The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the following initial DP problem:

  P1. (1) pow#(b, e) => condLoop#(e > 0, b, e, 1)
      (2) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (3) condMod#(false, b, e, r) => sqBase#(b, e, r)
      (4) condMod#(true, b, e, r) => sqBase#(b, e, r * b)
      (5) sqBase#(b, e, r) => halfExp#(b * b, e, r)
      (6) halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r)

***** We apply the Graph Processor on P1.

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

  P2. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (2) condMod#(false, b, e, r) => sqBase#(b, e, r)
      (3) condMod#(true, b, e, r) => sqBase#(b, e, r * b)
      (4) sqBase#(b, e, r) => halfExp#(b * b, e, r)
      (5) halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r)

***** We apply the Theory Arguments Processor on P2.

We use the following theory arguments function:

  condLoop# : [1, 2, 3, 4]
  condMod#  : [1, 2, 3, 4]
  halfExp#  : [1, 2, 3]
  sqBase#   : [1, 2, 3]

This yields the following new DP problems:

  P3. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (2) condMod#(false, b, e, r) => sqBase#(b, e, r) { b, e, r }
      (3) condMod#(true, b, e, r) => sqBase#(b, e, r * b) { b, e, r }
      (4) sqBase#(b, e, r) => halfExp#(b * b, e, r) { b, e, r }
      (5) halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r) { b, e, r }

  P4. (1) condMod#(false, b, e, r) => sqBase#(b, e, r)
      (2) condMod#(true, b, e, r) => sqBase#(b, e, r * b)
      (3) sqBase#(b, e, r) => halfExp#(b * b, e, r)
      (4) halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r)

***** We apply the Theory Arguments Processor on P3.

We use the following theory arguments function:

  condLoop# : [1, 2, 3, 4]
  condMod#  : [1, 2, 3, 4]
  halfExp#  : [1, 2, 3]
  sqBase#   : [1, 2, 3]

This yields the following new DP problems:

  P5. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r) { b, e, r }
      (2) condMod#(false, b, e, r) => sqBase#(b, e, r) { b, e, r }
      (3) condMod#(true, b, e, r) => sqBase#(b, e, r * b) { b, e, r }
      (4) sqBase#(b, e, r) => halfExp#(b * b, e, r) { b, e, r }
      (5) halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r) { b, e, r }

  P6. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)

***** We apply the Graph Processor on P4.

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

***** No progress could be made on DP problem P5.