The Coq Standard Library

Require Import Le.
Require Import Lt.
Require Import Gt.
Require Import Decidable.

Local Open Scope nat_scope.

Implicit Types m n x y : nat.

Definition zerop n : {n = 0} + {0 < n }.

Definition lt_eq_lt_dec n m : {n < m } + {n = m } + {m < n }.

Definition gt_eq_gt_dec n m : {m > n } + {n = m } + {n > m }.

Definition le_lt_dec n m : {n <= m } + {m < n }.

Definition le_le_S_dec n m : {n <= m } + {S m <= n }.

Definition le_ge_dec n m : {n <= m } + {n >= m }.

Definition le_gt_dec n m : {n <= m } + {n > m }.

Definition le_lt_eq_dec n m : n <= m -> {n < m } + {n = m }.

Theorem le_dec : forall n m, {n <= m } + {~ n <= m }.

Theorem lt_dec : forall n m, {n < m } + {~ n < m }.

Theorem gt_dec : forall n m, {n > m } + {~ n > m }.

Theorem ge_dec : forall n m, {n >= m } + {~ n >= m }.

Theorem dec_le : forall n m, decidable (n <= m).

Theorem dec_lt : forall n m, decidable (n < m).

Theorem dec_gt : forall n m, decidable (n > m).

Theorem dec_ge : forall n m, decidable (n >= m).

Theorem not_eq : forall n m, n <> m -> n < m \/ m < n.

Theorem not_le : forall n m, ~ n <= m -> n > m.

Theorem not_gt : forall n m, ~ n > m -> n <= m.

Theorem not_ge : forall n m, ~ n >= m -> n < m.

Theorem not_lt : forall n m, ~ n < m -> n >= m.

Fixpoint nat_compare n m :=
  match n, m with
   | O, O => Eq
   | O, S _ => Lt
   | S _, O => Gt
   | S n´, S m´ => nat_compare n´ m´
  end.

Lemma nat_compare_S : forall n m, nat_compare (S n) (S m) = nat_compare n m.

Lemma nat_compare_eq_iff : forall n m, nat_compare n m = Eq <-> n = m.

Lemma nat_compare_eq : forall n m, nat_compare n m = Eq -> n = m.

Lemma nat_compare_lt : forall n m, n <m <-> nat_compare n m = Lt.

Lemma nat_compare_gt : forall n m, n >m <-> nat_compare n m = Gt.

Lemma nat_compare_le : forall n m, n <=m <-> nat_compare n m <> Gt.

Lemma nat_compare_ge : forall n m, n >=m <-> nat_compare n m <> Lt.

Lemma nat_compare_spec :
  forall x y, CompareSpec (x =y) (x <y) (y <x) (nat_compare x y).

Lemma nat_compare_Lt_lt : forall n m, nat_compare n m = Lt -> n <m.

Lemma nat_compare_Gt_gt : forall n m, nat_compare n m = Gt -> n >m.

Definition nat_compare_alt (n m:nat) :=
  match lt_eq_lt_dec n m with
    | inleft (left _) => Lt
    | inleft (right _) => Eq
    | inright _ => Gt
  end.

Lemma nat_compare_equiv: forall n m,
nat_compare n m = nat_compare_alt n m.

Fixpoint leb (m:nat) : nat -> bool :=
  match m with
    | O => fun _:nat => true
    | S m´ =>
      fun n:nat => match n with
                     | O => false
                     | S n´ => leb m´ n´
                   end
  end.

Lemma leb_correct : forall m n, m <= n -> leb m n = true.

Lemma leb_complete : forall m n, leb m n = true -> m <= n.

Lemma leb_iff : forall m n, leb m n = true <-> m <= n.

Lemma leb_correct_conv : forall m n, m < n -> leb n m = false.

Lemma leb_complete_conv : forall m n, leb n m = false -> m < n.

Lemma leb_iff_conv : forall m n, leb n m = false <-> m < n.

Lemma leb_compare : forall n m, leb n m = true <-> nat_compare n m <> Gt.

Library Coq.Arith.Compare_dec