MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), N)) -> mark(N) , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) , active(U21(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(U21(and(isNat(M), isNat(N)), M, N)) , active(plus(N, 0())) -> mark(U11(isNat(N), N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNat(s(V1))) -> mark(isNat(V1)) , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) , active(isNat(0())) -> mark(tt()) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNat(ok(X)) -> ok(isNat(X)) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) , proper(s(X)) -> s(proper(X)) , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNat(X)) -> isNat(proper(X)) , proper(0()) -> ok(0()) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) , active^#(U11(tt(), N)) -> c_2(N) , active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3)) , active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M))) , active^#(s(X)) -> c_5(s^#(active(X))) , active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2))) , active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2)) , active^#(plus(N, s(M))) -> c_8(U21^#(and(isNat(M), isNat(N)), M, N)) , active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N)) , active^#(and(X1, X2)) -> c_10(and^#(active(X1), X2)) , active^#(and(tt(), X)) -> c_11(X) , active^#(isNat(s(V1))) -> c_12(isNat^#(V1)) , active^#(isNat(plus(V1, V2))) -> c_13(and^#(isNat(V1), isNat(V2))) , active^#(isNat(0())) -> c_14() , U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) , U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) , U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) , s^#(mark(X)) -> c_19(s^#(X)) , s^#(ok(X)) -> c_20(s^#(X)) , plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) , plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) , plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) , and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) , isNat^#(ok(X)) -> c_26(isNat^#(X)) , proper^#(U11(X1, X2)) -> c_27(U11^#(proper(X1), proper(X2))) , proper^#(tt()) -> c_28() , proper^#(U21(X1, X2, X3)) -> c_29(U21^#(proper(X1), proper(X2), proper(X3))) , proper^#(s(X)) -> c_30(s^#(proper(X))) , proper^#(plus(X1, X2)) -> c_31(plus^#(proper(X1), proper(X2))) , proper^#(and(X1, X2)) -> c_32(and^#(proper(X1), proper(X2))) , proper^#(isNat(X)) -> c_33(isNat^#(proper(X))) , proper^#(0()) -> c_34() , top^#(mark(X)) -> c_35(top^#(proper(X))) , top^#(ok(X)) -> c_36(top^#(active(X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) , active^#(U11(tt(), N)) -> c_2(N) , active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3)) , active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M))) , active^#(s(X)) -> c_5(s^#(active(X))) , active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2))) , active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2)) , active^#(plus(N, s(M))) -> c_8(U21^#(and(isNat(M), isNat(N)), M, N)) , active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N)) , active^#(and(X1, X2)) -> c_10(and^#(active(X1), X2)) , active^#(and(tt(), X)) -> c_11(X) , active^#(isNat(s(V1))) -> c_12(isNat^#(V1)) , active^#(isNat(plus(V1, V2))) -> c_13(and^#(isNat(V1), isNat(V2))) , active^#(isNat(0())) -> c_14() , U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) , U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) , U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) , s^#(mark(X)) -> c_19(s^#(X)) , s^#(ok(X)) -> c_20(s^#(X)) , plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) , plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) , plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) , and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) , isNat^#(ok(X)) -> c_26(isNat^#(X)) , proper^#(U11(X1, X2)) -> c_27(U11^#(proper(X1), proper(X2))) , proper^#(tt()) -> c_28() , proper^#(U21(X1, X2, X3)) -> c_29(U21^#(proper(X1), proper(X2), proper(X3))) , proper^#(s(X)) -> c_30(s^#(proper(X))) , proper^#(plus(X1, X2)) -> c_31(plus^#(proper(X1), proper(X2))) , proper^#(and(X1, X2)) -> c_32(and^#(proper(X1), proper(X2))) , proper^#(isNat(X)) -> c_33(isNat^#(proper(X))) , proper^#(0()) -> c_34() , top^#(mark(X)) -> c_35(top^#(proper(X))) , top^#(ok(X)) -> c_36(top^#(active(X))) } Strict Trs: { active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), N)) -> mark(N) , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) , active(U21(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(U21(and(isNat(M), isNat(N)), M, N)) , active(plus(N, 0())) -> mark(U11(isNat(N), N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNat(s(V1))) -> mark(isNat(V1)) , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) , active(isNat(0())) -> mark(tt()) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNat(ok(X)) -> ok(isNat(X)) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) , proper(s(X)) -> s(proper(X)) , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNat(X)) -> isNat(proper(X)) , proper(0()) -> ok(0()) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: runtime complexity Answer: MAYBE Consider the dependency graph: 1: active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) :16 -->_1 U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) :15 2: active^#(U11(tt(), N)) -> c_2(N) -->_1 top^#(ok(X)) -> c_36(top^#(active(X))) :36 -->_1 top^#(mark(X)) -> c_35(top^#(proper(X))) :35 -->_1 proper^#(isNat(X)) -> c_33(isNat^#(proper(X))) :33 -->_1 proper^#(and(X1, X2)) -> c_32(and^#(proper(X1), proper(X2))) :32 -->_1 proper^#(plus(X1, X2)) -> c_31(plus^#(proper(X1), proper(X2))) :31 -->_1 proper^#(s(X)) -> c_30(s^#(proper(X))) :30 -->_1 proper^#(U21(X1, X2, X3)) -> c_29(U21^#(proper(X1), proper(X2), proper(X3))) :29 -->_1 proper^#(U11(X1, X2)) -> c_27(U11^#(proper(X1), proper(X2))) :27 -->_1 isNat^#(ok(X)) -> c_26(isNat^#(X)) :26 -->_1 and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) :25 -->_1 and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) :24 -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 -->_1 s^#(ok(X)) -> c_20(s^#(X)) :20 -->_1 s^#(mark(X)) -> c_19(s^#(X)) :19 -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) :18 -->_1 U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) :17 -->_1 U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) :16 -->_1 U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) :15 -->_1 active^#(isNat(plus(V1, V2))) -> c_13(and^#(isNat(V1), isNat(V2))) :13 -->_1 active^#(isNat(s(V1))) -> c_12(isNat^#(V1)) :12 -->_1 active^#(and(tt(), X)) -> c_11(X) :11 -->_1 active^#(and(X1, X2)) -> c_10(and^#(active(X1), X2)) :10 -->_1 active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N)) :9 -->_1 active^#(plus(N, s(M))) -> c_8(U21^#(and(isNat(M), isNat(N)), M, N)) :8 -->_1 active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2)) :7 -->_1 active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2))) :6 -->_1 active^#(s(X)) -> c_5(s^#(active(X))) :5 -->_1 active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M))) :4 -->_1 active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3)) :3 -->_1 proper^#(0()) -> c_34() :34 -->_1 proper^#(tt()) -> c_28() :28 -->_1 active^#(isNat(0())) -> c_14() :14 -->_1 active^#(U11(tt(), N)) -> c_2(N) :2 -->_1 active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) :1 3: active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3)) -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) :18 -->_1 U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) :17 4: active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M))) -->_1 s^#(ok(X)) -> c_20(s^#(X)) :20 -->_1 s^#(mark(X)) -> c_19(s^#(X)) :19 5: active^#(s(X)) -> c_5(s^#(active(X))) -->_1 s^#(ok(X)) -> c_20(s^#(X)) :20 -->_1 s^#(mark(X)) -> c_19(s^#(X)) :19 6: active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2))) -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 7: active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2)) -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 8: active^#(plus(N, s(M))) -> c_8(U21^#(and(isNat(M), isNat(N)), M, N)) -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) :18 -->_1 U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) :17 9: active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N)) -->_1 U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) :16 -->_1 U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) :15 10: active^#(and(X1, X2)) -> c_10(and^#(active(X1), X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) :25 -->_1 and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) :24 11: active^#(and(tt(), X)) -> c_11(X) -->_1 top^#(ok(X)) -> c_36(top^#(active(X))) :36 -->_1 top^#(mark(X)) -> c_35(top^#(proper(X))) :35 -->_1 proper^#(isNat(X)) -> c_33(isNat^#(proper(X))) :33 -->_1 proper^#(and(X1, X2)) -> c_32(and^#(proper(X1), proper(X2))) :32 -->_1 proper^#(plus(X1, X2)) -> c_31(plus^#(proper(X1), proper(X2))) :31 -->_1 proper^#(s(X)) -> c_30(s^#(proper(X))) :30 -->_1 proper^#(U21(X1, X2, X3)) -> c_29(U21^#(proper(X1), proper(X2), proper(X3))) :29 -->_1 proper^#(U11(X1, X2)) -> c_27(U11^#(proper(X1), proper(X2))) :27 -->_1 isNat^#(ok(X)) -> c_26(isNat^#(X)) :26 -->_1 and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) :25 -->_1 and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) :24 -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 -->_1 s^#(ok(X)) -> c_20(s^#(X)) :20 -->_1 s^#(mark(X)) -> c_19(s^#(X)) :19 -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) :18 -->_1 U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) :17 -->_1 U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) :16 -->_1 U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) :15 -->_1 active^#(isNat(plus(V1, V2))) -> c_13(and^#(isNat(V1), isNat(V2))) :13 -->_1 active^#(isNat(s(V1))) -> c_12(isNat^#(V1)) :12 -->_1 proper^#(0()) -> c_34() :34 -->_1 proper^#(tt()) -> c_28() :28 -->_1 active^#(isNat(0())) -> c_14() :14 -->_1 active^#(and(tt(), X)) -> c_11(X) :11 -->_1 active^#(and(X1, X2)) -> c_10(and^#(active(X1), X2)) :10 -->_1 active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N)) :9 -->_1 active^#(plus(N, s(M))) -> c_8(U21^#(and(isNat(M), isNat(N)), M, N)) :8 -->_1 active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2)) :7 -->_1 active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2))) :6 -->_1 active^#(s(X)) -> c_5(s^#(active(X))) :5 -->_1 active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M))) :4 -->_1 active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3)) :3 -->_1 active^#(U11(tt(), N)) -> c_2(N) :2 -->_1 active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) :1 12: active^#(isNat(s(V1))) -> c_12(isNat^#(V1)) -->_1 isNat^#(ok(X)) -> c_26(isNat^#(X)) :26 13: active^#(isNat(plus(V1, V2))) -> c_13(and^#(isNat(V1), isNat(V2))) -->_1 and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) :25 -->_1 and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) :24 14: active^#(isNat(0())) -> c_14() 15: U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) :16 -->_1 U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) :15 16: U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) :16 -->_1 U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) :15 17: U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) :18 -->_1 U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) :17 18: U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) :18 -->_1 U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) :17 19: s^#(mark(X)) -> c_19(s^#(X)) -->_1 s^#(ok(X)) -> c_20(s^#(X)) :20 -->_1 s^#(mark(X)) -> c_19(s^#(X)) :19 20: s^#(ok(X)) -> c_20(s^#(X)) -->_1 s^#(ok(X)) -> c_20(s^#(X)) :20 -->_1 s^#(mark(X)) -> c_19(s^#(X)) :19 21: plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 22: plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 23: plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 24: and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) :25 -->_1 and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) :24 25: and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) :25 -->_1 and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) :24 26: isNat^#(ok(X)) -> c_26(isNat^#(X)) -->_1 isNat^#(ok(X)) -> c_26(isNat^#(X)) :26 27: proper^#(U11(X1, X2)) -> c_27(U11^#(proper(X1), proper(X2))) -->_1 U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) :16 -->_1 U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) :15 28: proper^#(tt()) -> c_28() 29: proper^#(U21(X1, X2, X3)) -> c_29(U21^#(proper(X1), proper(X2), proper(X3))) -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) :18 -->_1 U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) :17 30: proper^#(s(X)) -> c_30(s^#(proper(X))) -->_1 s^#(ok(X)) -> c_20(s^#(X)) :20 -->_1 s^#(mark(X)) -> c_19(s^#(X)) :19 31: proper^#(plus(X1, X2)) -> c_31(plus^#(proper(X1), proper(X2))) -->_1 plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) :23 -->_1 plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) :22 -->_1 plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) :21 32: proper^#(and(X1, X2)) -> c_32(and^#(proper(X1), proper(X2))) -->_1 and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) :25 -->_1 and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) :24 33: proper^#(isNat(X)) -> c_33(isNat^#(proper(X))) -->_1 isNat^#(ok(X)) -> c_26(isNat^#(X)) :26 34: proper^#(0()) -> c_34() 35: top^#(mark(X)) -> c_35(top^#(proper(X))) -->_1 top^#(ok(X)) -> c_36(top^#(active(X))) :36 -->_1 top^#(mark(X)) -> c_35(top^#(proper(X))) :35 36: top^#(ok(X)) -> c_36(top^#(active(X))) -->_1 top^#(ok(X)) -> c_36(top^#(active(X))) :36 -->_1 top^#(mark(X)) -> c_35(top^#(proper(X))) :35 Only the nodes {15,16,17,18,19,20,21,23,22,24,25,26,28,34,35,36} are reachable from nodes {15,16,17,18,19,20,21,22,23,24,25,26,28,34,35,36} that start derivation from marked basic terms. The nodes not reachable are removed from the problem. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) , U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) , U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) , s^#(mark(X)) -> c_19(s^#(X)) , s^#(ok(X)) -> c_20(s^#(X)) , plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) , plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) , plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) , and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) , isNat^#(ok(X)) -> c_26(isNat^#(X)) , proper^#(tt()) -> c_28() , proper^#(0()) -> c_34() , top^#(mark(X)) -> c_35(top^#(proper(X))) , top^#(ok(X)) -> c_36(top^#(active(X))) } Strict Trs: { active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), N)) -> mark(N) , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) , active(U21(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(U21(and(isNat(M), isNat(N)), M, N)) , active(plus(N, 0())) -> mark(U11(isNat(N), N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNat(s(V1))) -> mark(isNat(V1)) , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) , active(isNat(0())) -> mark(tt()) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNat(ok(X)) -> ok(isNat(X)) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) , proper(s(X)) -> s(proper(X)) , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNat(X)) -> isNat(proper(X)) , proper(0()) -> ok(0()) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {13,14} by applications of Pre({13,14}) = {}. Here rules are labeled as follows: DPs: { 1: U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) , 2: U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) , 3: U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) , 4: U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) , 5: s^#(mark(X)) -> c_19(s^#(X)) , 6: s^#(ok(X)) -> c_20(s^#(X)) , 7: plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) , 8: plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) , 9: plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) , 10: and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) , 11: and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) , 12: isNat^#(ok(X)) -> c_26(isNat^#(X)) , 13: proper^#(tt()) -> c_28() , 14: proper^#(0()) -> c_34() , 15: top^#(mark(X)) -> c_35(top^#(proper(X))) , 16: top^#(ok(X)) -> c_36(top^#(active(X))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { U11^#(mark(X1), X2) -> c_15(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_16(U11^#(X1, X2)) , U21^#(mark(X1), X2, X3) -> c_17(U21^#(X1, X2, X3)) , U21^#(ok(X1), ok(X2), ok(X3)) -> c_18(U21^#(X1, X2, X3)) , s^#(mark(X)) -> c_19(s^#(X)) , s^#(ok(X)) -> c_20(s^#(X)) , plus^#(X1, mark(X2)) -> c_21(plus^#(X1, X2)) , plus^#(mark(X1), X2) -> c_22(plus^#(X1, X2)) , plus^#(ok(X1), ok(X2)) -> c_23(plus^#(X1, X2)) , and^#(mark(X1), X2) -> c_24(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_25(and^#(X1, X2)) , isNat^#(ok(X)) -> c_26(isNat^#(X)) , top^#(mark(X)) -> c_35(top^#(proper(X))) , top^#(ok(X)) -> c_36(top^#(active(X))) } Strict Trs: { active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), N)) -> mark(N) , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) , active(U21(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(U21(and(isNat(M), isNat(N)), M, N)) , active(plus(N, 0())) -> mark(U11(isNat(N), N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNat(s(V1))) -> mark(isNat(V1)) , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) , active(isNat(0())) -> mark(tt()) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNat(ok(X)) -> ok(isNat(X)) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) , proper(s(X)) -> s(proper(X)) , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNat(X)) -> isNat(proper(X)) , proper(0()) -> ok(0()) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Weak DPs: { proper^#(tt()) -> c_28() , proper^#(0()) -> c_34() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..