Question

目前，我已经开始研究关于Coq（VerifiedMathFoundations）中一阶逻辑的定理。我已经证明了推导定理，但后来我因正确性定理而陷入引理1。因此，我紧凑地制定了一个优雅的引理，并邀请社区来研究它。这是术语的充分依据的不完整证明。如何正确消除一对“承认”？

(* PUBLIC DOMAIN *)
Require Export Coq.Vectors.Vector.
Require Export Coq.Lists.List.
Require Import Bool.Bool.
Require Import Logic.FunctionalExtensionality.
Require Import Coq.Program.Wf.

Definition SetVars  := nat.
Definition FuncSymb := nat.
Definition PredSymb := nat.
Record FSV := {
 fs : FuncSymb;
 fsv : nat;
}.
Record PSV := MPSV{
 ps : PredSymb;
 psv : nat;
}.
Inductive Terms : Type :=
| FVC :> SetVars -> Terms
| FSC (f:FSV) : (Vector.t Terms (fsv f)) -> Terms.

Definition rela : forall (x y:Terms), Prop.
Proof.
fix rela 2.
intros x y.
destruct y as [s|f t].
+ exact False.
+ refine (or _ _).
  exact (Vector.In x t).
  simple refine (@Vector.fold_left Terms Prop _ False (fsv f) t).
  intros Q e.
  exact (or Q (rela x e)).
Defined.

Definition snglV {A} (a:A) := Vector.cons A a 0 (Vector.nil A).

Definition wfr : @well_founded Terms rela.
Proof.
clear.
unfold well_founded.
assert (H : forall (n:Terms) (a:Terms), (rela a n) -> Acc rela a).
{ fix iHn 1.
  destruct n.
  + simpl. intros a b; destruct b.
  + simpl. intros a Q. destruct Q as [L|R].
    * admit.  (* smth like apply Acc_intro. intros m Hm. apply (iHn a). exact Hm. *)
    * admit.  (* like in /Arith/Wf_nat.v *)
}
intros a.
simple refine (H _ _ _).
exact (FSC (Build_FSV 0 1) (snglV a)).
simpl.
apply or_introl.
constructor.
Defined.

也可以在这里使用：pastebin。

更新：至少需要传递性才能具有良好的基础。我也开始了一个证明，但没有完成。

Fixpoint Tra (a b c:Terms) (Hc : rela c b) (Hb : rela b a) {struct a}: rela c a.
Proof.
destruct a.
+ simpl in * |- *.
  exact Hb.
+ simpl in * |- *.
  destruct Hb.
  - apply or_intror.
    revert f t H .
    fix RECU 1.
    intros f t H.
    (* ... *)
Admitted.

Answer 1

您可以通过在Terms上定义高度函数并显示出降低rela意味着降低高度来做到这一点：

Require Export Coq.Vectors.Vector.
Require Export Coq.Lists.List.
Require Import Bool.Bool.
Require Import Logic.FunctionalExtensionality.
Require Import Coq.Program.Wf.

Definition SetVars  := nat.
Definition FuncSymb := nat.
Definition PredSymb := nat.
Record FSV := {
 fs : FuncSymb;
 fsv : nat;
}.
Record PSV := MPSV{
 ps : PredSymb;
 psv : nat;
}.

Unset Elimination Schemes.
Inductive Terms : Type :=
| FVC :> SetVars -> Terms
| FSC (f:FSV) : (Vector.t Terms (fsv f)) -> Terms.
Set Elimination Schemes.

Definition Terms_rect (T : Terms -> Type)
                      (H_FVC : forall sv, T (FVC sv))
                      (H_FSC : forall f v, (forall n, T (Vector.nth v n)) -> T (FSC f v)) :=
  fix loopt (t : Terms) : T t :=
    match t with
    | FVC sv  => H_FVC sv
    | FSC f v =>
      let fix loopv s (v : Vector.t Terms s) : forall n, T (Vector.nth v n) :=
        match v with
        | @Vector.nil _ => Fin.case0 _
        | @Vector.cons _ t _ v => fun n => Fin.caseS' n (fun n => T (Vector.nth (Vector.cons _ t _ v) n))
                                                      (loopt t)
                                                      (loopv _ v)
        end in
      H_FSC f v (loopv _ v)
    end.

Definition Terms_ind := Terms_rect.

Fixpoint height (t : Terms) : nat :=
  match t with
  | FVC _ => 0
  | FSC f v => S (Vector.fold_right (fun t acc => Nat.max acc (height t)) v 0)
  end.

Definition rela : forall (x y:Terms), Prop.
Proof.
fix rela 2.
intros x y.
destruct y as [s|f t].
+ exact False.
+ refine (or _ _).
  exact (Vector.In x t).
  simple refine (@Vector.fold_left Terms Prop _ False (fsv f) t).
  intros Q e.
  exact (or Q (rela x e)).
Defined.

Require Import Lia.

Definition wfr : @well_founded Terms rela.
Proof.
apply (Wf_nat.well_founded_lt_compat _ height).
intros t1 t2. induction t2 as [sv2|f2 v2 IH]; simpl; try easy.
intros [t_v|t_sub]; apply Lt.le_lt_n_Sm.
{ clear IH. induction t_v; simpl; lia. }
revert v2 IH t_sub; generalize (fsv f2); clear f2.
intros k v2 IH t_sub.
enough (H : exists n, rela t1 (Vector.nth v2 n)).
{ destruct H as [n H]. apply IH in H. clear IH t_sub.
  transitivity (height (Vector.nth v2 n)); try lia; clear H.
  induction v2 as [|t2 m v2 IHv2].
  - inversion n.
  - apply (Fin.caseS' n); clear n; simpl; try lia.
    intros n. specialize (IHv2 n). lia. }
clear IH.
assert (H : Vector.fold_right (fun t Q => Q \/ rela t1 t) v2 False).
{ revert t_sub; generalize False.
  induction v2 as [|t2 n v2]; simpl in *; trivial.
  intros P H; specialize (IHv2 _ H); clear H.
  induction v2 as [|t2' n v2 IHv2']; simpl in *; tauto. }
clear t_sub.
induction v2 as [|t2 k v2 IH]; simpl in *; try easy.
destruct H as [H|H].
- apply IH in H.
  destruct H as [n Hn].
  now exists (Fin.FS n).
- now exists Fin.F1.
Qed.

（请注意使用自定义归纳原理，由于嵌套了归纳法，因此需要使用自定义归纳原理。）

但是，这种发展方式太复杂了。避免某些陷阱会大大简化它：

Coq标准向量库太难使用。由于嵌套的电感，使此问题更加严重。最好使用普通列表，并在术语上使用单独的格式正确的谓词。

在证明模式下定义诸如rela之类的关系将使其更难以阅读。例如，考虑以下更简单的选择：

Fixpoint rela x y :=
  match y with
  | FVC _ => False
  | FSC f v =>
    Vector.In x v \/
    Vector.fold_right (fun z P => rela x z \/ P) v False
  end.

向左折叠的还原行为较差，因为它迫使我们对累加器参数进行归纳以使归纳法得以通过。这就是为什么在我的证明中我不得不切换到fold_right的原因。

如何证明一阶语言的术语是有根据的？

1 个答案: