YES(?, 18*a + 24*a*b + 8*b + 10) Initial complexity problem: 1: T: (1, 1) evalfstart(a, b, c) -> evalfentryin(a, b, c) (?, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (?, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (?, 1) evalfbb4in(a, b, c) -> evalfreturnin(a, b, c) [ 0 >= b ] (?, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (?, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (?, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (?, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) (?, 1) evalfreturnin(a, b, c) -> evalfstop(a, b, c) 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, c) -> evalfentryin(a, b, c) (?, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (?, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (?, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (?, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (?, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (?, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) start location: evalfstart leaf cost: 2 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) evalfstart(a, b, c) -> evalfentryin(a, b, c) (1, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (?, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (?, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (?, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (?, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (?, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) start location: evalfstart leaf cost: 2 A polynomial rank function with Pol(evalfstart) = 3*V_1 + 1 Pol(evalfentryin) = 3*V_1 + 1 Pol(evalfbb4in) = 3*V_2 + 1 Pol(evalfbb2in) = 3*V_2 Pol(evalfbb1in) = 3*V_2 Pol(evalfbb3in) = 3*V_2 - 1 orients all transitions weakly and the transition evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] strictly and produces the following problem: 4: T: (1, 1) evalfstart(a, b, c) -> evalfentryin(a, b, c) (1, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (3*a + 1, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (?, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (?, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (?, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (?, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) start location: evalfstart leaf cost: 2 A polynomial rank function with Pol(evalfbb3in) = 1 Pol(evalfbb4in) = 0 Pol(evalfbb2in) = 2 Pol(evalfbb1in) = 2 and size complexities S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-0) = b S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-1) = ? S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-2) = ? S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-0) = b S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-1) = ? S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-2) = ? S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-0) = b S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-1) = ? S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-2) = ? S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-0) = b S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-1) = ? S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-2) = ? S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-0) = b S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-1) = ? S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-2) = b S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-0) = b S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-1) = a S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-2) = c S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-0) = a S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-1) = b S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-2) = c orients the transitions evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) weakly and the transition evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) strictly and produces the following problem: 5: T: (1, 1) evalfstart(a, b, c) -> evalfentryin(a, b, c) (1, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (3*a + 1, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (?, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (?, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (?, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (6*a + 2, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) start location: evalfstart leaf cost: 2 A polynomial rank function with Pol(evalfbb2in) = 1 Pol(evalfbb3in) = 0 Pol(evalfbb1in) = 1 and size complexities S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-0) = b S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-1) = 7*a + 98 S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-2) = ? S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-0) = b S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-1) = 7*a + 98 S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-2) = ? S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-0) = b S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-1) = 7*a + 98 S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-2) = ? S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-0) = b S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-1) = 7*a + 98 S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-2) = ? S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-0) = b S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-1) = 7*a + 98 S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-2) = b S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-0) = b S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-1) = a S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-2) = c S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-0) = a S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-1) = b S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-2) = c orients the transitions evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) weakly and the transition evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] strictly and produces the following problem: 6: T: (1, 1) evalfstart(a, b, c) -> evalfentryin(a, b, c) (1, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (3*a + 1, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (?, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (3*a + 1, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (?, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (6*a + 2, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) start location: evalfstart leaf cost: 2 A polynomial rank function with Pol(evalfbb2in) = -2*V_1 + 2*V_3 + 1 Pol(evalfbb1in) = -2*V_1 + 2*V_3 and size complexities S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-0) = b S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-1) = 7*a + 98 S("evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c)", 0-2) = ? S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-0) = b S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-1) = 7*a + 98 S("evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1)", 0-2) = ? S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-0) = b S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-1) = 7*a + 98 S("evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ]", 0-2) = ? S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-0) = b S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-1) = 7*a + 98 S("evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ]", 0-2) = ? S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-0) = b S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-1) = 7*a + 98 S("evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ]", 0-2) = b S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-0) = b S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-1) = a S("evalfentryin(a, b, c) -> evalfbb4in(b, a, c)", 0-2) = c S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-0) = a S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-1) = b S("evalfstart(a, b, c) -> evalfentryin(a, b, c)", 0-2) = c orients the transitions evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) weakly and the transition evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] strictly and produces the following problem: 7: T: (1, 1) evalfstart(a, b, c) -> evalfentryin(a, b, c) (1, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (3*a + 1, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (12*a*b + 4*b + 3*a + 1, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (3*a + 1, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (?, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (6*a + 2, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) start location: evalfstart leaf cost: 2 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (1, 1) evalfstart(a, b, c) -> evalfentryin(a, b, c) (1, 1) evalfentryin(a, b, c) -> evalfbb4in(b, a, c) (3*a + 1, 1) evalfbb4in(a, b, c) -> evalfbb2in(a, b, a) [ b >= 1 ] (12*a*b + 4*b + 3*a + 1, 1) evalfbb2in(a, b, c) -> evalfbb1in(a, b, c) [ c >= a ] (3*a + 1, 1) evalfbb2in(a, b, c) -> evalfbb3in(a, b, c) [ a >= c + 1 ] (12*a*b + 4*b + 3*a + 1, 1) evalfbb1in(a, b, c) -> evalfbb2in(a, b, c - 1) (6*a + 2, 1) evalfbb3in(a, b, c) -> evalfbb4in(a, b - 1, c) start location: evalfstart leaf cost: 2 Complexity upper bound 18*a + 24*a*b + 8*b + 10 Time: 0.189 sec (SMT: 0.176 sec)