YES(?, 15*b + 6) Initial complexity problem: 1: T: (1, 1) evalspeedpldi4start(a, b) -> evalspeedpldi4entryin(a, b) (?, 1) evalspeedpldi4entryin(a, b) -> evalspeedpldi4returnin(a, b) [ 0 >= a ] (?, 1) evalspeedpldi4entryin(a, b) -> evalspeedpldi4returnin(a, b) [ a >= b ] (?, 1) evalspeedpldi4entryin(a, b) -> evalspeedpldi4bb5in(a, b) [ a >= 1 /\ b >= a + 1 ] (?, 1) evalspeedpldi4bb5in(a, b) -> evalspeedpldi4bb2in(a, b) [ b >= 1 ] (?, 1) evalspeedpldi4bb5in(a, b) -> evalspeedpldi4returnin(a, b) [ 0 >= b ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb3in(a, b) [ a >= b + 1 ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb4in(a, b) [ b >= a ] (?, 1) evalspeedpldi4bb3in(a, b) -> evalspeedpldi4bb5in(a, b - 1) (?, 1) evalspeedpldi4bb4in(a, b) -> evalspeedpldi4bb5in(a, b - a) (?, 1) evalspeedpldi4returnin(a, b) -> evalspeedpldi4stop(a, b) start location: evalspeedpldi4start leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) evalspeedpldi4start(a, b) -> evalspeedpldi4entryin(a, b) (?, 1) evalspeedpldi4entryin(a, b) -> evalspeedpldi4bb5in(a, b) [ a >= 1 /\ b >= a + 1 ] (?, 1) evalspeedpldi4bb5in(a, b) -> evalspeedpldi4bb2in(a, b) [ b >= 1 ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb3in(a, b) [ a >= b + 1 ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb4in(a, b) [ b >= a ] (?, 1) evalspeedpldi4bb3in(a, b) -> evalspeedpldi4bb5in(a, b - 1) (?, 1) evalspeedpldi4bb4in(a, b) -> evalspeedpldi4bb5in(a, b - a) start location: evalspeedpldi4start leaf cost: 4 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) evalspeedpldi4start(a, b) -> evalspeedpldi4entryin(a, b) (1, 1) evalspeedpldi4entryin(a, b) -> evalspeedpldi4bb5in(a, b) [ a >= 1 /\ b >= a + 1 ] (?, 1) evalspeedpldi4bb5in(a, b) -> evalspeedpldi4bb2in(a, b) [ b >= 1 ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb3in(a, b) [ a >= b + 1 ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb4in(a, b) [ b >= a ] (?, 1) evalspeedpldi4bb3in(a, b) -> evalspeedpldi4bb5in(a, b - 1) (?, 1) evalspeedpldi4bb4in(a, b) -> evalspeedpldi4bb5in(a, b - a) start location: evalspeedpldi4start leaf cost: 4 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol evalspeedpldi4bb2in: X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ X_1 - 1 >= 0 For symbol evalspeedpldi4bb3in: X_1 - X_2 - 1 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ X_1 - 2 >= 0 For symbol evalspeedpldi4bb4in: X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ -X_1 + X_2 >= 0 /\ X_1 - 1 >= 0 For symbol evalspeedpldi4bb5in: X_1 - 1 >= 0 This yielded the following problem: 4: T: (?, 1) evalspeedpldi4bb4in(a, b) -> evalspeedpldi4bb5in(a, b - a) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ -a + b >= 0 /\ a - 1 >= 0 ] (?, 1) evalspeedpldi4bb3in(a, b) -> evalspeedpldi4bb5in(a, b - 1) [ a - b - 1 >= 0 /\ b - 1 >= 0 /\ a + b - 3 >= 0 /\ a - 2 >= 0 ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb4in(a, b) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ a - 1 >= 0 /\ b >= a ] (?, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb3in(a, b) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ a - 1 >= 0 /\ a >= b + 1 ] (?, 1) evalspeedpldi4bb5in(a, b) -> evalspeedpldi4bb2in(a, b) [ a - 1 >= 0 /\ b >= 1 ] (1, 1) evalspeedpldi4entryin(a, b) -> evalspeedpldi4bb5in(a, b) [ a >= 1 /\ b >= a + 1 ] (1, 1) evalspeedpldi4start(a, b) -> evalspeedpldi4entryin(a, b) start location: evalspeedpldi4start leaf cost: 4 A polynomial rank function with Pol(evalspeedpldi4bb4in) = 3*V_2 - 2 Pol(evalspeedpldi4bb5in) = 3*V_2 Pol(evalspeedpldi4bb3in) = 3*V_2 - 2 Pol(evalspeedpldi4bb2in) = 3*V_2 - 1 Pol(evalspeedpldi4entryin) = 3*V_2 Pol(evalspeedpldi4start) = 3*V_2 orients all transitions weakly and the transitions evalspeedpldi4bb5in(a, b) -> evalspeedpldi4bb2in(a, b) [ a - 1 >= 0 /\ b >= 1 ] evalspeedpldi4bb4in(a, b) -> evalspeedpldi4bb5in(a, b - a) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ -a + b >= 0 /\ a - 1 >= 0 ] evalspeedpldi4bb3in(a, b) -> evalspeedpldi4bb5in(a, b - 1) [ a - b - 1 >= 0 /\ b - 1 >= 0 /\ a + b - 3 >= 0 /\ a - 2 >= 0 ] evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb4in(a, b) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ a - 1 >= 0 /\ b >= a ] evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb3in(a, b) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ a - 1 >= 0 /\ a >= b + 1 ] strictly and produces the following problem: 5: T: (3*b, 1) evalspeedpldi4bb4in(a, b) -> evalspeedpldi4bb5in(a, b - a) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ -a + b >= 0 /\ a - 1 >= 0 ] (3*b, 1) evalspeedpldi4bb3in(a, b) -> evalspeedpldi4bb5in(a, b - 1) [ a - b - 1 >= 0 /\ b - 1 >= 0 /\ a + b - 3 >= 0 /\ a - 2 >= 0 ] (3*b, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb4in(a, b) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ a - 1 >= 0 /\ b >= a ] (3*b, 1) evalspeedpldi4bb2in(a, b) -> evalspeedpldi4bb3in(a, b) [ b - 1 >= 0 /\ a + b - 2 >= 0 /\ a - 1 >= 0 /\ a >= b + 1 ] (3*b, 1) evalspeedpldi4bb5in(a, b) -> evalspeedpldi4bb2in(a, b) [ a - 1 >= 0 /\ b >= 1 ] (1, 1) evalspeedpldi4entryin(a, b) -> evalspeedpldi4bb5in(a, b) [ a >= 1 /\ b >= a + 1 ] (1, 1) evalspeedpldi4start(a, b) -> evalspeedpldi4entryin(a, b) start location: evalspeedpldi4start leaf cost: 4 Complexity upper bound 15*b + 6 Time: 0.388 sec (SMT: 0.368 sec)