We consider termination of the LCTRS with only rule scheme Calc:

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

  Rules: eval_1(i, j) -> eval_2(i, 0) | i >= 0
         eval_2(i, j) -> eval_2(i, j + 1) | j <= i - 1
         eval_2(i, j) -> eval_1(i - 1, j) | j > i - 1

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, f, c), where:

  P1. (1) eval_1#(i, j) => eval_2#(i, 0) | i >= 0
      (2) eval_2#(i, j) => eval_2#(i, j + 1) | j <= i - 1
      (3) eval_2#(i, j) => eval_1#(i - 1, j) | j > i - 1

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

We use the following theory arguments function:

  eval_2# : [1, 2]

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

  P2. (1) eval_1#(i, j) => eval_2#(i, 0) | i >= 0 { i, j }
      (2) eval_2#(i, j) => eval_2#(i, j + 1) | j <= i - 1
      (3) eval_2#(i, j) => eval_1#(i - 1, j) | j > i - 1

  P3. (1) eval_1#(i, j) => eval_2#(i, 0) | i >= 0

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

We use the following integer mapping:

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

We thus have:

  (1) i >= 0     |= i     >  i - 1 (and i >= 0)
  (2) j <= i - 1 |= i - 1 >= i - 1
  (3) j > i - 1  |= i - 1 >= i - 1

We may remove the strictly oriented DPs, which yields:

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

  P4. (1) eval_2#(i, j) => eval_2#(i, j + 1) | j <= i - 1
      (2) eval_2#(i, j) => eval_1#(i - 1, j) | j > i - 1

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

We compute a graph approximation with the following edges:

  1: 1 2 
  2:

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

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

  P5. (1) eval_2#(i, j) => eval_2#(i, j + 1) | j <= i - 1

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

We use the following integer mapping:

  J(eval_2#) = arg_1 - 1 - arg_2

We thus have:

  (1) j <= i - 1 |= i - 1 - j > i - 1 - (j + 1) (and i - 1 - j >= 0)

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

Processor output: { }.