diff --git a/coq/equiv.v b/coq/equiv.v
index ff77ed6..9ad4a58 100644
--- a/coq/equiv.v
+++ b/coq/equiv.v
@@ -61,7 +61,12 @@ Admitted.
 
 Reserved Notation "S '-->distribute-ladder' T" (at level 40).
 Inductive type_distribute_ladder : type_term -> type_term -> Prop :=
-  | L_DistributeOverApp : forall x y y',
+  | L_DistributeOverSpec1 : forall x x' y,
+    (type_spec (type_ladder x x') y)
+    -->distribute-ladder
+    (type_ladder (type_spec x y) (type_spec x' y))
+
+  | L_DistributeOverSpec2 : forall x y y',
     (type_spec x (type_ladder y y'))
     -->distribute-ladder
     (type_ladder (type_spec x y) (type_spec x y'))
@@ -76,13 +81,28 @@ Inductive type_distribute_ladder : type_term -> type_term -> Prop :=
     -->distribute-ladder
     (type_ladder (type_fun x y) (type_fun x y'))
 
+  | L_DistributeOverMorph1 : forall x x' y,
+    (type_morph (type_ladder x x') y)
+    -->distribute-ladder
+    (type_ladder (type_morph x y) (type_morph x' y))
+
+  | L_DistributeOverMorph2 : forall x y y',
+    (type_morph x (type_ladder y y'))
+    -->distribute-ladder
+    (type_ladder (type_morph x y) (type_morph x y'))
+
 where "S '-->distribute-ladder' T" := (type_distribute_ladder S T).
 
 
 
 Reserved Notation "S '-->condense-ladder' T" (at level 40).
 Inductive type_condense_ladder : type_term -> type_term -> Prop :=
-  | L_CondenseOverApp : forall x y y',
+  | L_CondenseOverSpec1 : forall x x' y,
+    (type_ladder (type_spec x y) (type_spec x' y))
+    -->condense-ladder
+    (type_spec (type_ladder x x') y)
+
+  | L_CondenseOverSpec2 : forall x y y',
     (type_ladder (type_spec x y) (type_spec x y'))
     -->condense-ladder
     (type_spec x (type_ladder y y'))
@@ -97,6 +117,16 @@ Inductive type_condense_ladder : type_term -> type_term -> Prop :=
     -->condense-ladder
     (type_fun x (type_ladder y y'))
 
+  | L_CondenseOverMorph1 : forall x x' y,
+    (type_ladder (type_morph x y) (type_morph x' y))
+    -->condense-ladder
+    (type_morph (type_ladder x x') y)
+
+  | L_CondenseOverMorph2 : forall x y y',
+    (type_ladder (type_morph x y) (type_morph x y'))
+    -->condense-ladder
+    (type_morph x (type_ladder y y'))
+
 where "S '-->condense-ladder' T" := (type_condense_ladder S T).
 
 
@@ -110,9 +140,12 @@ Lemma distribute_inverse :
 Proof.
   intros.
   destruct H.
-  apply L_CondenseOverApp.
+  apply L_CondenseOverSpec1.
+  apply L_CondenseOverSpec2.
   apply L_CondenseOverFun1.
   apply L_CondenseOverFun2.
+  apply L_CondenseOverMorph1.
+  apply L_CondenseOverMorph2.
 Qed.
 
 (** Inversion Lemma:
@@ -125,9 +158,12 @@ Lemma condense_inverse :
 Proof.
   intros.
   destruct H.
-  apply L_DistributeOverApp.
+  apply L_DistributeOverSpec1.
+  apply L_DistributeOverSpec2.
   apply L_DistributeOverFun1.
   apply L_DistributeOverFun2.
+  apply L_DistributeOverMorph1.
+  apply L_DistributeOverMorph2.
 Qed.
 
 
@@ -331,7 +367,7 @@ Example example_type_eq :
                (type_spec (type_id "Seq") (type_id "Byte"))).
 Proof.
   apply L_Distribute.
-  apply L_DistributeOverApp.
+  apply L_DistributeOverSpec2.
 Qed.