YES(?, 17) Initial complexity problem: 1: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (?, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (?, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (?, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (?, 1) f28(a, b, c) -> f23(a + 1, b, c) (?, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f23(a, b, c) -> f38(a, b, c) [ a >= 1 ] (?, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (?, 1) f14(a, b, c) -> f8(a + 1, b, c) (?, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (?, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (?, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (?, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (?, 1) f28(a, b, c) -> f23(a + 1, b, c) (?, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (?, 1) f14(a, b, c) -> f8(a + 1, b, c) (?, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 1 Pol(f8) = 1 Pol(f14) = 1 Pol(f23) = -1 Pol(f28) = -1 orients all transitions weakly and the transition f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (?, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (?, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (?, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (?, 1) f28(a, b, c) -> f23(a + 1, b, c) (?, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (?, 1) f14(a, b, c) -> f8(a + 1, b, c) (?, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 1 Pol(f8) = 1 Pol(f14) = 1 Pol(f23) = -2*V_1 + 1 Pol(f28) = -2*V_1 orients all transitions weakly and the transitions f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] strictly and produces the following problem: 4: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (?, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (1, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (?, 1) f28(a, b, c) -> f23(a + 1, b, c) (?, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (?, 1) f14(a, b, c) -> f8(a + 1, b, c) (?, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 1 Pol(f8) = 1 Pol(f14) = 1 Pol(f23) = -3*V_1 + 1 Pol(f28) = -3*V_1 - 1 orients all transitions weakly and the transition f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] strictly and produces the following problem: 5: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (1, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (?, 1) f28(a, b, c) -> f23(a + 1, b, c) (?, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (?, 1) f14(a, b, c) -> f8(a + 1, b, c) (?, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 1 Repeatedly propagating knowledge in problem 5 produces the following problem: 6: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (?, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (1, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (2, 1) f28(a, b, c) -> f23(a + 1, b, c) (2, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (?, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (?, 1) f14(a, b, c) -> f8(a + 1, b, c) (?, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f8) = -3*V_1 + 1 Pol(f14) = -3*V_1 - 1 and size complexities S("f8(a, b, c) -> f23(0, b, c) [ a >= 1 ]", 0-0) = 0 S("f8(a, b, c) -> f23(0, b, c) [ a >= 1 ]", 0-1) = ? S("f8(a, b, c) -> f23(0, b, c) [ a >= 1 ]", 0-2) = c S("f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ]", 0-0) = ? S("f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ]", 0-1) = ? S("f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ]", 0-2) = c S("f14(a, b, c) -> f8(a + 1, b, c)", 0-0) = ? S("f14(a, b, c) -> f8(a + 1, b, c)", 0-1) = ? S("f14(a, b, c) -> f8(a + 1, b, c)", 0-2) = c S("f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ]", 0-0) = ? S("f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ]", 0-1) = ? S("f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ]", 0-2) = c S("f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ]", 0-0) = 5 S("f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ]", 0-1) = ? S("f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ]", 0-2) = ? S("f28(a, b, c) -> f23(a + 1, b, c)", 0-0) = 5 S("f28(a, b, c) -> f23(a + 1, b, c)", 0-1) = ? S("f28(a, b, c) -> f23(a + 1, b, c)", 0-2) = ? S("f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ]", 0-0) = 5 S("f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ]", 0-1) = ? S("f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ]", 0-2) = ? S("f23(a, b, c) -> f28(a, b, d) [ 0 >= a ]", 0-0) = 5 S("f23(a, b, c) -> f28(a, b, d) [ 0 >= a ]", 0-1) = ? S("f23(a, b, c) -> f28(a, b, d) [ 0 >= a ]", 0-2) = ? S("f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\\ 0 >= e + 1 ]", 0-0) = 5 S("f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\\ 0 >= e + 1 ]", 0-1) = ? S("f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\\ 0 >= e + 1 ]", 0-2) = ? S("f8(a, b, c) -> f14(a, a, c) [ 0 >= a ]", 0-0) = ? S("f8(a, b, c) -> f14(a, a, c) [ 0 >= a ]", 0-1) = ? S("f8(a, b, c) -> f14(a, a, c) [ 0 >= a ]", 0-2) = c S("f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\\ 0 >= d ]", 0-0) = ? S("f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\\ 0 >= d ]", 0-1) = ? S("f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\\ 0 >= d ]", 0-2) = c S("f0(a, b, c) -> f8(0, b, c)", 0-0) = 0 S("f0(a, b, c) -> f8(0, b, c)", 0-1) = b S("f0(a, b, c) -> f8(0, b, c)", 0-2) = c orients the transitions f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] f14(a, b, c) -> f8(a + 1, b, c) weakly and the transitions f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] strictly and produces the following problem: 7: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (1, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (1, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (1, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (2, 1) f28(a, b, c) -> f23(a + 1, b, c) (2, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (?, 1) f14(a, b, c) -> f8(a + 1, b, c) (?, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 1 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (1, 1) f0(a, b, c) -> f8(0, b, c) (1, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a /\ 0 >= d ] (1, 1) f8(a, b, c) -> f14(a, a, c) [ 0 >= a ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a /\ 0 >= e + 1 ] (1, 1) f23(a, b, c) -> f28(a, b, d) [ 0 >= a ] (1, 1) f23(a, b, c) -> f23(a + 1, b, c) [ 0 >= a ] (2, 1) f28(a, b, c) -> f23(a + 1, b, c) (2, 1) f28(a, b, c) -> f23(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f8(a + 1, a, c) [ 0 >= a ] (2, 1) f14(a, b, c) -> f8(a + 1, b, c) (2, 1) f14(a, b, c) -> f8(a + 1, b, c) [ 0 >= d + 1 ] (1, 1) f8(a, b, c) -> f23(0, b, c) [ a >= 1 ] start location: f0 leaf cost: 1 Complexity upper bound 17 Time: 0.412 sec (SMT: 0.388 sec)