+
+let rec colength (cs:trace) on cs ≝ match cs with
+[ nil ⇒ 0
+| cons b tl ⇒ match b with [ true ⇒ ⫯(colength tl) | false ⇒ colength tl ]
+].
+
+interpretation "colength (trace)"
+ 'Norm cs = (colength cs).
+
+(* basic properties *********************************************************)
+
+lemma colength_empty: ∥◊∥ = 0.
+// qed.
+
+lemma colength_true: ∀cs. ∥Ⓣ@cs∥ = ⫯∥cs∥.
+// qed.
+
+lemma colength_false: ∀cs. ∥Ⓕ@cs∥ = ∥cs∥.
+// qed.
+
+lemma colength_cons: ∀cs1,cs2. ∥cs1∥ = ∥cs2∥ →
+ ∀b. ∥b@cs1∥ = ∥b@cs2∥.
+#cs1 #cs2 #H * /2 width=1 by/
+qed.
+
+lemma colength_le: ∀cs. ∥cs∥ ≤ |cs|.
+#cs elim cs -cs //
+* /2 width=1 by le_S_S, le_S/
+qed.