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

  Signature: cons  :: nat -> list -> list
             map   :: (nat -> nat) -> list -> list
             merge :: list -> list -> list -> list
             nil   :: list

  Rules: merge(nil, nil, X) -> X
         merge(nil, cons(Y, U), V) -> merge(U, nil, cons(Y, V))
         merge(cons(W, P), X1, Y1) -> merge(X1, P, cons(W, Y1))
         map(G1, nil) -> nil
         map(H1, cons(W1, P1)) -> cons(H1(W1), map(H1, P1))

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_?, f, c), where:

  P1. (1) merge#(nil, cons(Y, U), V) => merge#(U, nil, cons(Y, V))
      (2) merge#(cons(W, P), X1, Y1) => merge#(X1, P, cons(W, Y1))
      (3) map#(H1, cons(W1, P1)) => map#(H1, P1)

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

We compute a graph approximation with the following edges:

  1: 2 
  2: 1 2 
  3: 3 

There are 2 SCCs.

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

  P2. (1) merge#(nil, cons(Y, U), V) => merge#(U, nil, cons(Y, V))
      (2) merge#(cons(W, P), X1, Y1) => merge#(X1, P, cons(W, Y1))

  P3. (1) map#(H1, cons(W1, P1)) => map#(H1, P1)

***** No progress could be made on DP problem D2 = (P2, R UNION R_?, f, c).