MAYBE 899.50/297.13 MAYBE 899.50/297.13 899.50/297.13 We are left with following problem, upon which TcT provides the 899.50/297.13 certificate MAYBE. 899.50/297.13 899.50/297.13 Strict Trs: 899.50/297.13 { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 899.50/297.13 , U12(tt(), V2) -> U13(isNat(activate(V2))) 899.50/297.13 , isNat(X) -> n__isNat(X) 899.50/297.13 , isNat(n__0()) -> tt() 899.50/297.13 , isNat(n__plus(V1, V2)) -> 899.50/297.13 U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.13 activate(V1), 899.50/297.13 activate(V2)) 899.50/297.13 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 899.50/297.13 , activate(X) -> X 899.50/297.13 , activate(n__0()) -> 0() 899.50/297.13 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 899.50/297.13 , activate(n__isNatKind(X)) -> isNatKind(X) 899.50/297.13 , activate(n__s(X)) -> s(activate(X)) 899.50/297.13 , activate(n__and(X1, X2)) -> and(activate(X1), X2) 899.50/297.13 , activate(n__isNat(X)) -> isNat(X) 899.50/297.13 , U13(tt()) -> tt() 899.50/297.13 , U21(tt(), V1) -> U22(isNat(activate(V1))) 899.50/297.13 , U22(tt()) -> tt() 899.50/297.13 , U31(tt(), N) -> activate(N) 899.50/297.13 , U41(tt(), M, N) -> s(plus(activate(N), activate(M))) 899.50/297.13 , s(X) -> n__s(X) 899.50/297.13 , plus(X1, X2) -> n__plus(X1, X2) 899.50/297.13 , plus(N, s(M)) -> 899.50/297.13 U41(and(and(isNat(M), n__isNatKind(M)), 899.50/297.13 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.13 M, 899.50/297.13 N) 899.50/297.13 , plus(N, 0()) -> U31(and(isNat(N), n__isNatKind(N)), N) 899.50/297.13 , and(X1, X2) -> n__and(X1, X2) 899.50/297.13 , and(tt(), X) -> activate(X) 899.50/297.13 , isNatKind(X) -> n__isNatKind(X) 899.50/297.13 , isNatKind(n__0()) -> tt() 899.50/297.13 , isNatKind(n__plus(V1, V2)) -> 899.50/297.13 and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 899.50/297.13 , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 899.50/297.13 , 0() -> n__0() } 899.50/297.13 Obligation: 899.50/297.13 runtime complexity 899.50/297.13 Answer: 899.50/297.13 MAYBE 899.50/297.13 899.50/297.13 None of the processors succeeded. 899.50/297.13 899.50/297.13 Details of failed attempt(s): 899.50/297.13 ----------------------------- 899.50/297.13 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 899.50/297.13 following reason: 899.50/297.13 899.50/297.13 Computation stopped due to timeout after 297.0 seconds. 899.50/297.13 899.50/297.13 2) 'Best' failed due to the following reason: 899.50/297.13 899.50/297.13 None of the processors succeeded. 899.50/297.13 899.50/297.13 Details of failed attempt(s): 899.50/297.13 ----------------------------- 899.50/297.13 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 899.50/297.13 seconds)' failed due to the following reason: 899.50/297.13 899.50/297.13 Computation stopped due to timeout after 148.0 seconds. 899.50/297.13 899.50/297.13 2) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 899.50/297.13 failed due to the following reason: 899.50/297.13 899.50/297.13 None of the processors succeeded. 899.50/297.13 899.50/297.13 Details of failed attempt(s): 899.50/297.13 ----------------------------- 899.50/297.13 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 899.50/297.13 failed due to the following reason: 899.50/297.13 899.50/297.13 match-boundness of the problem could not be verified. 899.50/297.13 899.50/297.13 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 899.50/297.13 failed due to the following reason: 899.50/297.13 899.50/297.13 match-boundness of the problem could not be verified. 899.50/297.13 899.50/297.13 899.50/297.13 3) 'Best' failed due to the following reason: 899.50/297.13 899.50/297.13 None of the processors succeeded. 899.50/297.13 899.50/297.13 Details of failed attempt(s): 899.50/297.13 ----------------------------- 899.50/297.13 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 899.50/297.13 following reason: 899.50/297.13 899.50/297.13 The processor is inapplicable, reason: 899.50/297.13 Processor only applicable for innermost runtime complexity analysis 899.50/297.13 899.50/297.13 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 899.50/297.13 to the following reason: 899.50/297.13 899.50/297.13 The processor is inapplicable, reason: 899.50/297.13 Processor only applicable for innermost runtime complexity analysis 899.50/297.13 899.50/297.13 899.50/297.13 899.50/297.13 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 899.50/297.13 the following reason: 899.50/297.13 899.50/297.13 We add the following weak dependency pairs: 899.50/297.13 899.50/297.13 Strict DPs: 899.50/297.13 { U11^#(tt(), V1, V2) -> 899.50/297.13 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.13 , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.13 , U13^#(tt()) -> c_14() 899.50/297.13 , isNat^#(X) -> c_3(X) 899.50/297.13 , isNat^#(n__0()) -> c_4() 899.50/297.13 , isNat^#(n__plus(V1, V2)) -> 899.50/297.13 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.13 activate(V1), 899.50/297.13 activate(V2))) 899.50/297.13 , isNat^#(n__s(V1)) -> 899.50/297.13 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.13 , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.13 , activate^#(X) -> c_7(X) 899.50/297.13 , activate^#(n__0()) -> c_8(0^#()) 899.50/297.13 , activate^#(n__plus(X1, X2)) -> 899.50/297.13 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.13 , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.13 , activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.13 , activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.13 , activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.13 , 0^#() -> c_29() 899.50/297.13 , plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.13 , plus^#(N, s(M)) -> 899.50/297.13 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.13 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.13 M, 899.50/297.13 N)) 899.50/297.13 , plus^#(N, 0()) -> c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.13 , isNatKind^#(X) -> c_25(X) 899.50/297.13 , isNatKind^#(n__0()) -> c_26() 899.50/297.13 , isNatKind^#(n__plus(V1, V2)) -> 899.50/297.13 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.13 , isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.13 , s^#(X) -> c_19(X) 899.50/297.13 , and^#(X1, X2) -> c_23(X1, X2) 899.50/297.13 , and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.13 , U22^#(tt()) -> c_16() 899.50/297.13 , U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.13 , U41^#(tt(), M, N) -> c_18(s^#(plus(activate(N), activate(M)))) } 899.50/297.13 899.50/297.13 and mark the set of starting terms. 899.50/297.13 899.50/297.13 We are left with following problem, upon which TcT provides the 899.50/297.13 certificate MAYBE. 899.50/297.13 899.50/297.13 Strict DPs: 899.50/297.13 { U11^#(tt(), V1, V2) -> 899.50/297.13 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.13 , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.13 , U13^#(tt()) -> c_14() 899.50/297.13 , isNat^#(X) -> c_3(X) 899.50/297.13 , isNat^#(n__0()) -> c_4() 899.50/297.13 , isNat^#(n__plus(V1, V2)) -> 899.50/297.13 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.13 activate(V1), 899.50/297.13 activate(V2))) 899.50/297.13 , isNat^#(n__s(V1)) -> 899.50/297.13 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.13 , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.13 , activate^#(X) -> c_7(X) 899.50/297.13 , activate^#(n__0()) -> c_8(0^#()) 899.50/297.13 , activate^#(n__plus(X1, X2)) -> 899.50/297.13 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.13 , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.13 , activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.13 , activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.13 , activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.13 , 0^#() -> c_29() 899.50/297.13 , plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.13 , plus^#(N, s(M)) -> 899.50/297.13 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.13 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.13 M, 899.50/297.13 N)) 899.50/297.13 , plus^#(N, 0()) -> c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.13 , isNatKind^#(X) -> c_25(X) 899.50/297.13 , isNatKind^#(n__0()) -> c_26() 899.50/297.13 , isNatKind^#(n__plus(V1, V2)) -> 899.50/297.13 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.13 , isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.13 , s^#(X) -> c_19(X) 899.50/297.13 , and^#(X1, X2) -> c_23(X1, X2) 899.50/297.13 , and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.13 , U22^#(tt()) -> c_16() 899.50/297.13 , U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.13 , U41^#(tt(), M, N) -> c_18(s^#(plus(activate(N), activate(M)))) } 899.50/297.13 Strict Trs: 899.50/297.13 { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 899.50/297.13 , U12(tt(), V2) -> U13(isNat(activate(V2))) 899.50/297.13 , isNat(X) -> n__isNat(X) 899.50/297.13 , isNat(n__0()) -> tt() 899.50/297.13 , isNat(n__plus(V1, V2)) -> 899.50/297.13 U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.13 activate(V1), 899.50/297.13 activate(V2)) 899.50/297.13 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 899.50/297.13 , activate(X) -> X 899.50/297.13 , activate(n__0()) -> 0() 899.50/297.13 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 899.50/297.13 , activate(n__isNatKind(X)) -> isNatKind(X) 899.50/297.13 , activate(n__s(X)) -> s(activate(X)) 899.50/297.13 , activate(n__and(X1, X2)) -> and(activate(X1), X2) 899.50/297.13 , activate(n__isNat(X)) -> isNat(X) 899.50/297.13 , U13(tt()) -> tt() 899.50/297.13 , U21(tt(), V1) -> U22(isNat(activate(V1))) 899.50/297.13 , U22(tt()) -> tt() 899.50/297.13 , U31(tt(), N) -> activate(N) 899.50/297.14 , U41(tt(), M, N) -> s(plus(activate(N), activate(M))) 899.50/297.14 , s(X) -> n__s(X) 899.50/297.14 , plus(X1, X2) -> n__plus(X1, X2) 899.50/297.14 , plus(N, s(M)) -> 899.50/297.14 U41(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N) 899.50/297.14 , plus(N, 0()) -> U31(and(isNat(N), n__isNatKind(N)), N) 899.50/297.14 , and(X1, X2) -> n__and(X1, X2) 899.50/297.14 , and(tt(), X) -> activate(X) 899.50/297.14 , isNatKind(X) -> n__isNatKind(X) 899.50/297.14 , isNatKind(n__0()) -> tt() 899.50/297.14 , isNatKind(n__plus(V1, V2)) -> 899.50/297.14 and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 899.50/297.14 , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 899.50/297.14 , 0() -> n__0() } 899.50/297.14 Obligation: 899.50/297.14 runtime complexity 899.50/297.14 Answer: 899.50/297.14 MAYBE 899.50/297.14 899.50/297.14 We estimate the number of application of {3,5,16,21,27} by 899.50/297.14 applications of Pre({3,5,16,21,27}) = 899.50/297.14 {2,4,8,9,10,12,15,17,20,23,24,25}. Here rules are labeled as 899.50/297.14 follows: 899.50/297.14 899.50/297.14 DPs: 899.50/297.14 { 1: U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , 2: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , 3: U13^#(tt()) -> c_14() 899.50/297.14 , 4: isNat^#(X) -> c_3(X) 899.50/297.14 , 5: isNat^#(n__0()) -> c_4() 899.50/297.14 , 6: isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , 7: isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , 8: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , 9: activate^#(X) -> c_7(X) 899.50/297.14 , 10: activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , 11: activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , 12: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , 13: activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , 14: activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , 15: activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , 16: 0^#() -> c_29() 899.50/297.14 , 17: plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , 18: plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , 19: plus^#(N, 0()) -> 899.50/297.14 c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , 20: isNatKind^#(X) -> c_25(X) 899.50/297.14 , 21: isNatKind^#(n__0()) -> c_26() 899.50/297.14 , 22: isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , 23: isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , 24: s^#(X) -> c_19(X) 899.50/297.14 , 25: and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , 26: and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , 27: U22^#(tt()) -> c_16() 899.50/297.14 , 28: U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , 29: U41^#(tt(), M, N) -> 899.50/297.14 c_18(s^#(plus(activate(N), activate(M)))) } 899.50/297.14 899.50/297.14 We are left with following problem, upon which TcT provides the 899.50/297.14 certificate MAYBE. 899.50/297.14 899.50/297.14 Strict DPs: 899.50/297.14 { U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , isNat^#(X) -> c_3(X) 899.50/297.14 , isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , activate^#(X) -> c_7(X) 899.50/297.14 , activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , plus^#(N, 0()) -> c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , isNatKind^#(X) -> c_25(X) 899.50/297.14 , isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , s^#(X) -> c_19(X) 899.50/297.14 , and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , U41^#(tt(), M, N) -> c_18(s^#(plus(activate(N), activate(M)))) } 899.50/297.14 Strict Trs: 899.50/297.14 { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 899.50/297.14 , U12(tt(), V2) -> U13(isNat(activate(V2))) 899.50/297.14 , isNat(X) -> n__isNat(X) 899.50/297.14 , isNat(n__0()) -> tt() 899.50/297.14 , isNat(n__plus(V1, V2)) -> 899.50/297.14 U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2)) 899.50/297.14 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 899.50/297.14 , activate(X) -> X 899.50/297.14 , activate(n__0()) -> 0() 899.50/297.14 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 899.50/297.14 , activate(n__isNatKind(X)) -> isNatKind(X) 899.50/297.14 , activate(n__s(X)) -> s(activate(X)) 899.50/297.14 , activate(n__and(X1, X2)) -> and(activate(X1), X2) 899.50/297.14 , activate(n__isNat(X)) -> isNat(X) 899.50/297.14 , U13(tt()) -> tt() 899.50/297.14 , U21(tt(), V1) -> U22(isNat(activate(V1))) 899.50/297.14 , U22(tt()) -> tt() 899.50/297.14 , U31(tt(), N) -> activate(N) 899.50/297.14 , U41(tt(), M, N) -> s(plus(activate(N), activate(M))) 899.50/297.14 , s(X) -> n__s(X) 899.50/297.14 , plus(X1, X2) -> n__plus(X1, X2) 899.50/297.14 , plus(N, s(M)) -> 899.50/297.14 U41(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N) 899.50/297.14 , plus(N, 0()) -> U31(and(isNat(N), n__isNatKind(N)), N) 899.50/297.14 , and(X1, X2) -> n__and(X1, X2) 899.50/297.14 , and(tt(), X) -> activate(X) 899.50/297.14 , isNatKind(X) -> n__isNatKind(X) 899.50/297.14 , isNatKind(n__0()) -> tt() 899.50/297.14 , isNatKind(n__plus(V1, V2)) -> 899.50/297.14 and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 899.50/297.14 , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 899.50/297.14 , 0() -> n__0() } 899.50/297.14 Weak DPs: 899.50/297.14 { U13^#(tt()) -> c_14() 899.50/297.14 , isNat^#(n__0()) -> c_4() 899.50/297.14 , 0^#() -> c_29() 899.50/297.14 , isNatKind^#(n__0()) -> c_26() 899.50/297.14 , U22^#(tt()) -> c_16() } 899.50/297.14 Obligation: 899.50/297.14 runtime complexity 899.50/297.14 Answer: 899.50/297.14 MAYBE 899.50/297.14 899.50/297.14 We estimate the number of application of {2,6,8} by applications of 899.50/297.14 Pre({2,6,8}) = {1,3,5,7,14,17,20,21,22,23}. Here rules are labeled 899.50/297.14 as follows: 899.50/297.14 899.50/297.14 DPs: 899.50/297.14 { 1: U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , 2: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , 3: isNat^#(X) -> c_3(X) 899.50/297.14 , 4: isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , 5: isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , 6: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , 7: activate^#(X) -> c_7(X) 899.50/297.14 , 8: activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , 9: activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , 10: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , 11: activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , 12: activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , 13: activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , 14: plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , 15: plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , 16: plus^#(N, 0()) -> 899.50/297.14 c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , 17: isNatKind^#(X) -> c_25(X) 899.50/297.14 , 18: isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , 19: isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , 20: s^#(X) -> c_19(X) 899.50/297.14 , 21: and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , 22: and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , 23: U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , 24: U41^#(tt(), M, N) -> 899.50/297.14 c_18(s^#(plus(activate(N), activate(M)))) 899.50/297.14 , 25: U13^#(tt()) -> c_14() 899.50/297.14 , 26: isNat^#(n__0()) -> c_4() 899.50/297.14 , 27: 0^#() -> c_29() 899.50/297.14 , 28: isNatKind^#(n__0()) -> c_26() 899.50/297.14 , 29: U22^#(tt()) -> c_16() } 899.50/297.14 899.50/297.14 We are left with following problem, upon which TcT provides the 899.50/297.14 certificate MAYBE. 899.50/297.14 899.50/297.14 Strict DPs: 899.50/297.14 { U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , isNat^#(X) -> c_3(X) 899.50/297.14 , isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , activate^#(X) -> c_7(X) 899.50/297.14 , activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , plus^#(N, 0()) -> c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , isNatKind^#(X) -> c_25(X) 899.50/297.14 , isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , s^#(X) -> c_19(X) 899.50/297.14 , and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , U41^#(tt(), M, N) -> c_18(s^#(plus(activate(N), activate(M)))) } 899.50/297.14 Strict Trs: 899.50/297.14 { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 899.50/297.14 , U12(tt(), V2) -> U13(isNat(activate(V2))) 899.50/297.14 , isNat(X) -> n__isNat(X) 899.50/297.14 , isNat(n__0()) -> tt() 899.50/297.14 , isNat(n__plus(V1, V2)) -> 899.50/297.14 U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2)) 899.50/297.14 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 899.50/297.14 , activate(X) -> X 899.50/297.14 , activate(n__0()) -> 0() 899.50/297.14 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 899.50/297.14 , activate(n__isNatKind(X)) -> isNatKind(X) 899.50/297.14 , activate(n__s(X)) -> s(activate(X)) 899.50/297.14 , activate(n__and(X1, X2)) -> and(activate(X1), X2) 899.50/297.14 , activate(n__isNat(X)) -> isNat(X) 899.50/297.14 , U13(tt()) -> tt() 899.50/297.14 , U21(tt(), V1) -> U22(isNat(activate(V1))) 899.50/297.14 , U22(tt()) -> tt() 899.50/297.14 , U31(tt(), N) -> activate(N) 899.50/297.14 , U41(tt(), M, N) -> s(plus(activate(N), activate(M))) 899.50/297.14 , s(X) -> n__s(X) 899.50/297.14 , plus(X1, X2) -> n__plus(X1, X2) 899.50/297.14 , plus(N, s(M)) -> 899.50/297.14 U41(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N) 899.50/297.14 , plus(N, 0()) -> U31(and(isNat(N), n__isNatKind(N)), N) 899.50/297.14 , and(X1, X2) -> n__and(X1, X2) 899.50/297.14 , and(tt(), X) -> activate(X) 899.50/297.14 , isNatKind(X) -> n__isNatKind(X) 899.50/297.14 , isNatKind(n__0()) -> tt() 899.50/297.14 , isNatKind(n__plus(V1, V2)) -> 899.50/297.14 and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 899.50/297.14 , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 899.50/297.14 , 0() -> n__0() } 899.50/297.14 Weak DPs: 899.50/297.14 { U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , U13^#(tt()) -> c_14() 899.50/297.14 , isNat^#(n__0()) -> c_4() 899.50/297.14 , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , 0^#() -> c_29() 899.50/297.14 , isNatKind^#(n__0()) -> c_26() 899.50/297.14 , U22^#(tt()) -> c_16() } 899.50/297.14 Obligation: 899.50/297.14 runtime complexity 899.50/297.14 Answer: 899.50/297.14 MAYBE 899.50/297.14 899.50/297.14 We estimate the number of application of {1,4} by applications of 899.50/297.14 Pre({1,4}) = {2,3,5,10,11,14,17,18}. Here rules are labeled as 899.50/297.14 follows: 899.50/297.14 899.50/297.14 DPs: 899.50/297.14 { 1: U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , 2: isNat^#(X) -> c_3(X) 899.50/297.14 , 3: isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , 4: isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , 5: activate^#(X) -> c_7(X) 899.50/297.14 , 6: activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , 7: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , 8: activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , 9: activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , 10: activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , 11: plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , 12: plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , 13: plus^#(N, 0()) -> 899.50/297.14 c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , 14: isNatKind^#(X) -> c_25(X) 899.50/297.14 , 15: isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , 16: isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , 17: s^#(X) -> c_19(X) 899.50/297.14 , 18: and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , 19: and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , 20: U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , 21: U41^#(tt(), M, N) -> 899.50/297.14 c_18(s^#(plus(activate(N), activate(M)))) 899.50/297.14 , 22: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , 23: U13^#(tt()) -> c_14() 899.50/297.14 , 24: isNat^#(n__0()) -> c_4() 899.50/297.14 , 25: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , 26: activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , 27: 0^#() -> c_29() 899.50/297.14 , 28: isNatKind^#(n__0()) -> c_26() 899.50/297.14 , 29: U22^#(tt()) -> c_16() } 899.50/297.14 899.50/297.14 We are left with following problem, upon which TcT provides the 899.50/297.14 certificate MAYBE. 899.50/297.14 899.50/297.14 Strict DPs: 899.50/297.14 { isNat^#(X) -> c_3(X) 899.50/297.14 , isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , activate^#(X) -> c_7(X) 899.50/297.14 , activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , plus^#(N, 0()) -> c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , isNatKind^#(X) -> c_25(X) 899.50/297.14 , isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , s^#(X) -> c_19(X) 899.50/297.14 , and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , U41^#(tt(), M, N) -> c_18(s^#(plus(activate(N), activate(M)))) } 899.50/297.14 Strict Trs: 899.50/297.14 { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 899.50/297.14 , U12(tt(), V2) -> U13(isNat(activate(V2))) 899.50/297.14 , isNat(X) -> n__isNat(X) 899.50/297.14 , isNat(n__0()) -> tt() 899.50/297.14 , isNat(n__plus(V1, V2)) -> 899.50/297.14 U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2)) 899.50/297.14 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 899.50/297.14 , activate(X) -> X 899.50/297.14 , activate(n__0()) -> 0() 899.50/297.14 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 899.50/297.14 , activate(n__isNatKind(X)) -> isNatKind(X) 899.50/297.14 , activate(n__s(X)) -> s(activate(X)) 899.50/297.14 , activate(n__and(X1, X2)) -> and(activate(X1), X2) 899.50/297.14 , activate(n__isNat(X)) -> isNat(X) 899.50/297.14 , U13(tt()) -> tt() 899.50/297.14 , U21(tt(), V1) -> U22(isNat(activate(V1))) 899.50/297.14 , U22(tt()) -> tt() 899.50/297.14 , U31(tt(), N) -> activate(N) 899.50/297.14 , U41(tt(), M, N) -> s(plus(activate(N), activate(M))) 899.50/297.14 , s(X) -> n__s(X) 899.50/297.14 , plus(X1, X2) -> n__plus(X1, X2) 899.50/297.14 , plus(N, s(M)) -> 899.50/297.14 U41(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N) 899.50/297.14 , plus(N, 0()) -> U31(and(isNat(N), n__isNatKind(N)), N) 899.50/297.14 , and(X1, X2) -> n__and(X1, X2) 899.50/297.14 , and(tt(), X) -> activate(X) 899.50/297.14 , isNatKind(X) -> n__isNatKind(X) 899.50/297.14 , isNatKind(n__0()) -> tt() 899.50/297.14 , isNatKind(n__plus(V1, V2)) -> 899.50/297.14 and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 899.50/297.14 , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 899.50/297.14 , 0() -> n__0() } 899.50/297.14 Weak DPs: 899.50/297.14 { U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , U13^#(tt()) -> c_14() 899.50/297.14 , isNat^#(n__0()) -> c_4() 899.50/297.14 , isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , 0^#() -> c_29() 899.50/297.14 , isNatKind^#(n__0()) -> c_26() 899.50/297.14 , U22^#(tt()) -> c_16() } 899.50/297.14 Obligation: 899.50/297.14 runtime complexity 899.50/297.14 Answer: 899.50/297.14 MAYBE 899.50/297.14 899.50/297.14 We estimate the number of application of {2} by applications of 899.50/297.14 Pre({2}) = {1,3,8,9,12,15,16}. Here rules are labeled as follows: 899.50/297.14 899.50/297.14 DPs: 899.50/297.14 { 1: isNat^#(X) -> c_3(X) 899.50/297.14 , 2: isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , 3: activate^#(X) -> c_7(X) 899.50/297.14 , 4: activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , 5: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , 6: activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , 7: activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , 8: activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , 9: plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , 10: plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , 11: plus^#(N, 0()) -> 899.50/297.14 c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , 12: isNatKind^#(X) -> c_25(X) 899.50/297.14 , 13: isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , 14: isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , 15: s^#(X) -> c_19(X) 899.50/297.14 , 16: and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , 17: and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , 18: U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , 19: U41^#(tt(), M, N) -> 899.50/297.14 c_18(s^#(plus(activate(N), activate(M)))) 899.50/297.14 , 20: U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , 21: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , 22: U13^#(tt()) -> c_14() 899.50/297.14 , 23: isNat^#(n__0()) -> c_4() 899.50/297.14 , 24: isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , 25: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , 26: activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , 27: 0^#() -> c_29() 899.50/297.14 , 28: isNatKind^#(n__0()) -> c_26() 899.50/297.14 , 29: U22^#(tt()) -> c_16() } 899.50/297.14 899.50/297.14 We are left with following problem, upon which TcT provides the 899.50/297.14 certificate MAYBE. 899.50/297.14 899.50/297.14 Strict DPs: 899.50/297.14 { isNat^#(X) -> c_3(X) 899.50/297.14 , activate^#(X) -> c_7(X) 899.50/297.14 , activate^#(n__plus(X1, X2)) -> 899.50/297.14 c_9(plus^#(activate(X1), activate(X2))) 899.50/297.14 , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X)) 899.50/297.14 , activate^#(n__s(X)) -> c_11(s^#(activate(X))) 899.50/297.14 , activate^#(n__and(X1, X2)) -> c_12(and^#(activate(X1), X2)) 899.50/297.14 , activate^#(n__isNat(X)) -> c_13(isNat^#(X)) 899.50/297.14 , plus^#(X1, X2) -> c_20(X1, X2) 899.50/297.14 , plus^#(N, s(M)) -> 899.50/297.14 c_21(U41^#(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N)) 899.50/297.14 , plus^#(N, 0()) -> c_22(U31^#(and(isNat(N), n__isNatKind(N)), N)) 899.50/297.14 , isNatKind^#(X) -> c_25(X) 899.50/297.14 , isNatKind^#(n__plus(V1, V2)) -> 899.50/297.14 c_27(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2)))) 899.50/297.14 , isNatKind^#(n__s(V1)) -> c_28(isNatKind^#(activate(V1))) 899.50/297.14 , s^#(X) -> c_19(X) 899.50/297.14 , and^#(X1, X2) -> c_23(X1, X2) 899.50/297.14 , and^#(tt(), X) -> c_24(activate^#(X)) 899.50/297.14 , U31^#(tt(), N) -> c_17(activate^#(N)) 899.50/297.14 , U41^#(tt(), M, N) -> c_18(s^#(plus(activate(N), activate(M)))) } 899.50/297.14 Strict Trs: 899.50/297.14 { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 899.50/297.14 , U12(tt(), V2) -> U13(isNat(activate(V2))) 899.50/297.14 , isNat(X) -> n__isNat(X) 899.50/297.14 , isNat(n__0()) -> tt() 899.50/297.14 , isNat(n__plus(V1, V2)) -> 899.50/297.14 U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2)) 899.50/297.14 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 899.50/297.14 , activate(X) -> X 899.50/297.14 , activate(n__0()) -> 0() 899.50/297.14 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 899.50/297.14 , activate(n__isNatKind(X)) -> isNatKind(X) 899.50/297.14 , activate(n__s(X)) -> s(activate(X)) 899.50/297.14 , activate(n__and(X1, X2)) -> and(activate(X1), X2) 899.50/297.14 , activate(n__isNat(X)) -> isNat(X) 899.50/297.14 , U13(tt()) -> tt() 899.50/297.14 , U21(tt(), V1) -> U22(isNat(activate(V1))) 899.50/297.14 , U22(tt()) -> tt() 899.50/297.14 , U31(tt(), N) -> activate(N) 899.50/297.14 , U41(tt(), M, N) -> s(plus(activate(N), activate(M))) 899.50/297.14 , s(X) -> n__s(X) 899.50/297.14 , plus(X1, X2) -> n__plus(X1, X2) 899.50/297.14 , plus(N, s(M)) -> 899.50/297.14 U41(and(and(isNat(M), n__isNatKind(M)), 899.50/297.14 n__and(n__isNat(N), n__isNatKind(N))), 899.50/297.14 M, 899.50/297.14 N) 899.50/297.14 , plus(N, 0()) -> U31(and(isNat(N), n__isNatKind(N)), N) 899.50/297.14 , and(X1, X2) -> n__and(X1, X2) 899.50/297.14 , and(tt(), X) -> activate(X) 899.50/297.14 , isNatKind(X) -> n__isNatKind(X) 899.50/297.14 , isNatKind(n__0()) -> tt() 899.50/297.14 , isNatKind(n__plus(V1, V2)) -> 899.50/297.14 and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 899.50/297.14 , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 899.50/297.14 , 0() -> n__0() } 899.50/297.14 Weak DPs: 899.50/297.14 { U11^#(tt(), V1, V2) -> 899.50/297.14 c_1(U12^#(isNat(activate(V1)), activate(V2))) 899.50/297.14 , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2)))) 899.50/297.14 , U13^#(tt()) -> c_14() 899.50/297.14 , isNat^#(n__0()) -> c_4() 899.50/297.14 , isNat^#(n__plus(V1, V2)) -> 899.50/297.14 c_5(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), 899.50/297.14 activate(V1), 899.50/297.14 activate(V2))) 899.50/297.14 , isNat^#(n__s(V1)) -> 899.50/297.14 c_6(U21^#(isNatKind(activate(V1)), activate(V1))) 899.50/297.14 , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1)))) 899.50/297.14 , activate^#(n__0()) -> c_8(0^#()) 899.50/297.14 , 0^#() -> c_29() 899.50/297.14 , isNatKind^#(n__0()) -> c_26() 899.50/297.14 , U22^#(tt()) -> c_16() } 899.50/297.14 Obligation: 899.50/297.14 runtime complexity 899.50/297.14 Answer: 899.50/297.14 MAYBE 899.50/297.14 899.50/297.14 Empty strict component of the problem is NOT empty. 899.50/297.14 899.50/297.14 899.50/297.14 Arrrr.. 899.71/297.25 EOF