Coq最佳实践：相互递归，只有一个函数在结构上递减

Question

考虑以下用于无类型 lambda 演算的玩具表示：

Require Import String.
Open Scope string_scope.

Inductive term : Set :=
| Var : string -> term
| Abs : string -> term -> term
| App : term -> term -> term.

Fixpoint print (term : term) :=
  match term return string with
  | Var id => id
  | Abs id term => "\" ++ id ++ " " ++ print term
  | App term1 term2 => print_inner term1 ++ " " ++ print_inner term2
  end
with print_inner (term : term) :=
  match term return string with
  | Var id => id
  | term => "(" ++ print term ++ ")"
  end.

类型检查 print 失败并出现以下错误：

Recursive definition of print_inner is ill-formed.
[...]
Recursive call to print has principal argument equal to "term" instead of "t".

最 readable/ergonomic/efficient 的实施方式是什么？

Answer 1

您可以使用嵌套递归函数：

Fixpoint print (tm : term) : string :=
  match tm return string with
  | Var id => id
  | Abs id body => "\" ++ id ++ ". " ++ print body
  | App tm1 tm2 =>
     let fix print_inner (tm : term) : string :=
         match tm return string with
         | Var id => id
         | _ => "(" ++ print tm ++ ")"
         end
     in
     print_inner tm1 ++ " " ++ print_inner tm2
  end.

这种方法可以扩展到处理漂亮的打印——通常的约定是不在表达式中打印括号，如 x y z（应用程序关联到左边）或打印 \x. \y. x y 为 \xy. x y:

Definition in_parens (stm : string) : string := "(" ++ stm ++ ")".

Fixpoint pprint (tm : term) : string :=
  match tm with
  | Var id => id
  | Abs id tm1 =>
    let fix pprint_nested_abs (tm : term) : string :=
        match tm with
        | Abs id tm1 => id ++ pprint_nested_abs tm1
        | _ => ". " ++ pprint tm
        end
    in
    "\" ++ id ++ pprint_nested_abs tm1

  (* e.g. (\x. x x) (\x. x x) *)
  | App ((Abs _ _) as tm1) ((Abs _ _) as tm2) =>     
      in_parens (pprint tm1) ++ " " ++ in_parens (pprint tm2)

  (* variable scopes *)
  | App ((Abs _ _) as tm1) tm2 => in_parens (pprint tm1) ++ " " ++ pprint tm2

  (* `x \x. x` looks ugly, `x (\x. x)` is better; also handle `x (y z)` *) 
  | App tm1 ((Abs _ _) as tm2) | App tm1 (App _ _ as tm2) =>
      pprint tm1 ++ " " ++ in_parens (pprint tm2)

  | App tm1 tm2 => pprint tm1 ++ " " ++ pprint tm2
  end.

顺便说一句，CPDT 在相互递归与嵌套递归方面有 some material，但设置不同。

Answer 2

您还可以将递归调用的想法与 print_inner 执行的案例分析分离，如下所示：

Definition print_inner (term : term) (sterm : string) : string :=
 match term with
 | Var id => id
 | _      => "(" ++ sterm ++ ")"
 end.

Fixpoint print (term : term) :=
  match term return string with
  | Var id => id
  | Abs id term => "\" ++ id ++ " " ++ print term
  | App term1 term2 => print_inner term1 (print term1)
                    ++ " " ++ print_inner term2 (print term2)
  end.

或者，您可以根据构造函数的固定级别使用不同的算法来决定是否省略括号。

Coq最佳实践：相互递归，只有一个函数在结构上递减

Coq best practice: mutual recursion, only one function is structurally decreasing

coq

mutual-recursion

totality