YES(?, 4*a + 6) Initial complexity problem: 1: T: (1, 1) evalndecrstart(a) -> evalndecrentryin(a) (?, 1) evalndecrentryin(a) -> evalndecrbb1in(a - 1) (?, 1) evalndecrbb1in(a) -> evalndecrbbin(a) [ a >= 2 ] (?, 1) evalndecrbb1in(a) -> evalndecrreturnin(a) [ 1 >= a ] (?, 1) evalndecrbbin(a) -> evalndecrbb1in(a - 1) (?, 1) evalndecrreturnin(a) -> evalndecrstop(a) start location: evalndecrstart leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) evalndecrstart(a) -> evalndecrentryin(a) (?, 1) evalndecrentryin(a) -> evalndecrbb1in(a - 1) (?, 1) evalndecrbb1in(a) -> evalndecrbbin(a) [ a >= 2 ] (?, 1) evalndecrbbin(a) -> evalndecrbb1in(a - 1) start location: evalndecrstart leaf cost: 2 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) evalndecrstart(a) -> evalndecrentryin(a) (1, 1) evalndecrentryin(a) -> evalndecrbb1in(a - 1) (?, 1) evalndecrbb1in(a) -> evalndecrbbin(a) [ a >= 2 ] (?, 1) evalndecrbbin(a) -> evalndecrbb1in(a - 1) start location: evalndecrstart leaf cost: 2 A polynomial rank function with Pol(evalndecrstart) = 2*V_1 - 1 Pol(evalndecrentryin) = 2*V_1 - 1 Pol(evalndecrbb1in) = 2*V_1 + 1 Pol(evalndecrbbin) = 2*V_1 orients all transitions weakly and the transition evalndecrbb1in(a) -> evalndecrbbin(a) [ a >= 2 ] strictly and produces the following problem: 4: T: (1, 1) evalndecrstart(a) -> evalndecrentryin(a) (1, 1) evalndecrentryin(a) -> evalndecrbb1in(a - 1) (2*a + 1, 1) evalndecrbb1in(a) -> evalndecrbbin(a) [ a >= 2 ] (?, 1) evalndecrbbin(a) -> evalndecrbb1in(a - 1) start location: evalndecrstart leaf cost: 2 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (1, 1) evalndecrstart(a) -> evalndecrentryin(a) (1, 1) evalndecrentryin(a) -> evalndecrbb1in(a - 1) (2*a + 1, 1) evalndecrbb1in(a) -> evalndecrbbin(a) [ a >= 2 ] (2*a + 1, 1) evalndecrbbin(a) -> evalndecrbb1in(a - 1) start location: evalndecrstart leaf cost: 2 Complexity upper bound 4*a + 6 Time: 0.074 sec (SMT: 0.071 sec)