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

  Signature: eval_1 :: Int -> Int -> Int -> o
             eval_2 :: Int -> Int -> Int -> o

  Rules: eval_1(x, y, z) -> eval_2(x, y, z) | x > z
         eval_2(x, y, z) -> eval_2(x, y - 1, z) | x > z /\ y > z
         eval_2(x, y, z) -> eval_1(x - 1, y, z) | x > z /\ z >= 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 UNION R_?, f, c), where:

  P1. (1) eval_1#(x, y, z) => eval_2#(x, y, z) | x > z
      (2) eval_2#(x, y, z) => eval_2#(x, y - 1, z) | x > z /\ y > z
      (3) eval_2#(x, y, z) => eval_1#(x - 1, y, z) | x > z /\ z >= y

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

We use the following theory arguments function:

  eval_2# : [1, 2, 3]

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

  P2. (1) eval_1#(x, y, z) => eval_2#(x, y, z) | x > z { x, y, z }
      (2) eval_2#(x, y, z) => eval_2#(x, y - 1, z) | x > z /\ y > z
      (3) eval_2#(x, y, z) => eval_1#(x - 1, y, z) | x > z /\ z >= y

  P3. (1) eval_1#(x, y, z) => eval_2#(x, y, z) | x > z

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

We use the following integer mapping:

  J(eval_1#) = arg_1 - arg_3 - 1
  J(eval_2#) = arg_1 - arg_3 - 1

We thus have:

  (1) x > z           |= x - z - 1 >= x - z - 1
  (2) x > z /\ y > z  |= x - z - 1 >= x - z - 1
  (3) x > z /\ z >= y |= x - z - 1 >  x - 1 - z - 1 (and x - z - 1 >= 0)

We may remove the strictly oriented DPs, which yields:

Processor output: { D4 = (P4, R UNION R_?, f, c) }, where:

  P4. (1) eval_1#(x, y, z) => eval_2#(x, y, z) | x > z { x, y, z }
      (2) eval_2#(x, y, z) => eval_2#(x, y - 1, z) | x > z /\ y > z

***** We apply the Graph Processor on D3 = (P3, R UNION R_?, f, c).

We compute a graph approximation with the following edges:

  1:

As there are no SCCs, this DP problem is removed.

Processor output: { }.

***** We apply the Graph Processor on D4 = (P4, R UNION R_?, f, c).

We compute a graph approximation with the following edges:

  1: 2 
  2: 2 

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

Processor output: { D5 = (P5, R UNION R_?, f, c) }, where:

  P5. (1) eval_2#(x, y, z) => eval_2#(x, y - 1, z) | x > z /\ y > z

***** We apply the Integer Function Processor on D5 = (P5, R UNION R_?, f, c).

We use the following integer mapping:

  J(eval_2#) = arg_2 - arg_3

We thus have:

  (1) x > z /\ y > z |= y - z > y - 1 - z (and y - z >= 0)

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

Processor output: { }.