diff --git a/src/egraph.rs b/src/egraph.rs
index b8688153..4fad79de 100644
--- a/src/egraph.rs
+++ b/src/egraph.rs
@@ -231,8 +231,8 @@ impl<L: Language, N: Analysis<L>> EGraph<L, N> {
         let right_unions = other.get_union_equalities();
         for (left, right, why) in right_unions {
             self.union_instantiations(
-                &other.id_to_pattern(left, &Default::default()).0.ast,
-                &other.id_to_pattern(right, &Default::default()).0.ast,
+                &other.id_to_pattern(left, |_| None).0.ast,
+                &other.id_to_pattern(right, |_| None).0.ast,
                 &Default::default(),
                 why,
             );
@@ -376,11 +376,15 @@ impl<L: Language, N: Analysis<L>> EGraph<L, N> {
     /// When an eclass listed in the given substitutions is found, it creates a variable.
     /// It also adds this variable and the corresponding Id value to the resulting [`Subst`]
     /// Otherwise it behaves like [`id_to_expr`](EGraph::id_to_expr).
-    pub fn id_to_pattern(&self, id: Id, substitutions: &HashMap<Id, Id>) -> (Pattern<L>, Subst) {
+    pub fn id_to_pattern(
+        &self,
+        id: Id,
+        mut substitutions: impl FnMut(Id) -> Option<Id>,
+    ) -> (Pattern<L>, Subst) {
         let mut res = Default::default();
         let mut subst = Default::default();
         let mut cache = Default::default();
-        self.id_to_pattern_internal(&mut res, id, substitutions, &mut subst, &mut cache);
+        self.id_to_pattern_internal(&mut res, id, &mut substitutions, &mut subst, &mut cache);
         (Pattern::new(res), subst)
     }
 
@@ -388,16 +392,16 @@ impl<L: Language, N: Analysis<L>> EGraph<L, N> {
         &self,
         res: &mut PatternAst<L>,
         node_id: Id,
-        var_substitutions: &HashMap<Id, Id>,
+        var_substitutions: &mut impl FnMut(Id) -> Option<Id>,
         subst: &mut Subst,
         cache: &mut HashMap<Id, Id>,
     ) -> Id {
         if let Some(existing) = cache.get(&node_id) {
             return *existing;
         }
-        let res_id = if let Some(existing) = var_substitutions.get(&node_id) {
+        let res_id = if let Some(existing) = var_substitutions(node_id) {
             let var = format!("?{}", node_id).parse().unwrap();
-            subst.insert(var, *existing);
+            subst.insert(var, existing);
             res.add(ENodeOrVar::Var(var))
         } else {
             let new_node = self.id_to_node(node_id).clone().map_children(|child| {