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

  Signature: 0       :: d
             cons    :: d -> c -> c
             false   :: a
             height  :: d -> d
             if      :: a -> d -> d
             le      :: d -> d -> a
             map     :: (d -> d) -> c -> c
             maxlist :: d -> c -> d
             nil     :: c
             node    :: b -> c -> d
             s       :: d -> d
             true    :: a

  Rules: map(F, nil) -> nil
         map(Z, cons(U, V)) -> cons(Z(U), map(Z, V))
         le(0, W) -> true
         le(s(P), 0) -> false
         le(s(X1), s(Y1)) -> le(X1, Y1)
         maxlist(U1, cons(V1, W1)) -> if(le(U1, V1), maxlist(V1, W1))
         maxlist(P1, nil) -> P1
         height(node(X2, Y2)) -> s(maxlist(0, map(height, Y2)))

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

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) le#(s(X1), s(Y1)) => le#(X1, Y1)
      (3) maxlist#(U1, cons(V1, W1)) => le#(U1, V1)
      (4) maxlist#(U1, cons(V1, W1)) => maxlist#(V1, W1)
      (5) height#(node(X2, Y2)) => height#(fresh1)
      (6) height#(node(X2, Y2)) => map#(height, Y2)
      (7) height#(node(X2, Y2)) => maxlist#(0, map(height, Y2))

***** 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: 2 
  4: 3 4 
  5: 5 6 7 
  6: 1 
  7: 3 4 

There are 4 SCCs.

Processor output: { D2 = (P2, R UNION R_?, i, c) ; D3 = (P3, R UNION R_?, i, c) ; D4 = (P4, R UNION R_?, i, c) ; D5 = (P5, R UNION R_?, i, c) }, where:

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

  P3. (1) le#(s(X1), s(Y1)) => le#(X1, Y1)

  P4. (1) maxlist#(U1, cons(V1, W1)) => maxlist#(V1, W1)

  P5. (1) height#(node(X2, Y2)) => height#(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 Subterm Criterion Processor on D3 = (P3, R UNION R_?, i, c).

We use the following projection function:

  nu(le#) = 1

We thus have:

  (1) s(X1) |>| X1

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

Processor output: { }.

***** We apply the Subterm Criterion Processor on D4 = (P4, R UNION R_?, i, c).

We use the following projection function:

  nu(maxlist#) = 2

We thus have:

  (1) cons(V1, W1) |>| W1

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

Processor output: { }.

***** We apply the Usable Rules Processor on D5 = (P5, R UNION R_?, i, c).

We obtain 0 usable rules (out of 8 rules in the input problem).

Processor output: { D6 = (P5, {}, i, c) }.

***** No progress could be made on DP problem D6 = (P5, {}, i, c).