YES(?, 9*a*b + b^2 + 25*b + 8*a + 8*a^2 + 10) Initial complexity problem: 1: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ 0 >= a + 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ 0 >= a + 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ a >= 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ a >= 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ 0 >= b + 1 ] start location: f0 leaf cost: 0 Testing for unsatisfiable constraints removes the following transitions from problem 1: f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ 0 >= a + 1 /\ a + b >= 2 ] f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ 0 >= a + 1 /\ a + b >= 2 ] f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ b >= 1 ] f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ b >= 1 ] f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ a >= 1 /\ 0 >= b + a + 1 ] f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ a >= 1 /\ 0 >= b + a + 1 ] f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ 0 >= b + 1 ] f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ 0 >= b + 1 ] We thus obtain the following problem: 2: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 + V_2 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transition f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] weakly and the transition f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = V_2 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transition f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] weakly and the transition f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] strictly and produces the following problem: 4: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (a*b + b^2 + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 + V_2 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transition f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] weakly and the transition f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] strictly and produces the following problem: 5: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (a*b + b^2 + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (a + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = -6*V_2 - 2 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transitions f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] weakly and the transition f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] strictly and produces the following problem: 6: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (a*b + b^2 + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (a + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (20*b + 6*a^2 + 6*a*b + 2*a + 6, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transition f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] weakly and the transition f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] strictly and produces the following problem: 7: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (a*b + b^2 + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (a + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (a^2 + a*b + a, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (20*b + 6*a^2 + 6*a*b + 2*a + 6, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 - V_2 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transition f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] weakly and the transition f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] strictly and produces the following problem: 8: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (a*b + b^2 + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (a + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (a^2 + a*b + a, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (a + b, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (20*b + 6*a^2 + 6*a*b + 2*a + 6, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (?, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transition f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] weakly and the transition f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] strictly and produces the following problem: 9: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (a*b + b^2 + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (a + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (a^2 + a*b + a, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (?, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (a + b, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (20*b + 6*a^2 + 6*a*b + 2*a + 6, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (a^2 + a*b + a, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 - V_2 and size complexities S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-0) = a S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ 0 >= a + 2 /\\ b >= 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-0) = b S("f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\\ b >= a + 1 /\\ a >= 0 /\\ b >= 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ 0 >= b + 2 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = b S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-0) = a S("f2(a, b) -> f2(a, b + 1) [ b >= a /\\ b >= 0 /\\ 0 >= a + 1 /\\ 0 >= b + a + 1 ]", 0-1) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-0) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ 0 >= a /\\ 0 >= b + 1 ]", 0-1) = b S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-0) = a S("f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\\ a >= b + 1 /\\ a >= 2 /\\ 0 >= b + 1 ]", 0-1) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ 0 >= b /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = a S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-0) = a + b S("f2(a, b) -> f2(a, b - 1) [ a >= b /\\ b >= 2 /\\ a >= 1 /\\ a + b >= 2 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\\ b >= 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ 0 >= b + 1 ]", 0-1) = b S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-0) = a S("f0(a, b) -> f2(a, b) [ a >= 1 /\\ b >= 1 ]", 0-1) = b orients the transition f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] weakly and the transition f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] strictly and produces the following problem: 10: T: (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ a >= 1 /\ 0 >= b + 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ b >= 1 ] (1, 1) f0(a, b) -> f2(a, b) [ 0 >= a + 1 /\ 0 >= b + 1 ] (a*b + b^2 + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ b >= 2 /\ a >= 1 /\ a + b >= 2 ] (a + b, 1) f2(a, b) -> f2(a, b - 1) [ a >= b /\ 0 >= b /\ a >= 1 /\ a + b >= 2 ] (a^2 + a*b + a, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ a >= 2 /\ 0 >= b + 1 ] (a + b, 1) f2(a, b) -> f2(a - 1, b) [ 1 >= a + b /\ a >= b + 1 /\ 0 >= a /\ 0 >= b + 1 ] (a + b, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ b >= 0 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (20*b + 6*a^2 + 6*a*b + 2*a + 6, 1) f2(a, b) -> f2(a, b + 1) [ b >= a /\ 0 >= b + 2 /\ 0 >= a + 1 /\ 0 >= b + a + 1 ] (a + b, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ a >= 0 /\ b >= 1 ] (a^2 + a*b + a, 1) f2(a, b) -> f2(a + 1, b) [ b + a >= 0 /\ b >= a + 1 /\ 0 >= a + 2 /\ b >= 1 ] start location: f0 leaf cost: 0 Complexity upper bound 9*a*b + b^2 + 25*b + 8*a + 8*a^2 + 10 Time: 1.572 sec (SMT: 1.485 sec)