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Require Import OrderedType.

An alternative (but equivalent) presentation for an Ordered Type

inferface.

NB: comparison, defined in Datatypes.v is Eq|Lt|Gt whereas compare, defined in OrderedType.v is EQ _ | LT _ | GT _

Module Type OrderedTypeAlt.

 Parameter t : Type.

 Parameter compare : t -> t -> comparison.

 Infix "?=" := compare (at level 70, no associativity).

 Parameter compare_sym :
   forall x y, (y?=x) = CompOpp (x?=y).
 Parameter compare_trans :
   forall c x y z, (x?=y) = c -> (y?=z) = c -> (x?=z) = c.

End OrderedTypeAlt.

From this new presentation to the original one.

Module OrderedType_from_Alt (O:OrderedTypeAlt) <: OrderedType.
 Import O.

 Definition t := t.

 Definition eq x y := (x?=y) = Eq.
 Definition lt x y := (x?=y) = Lt.

 Lemma eq_refl : forall x, eq x x.forall x : O.t, eq x x
 Proof.forall x : O.t, eq x x
 intro x.x:O.t
eq x x
 unfold eq.x:O.t
(x ?= x) = Eq
 assert (H:=compare_sym x x).x:O.t
H:(x ?= x) = CompOpp (x ?= x)
(x ?= x) = Eq
 destruct (x ?= x); simpl in *; try discriminate; auto.
 Qed.

 Lemma eq_sym : forall x y, eq x y -> eq y x.forall x y : O.t, eq x y -> eq y x
 Proof.forall x y : O.t, eq x y -> eq y x
 unfold eq; intros.x, y:O.t
H:(x ?= y) = Eq
(y ?= x) = Eq
 rewrite compare_sym.x, y:O.t
H:(x ?= y) = Eq
CompOpp (x ?= y) = Eq
 rewrite H; simpl; auto.
 Qed.

 Definition eq_trans := (compare_trans Eq).

 Definition lt_trans := (compare_trans Lt).

 Lemma lt_not_eq : forall x y, lt x y -> ~eq x y.forall x y : O.t, lt x y -> ~ eq x y
 Proof.forall x y : O.t, lt x y -> ~ eq x y
 unfold eq, lt; intros.x, y:O.t
H:(x ?= y) = Lt
(x ?= y) <> Eq
 rewrite H; discriminate.
 Qed.

 Definition compare : forall x y, Compare lt eq x y.forall x y : O.t, Compare lt eq x y
 Proof.forall x y : O.t, Compare lt eq x y
 intros.x, y:O.t
Compare lt eq x y
 case_eq (x ?= y); intros.x, y:O.t
H:(x ?= y) = Eq
Compare lt eq x y
x, y:O.t
H:(x ?= y) = Lt
Compare lt eq x y
x, y:O.t
H:(x ?= y) = Gt
Compare lt eq x y
 apply EQ; auto.x, y:O.t
H:(x ?= y) = Lt
Compare lt eq x y
x, y:O.t
H:(x ?= y) = Gt
Compare lt eq x y
 apply LT; auto.x, y:O.t
H:(x ?= y) = Gt
Compare lt eq x y
 apply GT; red.x, y:O.t
H:(x ?= y) = Gt
(y ?= x) = Lt
 rewrite compare_sym; rewrite H; auto.
 Defined.

 Definition eq_dec : forall x y, { eq x y } + { ~ eq x y }.forall x y : O.t, {eq x y} + {~ eq x y}
 Proof.forall x y : O.t, {eq x y} + {~ eq x y}
 intros; unfold eq.x, y:O.t
{(x ?= y) = Eq} + {(x ?= y) <> Eq}
 case (x ?= y); [ left | right | right ]; auto; discriminate.
 Defined.

End OrderedType_from_Alt.

From the original presentation to this alternative one.

Module OrderedType_to_Alt (O:OrderedType) <: OrderedTypeAlt.
 Import O.
 Module MO:=OrderedTypeFacts(O).
 Import MO.

 Definition t := t.

 Definition compare x y := match compare x y with
   | LT _ => Lt
   | EQ _ => Eq
   | GT _ => Gt
  end.

 Infix "?=" := compare (at level 70, no associativity).

 Lemma compare_sym :
   forall x y, (y?=x) = CompOpp (x?=y).forall x y : O.t, (y ?= x) = CompOpp (x ?= y)
 Proof.forall x y : O.t, (y ?= x) = CompOpp (x ?= y)
 intros x y; unfold compare.x, y:O.t
match O.compare y x with
| LT _ => Lt
| EQ _ => Eq
| GT _ => Gt
end =
CompOpp
  match O.compare x y with
  | LT _ => Lt
  | EQ _ => Eq
  | GT _ => Gt
  end
 destruct O.compare; elim_comp; simpl; auto.
 Qed.

 Lemma compare_trans :
   forall c x y z, (x?=y) = c -> (y?=z) = c -> (x?=z) = c.forall (c : comparison) (x y z : O.t),
(x ?= y) = c -> (y ?= z) = c -> (x ?= z) = c
 Proof.forall (c : comparison) (x y z : O.t),
(x ?= y) = c -> (y ?= z) = c -> (x ?= z) = c
 intros c x y z.c:comparison
x, y, z:O.t
(x ?= y) = c -> (y ?= z) = c -> (x ?= z) = c
 destruct c; unfold compare;
 do 2 (destruct O.compare; intros; try discriminate);
 elim_comp; auto.
 Qed.

End OrderedType_to_Alt.