diff --git a/sledge/lib/equality.ml b/sledge/lib/equality.ml
index fa3d58546..b4918b276 100644
--- a/sledge/lib/equality.ml
+++ b/sledge/lib/equality.ml
@@ -89,10 +89,20 @@ end = struct
 
   (** compose two substitutions *)
   let compose r s =
-    let r' = Term.Map.map ~f:(norm s) r in
-    Term.Map.merge_skewed r' s ~combine:(fun ~key v1 v2 ->
-        if Term.equal v1 v2 then v1
-        else fail "domains intersect: %a" Term.pp key () )
+    [%Trace.call fun {pf} -> pf "%a@ %a" pp r pp s]
+    ;
+    let r' = Term.Map.map_endo ~f:(norm s) r in
+    Term.Map.merge_endo r' s ~f:(fun ~key -> function
+      | `Both (data_r, data_s) ->
+          assert (
+            Term.equal data_s data_r
+            || fail "domains intersect: %a" Term.pp key () ) ;
+          Some data_r
+      | `Left data | `Right data -> Some data )
+    |>
+    [%Trace.retn fun {pf} r' ->
+      pf "%a" pp_diff (r, r') ;
+      assert (r' != r ==> not (equal r' r))]
 
   (** compose a substitution with a mapping *)
   let compose1 ~key ~data s =
diff --git a/sledge/lib/import/map.ml b/sledge/lib/import/map.ml
index 65117e08c..bead4fd89 100644
--- a/sledge/lib/import/map.ml
+++ b/sledge/lib/import/map.ml
@@ -30,6 +30,19 @@ end) : S with type key = Key.t = struct
 
   let map_endo t ~f = map_endo map t ~f
 
+  let merge_endo t u ~f =
+    let change = ref false in
+    let t' =
+      merge t u ~f:(fun ~key side ->
+          let f_side = f ~key side in
+          ( match (side, f_side) with
+          | (`Both (data, _) | `Left data), Some data' when data' == data ->
+              ()
+          | _ -> change := true ) ;
+          f_side )
+    in
+    if !change then t' else t
+
   let fold_until m ~init ~f ~finish =
     let fold m ~init ~f =
       let f ~key ~data s = f s (key, data) in
diff --git a/sledge/lib/import/map_intf.ml b/sledge/lib/import/map_intf.ml
index e2166afe7..747de3d8d 100644
--- a/sledge/lib/import/map_intf.ml
+++ b/sledge/lib/import/map_intf.ml
@@ -24,6 +24,18 @@ module type S = sig
   (** Like map, but specialized to require [f] to be an endofunction, which
       enables preserving [==] if [f] preserves [==] of every element. *)
 
+  val merge_endo :
+       'a t
+    -> 'b t
+    -> f:
+         (   key:key
+          -> [`Both of 'a * 'b | `Left of 'a | `Right of 'b]
+          -> 'a option)
+    -> 'a t
+  (** Like merge, but specialized to require [f] to preserve the type of the
+      left argument, which enables preserving [==] if [f] preserves [==] of
+      every element. *)
+
   val merge_skewed :
     'a t -> 'a t -> combine:(key:key -> 'a -> 'a -> 'a) -> 'a t