Mutualy recursive function and termination checker in Coq

Question

EDIT

Require Import Bool List ZArith.
  Variable A: Type.
    Inductive error :=
    | Todo.
    Inductive result (A : Type) : Type :=
        Ok : A -> result A | Ko : error -> result A.
    Variable bool_of_result : result A -> bool.
    Variable rules : Type.
    Variable boolean : Type.
    Variable positiveInteger : Type.
    Variable OK: result unit.
    Definition dps := rules.
    Inductive dpProof := 
      | DpProof_depGraphProc : list 
       (dps * boolean * option (list positiveInteger) * option dpProof) -> dpProof.
    Fixpoint dpProof' (R D: rules) (p: dpProof) {struct p}:=
    match p with
      | DpProof_depGraphProc cs => dpGraphProc R D cs
     end   
   with dpGraphProc (R D: rules ) cs {struct cs} :=
    match cs with
    | nil => Ko unit Todo
    | (_, _, _, op) :: cs' => 
      match op with
       | None => Ko unit Todo
       | Some p2 => dpProof' R D p2
      end
 end.

I got an error message saying that: Recursive call to dpProof has principal argument equal to

"p2" instead of "cs'".
Recursive definition is:
"fun (R D : rules)
   (cs : list
           (dps * boolean * option (list positiveInteger) *
            option dpProof)) =>
 match cs with
 | nil => Ko unit Todo
 | (_, _, _, Some p2) :: _ => dpProof' R D p2
 | (_, _, _, None) :: _ => OK
 end".

If I do not use the mutual recursive and use the nested fixpoint, it will combine and pass the checker of termination. Here is the code that successfully combined.

Fixpoint dpProof' (R D: rules) (p: dpProof) {struct p}:=
      match p with
      | DpProof_depGraphProc cs =>
        match cs with
          | nil => Ko _ Todo
          | (_, _, _, op) :: cs' => 
            match op with
              | None => Ko unit Todo
              | Some p2 => dpProof' R D p2
            end
        end end.

I would like to understand deeper about the reason why it cannot pass the termination checker? Is it because they cannot guess the argument descreasing? Is there any way that I can use the mutually recursive to express my function dpGraphProc?

Also How can I write the function dpGraphProc that check in the whole list? Here I do not know how to use the argument cs'.

Answer 1

Mutual recursion is to be used either with a single inductive data-type or with different inductive data-types that have been defined together in a single inductive definition. In your case, you are using polymorphic data-types prod (the type of pairs), list, and option which were already defined before dpProof.

The nested fixpoint approach does not have the restriction.