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

  Signature: 0      :: nat
             bool   :: nat -> boolean
             cons   :: nat -> list -> list
             consif :: boolean -> nat -> list -> list
             false  :: boolean
             filter :: (nat -> boolean) -> list -> list
             nil    :: list
             rand   :: nat -> nat
             s      :: nat -> nat
             true   :: boolean

  Rules: rand(X) -> X
         rand(s(X)) -> rand(X)
         bool(0) -> false
         bool(s(0)) -> true
         filter(F, nil) -> nil
         filter(F, cons(H, T)) -> consif(F(H), H, filter(F, T))
         consif(true, H, T) -> cons(H, T)
         consif(false, H, T) -> T

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) rand#(s(X)) => rand#(X)
      (2) filter#(F, cons(H, T)) => filter#(F, T)
      (3) filter#(F, cons(H, T)) => consif#(F(H), H, filter(F, T))

***** 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:

There are 2 SCCs.

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

  P2. (1) rand#(s(X)) => rand#(X)

  P3. (1) filter#(F, cons(H, T)) => filter#(F, T)

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

We use the following projection function:

  nu(rand#) = 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: { }.

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

We use the following projection function:

  nu(filter#) = 2

We thus have:

  (1) cons(H, T) |>| T

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

Processor output: { }.