MAYBE Initial complexity problem: 1: T: (?, 1) f11(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f11(a, b, b, b, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) [ a >= b + 1 ] (?, 1) f11(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f11(a, b, b, b, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) [ b >= a + 1 ] (?, 1) f6(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f6(a, b, c, d, e, h1, g + 1, h - 1, g + 1, h - 1, f1, g1, h - 1, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) [ e >= f + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f6(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f6(a, b, c, d, e, h1, g + 1, h - 1, g + 1, h - 1, f1, g1, h - 1, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) [ f >= e + 1 /\ g >= 0 /\ h >= 1 ] (1, 1) f26(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f6(a, b, c, d, e, l1, 1, 4, 1, 4, k, l, m, f1, 2, 3, g1, h1, 4, j1, 0, k1, m1, m1, m1, m1, n1, o1, p1, 4, e1) [ e >= i1 + 1 ] (1, 1) f26(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f6(a, b, c, d, e, l1, 1, 4, 1, 4, k, l, m, f1, 2, 3, g1, h1, 4, j1, 0, k1, m1, m1, m1, m1, n1, o1, p1, 4, e1) [ i1 >= e + 1 ] (?, 1) f6(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f11(a, j1, j1, j1, f, f, g, h, i, j, k, l, m, f1, o, p, q, r, s, t, u, v, w, x, y, g1, a1, b1, c1, d1, h1) [ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (?, 1) f6(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, a1, b1, c1, d1, e1) -> f11(a, j1, j1, j1, f, f, g, h, i, j, k, l, m, f1, o, p, q, r, s, t, u, v, w, x, y, g1, a1, b1, c1, d1, h1) [ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] start location: f26 leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [a, b, e, f, g, h]. We thus obtain the following problem: 2: T: (?, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (?, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ i1 >= e + 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ e >= i1 + 1 ] (?, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ f >= e + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ e >= f + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ b >= a + 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ a >= b + 1 ] start location: f26 leaf cost: 0 A polynomial rank function with Pol(f6) = 1 Pol(f11) = -1 Pol(f26) = 1 orients all transitions weakly and the transition f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] strictly and produces the following problem: 3: T: (?, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ i1 >= e + 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ e >= i1 + 1 ] (?, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ f >= e + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ e >= f + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ b >= a + 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ a >= b + 1 ] start location: f26 leaf cost: 0 A polynomial rank function with Pol(f6) = 1 Pol(f11) = -1 Pol(f26) = 1 orients all transitions weakly and the transition f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] strictly and produces the following problem: 4: T: (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ i1 >= e + 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ e >= i1 + 1 ] (?, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ f >= e + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ e >= f + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ b >= a + 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ a >= b + 1 ] start location: f26 leaf cost: 0 A polynomial rank function with Pol(f6) = V_5 + 2*V_6 - 1 Pol(f11) = -V_3 + V_4 + V_5 + 2*V_6 - 1 Pol(f26) = 8 orients all transitions weakly and the transitions f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ f >= e + 1 /\ g >= 0 /\ h >= 1 ] f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ e >= f + 1 /\ g >= 0 /\ h >= 1 ] strictly and produces the following problem: 5: T: (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ i1 >= e + 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ e >= i1 + 1 ] (8, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ f >= e + 1 /\ g >= 0 /\ h >= 1 ] (8, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ e >= f + 1 /\ g >= 0 /\ h >= 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ b >= a + 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ a >= b + 1 ] start location: f26 leaf cost: 0 Applied AI with 'oct' on problem 5 to obtain the following invariants: For symbol f11: -X_6 + 4 >= 0 /\ X_5 - X_6 + 3 >= 0 /\ -X_5 - X_6 + 5 >= 0 /\ X_6 - 1 >= 0 /\ X_5 + X_6 - 5 >= 0 /\ -X_5 + X_6 + 3 >= 0 /\ -X_5 + 4 >= 0 /\ X_5 - 1 >= 0 /\ X_3 - X_4 >= 0 /\ -X_3 + X_4 >= 0 For symbol f6: -X_6 + 4 >= 0 /\ X_5 - X_6 + 3 >= 0 /\ -X_5 - X_6 + 5 >= 0 /\ X_5 + X_6 - 5 >= 0 /\ X_5 - 1 >= 0 This yielded the following problem: 6: T: (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ h - 1 >= 0 /\ g + h - 5 >= 0 /\ -g + h + 3 >= 0 /\ -g + 4 >= 0 /\ g - 1 >= 0 /\ e - f >= 0 /\ -e + f >= 0 /\ a >= b + 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ h - 1 >= 0 /\ g + h - 5 >= 0 /\ -g + h + 3 >= 0 /\ -g + 4 >= 0 /\ g - 1 >= 0 /\ e - f >= 0 /\ -e + f >= 0 /\ b >= a + 1 ] (8, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ e >= f + 1 /\ g >= 0 /\ h >= 1 ] (8, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ f >= e + 1 /\ g >= 0 /\ h >= 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ e >= i1 + 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ i1 >= e + 1 ] (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] start location: f26 leaf cost: 0 By chaining the transition f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] with all transitions in problem 6, the following new transition is obtained: f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e /\ h - 1 >= 0 /\ -g + h + 3 >= 0 /\ -g + 4 >= 0 /\ 0 >= 0 ] We thus obtain the following problem: 7: T: (1, 2) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ a >= j1 + 1 /\ g >= 0 /\ h >= 1 /\ f = e /\ h - 1 >= 0 /\ -g + h + 3 >= 0 /\ -g + 4 >= 0 /\ 0 >= 0 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ h - 1 >= 0 /\ g + h - 5 >= 0 /\ -g + h + 3 >= 0 /\ -g + 4 >= 0 /\ g - 1 >= 0 /\ e - f >= 0 /\ -e + f >= 0 /\ a >= b + 1 ] (?, 1) f11(a, b, e, f, g, h) -> f11(a, b, e, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ h - 1 >= 0 /\ g + h - 5 >= 0 /\ -g + h + 3 >= 0 /\ -g + 4 >= 0 /\ g - 1 >= 0 /\ e - f >= 0 /\ -e + f >= 0 /\ b >= a + 1 ] (8, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ e >= f + 1 /\ g >= 0 /\ h >= 1 ] (8, 1) f6(a, b, e, f, g, h) -> f6(a, b, e, h1, g + 1, h - 1) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ f >= e + 1 /\ g >= 0 /\ h >= 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ e >= i1 + 1 ] (1, 1) f26(a, b, e, f, g, h) -> f6(a, b, e, l1, 1, 4) [ i1 >= e + 1 ] (1, 1) f6(a, b, e, f, g, h) -> f11(a, j1, f, f, g, h) [ -h + 4 >= 0 /\ g - h + 3 >= 0 /\ -g - h + 5 >= 0 /\ g + h - 5 >= 0 /\ g - 1 >= 0 /\ j1 >= a + 1 /\ g >= 0 /\ h >= 1 /\ f = e ] start location: f26 leaf cost: 0 Complexity upper bound ? Time: 1.293 sec (SMT: 1.202 sec)