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

  Signature: cons    :: c -> b -> b
             map     :: (c -> c) -> b -> b
             nil     :: b
             node    :: a -> b -> c
             treemap :: (a -> a) -> c -> c

  Rules: map(F, nil) -> nil
         map(Z, cons(U, V)) -> cons(Z(U), map(Z, V))
         treemap(I, node(P, X1)) -> node(I(P), map(treemap(I), X1))

The system is accessible function passing by a sort ordering with b = c ≻ a.

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

  P1. (1) map#(Z, cons(U, V)) => map#(Z, V)
      (2) treemap#(I, node(P, X1)) => treemap#(I, fresh1)
      (3) treemap#(I, node(P, X1)) => map#(treemap(I), X1)

***** 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 3 
  3: 1 

There are 2 SCCs.

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

  P2. (1) map#(Z, cons(U, V)) => map#(Z, V)

  P3. (1) treemap#(I, node(P, X1)) => treemap#(I, fresh1)

***** 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(U, V) |>| V

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

Processor output: { }.

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

We obtain 0 usable rules (out of 3 rules in the input problem).

Processor output: { D4 = (P3, {}, i, c) }.

***** No progress could be made on DP problem D4 = (P3, {}, i, c).