We consider termination of the STRS with no additional rule schemes:

  Signature: 0 :: nat
             a :: nat -> nat
             f :: (nat -> nat) -> nat -> nat -> nat
             g :: nat -> nat -> (nat -> nat) -> nat -> nat
             h :: nat -> nat
             i :: nat -> nat
             s :: nat -> nat

  Rules: f(F, Y, U) -> g(Y, U, F, Y)
         g(s(Y), s(U), F, V) -> g(Y, U, F, V)
         g(0, 0, F, V) -> f(F, s(V), F(h(V)))
         h(0) -> a(0)
         h(s(Y)) -> a(h(Y))
         i(0) -> 0
         i(a(Y)) -> s(i(Y))

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, i, c), where:

  P1. (1) f#(F, Y, U) => g#(Y, U, F, Y)
      (2) g#(s(Y), s(U), F, V) => g#(Y, U, F, V)
      (3) g#(0, 0, F, V) => h#(V)
      (4) g#(0, 0, F, V) => f#(F, s(V), F(h(V)))
      (5) h#(s(Y)) => h#(Y)
      (6) i#(a(Y)) => i#(Y)

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

We compute a graph approximation with the following edges:

  1: 2 3 4 
  2: 2 3 4 
  3: 5 
  4: 1 
  5: 5 
  6: 6 

There are 3 SCCs.

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

  P2. (1) h#(s(Y)) => h#(Y)

  P3. (1) f#(F, Y, U) => g#(Y, U, F, Y)
      (2) g#(s(Y), s(U), F, V) => g#(Y, U, F, V)
      (3) g#(0, 0, F, V) => f#(F, s(V), F(h(V)))

  P4. (1) i#(a(Y)) => i#(Y)

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

We use the following projection function:

  nu(h#) = 1

We thus have:

  (1) s(Y) |>| Y

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

Processor output: { }.

***** No progress could be made on DP problem D3 = (P3, R, i, c).