Definition lt (n m:nat) := le (S n) m.(** Here are a few more simple relations on numbers: *)

Inductive square_of : nat -> nat -> Prop :=
  | sq : forall n:nat, square_of n (n * n).(** Here are a few more simple relations on numbers: *)

Inductive square_of : nat -> nat -> Prop :=
  | sq : forall n:nat, square_of n (n * n).Inductive next_nat : nat -> nat -> Prop :=
  | nn : forall n:nat, next_nat n (S n).Inductive next_nat : nat -> nat -> Prop :=
  | nn : forall n:nat, next_nat n (S n).Inductive next_even : nat -> nat -> Prop :=
  | ne_1 : forall n, ev (S n) -> next_even n (S n)
  | ne_2 : forall n, ev (S (S n)) -> next_even n (S (S n)).Inductive next_even : nat -> nat -> Prop :=
  | ne_1 : forall n, ev (S n) -> next_even n (S n)
  | ne_2 : forall n, ev (S (S n)) -> next_even n (S (S n)).Inductive next_even : nat -> nat -> Prop :=
  | ne_1 : forall n, ev (S n) -> next_even n (S n)
  | ne_2 : forall n, ev (S (S n)) -> next_even n (S (S n)).

Theorem O_le_n : forall n, 0 <= n.