MAYBE Initial complexity problem: 1: T: (?, 1) f10(a, b, c, d, e, f) -> f16(a, 0, g, g, e, f) [ 0 >= a ] (?, 1) f16(a, b, c, d, e, f) -> f16(a, b, c, d, e, f) [ d >= 1 ] (?, 1) f25(a, b, c, d, e, f) -> f25(a, b, c, d, e, f) (?, 1) f27(a, b, c, d, e, f) -> f30(a, b, c, d, e, f) (?, 1) f16(a, b, c, d, e, f) -> f10(g, b, c, d, 0, g) [ 0 >= d ] (?, 1) f10(a, b, c, d, e, f) -> f25(a, b, c, d, e, f) [ a >= 1 ] (1, 1) f0(a, b, c, d, e, f) -> f10(g, 0, c, d, 0, g) start location: f0 leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [a, d]. We thus obtain the following problem: 2: T: (1, 1) f0(a, d) -> f10(g, d) (?, 1) f10(a, d) -> f25(a, d) [ a >= 1 ] (?, 1) f16(a, d) -> f10(g, d) [ 0 >= d ] (?, 1) f27(a, d) -> f30(a, d) (?, 1) f25(a, d) -> f25(a, d) (?, 1) f16(a, d) -> f16(a, d) [ d >= 1 ] (?, 1) f10(a, d) -> f16(a, g) [ 0 >= a ] start location: f0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 2 produces the following problem: 3: T: (1, 1) f0(a, d) -> f10(g, d) (?, 1) f10(a, d) -> f25(a, d) [ a >= 1 ] (?, 1) f16(a, d) -> f10(g, d) [ 0 >= d ] (?, 1) f25(a, d) -> f25(a, d) (?, 1) f16(a, d) -> f16(a, d) [ d >= 1 ] (?, 1) f10(a, d) -> f16(a, g) [ 0 >= a ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 1 Pol(f10) = 1 Pol(f25) = 0 Pol(f16) = 1 orients all transitions weakly and the transition f10(a, d) -> f25(a, d) [ a >= 1 ] strictly and produces the following problem: 4: T: (1, 1) f0(a, d) -> f10(g, d) (1, 1) f10(a, d) -> f25(a, d) [ a >= 1 ] (?, 1) f16(a, d) -> f10(g, d) [ 0 >= d ] (?, 1) f25(a, d) -> f25(a, d) (?, 1) f16(a, d) -> f16(a, d) [ d >= 1 ] (?, 1) f10(a, d) -> f16(a, g) [ 0 >= a ] start location: f0 leaf cost: 1 Applied AI with 'oct' on problem 4 to obtain the following invariants: For symbol f16: -X_1 >= 0 For symbol f25: X_1 - 1 >= 0 This yielded the following problem: 5: T: (?, 1) f10(a, d) -> f16(a, g) [ 0 >= a ] (?, 1) f16(a, d) -> f16(a, d) [ -a >= 0 /\ d >= 1 ] (?, 1) f25(a, d) -> f25(a, d) [ a - 1 >= 0 ] (?, 1) f16(a, d) -> f10(g, d) [ -a >= 0 /\ 0 >= d ] (1, 1) f10(a, d) -> f25(a, d) [ a >= 1 ] (1, 1) f0(a, d) -> f10(g, d) start location: f0 leaf cost: 1 By chaining the transition f10(a, d) -> f25(a, d) [ a >= 1 ] with all transitions in problem 5, the following new transition is obtained: f10(a, d) -> f25(a, d) [ a >= 1 /\ a - 1 >= 0 ] We thus obtain the following problem: 6: T: (1, 2) f10(a, d) -> f25(a, d) [ a >= 1 /\ a - 1 >= 0 ] (?, 1) f10(a, d) -> f16(a, g) [ 0 >= a ] (?, 1) f16(a, d) -> f16(a, d) [ -a >= 0 /\ d >= 1 ] (?, 1) f25(a, d) -> f25(a, d) [ a - 1 >= 0 ] (?, 1) f16(a, d) -> f10(g, d) [ -a >= 0 /\ 0 >= d ] (1, 1) f0(a, d) -> f10(g, d) start location: f0 leaf cost: 1 Complexity upper bound ? Time: 0.289 sec (SMT: 0.270 sec)