YES(?,O(n^1)) We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict Trs: { active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) , active(U11(tt(), M, N)) -> mark(U12(tt(), M, N)) , active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) , active(U12(tt(), M, N)) -> mark(s(plus(N, M))) , active(s(X)) -> s(active(X)) , active(plus(X1, X2)) -> plus(X1, active(X2)) , active(plus(X1, X2)) -> plus(active(X1), X2) , active(plus(N, s(M))) -> mark(U11(tt(), M, N)) , active(plus(N, 0())) -> mark(N) , U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) , U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) , U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) , U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) , s(mark(X)) -> mark(s(X)) , s(ok(X)) -> ok(s(X)) , plus(X1, mark(X2)) -> mark(plus(X1, X2)) , plus(mark(X1), X2) -> mark(plus(X1, X2)) , plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) , proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) , proper(tt()) -> ok(tt()) , proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) , proper(s(X)) -> s(proper(X)) , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) , proper(0()) -> ok(0()) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The problem is match-bounded by 2. The enriched problem is compatible with the following automaton. { active_0(3) -> 1 , active_0(4) -> 1 , active_0(8) -> 1 , active_0(10) -> 1 , active_1(3) -> 17 , active_1(4) -> 17 , active_1(8) -> 17 , active_1(10) -> 17 , active_2(16) -> 18 , U11_0(3, 3, 3) -> 2 , U11_0(3, 3, 4) -> 2 , U11_0(3, 3, 8) -> 2 , U11_0(3, 3, 10) -> 2 , U11_0(3, 4, 3) -> 2 , U11_0(3, 4, 4) -> 2 , U11_0(3, 4, 8) -> 2 , U11_0(3, 4, 10) -> 2 , U11_0(3, 8, 3) -> 2 , U11_0(3, 8, 4) -> 2 , U11_0(3, 8, 8) -> 2 , U11_0(3, 8, 10) -> 2 , U11_0(3, 10, 3) -> 2 , U11_0(3, 10, 4) -> 2 , U11_0(3, 10, 8) -> 2 , U11_0(3, 10, 10) -> 2 , U11_0(4, 3, 3) -> 2 , U11_0(4, 3, 4) -> 2 , U11_0(4, 3, 8) -> 2 , U11_0(4, 3, 10) -> 2 , U11_0(4, 4, 3) -> 2 , U11_0(4, 4, 4) -> 2 , U11_0(4, 4, 8) -> 2 , U11_0(4, 4, 10) -> 2 , U11_0(4, 8, 3) -> 2 , U11_0(4, 8, 4) -> 2 , U11_0(4, 8, 8) -> 2 , U11_0(4, 8, 10) -> 2 , U11_0(4, 10, 3) -> 2 , U11_0(4, 10, 4) -> 2 , U11_0(4, 10, 8) -> 2 , U11_0(4, 10, 10) -> 2 , U11_0(8, 3, 3) -> 2 , U11_0(8, 3, 4) -> 2 , U11_0(8, 3, 8) -> 2 , U11_0(8, 3, 10) -> 2 , U11_0(8, 4, 3) -> 2 , U11_0(8, 4, 4) -> 2 , U11_0(8, 4, 8) -> 2 , U11_0(8, 4, 10) -> 2 , U11_0(8, 8, 3) -> 2 , U11_0(8, 8, 4) -> 2 , U11_0(8, 8, 8) -> 2 , U11_0(8, 8, 10) -> 2 , U11_0(8, 10, 3) -> 2 , U11_0(8, 10, 4) -> 2 , U11_0(8, 10, 8) -> 2 , U11_0(8, 10, 10) -> 2 , U11_0(10, 3, 3) -> 2 , U11_0(10, 3, 4) -> 2 , U11_0(10, 3, 8) -> 2 , U11_0(10, 3, 10) -> 2 , U11_0(10, 4, 3) -> 2 , U11_0(10, 4, 4) -> 2 , U11_0(10, 4, 8) -> 2 , U11_0(10, 4, 10) -> 2 , U11_0(10, 8, 3) -> 2 , U11_0(10, 8, 4) -> 2 , U11_0(10, 8, 8) -> 2 , U11_0(10, 8, 10) -> 2 , U11_0(10, 10, 3) -> 2 , U11_0(10, 10, 4) -> 2 , U11_0(10, 10, 8) -> 2 , U11_0(10, 10, 10) -> 2 , U11_1(3, 3, 3) -> 12 , U11_1(3, 3, 4) -> 12 , U11_1(3, 3, 8) -> 12 , U11_1(3, 3, 10) -> 12 , U11_1(3, 4, 3) -> 12 , U11_1(3, 4, 4) -> 12 , U11_1(3, 4, 8) -> 12 , U11_1(3, 4, 10) -> 12 , U11_1(3, 8, 3) -> 12 , U11_1(3, 8, 4) -> 12 , U11_1(3, 8, 8) -> 12 , U11_1(3, 8, 10) -> 12 , U11_1(3, 10, 3) -> 12 , U11_1(3, 10, 4) -> 12 , U11_1(3, 10, 8) -> 12 , U11_1(3, 10, 10) -> 12 , U11_1(4, 3, 3) -> 12 , U11_1(4, 3, 4) -> 12 , U11_1(4, 3, 8) -> 12 , U11_1(4, 3, 10) -> 12 , U11_1(4, 4, 3) -> 12 , U11_1(4, 4, 4) -> 12 , U11_1(4, 4, 8) -> 12 , U11_1(4, 4, 10) -> 12 , U11_1(4, 8, 3) -> 12 , U11_1(4, 8, 4) -> 12 , U11_1(4, 8, 8) -> 12 , U11_1(4, 8, 10) -> 12 , U11_1(4, 10, 3) -> 12 , U11_1(4, 10, 4) -> 12 , U11_1(4, 10, 8) -> 12 , U11_1(4, 10, 10) -> 12 , U11_1(8, 3, 3) -> 12 , U11_1(8, 3, 4) -> 12 , U11_1(8, 3, 8) -> 12 , U11_1(8, 3, 10) -> 12 , U11_1(8, 4, 3) -> 12 , U11_1(8, 4, 4) -> 12 , U11_1(8, 4, 8) -> 12 , U11_1(8, 4, 10) -> 12 , U11_1(8, 8, 3) -> 12 , U11_1(8, 8, 4) -> 12 , U11_1(8, 8, 8) -> 12 , U11_1(8, 8, 10) -> 12 , U11_1(8, 10, 3) -> 12 , U11_1(8, 10, 4) -> 12 , U11_1(8, 10, 8) -> 12 , U11_1(8, 10, 10) -> 12 , U11_1(10, 3, 3) -> 12 , U11_1(10, 3, 4) -> 12 , U11_1(10, 3, 8) -> 12 , U11_1(10, 3, 10) -> 12 , U11_1(10, 4, 3) -> 12 , U11_1(10, 4, 4) -> 12 , U11_1(10, 4, 8) -> 12 , U11_1(10, 4, 10) -> 12 , U11_1(10, 8, 3) -> 12 , U11_1(10, 8, 4) -> 12 , U11_1(10, 8, 8) -> 12 , U11_1(10, 8, 10) -> 12 , U11_1(10, 10, 3) -> 12 , U11_1(10, 10, 4) -> 12 , U11_1(10, 10, 8) -> 12 , U11_1(10, 10, 10) -> 12 , tt_0() -> 3 , tt_1() -> 16 , mark_0(3) -> 4 , mark_0(4) -> 4 , mark_0(8) -> 4 , mark_0(10) -> 4 , mark_1(12) -> 2 , mark_1(12) -> 12 , mark_1(13) -> 5 , mark_1(13) -> 13 , mark_1(14) -> 6 , mark_1(14) -> 14 , mark_1(15) -> 7 , mark_1(15) -> 15 , U12_0(3, 3, 3) -> 5 , U12_0(3, 3, 4) -> 5 , U12_0(3, 3, 8) -> 5 , U12_0(3, 3, 10) -> 5 , U12_0(3, 4, 3) -> 5 , U12_0(3, 4, 4) -> 5 , U12_0(3, 4, 8) -> 5 , U12_0(3, 4, 10) -> 5 , U12_0(3, 8, 3) -> 5 , U12_0(3, 8, 4) -> 5 , U12_0(3, 8, 8) -> 5 , U12_0(3, 8, 10) -> 5 , U12_0(3, 10, 3) -> 5 , U12_0(3, 10, 4) -> 5 , U12_0(3, 10, 8) -> 5 , U12_0(3, 10, 10) -> 5 , U12_0(4, 3, 3) -> 5 , U12_0(4, 3, 4) -> 5 , U12_0(4, 3, 8) -> 5 , U12_0(4, 3, 10) -> 5 , U12_0(4, 4, 3) -> 5 , U12_0(4, 4, 4) -> 5 , U12_0(4, 4, 8) -> 5 , U12_0(4, 4, 10) -> 5 , U12_0(4, 8, 3) -> 5 , U12_0(4, 8, 4) -> 5 , U12_0(4, 8, 8) -> 5 , U12_0(4, 8, 10) -> 5 , U12_0(4, 10, 3) -> 5 , U12_0(4, 10, 4) -> 5 , U12_0(4, 10, 8) -> 5 , U12_0(4, 10, 10) -> 5 , U12_0(8, 3, 3) -> 5 , U12_0(8, 3, 4) -> 5 , U12_0(8, 3, 8) -> 5 , U12_0(8, 3, 10) -> 5 , U12_0(8, 4, 3) -> 5 , U12_0(8, 4, 4) -> 5 , U12_0(8, 4, 8) -> 5 , U12_0(8, 4, 10) -> 5 , U12_0(8, 8, 3) -> 5 , U12_0(8, 8, 4) -> 5 , U12_0(8, 8, 8) -> 5 , U12_0(8, 8, 10) -> 5 , U12_0(8, 10, 3) -> 5 , U12_0(8, 10, 4) -> 5 , U12_0(8, 10, 8) -> 5 , U12_0(8, 10, 10) -> 5 , U12_0(10, 3, 3) -> 5 , U12_0(10, 3, 4) -> 5 , U12_0(10, 3, 8) -> 5 , U12_0(10, 3, 10) -> 5 , U12_0(10, 4, 3) -> 5 , U12_0(10, 4, 4) -> 5 , U12_0(10, 4, 8) -> 5 , U12_0(10, 4, 10) -> 5 , U12_0(10, 8, 3) -> 5 , U12_0(10, 8, 4) -> 5 , U12_0(10, 8, 8) -> 5 , U12_0(10, 8, 10) -> 5 , U12_0(10, 10, 3) -> 5 , U12_0(10, 10, 4) -> 5 , U12_0(10, 10, 8) -> 5 , U12_0(10, 10, 10) -> 5 , U12_1(3, 3, 3) -> 13 , U12_1(3, 3, 4) -> 13 , U12_1(3, 3, 8) -> 13 , U12_1(3, 3, 10) -> 13 , U12_1(3, 4, 3) -> 13 , U12_1(3, 4, 4) -> 13 , U12_1(3, 4, 8) -> 13 , U12_1(3, 4, 10) -> 13 , U12_1(3, 8, 3) -> 13 , U12_1(3, 8, 4) -> 13 , U12_1(3, 8, 8) -> 13 , U12_1(3, 8, 10) -> 13 , U12_1(3, 10, 3) -> 13 , U12_1(3, 10, 4) -> 13 , U12_1(3, 10, 8) -> 13 , U12_1(3, 10, 10) -> 13 , U12_1(4, 3, 3) -> 13 , U12_1(4, 3, 4) -> 13 , U12_1(4, 3, 8) -> 13 , U12_1(4, 3, 10) -> 13 , U12_1(4, 4, 3) -> 13 , U12_1(4, 4, 4) -> 13 , U12_1(4, 4, 8) -> 13 , U12_1(4, 4, 10) -> 13 , U12_1(4, 8, 3) -> 13 , U12_1(4, 8, 4) -> 13 , U12_1(4, 8, 8) -> 13 , U12_1(4, 8, 10) -> 13 , U12_1(4, 10, 3) -> 13 , U12_1(4, 10, 4) -> 13 , U12_1(4, 10, 8) -> 13 , U12_1(4, 10, 10) -> 13 , U12_1(8, 3, 3) -> 13 , U12_1(8, 3, 4) -> 13 , U12_1(8, 3, 8) -> 13 , U12_1(8, 3, 10) -> 13 , U12_1(8, 4, 3) -> 13 , U12_1(8, 4, 4) -> 13 , U12_1(8, 4, 8) -> 13 , U12_1(8, 4, 10) -> 13 , U12_1(8, 8, 3) -> 13 , U12_1(8, 8, 4) -> 13 , U12_1(8, 8, 8) -> 13 , U12_1(8, 8, 10) -> 13 , U12_1(8, 10, 3) -> 13 , U12_1(8, 10, 4) -> 13 , U12_1(8, 10, 8) -> 13 , U12_1(8, 10, 10) -> 13 , U12_1(10, 3, 3) -> 13 , U12_1(10, 3, 4) -> 13 , U12_1(10, 3, 8) -> 13 , U12_1(10, 3, 10) -> 13 , U12_1(10, 4, 3) -> 13 , U12_1(10, 4, 4) -> 13 , U12_1(10, 4, 8) -> 13 , U12_1(10, 4, 10) -> 13 , U12_1(10, 8, 3) -> 13 , U12_1(10, 8, 4) -> 13 , U12_1(10, 8, 8) -> 13 , U12_1(10, 8, 10) -> 13 , U12_1(10, 10, 3) -> 13 , U12_1(10, 10, 4) -> 13 , U12_1(10, 10, 8) -> 13 , U12_1(10, 10, 10) -> 13 , s_0(3) -> 6 , s_0(4) -> 6 , s_0(8) -> 6 , s_0(10) -> 6 , s_1(3) -> 14 , s_1(4) -> 14 , s_1(8) -> 14 , s_1(10) -> 14 , plus_0(3, 3) -> 7 , plus_0(3, 4) -> 7 , plus_0(3, 8) -> 7 , plus_0(3, 10) -> 7 , plus_0(4, 3) -> 7 , plus_0(4, 4) -> 7 , plus_0(4, 8) -> 7 , plus_0(4, 10) -> 7 , plus_0(8, 3) -> 7 , plus_0(8, 4) -> 7 , plus_0(8, 8) -> 7 , plus_0(8, 10) -> 7 , plus_0(10, 3) -> 7 , plus_0(10, 4) -> 7 , plus_0(10, 8) -> 7 , plus_0(10, 10) -> 7 , plus_1(3, 3) -> 15 , plus_1(3, 4) -> 15 , plus_1(3, 8) -> 15 , plus_1(3, 10) -> 15 , plus_1(4, 3) -> 15 , plus_1(4, 4) -> 15 , plus_1(4, 8) -> 15 , plus_1(4, 10) -> 15 , plus_1(8, 3) -> 15 , plus_1(8, 4) -> 15 , plus_1(8, 8) -> 15 , plus_1(8, 10) -> 15 , plus_1(10, 3) -> 15 , plus_1(10, 4) -> 15 , plus_1(10, 8) -> 15 , plus_1(10, 10) -> 15 , 0_0() -> 8 , 0_1() -> 16 , proper_0(3) -> 9 , proper_0(4) -> 9 , proper_0(8) -> 9 , proper_0(10) -> 9 , proper_1(3) -> 17 , proper_1(4) -> 17 , proper_1(8) -> 17 , proper_1(10) -> 17 , ok_0(3) -> 10 , ok_0(4) -> 10 , ok_0(8) -> 10 , ok_0(10) -> 10 , ok_1(12) -> 2 , ok_1(12) -> 12 , ok_1(13) -> 5 , ok_1(13) -> 13 , ok_1(14) -> 6 , ok_1(14) -> 14 , ok_1(15) -> 7 , ok_1(15) -> 15 , ok_1(16) -> 9 , ok_1(16) -> 17 , top_0(3) -> 11 , top_0(4) -> 11 , top_0(8) -> 11 , top_0(10) -> 11 , top_1(17) -> 11 , top_2(18) -> 11 } Hurray, we answered YES(?,O(n^1))