YES(?, 4*a + 4*b + 6) Initial complexity problem: 1: T: (1, 1) evalfstart(a, b) -> evalfentryin(a, b) (?, 1) evalfentryin(a, b) -> evalfbb1in(b, a) (?, 1) evalfbb1in(a, b) -> evalfbbin(a, b) [ a >= b ] (?, 1) evalfbb1in(a, b) -> evalfreturnin(a, b) [ b >= a + 1 ] (?, 1) evalfbbin(a, b) -> evalfbb1in(a, b + 1) (?, 1) evalfreturnin(a, b) -> evalfstop(a, b) start location: evalfstart leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) evalfstart(a, b) -> evalfentryin(a, b) (?, 1) evalfentryin(a, b) -> evalfbb1in(b, a) (?, 1) evalfbb1in(a, b) -> evalfbbin(a, b) [ a >= b ] (?, 1) evalfbbin(a, b) -> evalfbb1in(a, b + 1) start location: evalfstart leaf cost: 2 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) evalfstart(a, b) -> evalfentryin(a, b) (1, 1) evalfentryin(a, b) -> evalfbb1in(b, a) (?, 1) evalfbb1in(a, b) -> evalfbbin(a, b) [ a >= b ] (?, 1) evalfbbin(a, b) -> evalfbb1in(a, b + 1) start location: evalfstart leaf cost: 2 A polynomial rank function with Pol(evalfstart) = -2*V_1 + 2*V_2 + 1 Pol(evalfentryin) = -2*V_1 + 2*V_2 + 1 Pol(evalfbb1in) = 2*V_1 - 2*V_2 + 1 Pol(evalfbbin) = 2*V_1 - 2*V_2 orients all transitions weakly and the transition evalfbb1in(a, b) -> evalfbbin(a, b) [ a >= b ] strictly and produces the following problem: 4: T: (1, 1) evalfstart(a, b) -> evalfentryin(a, b) (1, 1) evalfentryin(a, b) -> evalfbb1in(b, a) (2*a + 2*b + 1, 1) evalfbb1in(a, b) -> evalfbbin(a, b) [ a >= b ] (?, 1) evalfbbin(a, b) -> evalfbb1in(a, b + 1) start location: evalfstart leaf cost: 2 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (1, 1) evalfstart(a, b) -> evalfentryin(a, b) (1, 1) evalfentryin(a, b) -> evalfbb1in(b, a) (2*a + 2*b + 1, 1) evalfbb1in(a, b) -> evalfbbin(a, b) [ a >= b ] (2*a + 2*b + 1, 1) evalfbbin(a, b) -> evalfbb1in(a, b + 1) start location: evalfstart leaf cost: 2 Complexity upper bound 4*a + 4*b + 6 Time: 0.083 sec (SMT: 0.079 sec)