diff --git a/src/mapping.v b/src/mapping.v
index b8c18e1..b62d28d 100644
--- a/src/mapping.v
+++ b/src/mapping.v
@@ -83,6 +83,25 @@ Proof.
 Qed.
 
 
+Lemma Permutation_mapping_cons3 {X: Type} :
+  forall (u v base: list X) a,
+     Permutation_mapping u v base
+    -> Permutation_mapping (a::u) (a::v) (a::base).
+Proof.
+  intros u v base a. intros H.
+  destruct H as [p]. destruct H as [l].
+  destruct H. destruct H0. exists (a::p). exists (0::(map S l)).
+  split. apply perm_skip. assumption.
+
+  assert (forall x (y: list X),
+    map (nth_error (a :: y)) (map S x) = map (nth_error y) x).
+  intro x. induction x; intro y. reflexivity. simpl. rewrite IHx.
+  reflexivity.
+
+  split; simpl; rewrite H2; [ rewrite H0 | rewrite H1]; reflexivity.
+Qed.
+
+
 Lemma Permutation_mapping_app {X: Type} :
   forall (u1 u2 v1 v2 base : list X),
     Permutation_mapping (u1 ++ u2) (v1 ++ v2) base
@@ -301,6 +320,22 @@ Proof.
 Qed.
 
 
+Lemma Permutation_mapping_subset {X: Type} :
+  forall (u v base u': list X),
+     Permutation_mapping u v base
+    -> incl u' u -> (exists v', incl v' v /\ Permutation_mapping u' v' base).
+Proof.
+  intros u v base u'. intros H I. destruct H as [p]. destruct H as [l].
+  destruct H. destruct H0.
+  assert (exists l', map Some u' = map (nth_error base) l').
+Admitted.
+
+
+
+
+
+
+
 
 Lemma Permutation_mapping_base {X: Type} :
   forall (u v base p: list X),