We consider universal computability of the STRS with no additional rule schemes:

  Signature: false :: a
             if    :: a -> b -> b -> b
             true  :: a
             until :: (b -> a) -> (b -> b) -> b -> b

  Rules: if(true, X, Y) -> X
         if(false, U, V) -> V
         until(J, I, X1) -> if(J(X1), X1, until(J, I, I(X1)))

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) until#(J, I, X1) => until#(J, I, I(X1))
      (2) until#(J, I, X1) => if#(J(X1), X1, until(J, I, I(X1)))

***** We apply the Graph Processor on D1 = (P1, R UNION 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: { D2 = (P2, R UNION R_?, f, c) }, where:

  P2. (1) until#(J, I, X1) => until#(J, I, I(X1))

***** No progress could be made on DP problem D2 = (P2, R UNION R_?, f, c).