Coq最佳实践: mutual recursion,只有一个函数在结构上减少

Question

考虑以下无类型 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".

实现此目的最可读/符合人体工程学/最有效的方法是什么？

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 pretty-print ——通常的约定是不打印 x y z 等表达式中的括号(应用程序与左侧关联)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有some material关于相互递归与嵌套递归，但在不同的设置中。