在 Coq 中使用我自己的 == 运算符重写策略

Question

我试图直接从字段的公理证明简单的字段属性。在对 Coq 的原生字段支持（就像这个）进行了一些实验之后，我决定最好简单地写下 10 条公理并使其自包含。当我需要使用rewrite我自己的==操作员时，我遇到了一个困难，这自然是行不通的。我意识到我必须添加一些==自反的、对称的和可传递的公理，但我想知道这是否就够了？或者也许有更简单的方法来使用rewrite用户定义==？这是我的 Coq 代码：

Variable (F:Type).
Variable (zero:F).
Variable (one :F).
Variable (add: F -> F -> F).
Variable (mul: F -> F -> F).
Variable (opposite: F -> F).
Variable (inverse : F -> F).
Variable (eq: F -> F -> Prop).

Axiom add_assoc: forall (a b c : F), (eq (add (add a b) c) (add a (add b c))).
Axiom mul_assoc: forall (a b c : F), (eq (mul (mul a b) c) (mul a (mul b c))).

Axiom add_comm : forall (a b : F), (eq (add a b) (add b a)).
Axiom mul_comm : forall (a b : F), (eq (mul a b) (mul b a)).

Axiom distr1 : forall (a b c : F), (eq (mul a (add b c)) (add (mul a b) (mul a c))).
Axiom distr2 : forall (a b c : F), (eq (mul (add a b) c) (add (mul a c) (mul b c))).

Axiom add_id1 : forall (a : F), (eq (add a zero) a).
Axiom mul_id1 : forall (a : F), (eq (mul a  one) a).
Axiom add_id2 : forall (a : F), (eq (add zero a) a).
Axiom mul_id2 : forall (a : F), (eq (mul one  a) a).

Axiom add_inv1 : forall (a : F), exists b, (eq (add a b) zero).
Axiom add_inv2 : forall (a : F), exists b, (eq (add b a) zero).

Axiom mul_inv1 : forall (a : F), exists b, (eq (mul a b) one).
Axiom mul_inv2 : forall (a : F), exists b, (eq (mul b a) one).

(*******************)
(* Field notations *)
(*******************)
Notation "0" := zero.
Notation "1" :=  one.
Infix    "+" :=  add.
Infix    "*" :=  mul.
(*******************)
(* Field notations *)
(*******************)
Infix "==" := eq (at level 70, no associativity).

Lemma mul_0_l: forall v, (0 * v == 0).
Proof.
  intros v.
  specialize add_id1 with (0 * v).
  intros H.

此时我有了假设H : 0 * v + 0 == 0 * v和目标 0 * v == 0。当我尝试时rewrite H，它自然会失败。

Answer 1

对于广义重写（用任意关系重写）：

1. 导入Setoid（加载一个覆盖rewrite策略的插件）。

2. 使用类将您的关系声明为等价关系（技术上rewrite也适用于较弱的假设，例如只有传递假设，但您还需要在步骤 3 中使用更细粒度的关系层次结构）Equivalence。

3. 使用类声明您的操作（add、mul等）尊重该等效性（例如，添加等效值必须产生等效值）Proper。这也需要Morphisms模块。

您需要第 3 步来重写子表达式。

Require Import Setoid Morphisms.

(* Define eq, add, etc. *)

Declare Instance Equivalence_eq : Equivalence eq.
Declare Instance Proper_add : Proper (eq ==> eq ==> eq) add.
Declare Instance Proper_mul : Proper (eq ==> eq ==> eq) mul.
(* etc. *)

Lemma mul_0_l: forall v, (0 * v == 0).
Proof.
  intros v.
  specialize add_id1 with (0 * v).
  intros H.
  rewrite <- H. (* Rewrite toplevel expression (allowed by Equivalence_eq) *)
  rewrite <- H. (* Rewrite subexpression (allowed by Proper_add and Equivalence_eq) *)