java.io.Serializable {
-
+internal abstract class Treap<@Treapable T, S : Treap<T, S>> : TreapKey<T>, java.io.Serializable {
+    abstract val left: S?
+    abstract val right: S?
     abstract val self: S
 
     /**
@@ -135,11 +133,11 @@ internal abstract class Treap<@Treapable T, S : Treap<T, S>>(
         case any hash functions have changed since this Treap was serialized.
      */
     protected fun readResolve(): Any? {
-        if (left != null) {
+        left?.let { left ->
             check(left.compareKeyTo(this) < 0) { "Treap key comparison logic changed: ${left.treapKey} >= ${this.treapKey}" }
             check(left.comparePriorityTo(this) < 0) { "Treap key priority hash logic changed: ${left.treapKey} >= ${this.treapKey} "}
         }
-        if (right != null) {
+        right?.let { right -> 
             check(right.compareKeyTo(this) > 0) { "Treap key comparison logic changed: ${right.treapKey} <= ${this.treapKey}" }
             check(right.comparePriorityTo(this) < 0) { "Treap key priority hash logic changed: ${right.treapKey} >= ${this.treapKey} "}
         }
diff --git a/collect/src/main/kotlin/com/certora/collect/TreapMap.kt b/collect/src/main/kotlin/com/certora/collect/TreapMap.kt
index 7931468..d7b0d5f 100644
--- a/collect/src/main/kotlin/com/certora/collect/TreapMap.kt
+++ b/collect/src/main/kotlin/com/certora/collect/TreapMap.kt
@@ -13,6 +13,8 @@ public sealed interface TreapMap<K, V> : PersistentMap<K, V> {
     override fun putAll(m: Map<out K, @UnsafeVariance V>): TreapMap<K, V>
     override fun clear(): TreapMap<K, V>
 
+    override val keys: TreapSet<K>
+
     /**
         A [PersistentMap.Builder] that produces a [TreapMap].
     */