We consider universal computability of the STRS with no additional rule schemes:

  Signature: 0      :: nat
             cons   :: nat -> list -> list
             double :: nat -> nat
             map    :: (nat -> nat) -> list -> list
             nil    :: list
             s      :: nat -> nat

  Rules: map(F, nil) -> nil
         map(F, cons(x, xs)) -> cons(F(x), map(F, xs))
         double(0) -> 0
         double(s(x)) -> s(s(double(x)))

The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the initial DP problem D1 = (P1, R UNION R_?, i, c), where:

  P1. (1) map#(F, cons(x, xs)) => map#(F, xs)
      (2) double#(s(x)) => double#(x)

***** We apply the Graph Processor on D1 = (P1, R UNION R_?, i, c).

We compute a graph approximation with the following edges:

  1: 1 
  2: 2 

There are 2 SCCs.

Processor output: { D2 = (P2, R UNION R_?, i, c) ; D3 = (P3, R UNION R_?, i, c) }, where:

  P2. (1) map#(F, cons(x, xs)) => map#(F, xs)

  P3. (1) double#(s(x)) => double#(x)

***** We apply the Subterm Criterion Processor on D2 = (P2, R UNION R_?, i, c).

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(x, xs) |>| xs

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.

***** We apply the Subterm Criterion Processor on D3 = (P3, R UNION R_?, i, c).

We use the following projection function:

  nu(double#) = 1

We thus have:

  (1) s(x) |>| x

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.