coq - 如何证明交换其参数的关系的可判定性？

Question

我有一种情况，我定义了一个归纳数据类型t和一个偏序le（cf le_refl、le_trans和le_antisym）。在这种情况下，顺序具有这种特殊性le_C，即在归纳假设中交换了参数的顺序。

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

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)}

归纳假设是指le t1 t2当我需要时le t2 t1。

当我想到这一点时，这是有道理的，这个二进制函数既不是在第一个参数上也不是在第二个参数上的原始递归，而是在两个参数的对上。我的印象是我应该以某种方式同时对两个论点进行归纳，但看不到如何做到这一点。

我确实设法定义了一个布尔函数leb并用它来证明le_dec，但我想知道，从学习的角度来看，如何直接用归纳法进行证明。

问题

1. 如何根据定义直接证明（即不先定义等效的布尔函数）？le_decle

最小的可执行示例

主要定义

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
  .

辅助引理

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.

偏序证明

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.

等效布尔函数

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.

我真正想证明的

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.

基于亚瑟答案的解决方案

Ltac destruct_exs_conjs :=
  repeat match goal with
  | H : exists _, _ |- _ => destruct H
  | H : _ /\ _ |- _ => destruct H
  end; subst.

Lemma le_dec_aux (t1 t2 : t) (n : nat) :
  height t1 + height t2 <= n ->
  {le t1 t2} + {~le t1 t2}.
Proof.
  revert t1 t2.
  induction n as [| n IH]; intros t1 t2 H.
  - destruct t1; simpl in H; omega.
  - destruct t1, t2.
    + eauto using le.
    + clear. right. intro contra. dependent induction contra.
      apply le_canonical_form_A_left in contra1; subst. eauto.
    + clear. right. intro contra. dependent induction contra.
      apply le_canonical_form_A_left in contra1; subst. eauto.
    + clear. right. intro contra. dependent induction contra.
      apply le_canonical_form_B_left in contra1; destruct_exs_conjs. eauto.
    + simpl in H.
      destruct (IH t1 t2); try omega.
      * eauto using le.
      * right. intro contra. apply le_inversion_B in contra. contradiction.
    + clear. right. intro contra. dependent induction contra.
      apply le_canonical_form_B_left in contra1; destruct_exs_conjs. eauto.
    + clear. right. intro contra. dependent induction contra.
      apply le_canonical_form_C_left in contra1; destruct_exs_conjs. eauto.
    + clear. right. intro contra. dependent induction contra.
      apply le_canonical_form_C_left in contra1; destruct_exs_conjs. eauto.
    + simpl in H.
      destruct (IH t2 t1); try omega.
      * eauto using le.
      * right. intro contra. apply le_inversion_C in contra. contradiction.
Qed.

Lemma le_dec' (t1 t2 : t) :
  { le t1 t2 } + { ~le t1 t2 }.
Proof.
  destruct (le_dec_aux t1 t2 (height t1 + height t2)); auto.
Qed.

Answer 1

与您用于定义leb函数的内容类似，您需要le_dec通过对元素高度的归纳来证明：

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。数学组件库广泛使用这种模式，使用专门的reflect谓词将一般命题连接到布尔计算，而不是sumbooltype {A} + {B}。

Answer 2

我使用有根据的递归尝试了@Arthur 建议的版本。这确实提供了很好的提取。

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.