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

  Signature: eval :: Int -> Int -> Int -> o

  Rules: eval(i, j, k) -> eval(j, i + 1, k - 1) | i <= 100 /\ j <= k

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) eval#(i, j, k) => eval#(j, i + 1, k - 1) | i <= 100 /\ j <= k

***** We apply the Usable Rules Processor on D1 = (P1, R UNION R_?, i, c).

We obtain 0 usable rules (out of 1 rules in the input problem).

Processor output: { D2 = (P1, {}, i, c) }.

***** We apply the Usable rules with respect to HORPO Processor on D2 = (P1, {}, i, c).

Constrained HORPO yields:

  eval#3(i, j, k) (>) eval#3(j, i + 1, k - 1) | i <= 100 /\ j <= k

We do this using the following settings:

* Disregarded arguments:

  eval#3 3 

* Precedence and permutation:

   eval#3  { 1 2 }

* Well-founded theory orderings:

  [>]_{Bool} = {(true,false)}
  [>]_{Int}  = {(x,y) | x < 1000 /\ x < y }

Processor output: { }.