从注入到“性质”中获得一种类型的可判定的总订单

Question

由于自然数支持可判定的全序，注入nat_of_ascii (a : ascii) : nat在ascii类型上产生可判定的全序。用Coq表达这一点的简洁、惯用的方式是什么？(有或没有类型类、模块等)

Answer 1

这样的过程是相当例行的，并将取决于您选择的库。对于order.v，基于数学-comp，这个过程完全是机械的--事实上，我们将在后面为类型开发一个通用结构，并将其注入到总订单中。

From Coq Require Import Ascii String ssreflect ssrfun ssrbool.
From mathcomp Require Import eqtype choice ssrnat.
Require Import order.

Import Order.Syntax.
Import Order.Theory.

Lemma ascii_of_natK : cancel nat_of_ascii ascii_of_nat.
Proof. exact: ascii_nat_embedding. Qed.

(* Declares ascii to be a member of the eq class *)
Definition ascii_eqMixin := CanEqMixin ascii_of_natK.
Canonical ascii_eqType := EqType _ ascii_eqMixin.

(* Declares ascii to be a member of the choice class *)
Definition ascii_choiceMixin := CanChoiceMixin ascii_of_natK.
Canonical ascii_choiceType := ChoiceType _ ascii_choiceMixin.

(* Specific stuff for the order library *)
Definition ascii_display : unit. Proof. exact: tt. Qed.

Open Scope order_scope.

(* We use the order from nat *)
Definition lea x y := nat_of_ascii x <= nat_of_ascii y.
Definition lta x y := ~~ (lea y x).

Lemma lea_ltNeq x y : lta x y = (x != y) && (lea x y).
Proof.
rewrite /lta /lea leNgt negbK lt_neqAle.
by rewrite (inj_eq (can_inj ascii_of_natK)).
Qed.
Lemma lea_refl : reflexive lea.
Proof. by move=> x; apply: le_refl. Qed.
Lemma lea_trans : transitive lea.
Proof. by move=> x y z; apply: le_trans. Qed.
Lemma lea_anti : antisymmetric lea.
Proof. by move=> x y /le_anti /(can_inj ascii_of_natK). Qed.
Lemma lea_total : total lea.
Proof. by move=> x y; apply: le_total. Qed.

(* We can now declare ascii to belong to the order class. We must declare its
   subclasses first. *)
Definition asciiPOrderMixin :=
  POrderMixin lea_ltNeq lea_refl lea_anti lea_trans.

Canonical asciiPOrderType  := POrderType ascii_display ascii asciiPOrderMixin.

Definition asciiLatticeMixin := Order.TotalLattice.Mixin lea_total.
Canonical asciiLatticeType := LatticeType ascii asciiLatticeMixin.
Canonical asciiOrderType := OrderType ascii lea_total.

请注意，为ascii提供一个order实例可以让我们访问一个很大的总订单理论，加上运算符等等.然而，total本身的定义相当简单：

"<= is total" == x <= y || y <= x

其中<=是一个“可判定的关系”，当然，我们假定，对于特定类型，平等的可判定性。具体来说，对于任意的关系：

Definition total (T: Type) (r : T -> T -> bool) := forall x y, r x y || r y x.

因此，如果T是和order，并且满足total，那么您就完成了。

更广泛地说，您可以定义一个泛型原则来使用注入构建这样的类型：

Section InjOrder.

Context {display : unit}.
Local Notation orderType := (orderType display).
Variable (T : orderType) (U : eqType) (f : U -> T) (f_inj : injective f).

Open Scope order_scope.

Let le x y := f x <= f y.
Let lt x y := ~~ (f y <= f x).
Lemma CO_le_ltNeq x y: lt x y = (x != y) && (le x y).
Proof. by rewrite /lt /le leNgt negbK lt_neqAle (inj_eq f_inj). Qed.
Lemma CO_le_refl : reflexive le. Proof. by move=> x; apply: le_refl. Qed.
Lemma CO_le_trans : transitive le. Proof. by move=> x y z; apply: le_trans. Qed.
Lemma CO_le_anti : antisymmetric le. Proof. by move=> x y /le_anti /f_inj. Qed.

Definition InjOrderMixin : porderMixin U :=
  POrderMixin CO_le_ltNeq CO_le_refl CO_le_anti CO_le_trans.
End InjOrder.

然后，按照以下方式重写ascii实例：

Definition ascii_display : unit. Proof. exact: tt. Qed.
Definition ascii_porderMixin := InjOrderMixin (can_inj ascii_of_natK).
Canonical asciiPOrderType := POrderType ascii_display ascii ascii_porderMixin.

Lemma lea_total : @total ascii (<=%O)%O.
Proof. by move=> x y; apply: le_total. Qed.

Definition asciiLatticeMixin := Order.TotalLattice.Mixin lea_total.
Canonical asciiLatticeType := LatticeType ascii asciiLatticeMixin.
Canonical asciiOrderType := OrderType ascii lea_total.