MAYBE 859.88/297.03 MAYBE 859.88/297.03 859.88/297.03 We are left with following problem, upon which TcT provides the 859.88/297.03 certificate MAYBE. 859.88/297.03 859.88/297.03 Strict Trs: 859.88/297.03 { U11(tt(), N) -> activate(N) 859.88/297.03 , activate(X) -> X 859.88/297.03 , activate(n__0()) -> 0() 859.88/297.03 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 859.88/297.03 , activate(n__isNat(X)) -> isNat(X) 859.88/297.03 , activate(n__s(X)) -> s(activate(X)) 859.88/297.03 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 859.88/297.03 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 859.88/297.03 , s(X) -> n__s(X) 859.88/297.03 , plus(X1, X2) -> n__plus(X1, X2) 859.88/297.03 , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , plus(N, 0()) -> U11(isNat(N), N) 859.88/297.03 , U31(tt()) -> 0() 859.88/297.03 , 0() -> n__0() 859.88/297.03 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 859.88/297.03 , x(X1, X2) -> n__x(X1, X2) 859.88/297.03 , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , x(N, 0()) -> U31(isNat(N)) 859.88/297.03 , and(tt(), X) -> activate(X) 859.88/297.03 , isNat(X) -> n__isNat(X) 859.88/297.03 , isNat(n__0()) -> tt() 859.88/297.03 , isNat(n__plus(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) 859.88/297.03 , isNat(n__s(V1)) -> isNat(activate(V1)) 859.88/297.03 , isNat(n__x(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) } 859.88/297.03 Obligation: 859.88/297.03 runtime complexity 859.88/297.03 Answer: 859.88/297.03 MAYBE 859.88/297.03 859.88/297.03 None of the processors succeeded. 859.88/297.03 859.88/297.03 Details of failed attempt(s): 859.88/297.03 ----------------------------- 859.88/297.03 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 859.88/297.03 following reason: 859.88/297.03 859.88/297.03 Computation stopped due to timeout after 297.0 seconds. 859.88/297.03 859.88/297.03 2) 'Best' failed due to the following reason: 859.88/297.03 859.88/297.03 None of the processors succeeded. 859.88/297.03 859.88/297.03 Details of failed attempt(s): 859.88/297.03 ----------------------------- 859.88/297.03 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 859.88/297.03 seconds)' failed due to the following reason: 859.88/297.03 859.88/297.03 Computation stopped due to timeout after 148.0 seconds. 859.88/297.03 859.88/297.03 2) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 859.88/297.03 failed due to the following reason: 859.88/297.03 859.88/297.03 None of the processors succeeded. 859.88/297.03 859.88/297.03 Details of failed attempt(s): 859.88/297.03 ----------------------------- 859.88/297.03 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 859.88/297.03 failed due to the following reason: 859.88/297.03 859.88/297.03 match-boundness of the problem could not be verified. 859.88/297.03 859.88/297.03 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 859.88/297.03 failed due to the following reason: 859.88/297.03 859.88/297.03 match-boundness of the problem could not be verified. 859.88/297.03 859.88/297.03 859.88/297.03 3) 'Best' failed due to the following reason: 859.88/297.03 859.88/297.03 None of the processors succeeded. 859.88/297.03 859.88/297.03 Details of failed attempt(s): 859.88/297.03 ----------------------------- 859.88/297.03 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 859.88/297.03 following reason: 859.88/297.03 859.88/297.03 The processor is inapplicable, reason: 859.88/297.03 Processor only applicable for innermost runtime complexity analysis 859.88/297.03 859.88/297.03 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 859.88/297.03 to the following reason: 859.88/297.03 859.88/297.03 The processor is inapplicable, reason: 859.88/297.03 Processor only applicable for innermost runtime complexity analysis 859.88/297.03 859.88/297.03 859.88/297.03 859.88/297.03 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 859.88/297.03 the following reason: 859.88/297.03 859.88/297.03 We add the following weak dependency pairs: 859.88/297.03 859.88/297.03 Strict DPs: 859.88/297.03 { U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , activate^#(X) -> c_2(X) 859.88/297.03 , activate^#(n__0()) -> c_3(0^#()) 859.88/297.03 , activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , 0^#() -> c_14() 859.88/297.03 , plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , isNat^#(X) -> c_20(X) 859.88/297.03 , isNat^#(n__0()) -> c_21() 859.88/297.03 , isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , s^#(X) -> c_9(X) 859.88/297.03 , x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.03 , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , U31^#(tt()) -> c_13(0^#()) 859.88/297.03 , U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , and^#(tt(), X) -> c_19(activate^#(X)) } 859.88/297.03 859.88/297.03 and mark the set of starting terms. 859.88/297.03 859.88/297.03 We are left with following problem, upon which TcT provides the 859.88/297.03 certificate MAYBE. 859.88/297.03 859.88/297.03 Strict DPs: 859.88/297.03 { U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , activate^#(X) -> c_2(X) 859.88/297.03 , activate^#(n__0()) -> c_3(0^#()) 859.88/297.03 , activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , 0^#() -> c_14() 859.88/297.03 , plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , isNat^#(X) -> c_20(X) 859.88/297.03 , isNat^#(n__0()) -> c_21() 859.88/297.03 , isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , s^#(X) -> c_9(X) 859.88/297.03 , x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.03 , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , U31^#(tt()) -> c_13(0^#()) 859.88/297.03 , U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , and^#(tt(), X) -> c_19(activate^#(X)) } 859.88/297.03 Strict Trs: 859.88/297.03 { U11(tt(), N) -> activate(N) 859.88/297.03 , activate(X) -> X 859.88/297.03 , activate(n__0()) -> 0() 859.88/297.03 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 859.88/297.03 , activate(n__isNat(X)) -> isNat(X) 859.88/297.03 , activate(n__s(X)) -> s(activate(X)) 859.88/297.03 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 859.88/297.03 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 859.88/297.03 , s(X) -> n__s(X) 859.88/297.03 , plus(X1, X2) -> n__plus(X1, X2) 859.88/297.03 , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , plus(N, 0()) -> U11(isNat(N), N) 859.88/297.03 , U31(tt()) -> 0() 859.88/297.03 , 0() -> n__0() 859.88/297.03 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 859.88/297.03 , x(X1, X2) -> n__x(X1, X2) 859.88/297.03 , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , x(N, 0()) -> U31(isNat(N)) 859.88/297.03 , and(tt(), X) -> activate(X) 859.88/297.03 , isNat(X) -> n__isNat(X) 859.88/297.03 , isNat(n__0()) -> tt() 859.88/297.03 , isNat(n__plus(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) 859.88/297.03 , isNat(n__s(V1)) -> isNat(activate(V1)) 859.88/297.03 , isNat(n__x(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) } 859.88/297.03 Obligation: 859.88/297.03 runtime complexity 859.88/297.03 Answer: 859.88/297.03 MAYBE 859.88/297.03 859.88/297.03 We estimate the number of application of {8,13} by applications of 859.88/297.03 Pre({8,13}) = {2,3,5,9,12,15,17,18,22}. Here rules are labeled as 859.88/297.03 follows: 859.88/297.03 859.88/297.03 DPs: 859.88/297.03 { 1: U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , 2: activate^#(X) -> c_2(X) 859.88/297.03 , 3: activate^#(n__0()) -> c_3(0^#()) 859.88/297.03 , 4: activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , 5: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , 6: activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , 7: activate^#(n__x(X1, X2)) -> 859.88/297.03 c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , 8: 0^#() -> c_14() 859.88/297.03 , 9: plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , 10: plus^#(N, s(M)) -> 859.88/297.03 c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , 11: plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , 12: isNat^#(X) -> c_20(X) 859.88/297.03 , 13: isNat^#(n__0()) -> c_21() 859.88/297.03 , 14: isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , 15: isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , 16: isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , 17: s^#(X) -> c_9(X) 859.88/297.03 , 18: x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , 19: x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , 20: x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.03 , 21: U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , 22: U31^#(tt()) -> c_13(0^#()) 859.88/297.03 , 23: U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , 24: and^#(tt(), X) -> c_19(activate^#(X)) } 859.88/297.03 859.88/297.03 We are left with following problem, upon which TcT provides the 859.88/297.03 certificate MAYBE. 859.88/297.03 859.88/297.03 Strict DPs: 859.88/297.03 { U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , activate^#(X) -> c_2(X) 859.88/297.03 , activate^#(n__0()) -> c_3(0^#()) 859.88/297.03 , activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , isNat^#(X) -> c_20(X) 859.88/297.03 , isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , s^#(X) -> c_9(X) 859.88/297.03 , x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.03 , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , U31^#(tt()) -> c_13(0^#()) 859.88/297.03 , U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , and^#(tt(), X) -> c_19(activate^#(X)) } 859.88/297.03 Strict Trs: 859.88/297.03 { U11(tt(), N) -> activate(N) 859.88/297.03 , activate(X) -> X 859.88/297.03 , activate(n__0()) -> 0() 859.88/297.03 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 859.88/297.03 , activate(n__isNat(X)) -> isNat(X) 859.88/297.03 , activate(n__s(X)) -> s(activate(X)) 859.88/297.03 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 859.88/297.03 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 859.88/297.03 , s(X) -> n__s(X) 859.88/297.03 , plus(X1, X2) -> n__plus(X1, X2) 859.88/297.03 , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , plus(N, 0()) -> U11(isNat(N), N) 859.88/297.03 , U31(tt()) -> 0() 859.88/297.03 , 0() -> n__0() 859.88/297.03 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 859.88/297.03 , x(X1, X2) -> n__x(X1, X2) 859.88/297.03 , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , x(N, 0()) -> U31(isNat(N)) 859.88/297.03 , and(tt(), X) -> activate(X) 859.88/297.03 , isNat(X) -> n__isNat(X) 859.88/297.03 , isNat(n__0()) -> tt() 859.88/297.03 , isNat(n__plus(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) 859.88/297.03 , isNat(n__s(V1)) -> isNat(activate(V1)) 859.88/297.03 , isNat(n__x(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) } 859.88/297.03 Weak DPs: 859.88/297.03 { 0^#() -> c_14() 859.88/297.03 , isNat^#(n__0()) -> c_21() } 859.88/297.03 Obligation: 859.88/297.03 runtime complexity 859.88/297.03 Answer: 859.88/297.03 MAYBE 859.88/297.03 859.88/297.03 We estimate the number of application of {3,20} by applications of 859.88/297.03 Pre({3,20}) = {1,2,8,11,15,16,18,22}. Here rules are labeled as 859.88/297.03 follows: 859.88/297.03 859.88/297.03 DPs: 859.88/297.03 { 1: U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , 2: activate^#(X) -> c_2(X) 859.88/297.03 , 3: activate^#(n__0()) -> c_3(0^#()) 859.88/297.03 , 4: activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , 5: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , 6: activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , 7: activate^#(n__x(X1, X2)) -> 859.88/297.03 c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , 8: plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , 9: plus^#(N, s(M)) -> 859.88/297.03 c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , 10: plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , 11: isNat^#(X) -> c_20(X) 859.88/297.03 , 12: isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , 13: isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , 14: isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , 15: s^#(X) -> c_9(X) 859.88/297.03 , 16: x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , 17: x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , 18: x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.03 , 19: U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , 20: U31^#(tt()) -> c_13(0^#()) 859.88/297.03 , 21: U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , 22: and^#(tt(), X) -> c_19(activate^#(X)) 859.88/297.03 , 23: 0^#() -> c_14() 859.88/297.03 , 24: isNat^#(n__0()) -> c_21() } 859.88/297.03 859.88/297.03 We are left with following problem, upon which TcT provides the 859.88/297.03 certificate MAYBE. 859.88/297.03 859.88/297.03 Strict DPs: 859.88/297.03 { U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , activate^#(X) -> c_2(X) 859.88/297.03 , activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , isNat^#(X) -> c_20(X) 859.88/297.03 , isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , s^#(X) -> c_9(X) 859.88/297.03 , x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.03 , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , and^#(tt(), X) -> c_19(activate^#(X)) } 859.88/297.03 Strict Trs: 859.88/297.03 { U11(tt(), N) -> activate(N) 859.88/297.03 , activate(X) -> X 859.88/297.03 , activate(n__0()) -> 0() 859.88/297.03 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 859.88/297.03 , activate(n__isNat(X)) -> isNat(X) 859.88/297.03 , activate(n__s(X)) -> s(activate(X)) 859.88/297.03 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 859.88/297.03 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 859.88/297.03 , s(X) -> n__s(X) 859.88/297.03 , plus(X1, X2) -> n__plus(X1, X2) 859.88/297.03 , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , plus(N, 0()) -> U11(isNat(N), N) 859.88/297.03 , U31(tt()) -> 0() 859.88/297.03 , 0() -> n__0() 859.88/297.03 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 859.88/297.03 , x(X1, X2) -> n__x(X1, X2) 859.88/297.03 , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , x(N, 0()) -> U31(isNat(N)) 859.88/297.03 , and(tt(), X) -> activate(X) 859.88/297.03 , isNat(X) -> n__isNat(X) 859.88/297.03 , isNat(n__0()) -> tt() 859.88/297.03 , isNat(n__plus(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) 859.88/297.03 , isNat(n__s(V1)) -> isNat(activate(V1)) 859.88/297.03 , isNat(n__x(V1, V2)) -> 859.88/297.03 and(isNat(activate(V1)), n__isNat(activate(V2))) } 859.88/297.03 Weak DPs: 859.88/297.03 { activate^#(n__0()) -> c_3(0^#()) 859.88/297.03 , 0^#() -> c_14() 859.88/297.03 , isNat^#(n__0()) -> c_21() 859.88/297.03 , U31^#(tt()) -> c_13(0^#()) } 859.88/297.03 Obligation: 859.88/297.03 runtime complexity 859.88/297.03 Answer: 859.88/297.03 MAYBE 859.88/297.03 859.88/297.03 We estimate the number of application of {17} by applications of 859.88/297.03 Pre({17}) = {2,6,7,10,14,15}. Here rules are labeled as follows: 859.88/297.03 859.88/297.03 DPs: 859.88/297.03 { 1: U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , 2: activate^#(X) -> c_2(X) 859.88/297.03 , 3: activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , 4: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , 5: activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , 6: activate^#(n__x(X1, X2)) -> 859.88/297.03 c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , 7: plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , 8: plus^#(N, s(M)) -> 859.88/297.03 c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , 9: plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , 10: isNat^#(X) -> c_20(X) 859.88/297.03 , 11: isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , 12: isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , 13: isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , 14: s^#(X) -> c_9(X) 859.88/297.03 , 15: x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , 16: x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , 17: x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.03 , 18: U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , 19: U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , 20: and^#(tt(), X) -> c_19(activate^#(X)) 859.88/297.03 , 21: activate^#(n__0()) -> c_3(0^#()) 859.88/297.03 , 22: 0^#() -> c_14() 859.88/297.03 , 23: isNat^#(n__0()) -> c_21() 859.88/297.03 , 24: U31^#(tt()) -> c_13(0^#()) } 859.88/297.03 859.88/297.03 We are left with following problem, upon which TcT provides the 859.88/297.03 certificate MAYBE. 859.88/297.03 859.88/297.03 Strict DPs: 859.88/297.03 { U11^#(tt(), N) -> c_1(activate^#(N)) 859.88/297.03 , activate^#(X) -> c_2(X) 859.88/297.03 , activate^#(n__plus(X1, X2)) -> 859.88/297.03 c_4(plus^#(activate(X1), activate(X2))) 859.88/297.03 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 859.88/297.03 , activate^#(n__s(X)) -> c_6(s^#(activate(X))) 859.88/297.03 , activate^#(n__x(X1, X2)) -> c_7(x^#(activate(X1), activate(X2))) 859.88/297.03 , plus^#(X1, X2) -> c_10(X1, X2) 859.88/297.03 , plus^#(N, s(M)) -> c_11(U21^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , plus^#(N, 0()) -> c_12(U11^#(isNat(N), N)) 859.88/297.03 , isNat^#(X) -> c_20(X) 859.88/297.03 , isNat^#(n__plus(V1, V2)) -> 859.88/297.03 c_22(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , isNat^#(n__s(V1)) -> c_23(isNat^#(activate(V1))) 859.88/297.03 , isNat^#(n__x(V1, V2)) -> 859.88/297.03 c_24(and^#(isNat(activate(V1)), n__isNat(activate(V2)))) 859.88/297.03 , s^#(X) -> c_9(X) 859.88/297.03 , x^#(X1, X2) -> c_16(X1, X2) 859.88/297.03 , x^#(N, s(M)) -> c_17(U41^#(and(isNat(M), n__isNat(N)), M, N)) 859.88/297.03 , U21^#(tt(), M, N) -> c_8(s^#(plus(activate(N), activate(M)))) 859.88/297.03 , U41^#(tt(), M, N) -> 859.88/297.03 c_15(plus^#(x(activate(N), activate(M)), activate(N))) 859.88/297.03 , and^#(tt(), X) -> c_19(activate^#(X)) } 859.88/297.03 Strict Trs: 859.88/297.03 { U11(tt(), N) -> activate(N) 859.88/297.03 , activate(X) -> X 859.88/297.03 , activate(n__0()) -> 0() 859.88/297.03 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 859.88/297.03 , activate(n__isNat(X)) -> isNat(X) 859.88/297.03 , activate(n__s(X)) -> s(activate(X)) 859.88/297.03 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 859.88/297.03 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 859.88/297.03 , s(X) -> n__s(X) 859.88/297.03 , plus(X1, X2) -> n__plus(X1, X2) 859.88/297.03 , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , plus(N, 0()) -> U11(isNat(N), N) 859.88/297.03 , U31(tt()) -> 0() 859.88/297.03 , 0() -> n__0() 859.88/297.03 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 859.88/297.03 , x(X1, X2) -> n__x(X1, X2) 859.88/297.03 , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 859.88/297.03 , x(N, 0()) -> U31(isNat(N)) 859.88/297.03 , and(tt(), X) -> activate(X) 859.88/297.03 , isNat(X) -> n__isNat(X) 859.88/297.03 , isNat(n__0()) -> tt() 859.88/297.03 , isNat(n__plus(V1, V2)) -> 859.88/297.04 and(isNat(activate(V1)), n__isNat(activate(V2))) 859.88/297.04 , isNat(n__s(V1)) -> isNat(activate(V1)) 859.88/297.04 , isNat(n__x(V1, V2)) -> 859.88/297.04 and(isNat(activate(V1)), n__isNat(activate(V2))) } 859.88/297.04 Weak DPs: 859.88/297.04 { activate^#(n__0()) -> c_3(0^#()) 859.88/297.04 , 0^#() -> c_14() 859.88/297.04 , isNat^#(n__0()) -> c_21() 859.88/297.04 , x^#(N, 0()) -> c_18(U31^#(isNat(N))) 859.88/297.04 , U31^#(tt()) -> c_13(0^#()) } 859.88/297.04 Obligation: 859.88/297.04 runtime complexity 859.88/297.04 Answer: 859.88/297.04 MAYBE 859.88/297.04 859.88/297.04 Empty strict component of the problem is NOT empty. 859.88/297.04 859.88/297.04 859.88/297.04 Arrrr.. 860.01/297.15 EOF