问题

Question

我遇到这样的情况，我定义了归纳数据类型t和其上的部分顺序le（参见{{1}，le_refl和le_trans）。在le_antisym情况下，顺序具有特殊性，即在归纳假设中交换参数的顺序。

因此，我没有成功证明这种排序关系是确定性的（参见le_C）。有问题的子目标如下。

le_dec

归纳假设是指1 subgoal t1 : t IHt1 : forall t2 : t, {le t1 t2} + {~ le t1 t2} t2 : t ______________________________________(1/1) {le (C t1) (C t2)} + {~ le (C t1) (C t2)}，而我需要le t1 t2。

当我考虑到这一点时，这是有意义的，这个二进制函数既不是最初在第一个参数上也不在第二个参数上递归的，而是在两个参数对上都是递归的。我的印象是，我应该以某种方式同时对两个参数进行归纳，但是看不到如何做到。

我确实定义了布尔函数le t2 t1并使用它来证明leb，但是我想从学习的角度出发，如何直接用归纳法做证明。 / p>

问题

如何基于le_dec的定义（即，无需先定义等效的布尔函数）直接证明le_dec？

最小的可执行文件示例

主要定义

le

助手引理

Inductive t : Set :=
  | A : t
  | B : t -> t
  | C : t -> t
  .

Inductive le : t -> t -> Prop :=
  | le_A :
      le A A

  | le_B : forall x y,
      le x y -> le (B x) (B y)

  | le_C : forall x y,
      le y x -> le (C x) (C y)

  | le_trans : forall t1 t2 t3,
      le t1 t2 -> le t2 t3 -> le t1 t3
  .

偏序证明

Require Import Coq.Program.Equality.

Lemma le_canonical_form_A_left (t1 : t) :
  le A t1 -> t1 = A.
Proof.
  intros LE. dependent induction LE; auto.
Qed.

Lemma le_canonical_form_B_left (t1 t2 : t) :
  le (B t1) t2 -> exists t3, t2 = B t3.
Proof.
  intros LE. dependent induction LE.
  - eauto.
  - destruct IHLE1 with t1 as [t4 ?]; clear IHLE1; trivial; subst.
    destruct IHLE2 with t4 as [t4' ?]; clear IHLE2; trivial; subst. eauto.
Qed.

Lemma le_canonical_form_C_left (t1 t2 : t) :
  le (C t1) t2 -> exists t3, t2 = C t3.
Proof.
  intros LE. dependent induction LE.
  - eauto.
  - destruct IHLE1 with t1 as [t4 ?]; clear IHLE1; trivial; subst.
    destruct IHLE2 with t4 as [t4' ?]; clear IHLE2; trivial; subst. eauto.
Qed.

Lemma le_inversion_B (t1 t2 : t) :
  le (B t1) (B t2) -> le t1 t2.
Proof.
  intros LE.
  dependent induction LE.
  - assumption.
  - apply le_canonical_form_B_left in LE1 as [t3 ?]; subst. eauto using le_trans.
Qed.

Lemma le_inversion_C (t1 t2 : t) :
  le (C t1) (C t2) -> le t2 t1.
Proof.
  intros LE.
  dependent induction LE.
  - assumption.
  - apply le_canonical_form_C_left in LE1 as [t3 ?]; subst. eauto using le_trans.
Qed.

Lemma le_inversion (t1 t2 : t) :
  le t1 t2 ->
  t1 = A /\ t2 = A \/
  (exists t1' t2', t1 = B t1' /\ t2 = B t2') \/
  (exists t1' t2', t1 = C t1' /\ t2 = C t2').
Proof.
  intros LE.
  destruct t1.
  - apply le_canonical_form_A_left in LE; subst. auto.
  - apply le_canonical_form_B_left in LE as [? ?]; subst. eauto 6.
  - apply le_canonical_form_C_left in LE as [? ?]; subst. eauto 6.
Qed.

等效布尔函数

Lemma le_refl (x : t) :
  le x x.
Proof.
  induction x; eauto using le.
Qed.    

Lemma le_antisym (t1 t2 : t) :
  le t1 t2 -> le t2 t1 -> t1 = t2.
Proof.
  induction 1; intros LE.
  - auto.
  - apply le_inversion_B in LE. f_equal; auto.
  - apply le_inversion_C in LE. f_equal; auto using eq_sym.
  - rewrite IHle1; eauto using le_trans.
Qed.

我实际上想证明的东西

Fixpoint height (x : t) : nat :=
  match x with
  | A => 1
  | B x' => 1 + height x'
  | C x' => 1 + height x'
  end.

Definition height_pair (p : t * t) : nat :=
  let (t1, t2) := p in height t1 + height t2.

Require Import Recdef.
Require Import Omega.

Function leb (p : t * t) { measure height_pair p } : bool :=
  match p with
  | (A, A) => true
  | (B x', B y') => leb (x', y')
  | (C x', C y') => leb (y', x')
  | _ => false
  end.
  - intros. subst. simpl. omega.
  - intros. subst. simpl. omega.
Defined.

Ltac inv H := inversion H; clear H; subst.

Lemma le_to_leb (t1 t2 : t) :
  le t1 t2 -> leb (t1, t2) = true.
Proof.
  remember (t1, t2) as p eqn:Heqn.
  revert Heqn.
  revert t1 t2.
  functional induction (leb p); intros t1 t2 Heqn LE; inv Heqn.
  - trivial.
  - apply IHb with x' y'; trivial.
    now apply le_inversion_B in LE.
  - apply IHb with y' x'; trivial.
    now apply le_inversion_C in LE.
  - exfalso. apply le_inversion in LE.
    intuition; subst.
    + easy.
    + destruct H0.
      destruct H.
      now (intuition; subst).
    + destruct H0.
      destruct H.
      now (intuition; subst).
Qed.

Lemma leb_to_le (t1 t2 : t) :
  leb (t1, t2) = true -> le t1 t2.
Proof.
  remember (t1, t2) as p eqn:Heqn.
  revert Heqn.
  revert t1 t2.
  functional induction (leb p); intros t1 t2 Heqn LEB; inv Heqn.
  - eauto using le.
  - eauto using le.
  - eauto using le.
  - discriminate LEB.
Qed.

Corollary le_iff_leb (t1 t2 : t) :
  le t1 t2 <-> leb (t1, t2) = true.
Proof.
  split.
  - apply le_to_leb.
  - apply leb_to_le.
Qed.

基于Arthur's answer的解决方案

Lemma le_dec (t1 t2 : t) :
  { le t1 t2 } + { ~le t1 t2 }.
Proof.
  revert t2.
  induction t1; intros t2.
  - destruct t2.
    + eauto using le.
    + right. intro contra. dependent induction contra.
      apply le_canonical_form_A_left in contra1; subst. eauto.
    + right. intro contra. dependent induction contra.
      apply le_canonical_form_A_left in contra1; subst. eauto.
  - destruct t2.
    + right. intro contra. clear IHt1. dependent induction contra.
      apply le_canonical_form_B_left in contra1 as [? ?]; subst. eauto.
    + destruct IHt1 with t2.
      * eauto using le.
      * right. intro contra. apply le_inversion_B in contra. contradiction.
    + right; intro contra. clear IHt1. dependent induction contra.
      apply le_canonical_form_B_left in contra1 as [? ?]; subst. eauto.
  - destruct t2.
    + right. intro contra. clear IHt1. dependent induction contra.
      apply le_canonical_form_C_left in contra1 as [? ?]; subst. eauto.
    + right. intro contra. clear IHt1. dependent induction contra.
      apply le_canonical_form_C_left in contra1 as [? ?]; subst. eauto.
    + destruct IHt1 with t2.
      * admit. (* Wrong assumption *)
      * admit. (* Wrong assumption *)
Restart.
  destruct (leb (t1, t2)) eqn:Heqn.
  - apply leb_to_le in Heqn. auto.
  - right. intro contra. apply le_to_leb in contra.
    rewrite Heqn in contra. discriminate.
Qed.

Answer 1

类似于用于定义leb函数的方法，您需要通过归纳元素的高度来证明le_dec：

Lemma le_dec_aux t1 t2 n : height t1 + height t2 <= n -> {le t1 t2} + {~le t1 t2}.
Proof.
revert t1 t2.
induction n as [|n IH].
(* ... *)

话虽如此，我认为使用布尔函数证明可判定性是完全可以的。 Mathematical Components库使用专门的reflect谓词（而不是sumbool类型{A} + {B}来将通用命题连接到布尔计算）广泛使用此模式。

Answer 2

我使用了有根据的递归尝试了@Arthur建议的版本。确实可以提供很好的提取效果。

Definition rel p1 p2 := height_pair p1 < height_pair p2.

Lemma rel_wf : well_founded rel.
Proof.
  apply well_founded_ltof.
Qed.

Lemma le_dec (t1 t2 : t) :
  { le t1 t2 } + { ~le t1 t2 }.
Proof.
  induction t1, t2 as [t1 t2]
    using (fun P => well_founded_induction_type_2 P rel_wf).
  destruct t1, t2;
    try (right; intros contra;
         (apply le_canonical_form_A_left in contra)
      || (apply le_canonical_form_B_left in contra; destruct contra)
      || (apply le_canonical_form_C_left in contra; destruct contra);
      discriminate).
  - left. apply le_A.
  - destruct (H t1 t2).
    + unfold rel, height_pair; simpl. omega.
    + left. apply le_B. assumption.
    + right. intros contra. apply le_inversion_B in contra. contradiction.
  - destruct (H t2 t1).
    + unfold rel, height_pair; simpl. omega.
    + left. apply le_C. assumption.
    + right. intros contra. apply le_inversion_C in contra. contradiction.
Qed.

Extraction Inline well_founded_induction_type_2 Fix_F_2.
  (* to have a nice extraction *)
Extraction le_dec.

但是请注意，您定义的顺序关系实际上只是相等关系，但是也许您描述了初始用例的简化。

Lemma le_is_eq : forall t1 t2, le t1 t2 -> t1 = t2.
Proof.
  intros.
  induction t1, t2 as [t1 t2]
    using (fun P => well_founded_induction_type_2 P rel_wf).
  destruct t1, t2;
    try ((apply le_canonical_form_A_left in H)
      || (apply le_canonical_form_B_left in H; destruct H)
      || (apply le_canonical_form_C_left in H; destruct H);
      discriminate).
  - reflexivity.
  - apply le_inversion_B in H.
    apply H0 in H.
    + congruence.
    + unfold rel, height_pair. simpl. omega.
  - apply le_inversion_C in H.
    apply H0 in H.
    + congruence.
    + unfold rel, height_pair. simpl. omega.
Qed.

如何证明交换其参数的关系的可判定性？

问题