MAYBE Initial complexity problem: 1: T: (?, 1) f49(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f49(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) (?, 1) f51(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f54(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) (?, 1) f11(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f49(a, b, 0, 0, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) [ a >= b ] (?, 1) f35(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f49(a, b, 0, 0, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) [ e >= 3 ] (?, 1) f35(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f49(a, b, 0, 0, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) [ 1 >= e ] (?, 1) f35(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f49(a, b, 0, 0, 2, g, g, h, i, j, k, l, m, n, o, p, q, r, s, t) [ e = 2 ] (?, 1) f11(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f49(a, b, 0, 0, e, f, u, v, w, x, d, u, u, n, o, p, q, r, s, t) [ u >= 1 /\ b >= a + 1 ] (?, 1) f11(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f35(a, b, c, d, y, f, u, v, w, x, d, u, u, u, p, 0, y, y, 0, t) [ b >= a + 1 /\ 0 >= u /\ y >= 2 ] (?, 1) f11(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f35(a, b, c, d, y, f, u, v, w, x, d, u, u, u, p, 0, y, y, 0, t) [ b >= a + 1 /\ 0 >= u /\ 0 >= y ] (?, 1) f11(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f11(a + 1, b, c, d, 1, f, u, v, w, x, d, u, u, u, p, p, 1, 1, 0, t) [ 0 >= u /\ b >= a + 1 ] (1, 1) f0(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t) -> f11(a, b, 0, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, 0) start location: f0 leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [a, b, e]. We thus obtain the following problem: 2: T: (1, 1) f0(a, b, e) -> f11(a, b, e) (?, 1) f11(a, b, e) -> f11(a + 1, b, 1) [ 0 >= u /\ b >= a + 1 ] (?, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ 0 >= y ] (?, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ y >= 2 ] (?, 1) f11(a, b, e) -> f49(a, b, e) [ u >= 1 /\ b >= a + 1 ] (?, 1) f35(a, b, e) -> f49(a, b, 2) [ e = 2 ] (?, 1) f35(a, b, e) -> f49(a, b, e) [ 1 >= e ] (?, 1) f35(a, b, e) -> f49(a, b, e) [ e >= 3 ] (?, 1) f11(a, b, e) -> f49(a, b, e) [ a >= b ] (?, 1) f51(a, b, e) -> f54(a, b, e) (?, 1) f49(a, b, e) -> f49(a, b, e) 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, b, e) -> f11(a, b, e) (?, 1) f11(a, b, e) -> f11(a + 1, b, 1) [ 0 >= u /\ b >= a + 1 ] (?, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ 0 >= y ] (?, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ y >= 2 ] (?, 1) f11(a, b, e) -> f49(a, b, e) [ u >= 1 /\ b >= a + 1 ] (?, 1) f35(a, b, e) -> f49(a, b, 2) [ e = 2 ] (?, 1) f35(a, b, e) -> f49(a, b, e) [ 1 >= e ] (?, 1) f35(a, b, e) -> f49(a, b, e) [ e >= 3 ] (?, 1) f11(a, b, e) -> f49(a, b, e) [ a >= b ] (?, 1) f49(a, b, e) -> f49(a, b, e) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 2 Pol(f11) = 2 Pol(f35) = 1 Pol(f49) = 0 orients all transitions weakly and the transitions f35(a, b, e) -> f49(a, b, 2) [ e = 2 ] f35(a, b, e) -> f49(a, b, e) [ 1 >= e ] f35(a, b, e) -> f49(a, b, e) [ e >= 3 ] f11(a, b, e) -> f49(a, b, e) [ u >= 1 /\ b >= a + 1 ] f11(a, b, e) -> f49(a, b, e) [ a >= b ] f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ y >= 2 ] f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ 0 >= y ] strictly and produces the following problem: 4: T: (1, 1) f0(a, b, e) -> f11(a, b, e) (?, 1) f11(a, b, e) -> f11(a + 1, b, 1) [ 0 >= u /\ b >= a + 1 ] (2, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ 0 >= y ] (2, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ y >= 2 ] (2, 1) f11(a, b, e) -> f49(a, b, e) [ u >= 1 /\ b >= a + 1 ] (2, 1) f35(a, b, e) -> f49(a, b, 2) [ e = 2 ] (2, 1) f35(a, b, e) -> f49(a, b, e) [ 1 >= e ] (2, 1) f35(a, b, e) -> f49(a, b, e) [ e >= 3 ] (2, 1) f11(a, b, e) -> f49(a, b, e) [ a >= b ] (?, 1) f49(a, b, e) -> f49(a, b, e) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = -V_1 + V_2 Pol(f11) = -V_1 + V_2 Pol(f35) = -V_1 + V_2 Pol(f49) = -V_1 + V_2 orients all transitions weakly and the transition f11(a, b, e) -> f11(a + 1, b, 1) [ 0 >= u /\ b >= a + 1 ] strictly and produces the following problem: 5: T: (1, 1) f0(a, b, e) -> f11(a, b, e) (a + b, 1) f11(a, b, e) -> f11(a + 1, b, 1) [ 0 >= u /\ b >= a + 1 ] (2, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ 0 >= y ] (2, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ y >= 2 ] (2, 1) f11(a, b, e) -> f49(a, b, e) [ u >= 1 /\ b >= a + 1 ] (2, 1) f35(a, b, e) -> f49(a, b, 2) [ e = 2 ] (2, 1) f35(a, b, e) -> f49(a, b, e) [ 1 >= e ] (2, 1) f35(a, b, e) -> f49(a, b, e) [ e >= 3 ] (2, 1) f11(a, b, e) -> f49(a, b, e) [ a >= b ] (?, 1) f49(a, b, e) -> f49(a, b, e) start location: f0 leaf cost: 1 Applied AI with 'oct' on problem 5 to obtain the following invariants: For symbol f35: -X_1 + X_2 - 1 >= 0 This yielded the following problem: 6: T: (?, 1) f49(a, b, e) -> f49(a, b, e) (2, 1) f11(a, b, e) -> f49(a, b, e) [ a >= b ] (2, 1) f35(a, b, e) -> f49(a, b, e) [ -a + b - 1 >= 0 /\ e >= 3 ] (2, 1) f35(a, b, e) -> f49(a, b, e) [ -a + b - 1 >= 0 /\ 1 >= e ] (2, 1) f35(a, b, e) -> f49(a, b, 2) [ -a + b - 1 >= 0 /\ e = 2 ] (2, 1) f11(a, b, e) -> f49(a, b, e) [ u >= 1 /\ b >= a + 1 ] (2, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ y >= 2 ] (2, 1) f11(a, b, e) -> f35(a, b, y) [ b >= a + 1 /\ 0 >= u /\ 0 >= y ] (a + b, 1) f11(a, b, e) -> f11(a + 1, b, 1) [ 0 >= u /\ b >= a + 1 ] (1, 1) f0(a, b, e) -> f11(a, b, e) start location: f0 leaf cost: 1 Complexity upper bound ? Time: 0.473 sec (SMT: 0.442 sec)