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

  Signature: condAcc :: Bool -> Int -> Int -> Int
             sqrt    :: Int -> Int
             sqrtAcc :: Int -> Int -> Int

  Rules: sqrt(x) -> sqrtAcc(x, 0)
         sqrtAcc(x, y) -> condAcc(y * y >= x \/ y < 0, x, y)
         condAcc(true, x, y) -> y
         condAcc(false, x, y) -> sqrtAcc(x, y + 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) sqrt#(x) => sqrtAcc#(x, 0)
      (2) sqrtAcc#(x, y) => condAcc#(y * y >= x \/ y < 0, x, y)
      (3) condAcc#(false, x, y) => sqrtAcc#(x, y + 1)

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

We compute a graph approximation with the following edges:

  1: 2 
  2: 3 
  3: 2 

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

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

  P2. (1) sqrtAcc#(x, y) => condAcc#(y * y >= x \/ y < 0, x, y)
      (2) condAcc#(false, x, y) => sqrtAcc#(x, y + 1)

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

We use the following theory arguments function:

  condAcc# : [1, 2, 3]
  sqrtAcc# : [1, 2]

Processor output: { D3 = (P3, R, f, c) ; D4 = (P4, R, f, c) }, where:

  P3. (1) sqrtAcc#(x, y) => condAcc#(y * y >= x \/ y < 0, x, y)
      (2) condAcc#(false, x, y) => sqrtAcc#(x, y + 1) { x, y }

  P4. (1) condAcc#(false, x, y) => sqrtAcc#(x, y + 1)

***** We apply the Theory Arguments Processor on D3 = (P3, R, f, c).

We use the following theory arguments function:

  condAcc# : [1, 2, 3]
  sqrtAcc# : [1, 2]

Processor output: { D5 = (P5, R, f, c) ; D6 = (P6, R, f, c) }, where:

  P5. (1) sqrtAcc#(x, y) => condAcc#(y * y >= x \/ y < 0, x, y) { x, y }
      (2) condAcc#(false, x, y) => sqrtAcc#(x, y + 1) { x, y }

  P6. (1) sqrtAcc#(x, y) => condAcc#(y * y >= x \/ y < 0, x, y)

***** We apply the Graph Processor on D4 = (P4, 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: { }.

***** No progress could be made on DP problem D5 = (P5, R, f, c).