-[nat]
+[\lambda x:nat.
+ [\lambda y:nat. Set]
+ match x:nat with [ O \Rightarrow nat | (S x) \Rightarrow bool ]]
+match (S O):nat with
+[ O \Rightarrow O
+| (S x) \Rightarrow false ]
+
+[\lambda z:nat. \lambda h:(le O z). (eq nat O O)]
+match (le_n O): le with
+[ le_n \Rightarrow (refl_equal nat O)
+| (le_S x y) \Rightarrow (refl_equal nat O) ]
+
+[\lambda z:nat. \lambda h:(le (plus (plus O O) (plus O O)) z). (eq nat (plus (plus O O) (plus O O)) (plus (plus O O) (plus O O)))]
+match (le_n (plus (plus O O) (plus O O))): le with
+[ le_n \Rightarrow (refl_equal nat (plus (plus O O) (plus O O)))
+| (le_S x y) \Rightarrow (refl_equal nat (plus (plus O O) (plus O O))) ]
+
+(*
+[\lambda z:nat. \lambda h:(le 1 z). (le 0 z)]
+match (le_S 2 (le_n 1)): le with
+[ le_n \Rightarrow (le_S 1 (le_n 0))
+| (le_S x y) \Rightarrow y ]
+*)
+
+[\lambda z:nat. \lambda h:(le 0 z). (le 0 (S z))]
+match (le_S 0 0 (le_n 0)): le with
+[ le_n \Rightarrow (le_S 0 0 (le_n 0))
+| (le_S x y) \Rightarrow (le_S 0 (S x) (le_S 0 x y)) ]
+
+
+[\lambda x:bool. nat]
match true:bool with
[ true \Rightarrow O
-| false \Rightarrow (S O)
-]
+| false \Rightarrow (S O) ]
-[nat]
+[\lambda x:nat. nat]
match O:nat with
[ O \Rightarrow O
-| (S x) \Rightarrow (S (S x))
-]
+| (S x) \Rightarrow (S (S x)) ]
-[nat]
+[\lambda x:list. list]
match nil:list with
[ nil \Rightarrow nil
-| (cons x y) \Rightarrow (cons x y)
-]
+| (cons x y) \Rightarrow (cons x y) ]
+
+\lambda x:False.
+ [\lambda h:False. True]
+ match x:False with []