YES(?, 104) Initial complexity problem: 1: T: (1, 1) f0(a, b, c, d, e, f, g) -> f5(0, b, c, d, e, f, g) (?, 1) f5(a, b, c, d, e, f, g) -> f5(a + 1, b, c, d, e, f, g) [ 99 >= a ] (?, 1) f17(a, b, c, d, e, f, g) -> f17(a, b, c, d, e, f, g) (?, 1) f17(a, b, c, d, e, f, g) -> f17(a, b + 1, c, d, e, f, g) [ 0 >= h + 1 ] (?, 1) f17(a, b, c, d, e, f, g) -> f17(a, b + 1, c, d, e, f, g) (?, 1) f32(a, b, c, d, e, f, g) -> f32(a, b, c, d, e, f, g) (?, 1) f32(a, b, c, d, e, f, g) -> f32(a, b, c + 1, d, e, f, g) [ 0 >= h + 1 ] (?, 1) f32(a, b, c, d, e, f, g) -> f32(a, b, c + 1, d, e, f, g) (?, 1) f32(a, b, c, d, e, f, g) -> f13(a, b, c, c, c, f, g) (?, 1) f17(a, b, c, d, e, f, g) -> f32(a, b, b, b, e, b, h) [ 0 >= i + 1 ] (?, 1) f17(a, b, c, d, e, f, g) -> f32(a, b, b, b, e, b, h) (?, 1) f17(a, b, c, d, e, f, g) -> f13(a, b, c, b, e, b, h) (?, 1) f5(a, b, c, d, e, f, g) -> f13(a, b, c, a - 2, e, f, g) [ a >= 100 ] (?, 1) f5(a, b, c, d, e, f, g) -> f17(a, a - 2, c, a - 2, e, f, g) [ 0 >= a + 1 /\ a >= 100 ] start location: f0 leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [a]. We thus obtain the following problem: 2: T: (?, 1) f5(a) -> f17(a) [ 0 >= a + 1 /\ a >= 100 ] (?, 1) f5(a) -> f13(a) [ a >= 100 ] (?, 1) f17(a) -> f13(a) (?, 1) f17(a) -> f32(a) (?, 1) f17(a) -> f32(a) [ 0 >= i + 1 ] (?, 1) f32(a) -> f13(a) (?, 1) f32(a) -> f32(a) (?, 1) f32(a) -> f32(a) [ 0 >= h + 1 ] (?, 1) f32(a) -> f32(a) (?, 1) f17(a) -> f17(a) (?, 1) f17(a) -> f17(a) [ 0 >= h + 1 ] (?, 1) f17(a) -> f17(a) (?, 1) f5(a) -> f5(a + 1) [ 99 >= a ] (1, 1) f0(a) -> f5(0) start location: f0 leaf cost: 0 Testing for unsatisfiable constraints removes the following transition from problem 2: f5(a) -> f17(a) [ 0 >= a + 1 /\ a >= 100 ] We thus obtain the following problem: 3: T: (?, 1) f5(a) -> f13(a) [ a >= 100 ] (?, 1) f17(a) -> f13(a) (?, 1) f17(a) -> f32(a) (?, 1) f17(a) -> f32(a) [ 0 >= i + 1 ] (?, 1) f32(a) -> f13(a) (?, 1) f32(a) -> f32(a) (?, 1) f32(a) -> f32(a) [ 0 >= h + 1 ] (?, 1) f32(a) -> f32(a) (?, 1) f17(a) -> f17(a) (?, 1) f17(a) -> f17(a) [ 0 >= h + 1 ] (?, 1) f17(a) -> f17(a) (?, 1) f5(a) -> f5(a + 1) [ 99 >= a ] (1, 1) f0(a) -> f5(0) start location: f0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 3 produces the following problem: 4: T: (?, 1) f17(a) -> f32(a) (?, 1) f17(a) -> f32(a) [ 0 >= i + 1 ] (?, 1) f32(a) -> f32(a) (?, 1) f32(a) -> f32(a) [ 0 >= h + 1 ] (?, 1) f32(a) -> f32(a) (?, 1) f17(a) -> f17(a) (?, 1) f17(a) -> f17(a) [ 0 >= h + 1 ] (?, 1) f17(a) -> f17(a) (?, 1) f5(a) -> f5(a + 1) [ 99 >= a ] (1, 1) f0(a) -> f5(0) start location: f0 leaf cost: 3 Testing for reachability in the complexity graph removes the following transitions from problem 4: f17(a) -> f32(a) f17(a) -> f32(a) [ 0 >= i + 1 ] f32(a) -> f32(a) f32(a) -> f32(a) [ 0 >= h + 1 ] f32(a) -> f32(a) f17(a) -> f17(a) f17(a) -> f17(a) [ 0 >= h + 1 ] f17(a) -> f17(a) We thus obtain the following problem: 5: T: (?, 1) f5(a) -> f5(a + 1) [ 99 >= a ] (1, 1) f0(a) -> f5(0) start location: f0 leaf cost: 3 A polynomial rank function with Pol(f5) = -V_1 + 100 Pol(f0) = 100 orients all transitions weakly and the transition f5(a) -> f5(a + 1) [ 99 >= a ] strictly and produces the following problem: 6: T: (100, 1) f5(a) -> f5(a + 1) [ 99 >= a ] (1, 1) f0(a) -> f5(0) start location: f0 leaf cost: 3 Complexity upper bound 104 Time: 0.233 sec (SMT: 0.223 sec)