MAYBE 847.46/297.03 MAYBE 847.46/297.03 847.46/297.03 We are left with following problem, upon which TcT provides the 847.46/297.03 certificate MAYBE. 847.46/297.03 847.46/297.03 Strict Trs: 847.46/297.03 { U11(tt(), V2) -> U12(isNat(activate(V2))) 847.46/297.03 , U12(tt()) -> tt() 847.46/297.03 , isNat(n__0()) -> tt() 847.46/297.03 , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) 847.46/297.03 , isNat(n__s(V1)) -> U21(isNat(activate(V1))) 847.46/297.03 , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) 847.46/297.03 , activate(X) -> X 847.46/297.03 , activate(n__0()) -> 0() 847.46/297.03 , activate(n__plus(X1, X2)) -> plus(X1, X2) 847.46/297.03 , activate(n__s(X)) -> s(X) 847.46/297.03 , activate(n__x(X1, X2)) -> x(X1, X2) 847.46/297.03 , U21(tt()) -> tt() 847.46/297.03 , U31(tt(), V2) -> U32(isNat(activate(V2))) 847.46/297.03 , U32(tt()) -> tt() 847.46/297.03 , U41(tt(), N) -> activate(N) 847.46/297.03 , U51(tt(), M, N) -> 847.46/297.03 U52(isNat(activate(N)), activate(M), activate(N)) 847.46/297.03 , U52(tt(), M, N) -> s(plus(activate(N), activate(M))) 847.46/297.03 , s(X) -> n__s(X) 847.46/297.03 , plus(X1, X2) -> n__plus(X1, X2) 847.46/297.03 , plus(N, s(M)) -> U51(isNat(M), M, N) 847.46/297.03 , plus(N, 0()) -> U41(isNat(N), N) 847.46/297.03 , U61(tt()) -> 0() 847.46/297.03 , 0() -> n__0() 847.46/297.03 , U71(tt(), M, N) -> 847.46/297.03 U72(isNat(activate(N)), activate(M), activate(N)) 847.46/297.03 , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 847.46/297.03 , x(X1, X2) -> n__x(X1, X2) 847.46/297.03 , x(N, s(M)) -> U71(isNat(M), M, N) 847.46/297.03 , x(N, 0()) -> U61(isNat(N)) } 847.46/297.03 Obligation: 847.46/297.03 runtime complexity 847.46/297.03 Answer: 847.46/297.03 MAYBE 847.46/297.03 847.46/297.03 None of the processors succeeded. 847.46/297.03 847.46/297.03 Details of failed attempt(s): 847.46/297.03 ----------------------------- 847.46/297.03 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 847.46/297.03 following reason: 847.46/297.03 847.46/297.03 Computation stopped due to timeout after 297.0 seconds. 847.46/297.03 847.46/297.03 2) 'Best' failed due to the following reason: 847.46/297.03 847.46/297.03 None of the processors succeeded. 847.46/297.03 847.46/297.03 Details of failed attempt(s): 847.46/297.03 ----------------------------- 847.46/297.03 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 847.46/297.03 seconds)' failed due to the following reason: 847.46/297.03 847.46/297.03 Computation stopped due to timeout after 148.0 seconds. 847.46/297.03 847.46/297.03 2) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 847.46/297.03 failed due to the following reason: 847.46/297.03 847.46/297.03 None of the processors succeeded. 847.46/297.03 847.46/297.03 Details of failed attempt(s): 847.46/297.03 ----------------------------- 847.46/297.03 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 847.46/297.03 failed due to the following reason: 847.46/297.03 847.46/297.03 match-boundness of the problem could not be verified. 847.46/297.03 847.46/297.03 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 847.46/297.03 failed due to the following reason: 847.46/297.03 847.46/297.03 match-boundness of the problem could not be verified. 847.46/297.03 847.46/297.03 847.46/297.03 3) 'Best' failed due to the following reason: 847.46/297.03 847.46/297.03 None of the processors succeeded. 847.46/297.03 847.46/297.03 Details of failed attempt(s): 847.46/297.03 ----------------------------- 847.46/297.03 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 847.46/297.03 following reason: 847.46/297.03 847.46/297.03 The processor is inapplicable, reason: 847.46/297.03 Processor only applicable for innermost runtime complexity analysis 847.46/297.03 847.46/297.03 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 847.46/297.03 to the following reason: 847.46/297.03 847.46/297.03 The processor is inapplicable, reason: 847.46/297.03 Processor only applicable for innermost runtime complexity analysis 847.46/297.03 847.46/297.03 847.46/297.03 847.46/297.03 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 847.46/297.03 the following reason: 847.46/297.03 847.46/297.03 We add the following weak dependency pairs: 847.46/297.03 847.46/297.03 Strict DPs: 847.46/297.03 { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.03 , U12^#(tt()) -> c_2() 847.46/297.03 , isNat^#(n__0()) -> c_3() 847.46/297.03 , isNat^#(n__plus(V1, V2)) -> 847.46/297.03 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.03 , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.03 , isNat^#(n__x(V1, V2)) -> 847.46/297.03 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.03 , U21^#(tt()) -> c_12() 847.46/297.03 , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.03 , activate^#(X) -> c_7(X) 847.46/297.03 , activate^#(n__0()) -> c_8(0^#()) 847.46/297.03 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.03 , activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.03 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.03 , 0^#() -> c_23() 847.46/297.03 , plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.03 , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.03 , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.03 , s^#(X) -> c_18(X) 847.46/297.03 , x^#(X1, X2) -> c_26(X1, X2) 847.46/297.03 , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.03 , x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.03 , U32^#(tt()) -> c_14() 847.46/297.03 , U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.03 , U51^#(tt(), M, N) -> 847.46/297.03 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.03 , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.03 , U61^#(tt()) -> c_22(0^#()) 847.46/297.03 , U71^#(tt(), M, N) -> 847.46/297.03 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.03 , U72^#(tt(), M, N) -> 847.46/297.03 c_25(plus^#(x(activate(N), activate(M)), activate(N))) } 847.46/297.03 847.46/297.03 and mark the set of starting terms. 847.46/297.03 847.46/297.03 We are left with following problem, upon which TcT provides the 847.46/297.03 certificate MAYBE. 847.46/297.03 847.46/297.03 Strict DPs: 847.46/297.03 { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.03 , U12^#(tt()) -> c_2() 847.46/297.03 , isNat^#(n__0()) -> c_3() 847.46/297.03 , isNat^#(n__plus(V1, V2)) -> 847.46/297.03 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.03 , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.03 , isNat^#(n__x(V1, V2)) -> 847.46/297.03 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.03 , U21^#(tt()) -> c_12() 847.46/297.03 , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.03 , activate^#(X) -> c_7(X) 847.46/297.03 , activate^#(n__0()) -> c_8(0^#()) 847.46/297.03 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.03 , activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.03 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.03 , 0^#() -> c_23() 847.46/297.03 , plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.03 , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.03 , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.03 , s^#(X) -> c_18(X) 847.46/297.03 , x^#(X1, X2) -> c_26(X1, X2) 847.46/297.03 , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.03 , x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.03 , U32^#(tt()) -> c_14() 847.46/297.03 , U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.03 , U51^#(tt(), M, N) -> 847.46/297.03 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.03 , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.03 , U61^#(tt()) -> c_22(0^#()) 847.46/297.03 , U71^#(tt(), M, N) -> 847.46/297.03 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.03 , U72^#(tt(), M, N) -> 847.46/297.03 c_25(plus^#(x(activate(N), activate(M)), activate(N))) } 847.46/297.03 Strict Trs: 847.46/297.03 { U11(tt(), V2) -> U12(isNat(activate(V2))) 847.46/297.03 , U12(tt()) -> tt() 847.46/297.03 , isNat(n__0()) -> tt() 847.46/297.03 , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) 847.46/297.03 , isNat(n__s(V1)) -> U21(isNat(activate(V1))) 847.46/297.03 , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) 847.46/297.03 , activate(X) -> X 847.46/297.03 , activate(n__0()) -> 0() 847.46/297.03 , activate(n__plus(X1, X2)) -> plus(X1, X2) 847.46/297.03 , activate(n__s(X)) -> s(X) 847.46/297.03 , activate(n__x(X1, X2)) -> x(X1, X2) 847.46/297.03 , U21(tt()) -> tt() 847.46/297.03 , U31(tt(), V2) -> U32(isNat(activate(V2))) 847.46/297.03 , U32(tt()) -> tt() 847.46/297.03 , U41(tt(), N) -> activate(N) 847.46/297.03 , U51(tt(), M, N) -> 847.46/297.03 U52(isNat(activate(N)), activate(M), activate(N)) 847.46/297.03 , U52(tt(), M, N) -> s(plus(activate(N), activate(M))) 847.46/297.03 , s(X) -> n__s(X) 847.46/297.03 , plus(X1, X2) -> n__plus(X1, X2) 847.46/297.03 , plus(N, s(M)) -> U51(isNat(M), M, N) 847.46/297.03 , plus(N, 0()) -> U41(isNat(N), N) 847.46/297.03 , U61(tt()) -> 0() 847.46/297.03 , 0() -> n__0() 847.46/297.03 , U71(tt(), M, N) -> 847.46/297.03 U72(isNat(activate(N)), activate(M), activate(N)) 847.46/297.03 , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 847.46/297.03 , x(X1, X2) -> n__x(X1, X2) 847.46/297.03 , x(N, s(M)) -> U71(isNat(M), M, N) 847.46/297.03 , x(N, 0()) -> U61(isNat(N)) } 847.46/297.03 Obligation: 847.46/297.03 runtime complexity 847.46/297.03 Answer: 847.46/297.03 MAYBE 847.46/297.03 847.46/297.03 We estimate the number of application of {2,3,7,14,22} by 847.46/297.03 applications of Pre({2,3,7,14,22}) = {1,5,8,9,10,15,18,19,26}. Here 847.46/297.03 rules are labeled as follows: 847.46/297.03 847.46/297.03 DPs: 847.46/297.03 { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.03 , 2: U12^#(tt()) -> c_2() 847.46/297.03 , 3: isNat^#(n__0()) -> c_3() 847.46/297.03 , 4: isNat^#(n__plus(V1, V2)) -> 847.46/297.03 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.03 , 5: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.04 , 6: isNat^#(n__x(V1, V2)) -> 847.46/297.04 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , 7: U21^#(tt()) -> c_12() 847.46/297.04 , 8: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.04 , 9: activate^#(X) -> c_7(X) 847.46/297.04 , 10: activate^#(n__0()) -> c_8(0^#()) 847.46/297.04 , 11: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.04 , 12: activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.04 , 13: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.04 , 14: 0^#() -> c_23() 847.46/297.04 , 15: plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.04 , 16: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.04 , 17: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.04 , 18: s^#(X) -> c_18(X) 847.46/297.04 , 19: x^#(X1, X2) -> c_26(X1, X2) 847.46/297.04 , 20: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.04 , 21: x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.04 , 22: U32^#(tt()) -> c_14() 847.46/297.04 , 23: U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.04 , 24: U51^#(tt(), M, N) -> 847.46/297.04 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , 25: U52^#(tt(), M, N) -> 847.46/297.04 c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.04 , 26: U61^#(tt()) -> c_22(0^#()) 847.46/297.04 , 27: U71^#(tt(), M, N) -> 847.46/297.04 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , 28: U72^#(tt(), M, N) -> 847.46/297.04 c_25(plus^#(x(activate(N), activate(M)), activate(N))) } 847.46/297.04 847.46/297.04 We are left with following problem, upon which TcT provides the 847.46/297.04 certificate MAYBE. 847.46/297.04 847.46/297.04 Strict DPs: 847.46/297.04 { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.04 , isNat^#(n__plus(V1, V2)) -> 847.46/297.04 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.04 , isNat^#(n__x(V1, V2)) -> 847.46/297.04 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.04 , activate^#(X) -> c_7(X) 847.46/297.04 , activate^#(n__0()) -> c_8(0^#()) 847.46/297.04 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.04 , activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.04 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.04 , plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.04 , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.04 , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.04 , s^#(X) -> c_18(X) 847.46/297.04 , x^#(X1, X2) -> c_26(X1, X2) 847.46/297.04 , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.04 , x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.04 , U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.04 , U51^#(tt(), M, N) -> 847.46/297.04 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.04 , U61^#(tt()) -> c_22(0^#()) 847.46/297.04 , U71^#(tt(), M, N) -> 847.46/297.04 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , U72^#(tt(), M, N) -> 847.46/297.04 c_25(plus^#(x(activate(N), activate(M)), activate(N))) } 847.46/297.04 Strict Trs: 847.46/297.04 { U11(tt(), V2) -> U12(isNat(activate(V2))) 847.46/297.04 , U12(tt()) -> tt() 847.46/297.04 , isNat(n__0()) -> tt() 847.46/297.04 , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) 847.46/297.04 , isNat(n__s(V1)) -> U21(isNat(activate(V1))) 847.46/297.04 , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) 847.46/297.04 , activate(X) -> X 847.46/297.04 , activate(n__0()) -> 0() 847.46/297.04 , activate(n__plus(X1, X2)) -> plus(X1, X2) 847.46/297.04 , activate(n__s(X)) -> s(X) 847.46/297.04 , activate(n__x(X1, X2)) -> x(X1, X2) 847.46/297.04 , U21(tt()) -> tt() 847.46/297.04 , U31(tt(), V2) -> U32(isNat(activate(V2))) 847.46/297.04 , U32(tt()) -> tt() 847.46/297.04 , U41(tt(), N) -> activate(N) 847.46/297.04 , U51(tt(), M, N) -> 847.46/297.04 U52(isNat(activate(N)), activate(M), activate(N)) 847.46/297.04 , U52(tt(), M, N) -> s(plus(activate(N), activate(M))) 847.46/297.04 , s(X) -> n__s(X) 847.46/297.04 , plus(X1, X2) -> n__plus(X1, X2) 847.46/297.04 , plus(N, s(M)) -> U51(isNat(M), M, N) 847.46/297.04 , plus(N, 0()) -> U41(isNat(N), N) 847.46/297.04 , U61(tt()) -> 0() 847.46/297.04 , 0() -> n__0() 847.46/297.04 , U71(tt(), M, N) -> 847.46/297.04 U72(isNat(activate(N)), activate(M), activate(N)) 847.46/297.04 , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 847.46/297.04 , x(X1, X2) -> n__x(X1, X2) 847.46/297.04 , x(N, s(M)) -> U71(isNat(M), M, N) 847.46/297.04 , x(N, 0()) -> U61(isNat(N)) } 847.46/297.04 Weak DPs: 847.46/297.04 { U12^#(tt()) -> c_2() 847.46/297.04 , isNat^#(n__0()) -> c_3() 847.46/297.04 , U21^#(tt()) -> c_12() 847.46/297.04 , 0^#() -> c_23() 847.46/297.04 , U32^#(tt()) -> c_14() } 847.46/297.04 Obligation: 847.46/297.04 runtime complexity 847.46/297.04 Answer: 847.46/297.04 MAYBE 847.46/297.04 847.46/297.04 We estimate the number of application of {1,3,5,7,21} by 847.46/297.04 applications of Pre({1,3,5,7,21}) = {2,4,6,11,14,15,17,18}. Here 847.46/297.04 rules are labeled as follows: 847.46/297.04 847.46/297.04 DPs: 847.46/297.04 { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.04 , 2: isNat^#(n__plus(V1, V2)) -> 847.46/297.04 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , 3: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.04 , 4: isNat^#(n__x(V1, V2)) -> 847.46/297.04 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , 5: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.04 , 6: activate^#(X) -> c_7(X) 847.46/297.04 , 7: activate^#(n__0()) -> c_8(0^#()) 847.46/297.04 , 8: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.04 , 9: activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.04 , 10: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.04 , 11: plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.04 , 12: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.04 , 13: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.04 , 14: s^#(X) -> c_18(X) 847.46/297.04 , 15: x^#(X1, X2) -> c_26(X1, X2) 847.46/297.04 , 16: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.04 , 17: x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.04 , 18: U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.04 , 19: U51^#(tt(), M, N) -> 847.46/297.04 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , 20: U52^#(tt(), M, N) -> 847.46/297.04 c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.04 , 21: U61^#(tt()) -> c_22(0^#()) 847.46/297.04 , 22: U71^#(tt(), M, N) -> 847.46/297.04 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , 23: U72^#(tt(), M, N) -> 847.46/297.04 c_25(plus^#(x(activate(N), activate(M)), activate(N))) 847.46/297.04 , 24: U12^#(tt()) -> c_2() 847.46/297.04 , 25: isNat^#(n__0()) -> c_3() 847.46/297.04 , 26: U21^#(tt()) -> c_12() 847.46/297.04 , 27: 0^#() -> c_23() 847.46/297.04 , 28: U32^#(tt()) -> c_14() } 847.46/297.04 847.46/297.04 We are left with following problem, upon which TcT provides the 847.46/297.04 certificate MAYBE. 847.46/297.04 847.46/297.04 Strict DPs: 847.46/297.04 { isNat^#(n__plus(V1, V2)) -> 847.46/297.04 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , isNat^#(n__x(V1, V2)) -> 847.46/297.04 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , activate^#(X) -> c_7(X) 847.46/297.04 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.04 , activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.04 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.04 , plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.04 , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.04 , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.04 , s^#(X) -> c_18(X) 847.46/297.04 , x^#(X1, X2) -> c_26(X1, X2) 847.46/297.04 , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.04 , x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.04 , U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.04 , U51^#(tt(), M, N) -> 847.46/297.04 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.04 , U71^#(tt(), M, N) -> 847.46/297.04 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , U72^#(tt(), M, N) -> 847.46/297.04 c_25(plus^#(x(activate(N), activate(M)), activate(N))) } 847.46/297.04 Strict Trs: 847.46/297.04 { U11(tt(), V2) -> U12(isNat(activate(V2))) 847.46/297.04 , U12(tt()) -> tt() 847.46/297.04 , isNat(n__0()) -> tt() 847.46/297.04 , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) 847.46/297.04 , isNat(n__s(V1)) -> U21(isNat(activate(V1))) 847.46/297.04 , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) 847.46/297.04 , activate(X) -> X 847.46/297.04 , activate(n__0()) -> 0() 847.46/297.04 , activate(n__plus(X1, X2)) -> plus(X1, X2) 847.46/297.04 , activate(n__s(X)) -> s(X) 847.46/297.04 , activate(n__x(X1, X2)) -> x(X1, X2) 847.46/297.04 , U21(tt()) -> tt() 847.46/297.04 , U31(tt(), V2) -> U32(isNat(activate(V2))) 847.46/297.04 , U32(tt()) -> tt() 847.46/297.04 , U41(tt(), N) -> activate(N) 847.46/297.04 , U51(tt(), M, N) -> 847.46/297.04 U52(isNat(activate(N)), activate(M), activate(N)) 847.46/297.04 , U52(tt(), M, N) -> s(plus(activate(N), activate(M))) 847.46/297.04 , s(X) -> n__s(X) 847.46/297.04 , plus(X1, X2) -> n__plus(X1, X2) 847.46/297.04 , plus(N, s(M)) -> U51(isNat(M), M, N) 847.46/297.04 , plus(N, 0()) -> U41(isNat(N), N) 847.46/297.04 , U61(tt()) -> 0() 847.46/297.04 , 0() -> n__0() 847.46/297.04 , U71(tt(), M, N) -> 847.46/297.04 U72(isNat(activate(N)), activate(M), activate(N)) 847.46/297.04 , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 847.46/297.04 , x(X1, X2) -> n__x(X1, X2) 847.46/297.04 , x(N, s(M)) -> U71(isNat(M), M, N) 847.46/297.04 , x(N, 0()) -> U61(isNat(N)) } 847.46/297.04 Weak DPs: 847.46/297.04 { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.04 , U12^#(tt()) -> c_2() 847.46/297.04 , isNat^#(n__0()) -> c_3() 847.46/297.04 , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.04 , U21^#(tt()) -> c_12() 847.46/297.04 , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.04 , activate^#(n__0()) -> c_8(0^#()) 847.46/297.04 , 0^#() -> c_23() 847.46/297.04 , U32^#(tt()) -> c_14() 847.46/297.04 , U61^#(tt()) -> c_22(0^#()) } 847.46/297.04 Obligation: 847.46/297.04 runtime complexity 847.46/297.04 Answer: 847.46/297.04 MAYBE 847.46/297.04 847.46/297.04 We estimate the number of application of {1,2,13} by applications 847.46/297.04 of Pre({1,2,13}) = {3,6,7,10,11}. Here rules are labeled as 847.46/297.04 follows: 847.46/297.04 847.46/297.04 DPs: 847.46/297.04 { 1: isNat^#(n__plus(V1, V2)) -> 847.46/297.04 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , 2: isNat^#(n__x(V1, V2)) -> 847.46/297.04 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , 3: activate^#(X) -> c_7(X) 847.46/297.04 , 4: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.04 , 5: activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.04 , 6: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.04 , 7: plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.04 , 8: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.04 , 9: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.04 , 10: s^#(X) -> c_18(X) 847.46/297.04 , 11: x^#(X1, X2) -> c_26(X1, X2) 847.46/297.04 , 12: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.04 , 13: x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.04 , 14: U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.04 , 15: U51^#(tt(), M, N) -> 847.46/297.04 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , 16: U52^#(tt(), M, N) -> 847.46/297.04 c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.04 , 17: U71^#(tt(), M, N) -> 847.46/297.04 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , 18: U72^#(tt(), M, N) -> 847.46/297.04 c_25(plus^#(x(activate(N), activate(M)), activate(N))) 847.46/297.04 , 19: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.04 , 20: U12^#(tt()) -> c_2() 847.46/297.04 , 21: isNat^#(n__0()) -> c_3() 847.46/297.04 , 22: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.04 , 23: U21^#(tt()) -> c_12() 847.46/297.04 , 24: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.04 , 25: activate^#(n__0()) -> c_8(0^#()) 847.46/297.04 , 26: 0^#() -> c_23() 847.46/297.04 , 27: U32^#(tt()) -> c_14() 847.46/297.04 , 28: U61^#(tt()) -> c_22(0^#()) } 847.46/297.04 847.46/297.04 We are left with following problem, upon which TcT provides the 847.46/297.04 certificate MAYBE. 847.46/297.04 847.46/297.04 Strict DPs: 847.46/297.04 { activate^#(X) -> c_7(X) 847.46/297.04 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 847.46/297.04 , activate^#(n__s(X)) -> c_10(s^#(X)) 847.46/297.04 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 847.46/297.04 , plus^#(X1, X2) -> c_19(X1, X2) 847.46/297.04 , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N)) 847.46/297.04 , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N)) 847.46/297.04 , s^#(X) -> c_18(X) 847.46/297.04 , x^#(X1, X2) -> c_26(X1, X2) 847.46/297.04 , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N)) 847.46/297.04 , U41^#(tt(), N) -> c_15(activate^#(N)) 847.46/297.04 , U51^#(tt(), M, N) -> 847.46/297.04 c_16(U52^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M)))) 847.46/297.04 , U71^#(tt(), M, N) -> 847.46/297.04 c_24(U72^#(isNat(activate(N)), activate(M), activate(N))) 847.46/297.04 , U72^#(tt(), M, N) -> 847.46/297.04 c_25(plus^#(x(activate(N), activate(M)), activate(N))) } 847.46/297.04 Strict Trs: 847.46/297.04 { U11(tt(), V2) -> U12(isNat(activate(V2))) 847.46/297.04 , U12(tt()) -> tt() 847.46/297.04 , isNat(n__0()) -> tt() 847.46/297.04 , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) 847.46/297.04 , isNat(n__s(V1)) -> U21(isNat(activate(V1))) 847.46/297.04 , isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) 847.46/297.04 , activate(X) -> X 847.46/297.04 , activate(n__0()) -> 0() 847.46/297.04 , activate(n__plus(X1, X2)) -> plus(X1, X2) 847.46/297.04 , activate(n__s(X)) -> s(X) 847.46/297.04 , activate(n__x(X1, X2)) -> x(X1, X2) 847.46/297.04 , U21(tt()) -> tt() 847.46/297.04 , U31(tt(), V2) -> U32(isNat(activate(V2))) 847.46/297.04 , U32(tt()) -> tt() 847.46/297.04 , U41(tt(), N) -> activate(N) 847.46/297.04 , U51(tt(), M, N) -> 847.46/297.04 U52(isNat(activate(N)), activate(M), activate(N)) 847.46/297.04 , U52(tt(), M, N) -> s(plus(activate(N), activate(M))) 847.46/297.04 , s(X) -> n__s(X) 847.46/297.04 , plus(X1, X2) -> n__plus(X1, X2) 847.46/297.04 , plus(N, s(M)) -> U51(isNat(M), M, N) 847.46/297.04 , plus(N, 0()) -> U41(isNat(N), N) 847.46/297.04 , U61(tt()) -> 0() 847.46/297.04 , 0() -> n__0() 847.46/297.04 , U71(tt(), M, N) -> 847.46/297.04 U72(isNat(activate(N)), activate(M), activate(N)) 847.46/297.04 , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 847.46/297.04 , x(X1, X2) -> n__x(X1, X2) 847.46/297.04 , x(N, s(M)) -> U71(isNat(M), M, N) 847.46/297.04 , x(N, 0()) -> U61(isNat(N)) } 847.46/297.04 Weak DPs: 847.46/297.04 { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2)))) 847.46/297.04 , U12^#(tt()) -> c_2() 847.46/297.04 , isNat^#(n__0()) -> c_3() 847.46/297.04 , isNat^#(n__plus(V1, V2)) -> 847.46/297.04 c_4(U11^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1)))) 847.46/297.04 , isNat^#(n__x(V1, V2)) -> 847.46/297.04 c_6(U31^#(isNat(activate(V1)), activate(V2))) 847.46/297.04 , U21^#(tt()) -> c_12() 847.46/297.04 , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2)))) 847.46/297.04 , activate^#(n__0()) -> c_8(0^#()) 847.46/297.04 , 0^#() -> c_23() 847.46/297.04 , x^#(N, 0()) -> c_28(U61^#(isNat(N))) 847.46/297.04 , U32^#(tt()) -> c_14() 847.46/297.04 , U61^#(tt()) -> c_22(0^#()) } 847.46/297.04 Obligation: 847.46/297.04 runtime complexity 847.46/297.04 Answer: 847.46/297.04 MAYBE 847.46/297.04 847.46/297.04 Empty strict component of the problem is NOT empty. 847.46/297.04 847.46/297.04 847.46/297.04 Arrrr.. 847.60/297.18 EOF