YES(?, 4*b + 6) Initial complexity problem: 1: T: (1, 1) evalSimpleSinglestart(a, b) -> evalSimpleSingleentryin(a, b) (?, 1) evalSimpleSingleentryin(a, b) -> evalSimpleSinglebb3in(0, b) (?, 1) evalSimpleSinglebb3in(a, b) -> evalSimpleSinglebbin(a, b) [ b >= a + 1 ] (?, 1) evalSimpleSinglebb3in(a, b) -> evalSimpleSinglereturnin(a, b) [ a >= b ] (?, 1) evalSimpleSinglebbin(a, b) -> evalSimpleSinglebb3in(a + 1, b) (?, 1) evalSimpleSinglereturnin(a, b) -> evalSimpleSinglestop(a, b) start location: evalSimpleSinglestart leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) evalSimpleSinglestart(a, b) -> evalSimpleSingleentryin(a, b) (?, 1) evalSimpleSingleentryin(a, b) -> evalSimpleSinglebb3in(0, b) (?, 1) evalSimpleSinglebb3in(a, b) -> evalSimpleSinglebbin(a, b) [ b >= a + 1 ] (?, 1) evalSimpleSinglebbin(a, b) -> evalSimpleSinglebb3in(a + 1, b) start location: evalSimpleSinglestart leaf cost: 2 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) evalSimpleSinglestart(a, b) -> evalSimpleSingleentryin(a, b) (1, 1) evalSimpleSingleentryin(a, b) -> evalSimpleSinglebb3in(0, b) (?, 1) evalSimpleSinglebb3in(a, b) -> evalSimpleSinglebbin(a, b) [ b >= a + 1 ] (?, 1) evalSimpleSinglebbin(a, b) -> evalSimpleSinglebb3in(a + 1, b) start location: evalSimpleSinglestart leaf cost: 2 A polynomial rank function with Pol(evalSimpleSinglestart) = 2*V_2 + 1 Pol(evalSimpleSingleentryin) = 2*V_2 + 1 Pol(evalSimpleSinglebb3in) = -2*V_1 + 2*V_2 + 1 Pol(evalSimpleSinglebbin) = -2*V_1 + 2*V_2 orients all transitions weakly and the transition evalSimpleSinglebb3in(a, b) -> evalSimpleSinglebbin(a, b) [ b >= a + 1 ] strictly and produces the following problem: 4: T: (1, 1) evalSimpleSinglestart(a, b) -> evalSimpleSingleentryin(a, b) (1, 1) evalSimpleSingleentryin(a, b) -> evalSimpleSinglebb3in(0, b) (2*b + 1, 1) evalSimpleSinglebb3in(a, b) -> evalSimpleSinglebbin(a, b) [ b >= a + 1 ] (?, 1) evalSimpleSinglebbin(a, b) -> evalSimpleSinglebb3in(a + 1, b) start location: evalSimpleSinglestart leaf cost: 2 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (1, 1) evalSimpleSinglestart(a, b) -> evalSimpleSingleentryin(a, b) (1, 1) evalSimpleSingleentryin(a, b) -> evalSimpleSinglebb3in(0, b) (2*b + 1, 1) evalSimpleSinglebb3in(a, b) -> evalSimpleSinglebbin(a, b) [ b >= a + 1 ] (2*b + 1, 1) evalSimpleSinglebbin(a, b) -> evalSimpleSinglebb3in(a + 1, b) start location: evalSimpleSinglestart leaf cost: 2 Complexity upper bound 4*b + 6 Time: 0.087 sec (SMT: 0.084 sec)