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)).

(* Specifications: *)
Lemma le_trans : forall m n o, m <= n -> n <= o -> m <= o.
Theorem O_le_n : forall n, 0 <= n.
Theorem n_le_m__Sn_le_Sm : forall n m, n <= m -> S n <= S m.
Theorem Sn_le_Sm__n_le_m : forall n m, S n <= S m -> n <= m.
Theorem le_plus_l : forall a b, a <= a + b.
Theorem plus_lt : forall n1 n2 m, n1 + n2 < m -> n1 < m /\ n2 < m.
Theorem lt_S : forall n m, n < m -> n < S m.
Theorem leb_complete : forall n m, leb n m = true -> n <= m.
Theorem leb_correct : forall n m, n <= m -> leb n m = true.
Theorem leb_true_trans : forall n m o, leb n m = true -> leb m o = true -> leb n o = true.
Theorem leb_iff : forall n m, leb n m = true <-> n <= m.