open nat

definition foo : nat → nat,
foo (0 + x) := x

definition foo : nat → nat → nat,
foo 0 _   := 0,
foo x ⌞y⌟ := x

definition foo : nat → nat → nat,
foo 0   _ := 0,
foo ⌞x⌟ x := x

inductive tree (A : Type) :=
node   : tree_list A → tree A,
leaf   : A → tree A
with tree_list :=
nil {} : tree_list A,
cons   : tree A → tree_list A → tree_list A

definition is_leaf {A : Type} : tree A → bool
with is_leaf_aux : tree_list A → bool,
is_leaf (tree.node _)            := bool.ff,
is_leaf (tree.leaf _)            := bool.tt,
is_leaf_aux tree_list.nil        := bool.ff,
is_leaf_aux (tree_list.cons _ _) := bool.ff

definition foo : nat → nat,
foo 0     := 0,
foo (x+1) := let y := x + 2 in y * y

example : foo 5 = 36 := rfl

definition boo : nat → nat,
boo (x + 1) := boo (x + 2),
boo 0       := 0