YES(?, 48*b + 16) Initial complexity problem: 1: T: (1, 1) evalNestedSinglestart(a, b, c) -> evalNestedSingleentryin(a, b, c) (?, 1) evalNestedSingleentryin(a, b, c) -> evalNestedSinglebb5in(0, b, c) (?, 1) evalNestedSinglebb5in(a, b, c) -> evalNestedSinglebb2in(a, b, a) [ b >= a + 1 ] (?, 1) evalNestedSinglebb5in(a, b, c) -> evalNestedSinglereturnin(a, b, c) [ a >= b ] (?, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb4in(a, b, c) [ c >= b ] (?, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb3in(a, b, c) [ b >= c + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ 0 >= d + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ d >= 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb4in(a, b, c) (?, 1) evalNestedSinglebb1in(a, b, c) -> evalNestedSinglebb2in(a, b, c + 1) (?, 1) evalNestedSinglebb4in(a, b, c) -> evalNestedSinglebb5in(c + 1, b, c) (?, 1) evalNestedSinglereturnin(a, b, c) -> evalNestedSinglestop(a, b, c) start location: evalNestedSinglestart leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) evalNestedSinglestart(a, b, c) -> evalNestedSingleentryin(a, b, c) (?, 1) evalNestedSingleentryin(a, b, c) -> evalNestedSinglebb5in(0, b, c) (?, 1) evalNestedSinglebb5in(a, b, c) -> evalNestedSinglebb2in(a, b, a) [ b >= a + 1 ] (?, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb4in(a, b, c) [ c >= b ] (?, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb3in(a, b, c) [ b >= c + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ 0 >= d + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ d >= 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb4in(a, b, c) (?, 1) evalNestedSinglebb1in(a, b, c) -> evalNestedSinglebb2in(a, b, c + 1) (?, 1) evalNestedSinglebb4in(a, b, c) -> evalNestedSinglebb5in(c + 1, b, c) start location: evalNestedSinglestart leaf cost: 2 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) evalNestedSinglestart(a, b, c) -> evalNestedSingleentryin(a, b, c) (1, 1) evalNestedSingleentryin(a, b, c) -> evalNestedSinglebb5in(0, b, c) (?, 1) evalNestedSinglebb5in(a, b, c) -> evalNestedSinglebb2in(a, b, a) [ b >= a + 1 ] (?, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb4in(a, b, c) [ c >= b ] (?, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb3in(a, b, c) [ b >= c + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ 0 >= d + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ d >= 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb4in(a, b, c) (?, 1) evalNestedSinglebb1in(a, b, c) -> evalNestedSinglebb2in(a, b, c + 1) (?, 1) evalNestedSinglebb4in(a, b, c) -> evalNestedSinglebb5in(c + 1, b, c) start location: evalNestedSinglestart leaf cost: 2 A polynomial rank function with Pol(evalNestedSinglestart) = 4*V_2 + 1 Pol(evalNestedSingleentryin) = 4*V_2 + 1 Pol(evalNestedSinglebb5in) = -4*V_1 + 4*V_2 + 1 Pol(evalNestedSinglebb2in) = 4*V_2 - 4*V_3 Pol(evalNestedSinglebb4in) = 4*V_2 - 4*V_3 - 2 Pol(evalNestedSinglebb3in) = 4*V_2 - 4*V_3 - 1 Pol(evalNestedSinglebb1in) = 4*V_2 - 4*V_3 - 2 orients all transitions weakly and the transitions evalNestedSinglebb5in(a, b, c) -> evalNestedSinglebb2in(a, b, a) [ b >= a + 1 ] evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb3in(a, b, c) [ b >= c + 1 ] strictly and produces the following problem: 4: T: (1, 1) evalNestedSinglestart(a, b, c) -> evalNestedSingleentryin(a, b, c) (1, 1) evalNestedSingleentryin(a, b, c) -> evalNestedSinglebb5in(0, b, c) (4*b + 1, 1) evalNestedSinglebb5in(a, b, c) -> evalNestedSinglebb2in(a, b, a) [ b >= a + 1 ] (?, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb4in(a, b, c) [ c >= b ] (4*b + 1, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb3in(a, b, c) [ b >= c + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ 0 >= d + 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ d >= 1 ] (?, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb4in(a, b, c) (?, 1) evalNestedSinglebb1in(a, b, c) -> evalNestedSinglebb2in(a, b, c + 1) (?, 1) evalNestedSinglebb4in(a, b, c) -> evalNestedSinglebb5in(c + 1, b, c) start location: evalNestedSinglestart leaf cost: 2 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (1, 1) evalNestedSinglestart(a, b, c) -> evalNestedSingleentryin(a, b, c) (1, 1) evalNestedSingleentryin(a, b, c) -> evalNestedSinglebb5in(0, b, c) (4*b + 1, 1) evalNestedSinglebb5in(a, b, c) -> evalNestedSinglebb2in(a, b, a) [ b >= a + 1 ] (8*b + 2, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb4in(a, b, c) [ c >= b ] (4*b + 1, 1) evalNestedSinglebb2in(a, b, c) -> evalNestedSinglebb3in(a, b, c) [ b >= c + 1 ] (4*b + 1, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ 0 >= d + 1 ] (4*b + 1, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb1in(a, b, c) [ d >= 1 ] (4*b + 1, 1) evalNestedSinglebb3in(a, b, c) -> evalNestedSinglebb4in(a, b, c) (8*b + 2, 1) evalNestedSinglebb1in(a, b, c) -> evalNestedSinglebb2in(a, b, c + 1) (12*b + 3, 1) evalNestedSinglebb4in(a, b, c) -> evalNestedSinglebb5in(c + 1, b, c) start location: evalNestedSinglestart leaf cost: 2 Complexity upper bound 48*b + 16 Time: 4.187 sec (SMT: 3.822 sec)