YES(?, 24) Initial complexity problem: 1: T: (1, 1) f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\ 3 >= a /\ 3 >= b /\ b >= 0 ] (?, 1) f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ] (?, 1) f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f1) = V_1 - 2*V_2 + V_3 and size complexities S("f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ]", 0-0) = 3 S("f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ]", 0-1) = ? S("f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ]", 0-2) = 2 S("f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ]", 0-0) = 3 S("f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ]", 0-1) = ? S("f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ]", 0-2) = 2 S("f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\\ 3 >= a /\\ 3 >= b /\\ b >= 0 ]", 0-0) = 3 S("f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\\ 3 >= a /\\ 3 >= b /\\ b >= 0 ]", 0-1) = 3 S("f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\\ 3 >= a /\\ 3 >= b /\\ b >= 0 ]", 0-2) = 2 orients the transition f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ] weakly and the transition f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ] strictly and produces the following problem: 2: T: (1, 1) f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\ 3 >= a /\ 3 >= b /\ b >= 0 ] (11, 1) f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ] (?, 1) f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ] start location: f0 leaf cost: 0 A polynomial rank function with Pol(f1) = -V_1 + 2*V_2 - V_3 - 1 and size complexities S("f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ]", 0-0) = 3 S("f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ]", 0-1) = ? S("f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ]", 0-2) = 2 S("f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ]", 0-0) = 3 S("f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ]", 0-1) = 14 S("f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ]", 0-2) = 2 S("f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\\ 3 >= a /\\ 3 >= b /\\ b >= 0 ]", 0-0) = 3 S("f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\\ 3 >= a /\\ 3 >= b /\\ b >= 0 ]", 0-1) = 3 S("f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\\ 3 >= a /\\ 3 >= b /\\ b >= 0 ]", 0-2) = 2 orients the transition f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ] weakly and the transition f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b, c) -> f1(a, b, 2) [ a >= 0 /\ 3 >= a /\ 3 >= b /\ b >= 0 ] (11, 1) f1(a, b, c) -> f1(a, b + 1, c) [ c + a >= 2*b + 1 ] (12, 1) f1(a, b, c) -> f1(a, b - 1, c) [ 2*b >= c + a + 2 ] start location: f0 leaf cost: 0 Complexity upper bound 24 Time: 0.189 sec (SMT: 0.177 sec)