Library Coq.Arith.Compare_dec
Require Import Le.
Require Import Lt.
Require Import Gt.
Require Import Decidable.
Open Local Scope nat_scope.
Implicit Types m n x y : nat.
Definition zerop n : {n = 0} + {0 < n}.
destruct n; auto with arith.
Defined.
Definition lt_eq_lt_dec n m : {n < m} + {n = m} + {m < n}.
induction n; simple destruct m; auto with arith.
intros m0; elim (IHn m0); auto with arith.
induction 1; auto with arith.
Defined.
Definition gt_eq_gt_dec n m : {m > n} + {n = m} + {n > m}.
exact lt_eq_lt_dec.
Defined.
Definition le_lt_dec n m : {n <= m} + {m < n}.
induction n.
auto with arith.
destruct m.
auto with arith.
elim (IHn m); auto with arith.
Defined.
Definition le_le_S_dec n m : {n <= m} + {S m <= n}.
exact le_lt_dec.
Defined.
Definition le_ge_dec n m : {n <= m} + {n >= m}.
intros; elim (le_lt_dec n m); auto with arith.
Defined.
Definition le_gt_dec n m : {n <= m} + {n > m}.
exact le_lt_dec.
Defined.
Definition le_lt_eq_dec n m : n <= m -> {n < m} + {n = m}.
intros; elim (lt_eq_lt_dec n m); auto with arith.
intros; absurd (m < n); auto with arith.
Defined.
Proofs of decidability
Theorem dec_le : forall n m, decidable (n <= m).
Proof.
intros x y; unfold decidable in |- *; elim (le_gt_dec x y);
[ auto with arith | intro; right; apply gt_not_le; assumption ].
Qed.
Theorem dec_lt : forall n m, decidable (n < m).
Proof.
intros x y; unfold lt in |- *; apply dec_le.
Qed.
Theorem dec_gt : forall n m, decidable (n > m).
Proof.
intros x y; unfold gt in |- *; apply dec_lt.
Qed.
Theorem dec_ge : forall n m, decidable (n >= m).
Proof.
intros x y; unfold ge in |- *; apply dec_le.
Qed.
Theorem not_eq : forall n m, n <> m -> n < m \/ m < n.
Proof.
intros x y H; elim (lt_eq_lt_dec x y);
[ intros H1; elim H1;
[ auto with arith | intros H2; absurd (x = y); assumption ]
| auto with arith ].
Qed.
Theorem not_le : forall n m, ~ n <= m -> n > m.
Proof.
intros x y H; elim (le_gt_dec x y);
[ intros H1; absurd (x <= y); assumption | trivial with arith ].
Qed.
Theorem not_gt : forall n m, ~ n > m -> n <= m.
Proof.
intros x y H; elim (le_gt_dec x y);
[ trivial with arith | intros H1; absurd (x > y); assumption ].
Qed.
Theorem not_ge : forall n m, ~ n >= m -> n < m.
Proof.
intros x y H; exact (not_le y x H).
Qed.
Theorem not_lt : forall n m, ~ n < m -> n >= m.
Proof.
intros x y H; exact (not_gt y x H).
Qed.
A ternary comparison function in the spirit of
Zcompare.
Definition nat_compare (n m:nat) :=
match lt_eq_lt_dec n m with
| inleft (left _) => Lt
| inleft (right _) => Eq
| inright _ => Gt
end.
Lemma nat_compare_S : forall n m, nat_compare (S n) (S m) = nat_compare n m.
Proof.
unfold nat_compare; intros.
simpl; destruct (lt_eq_lt_dec n m) as [[H|H]|H]; simpl; auto.
Qed.
Lemma nat_compare_eq : forall n m, nat_compare n m = Eq -> n = m.
Proof.
induction n; destruct m; simpl; auto.
unfold nat_compare; destruct (lt_eq_lt_dec 0 (S m)) as [[H|H]|H];
auto; intros; try discriminate.
unfold nat_compare; destruct (lt_eq_lt_dec (S n) 0) as [[H|H]|H];
auto; intros; try discriminate.
rewrite nat_compare_S; auto.
Qed.
Lemma nat_compare_lt : forall n m, n<m <-> nat_compare n m = Lt.
Proof.
induction n; destruct m; simpl.
unfold nat_compare; simpl; intuition; [inversion H | discriminate H].
split; auto with arith.
split; [inversion 1 |].
unfold nat_compare; destruct (lt_eq_lt_dec (S n) 0) as [[H|H]|H];
auto; intros; try discriminate.
rewrite nat_compare_S.
generalize (IHn m); clear IHn; intuition.
Qed.
Lemma nat_compare_gt : forall n m, n>m <-> nat_compare n m = Gt.
Proof.
induction n; destruct m; simpl.
unfold nat_compare; simpl; intuition; [inversion H | discriminate H].
split; [inversion 1 |].
unfold nat_compare; destruct (lt_eq_lt_dec 0 (S m)) as [[H|H]|H];
auto; intros; try discriminate.
split; auto with arith.
rewrite nat_compare_S.
generalize (IHn m); clear IHn; intuition.
Qed.
Lemma nat_compare_le : forall n m, n<=m <-> nat_compare n m <> Gt.
Proof.
split.
intros.
intro.
destruct (nat_compare_gt n m).
generalize (le_lt_trans _ _ _ H (H2 H0)).
exact (lt_irrefl n).
intros.
apply not_gt.
swap H.
destruct (nat_compare_gt n m); auto.
Qed.
Lemma nat_compare_ge : forall n m, n>=m <-> nat_compare n m <> Lt.
Proof.
split.
intros.
intro.
destruct (nat_compare_lt n m).
generalize (le_lt_trans _ _ _ H (H2 H0)).
exact (lt_irrefl m).
intros.
apply not_lt.
swap H.
destruct (nat_compare_lt n m); auto.
Qed.
A boolean version of
le over nat.
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:nat, m <= n -> leb m n = true.
Proof.
induction m as [| m IHm]. trivial.
destruct n. intro H. elim (le_Sn_O _ H).
intros. simpl in |- *. apply IHm. apply le_S_n. assumption.
Qed.
Lemma leb_complete : forall m n:nat, leb m n = true -> m <= n.
Proof.
induction m. trivial with arith.
destruct n. intro H. discriminate H.
auto with arith.
Qed.
Lemma leb_correct_conv : forall m n:nat, m < n -> leb n m = false.
Proof.
intros.
generalize (leb_complete n m).
destruct (leb n m); auto.
intros.
elim (lt_irrefl _ (lt_le_trans _ _ _ H (H0 (refl_equal true)))).
Qed.
Lemma leb_complete_conv : forall m n:nat, leb n m = false -> m < n.
Proof.
intros. elim (le_or_lt n m). intro. conditional trivial rewrite leb_correct in H. discriminate H.
trivial.
Qed.
Lemma leb_compare : forall n m, leb n m = true <-> nat_compare n m <> Gt.
Proof.
induction n; destruct m; simpl.
unfold nat_compare; simpl.
intuition; discriminate.
split; auto with arith.
unfold nat_compare; destruct (lt_eq_lt_dec 0 (S m)) as [[H|H]|H];
intuition; try discriminate.
inversion H.
split; try (intros; discriminate).
unfold nat_compare; destruct (lt_eq_lt_dec (S n) 0) as [[H|H]|H];
intuition; try discriminate.
inversion H.
rewrite nat_compare_S; auto.
Qed.