The Coq Proof Assistant

# Library Coq.Structures.OrderedTypeEx

Require Import OrderedType.
Require Import ZArith_base.
Require Import PeanoNat Peano_dec Compare_dec.
Require Import Ascii String.
Require Import NArith Ndec.

# Examples of Ordered Type structures.

First, a particular case of OrderedType where the equality is the usual one of Coq.
Module Type UsualOrderedType.
Parameter Inline t : Type.
Definition eq := @eq t.
Parameter Inline lt : t -> t -> Prop.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Axiom lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Axiom lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Parameter compare : forall x y : t, Compare lt eq x y.
Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }.
End UsualOrderedType.
a UsualOrderedType is in particular an OrderedType.
Module UOT_to_OT (U:UsualOrderedType) <: OrderedType := U.
nat is an ordered type with respect to the usual order on natural numbers.
Module Nat_as_OT <: UsualOrderedType.
Definition t := nat.
Definition eq := @eq nat.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt := lt.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare x y : Compare lt eq x y.
Definition eq_dec := eq_nat_dec.
End Nat_as_OT.
Z is an ordered type with respect to the usual order on integers.
Local Open Scope Z_scope.
Module Z_as_OT <: UsualOrderedType.
Definition t := Z.
Definition eq := @eq Z.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt (x y:Z) := (x<y).
Lemma lt_trans : forall x y z, x<y -> y<z -> x<z.
Lemma lt_not_eq : forall x y, x<y -> ~ x=y.
Definition compare x y : Compare lt eq x y.
Definition eq_dec := Z.eq_dec.
End Z_as_OT.
positive is an ordered type with respect to the usual order on natural numbers.
Local Open Scope positive_scope.
Module Positive_as_OT <: UsualOrderedType.
Definition t:=positive.
Definition eq:=@eq positive.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt := Pos.lt.
Definition lt_trans := Pos.lt_trans.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare x y : Compare lt eq x y.
Definition eq_dec := Pos.eq_dec.
End Positive_as_OT.
N is an ordered type with respect to the usual order on natural numbers.
Module N_as_OT <: UsualOrderedType.
Definition t:=N.
Definition eq:=@eq N.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt := N.lt.
Definition lt_trans := N.lt_trans.
Definition lt_not_eq := N.lt_neq.
Definition compare x y : Compare lt eq x y.
Definition eq_dec := N.eq_dec.
End N_as_OT.
From two ordered types, we can build a new OrderedType over their cartesian product, using the lexicographic order.
Module PairOrderedType(O1 O2:OrderedType) <: OrderedType.
Module MO1:=OrderedTypeFacts(O1).
Module MO2:=OrderedTypeFacts(O2).
Definition t := prod O1.t O2.t.
Definition eq x y := O1.eq (fst x) (fst y) /\ O2.eq (snd x) (snd y).
Definition lt x y :=
O1.lt (fst x) (fst y) \/
(O1.eq (fst x) (fst y) /\ O2.lt (snd x) (snd y)).
Lemma eq_refl : forall x : t, eq x x.
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare : forall x y : t, Compare lt eq x y.
Defined.
Definition eq_dec : forall x y : t, {eq x y} + {~ eq x y}.
End PairOrderedType.
Even if positive can be seen as an ordered type with respect to the usual order (see above), we can also use a lexicographic order over bits (lower bits are considered first). This is more natural when using positive as indexes for sets or maps (see FSetPositive and FMapPositive.
Module PositiveOrderedTypeBits <: UsualOrderedType.
Definition t:=positive.
Definition eq:=@eq positive.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Fixpoint bits_lt (p q:positive) : Prop :=
match p, q with
| xH, xI _ => True
| xH, _ => False
| xO p, xO q => bits_lt p q
| xO _, _ => True
| xI p, xI q => bits_lt p q
| xI _, _ => False
end.
Definition lt:=bits_lt.
Lemma bits_lt_trans :
forall x y z : positive, bits_lt x y -> bits_lt y z -> bits_lt x z.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma bits_lt_antirefl : forall x : positive, ~ bits_lt x x.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition compare : forall x y : t, Compare lt eq x y.
Lemma eq_dec (x y: positive): {x = y} + {x <> y}.
End PositiveOrderedTypeBits.
Module Ascii_as_OT <: UsualOrderedType.
Definition t := ascii.
Definition eq := @eq ascii.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition cmp : ascii -> ascii -> comparison := Ascii.compare.
Lemma cmp_eq (a b : ascii):
cmp a b = Eq <-> a = b.
Lemma cmp_lt_nat (a b : ascii):
cmp a b = Lt <-> (nat_of_ascii a < nat_of_ascii b)%nat.
Lemma cmp_antisym (a b : ascii):
cmp a b = CompOpp (cmp b a).
Definition lt (x y : ascii) := (N_of_ascii x < N_of_ascii y)%N.
Lemma lt_trans (x y z : ascii):
lt x y -> lt y z -> lt x z.
Lemma lt_not_eq (x y : ascii):
lt x y -> x <> y.
Definition compare (a b : ascii) : Compare lt eq a b :=
match cmp a b as z return _ = z -> _ with
| Lt => fun E => LT E
| Gt => fun E => GT (compare_helper_gt E)
| Eq => fun E => EQ (compare_helper_eq E)
end Logic.eq_refl.
Definition eq_dec (x y : ascii): {x = y} + { ~ (x = y)} := ascii_dec x y.
End Ascii_as_OT.
String is an ordered type with respect to the usual lexical order.
Module String_as_OT <: UsualOrderedType.
Definition t := string.
Definition eq := @eq string.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Inductive lts : string -> string -> Prop :=
| lts_empty : forall a s, lts EmptyString (String a s)
| lts_tail : forall a s1 s2, lts s1 s2 -> lts (String a s1) (String a s2)
| lts_head : forall (a b : ascii) s1 s2,
lt (nat_of_ascii a) (nat_of_ascii b) ->
lts (String a s1) (String b s2).
Definition lt := lts.
Lemma nat_of_ascii_inverse a b : nat_of_ascii a = nat_of_ascii b -> a = b.
Lemma lts_tail_unique a s1 s2 : lt (String a s1) (String a s2) ->
lt s1 s2.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
Definition cmp : string -> string -> comparison := String.compare.
Lemma cmp_eq (a b : string):
cmp a b = Eq <-> a = b.
Lemma cmp_antisym (a b : string):
cmp a b = CompOpp (cmp b a).
Lemma cmp_lt (a b : string):
cmp a b = Lt <-> lt a b.
Definition compare (a b : string) : Compare lt eq a b :=
match cmp a b as z return _ = z -> _ with
| Lt => fun E => LT (compare_helper_lt E)
| Gt => fun E => GT (compare_helper_gt E)
| Eq => fun E => EQ (compare_helper_eq E)
end Logic.eq_refl.
Definition eq_dec (x y : string): {x = y} + { ~ (x = y)} := string_dec x y.
End String_as_OT.