We consider universal computability of the LCTRS with only rule scheme Calc:

  Signature: mult :: Int -> Int -> Int

  Rules: mult(0, y) -> 0
         mult(x, y) -> mult(x - 1, y) + y | x > 0
         mult(x, y) -> -mult(-x, y) | 0 > x

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) mult#(x, y) => mult#(x - 1, y) | x > 0
      (2) mult#(x, y) => mult#(-x, y) | 0 > x

***** 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: 1 

There is only one SCC, so all DPs not inside the SCC can be removed.

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

  P2. (1) mult#(x, y) => mult#(x - 1, y) | x > 0

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

We use the following integer mapping:

  J(mult#) = arg_1

We thus have:

  (1) x > 0 |= x > x - 1 (and x >= 0)

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

Processor output: { }.