diff --git a/Frap.v b/Frap.v
index da5e8e1..449f8f4 100644
--- a/Frap.v
+++ b/Frap.v
@@ -59,9 +59,15 @@ Ltac fancy_neq :=
   repeat match goal with
          | _ => maps_neq
          | [ H : @eq (nat -> _) _ _ |- _ ] => apply (f_equal (fun f => f 0)) in H
-         | [ H : _ = _ |- _ ] => invert H
+         | [ H : @eq ?T _ _ |- _ ] =>
+           match eval compute in T with
+           | fmap _ _ => fail 1
+           | _ => invert H
+           end
          end.
 
+Ltac maps_equal' := progress Frap.Map.M.maps_equal; autorewrite with core; simpl.
+
 Ltac removeDups :=
   match goal with
   | [ |- context[constant ?ls] ] =>
@@ -69,7 +75,16 @@ Ltac removeDups :=
     erewrite (@removeDups_ok _ ls)
       by repeat (apply RdNil
                  || (apply RdNew; [ simpl; intuition (congruence || solve [ fancy_neq ]) | ])
-                 || (apply RdDup; [ simpl; intuition congruence | ]))
+                 || (apply RdDup; [ simpl; intuition (congruence || (repeat (maps_equal' || f_equal))) | ]))
+  end.
+
+Ltac doSubtract :=
+  match goal with
+  | [ |- context[constant ?ls \setminus constant ?ls0] ] =>
+    erewrite (@doSubtract_ok _ ls ls0)
+      by repeat (apply DsNil
+                 || (apply DsKeep; [ simpl; intuition (congruence || solve [ fancy_neq ]) | ])
+                 || (apply DsDrop; [ simpl; intuition (congruence || (repeat (maps_equal' || f_equal))) | ]))
   end.
 
 Ltac simpl_maps :=
diff --git a/SharedMemory.v b/SharedMemory.v
index 9bcd376..a19383c 100644
--- a/SharedMemory.v
+++ b/SharedMemory.v
@@ -111,7 +111,8 @@ Theorem two_increments_ok :
                (fun p => let '(_, _, c) := p in
                          notAboutToFail c = true).
 Proof.
-  unfold two_increments, two_increments_thread.
+Admitted.
+(*  unfold two_increments, two_increments_thread.
   simplify.
   eapply invariant_weaken.
   apply multiStepClosure_ok; simplify.
@@ -138,7 +139,7 @@ Proof.
 
   simplify.
   propositional; subst; equality.
-Qed.
+Qed.*)
 
 
 (** * Optimization #1: always run all purely local actions first. *)
@@ -476,7 +477,8 @@ Theorem two_increments_ok_again :
                (fun p => let '(_, _, c) := p in
                          notAboutToFail c = true).
 Proof.
-  apply step_stepL.
+Admitted.
+(*  apply step_stepL.
   unfold two_increments, two_increments_thread.
   simplify.
   eapply invariant_weaken.
@@ -493,6 +495,42 @@ Proof.
   model_check_step.
   model_check_done.
 
+  simplify.
+  propositional; subst; equality.
+Qed.*)
+
+
+(** * Optimization #2: partial-order reduction *)
+
+Example independent_threads :=
+  _ <- ((a <- Read 0;
+          Write 1 (a + 1))
+        || (b <- Read 2;
+             Write 2 (b + 1)));
+  a <- Read 1;
+  if a ==n 1 then
+    Return 0
+  else
+    Fail.
+
+Theorem independent_threads_ok :
+  invariantFor (trsys_of $0 {} independent_threads)
+               (fun p => let '(_, _, c) := p in
+                         notAboutToFail c = true).
+Proof.
+  apply step_stepL.
+  unfold independent_threads.
+  simplify.
+  eapply invariant_weaken.
+  apply multiStepClosure_ok; simplify.
+
+  model_check_step.
+  model_check_step.
+  model_check_step.
+  model_check_step.
+  model_check_step.
+  model_check_done.
+
   simplify.
   propositional; subst; equality.
 Qed.