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Diffstat (limited to 'lib/Ordered.v')
| -rw-r--r-- | lib/Ordered.v | 30 |
1 files changed, 30 insertions, 0 deletions
diff --git a/lib/Ordered.v b/lib/Ordered.v index f52a7ef..026671a 100644 --- a/lib/Ordered.v +++ b/lib/Ordered.v @@ -48,6 +48,36 @@ Definition eq_dec : forall x y, { eq x y } + { ~ eq x y } := peq. End OrderedPositive. +(** The ordered type of integers *) + +Module OrderedZ <: OrderedType. + +Definition t := Z. +Definition eq (x y: t) := x = y. +Definition lt := Zlt. + +Lemma eq_refl : forall x : t, eq x x. +Proof (@refl_equal t). +Lemma eq_sym : forall x y : t, eq x y -> eq y x. +Proof (@sym_equal t). +Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. +Proof (@trans_equal t). +Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z. +Proof Zlt_trans. +Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y. +Proof. unfold lt, eq, t; intros. omega. Qed. +Lemma compare : forall x y : t, Compare lt eq x y. +Proof. + intros. destruct (Z.compare x y) as [] eqn:E. + apply EQ. red. apply Z.compare_eq_iff. assumption. + apply LT. assumption. + apply GT. apply Z.compare_gt_iff. assumption. +Defined. + +Definition eq_dec : forall x y, { eq x y } + { ~ eq x y } := zeq. + +End OrderedZ. + (** The ordered type of machine integers *) Module OrderedInt <: OrderedType. |