MAYBE 839.39/297.05 MAYBE 839.39/297.05 839.39/297.05 We are left with following problem, upon which TcT provides the 839.39/297.05 certificate MAYBE. 839.39/297.05 839.39/297.05 Strict Trs: 839.39/297.05 { U11(tt(), V1, V2) -> 839.39/297.05 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.05 , U12(tt(), V1, V2) -> 839.39/297.05 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.05 , isNatKind(n__0()) -> tt() 839.39/297.05 , isNatKind(n__plus(V1, V2)) -> 839.39/297.05 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.05 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.05 , activate(X) -> X 839.39/297.05 , activate(n__0()) -> 0() 839.39/297.05 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.05 , activate(n__s(X)) -> s(activate(X)) 839.39/297.05 , U13(tt(), V1, V2) -> 839.39/297.05 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.05 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.05 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.05 , isNat(n__0()) -> tt() 839.39/297.05 , isNat(n__plus(V1, V2)) -> 839.39/297.05 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.05 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.05 , U16(tt()) -> tt() 839.39/297.05 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.05 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.05 , U23(tt()) -> tt() 839.39/297.05 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.05 , U32(tt()) -> tt() 839.39/297.05 , U41(tt()) -> tt() 839.39/297.05 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.05 , U52(tt(), N) -> activate(N) 839.39/297.05 , U61(tt(), M, N) -> 839.39/297.05 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.05 , U62(tt(), M, N) -> 839.39/297.05 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.05 , U63(tt(), M, N) -> 839.39/297.05 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.05 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.05 , s(X) -> n__s(X) 839.39/297.05 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.05 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.05 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.05 , 0() -> n__0() } 839.39/297.05 Obligation: 839.39/297.05 runtime complexity 839.39/297.05 Answer: 839.39/297.05 MAYBE 839.39/297.05 839.39/297.05 None of the processors succeeded. 839.39/297.05 839.39/297.05 Details of failed attempt(s): 839.39/297.05 ----------------------------- 839.39/297.05 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 839.39/297.05 following reason: 839.39/297.05 839.39/297.05 Computation stopped due to timeout after 297.0 seconds. 839.39/297.05 839.39/297.05 2) 'Best' failed due to the following reason: 839.39/297.05 839.39/297.05 None of the processors succeeded. 839.39/297.05 839.39/297.05 Details of failed attempt(s): 839.39/297.05 ----------------------------- 839.39/297.05 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 839.39/297.05 seconds)' failed due to the following reason: 839.39/297.05 839.39/297.05 Computation stopped due to timeout after 148.0 seconds. 839.39/297.05 839.39/297.05 2) 'Best' failed due to the following reason: 839.39/297.05 839.39/297.05 None of the processors succeeded. 839.39/297.05 839.39/297.05 Details of failed attempt(s): 839.39/297.05 ----------------------------- 839.39/297.05 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 839.39/297.05 following reason: 839.39/297.05 839.39/297.05 The processor is inapplicable, reason: 839.39/297.05 Processor only applicable for innermost runtime complexity analysis 839.39/297.05 839.39/297.05 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 839.39/297.05 to the following reason: 839.39/297.05 839.39/297.05 The processor is inapplicable, reason: 839.39/297.05 Processor only applicable for innermost runtime complexity analysis 839.39/297.05 839.39/297.05 839.39/297.05 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 839.39/297.05 failed due to the following reason: 839.39/297.05 839.39/297.05 None of the processors succeeded. 839.39/297.05 839.39/297.05 Details of failed attempt(s): 839.39/297.05 ----------------------------- 839.39/297.05 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 839.39/297.05 failed due to the following reason: 839.39/297.05 839.39/297.05 match-boundness of the problem could not be verified. 839.39/297.05 839.39/297.05 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 839.39/297.05 failed due to the following reason: 839.39/297.05 839.39/297.05 match-boundness of the problem could not be verified. 839.39/297.05 839.39/297.05 839.39/297.05 839.39/297.05 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 839.39/297.05 the following reason: 839.39/297.05 839.39/297.05 We add the following weak dependency pairs: 839.39/297.05 839.39/297.05 Strict DPs: 839.39/297.05 { U11^#(tt(), V1, V2) -> 839.39/297.05 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , U12^#(tt(), V1, V2) -> 839.39/297.05 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , U13^#(tt(), V1, V2) -> 839.39/297.05 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , isNatKind^#(n__0()) -> c_3() 839.39/297.05 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.05 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.05 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.05 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.05 , U41^#(tt()) -> c_22() 839.39/297.05 , activate^#(X) -> c_6(X) 839.39/297.05 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.05 , activate^#(n__plus(X1, X2)) -> 839.39/297.05 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.05 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.05 , 0^#() -> c_33() 839.39/297.05 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.05 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.05 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.05 , s^#(X) -> c_29(X) 839.39/297.05 , U14^#(tt(), V1, V2) -> 839.39/297.05 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.05 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.05 , U16^#(tt()) -> c_16() 839.39/297.05 , isNat^#(n__0()) -> c_13() 839.39/297.05 , isNat^#(n__plus(V1, V2)) -> 839.39/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , isNat^#(n__s(V1)) -> 839.39/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U21^#(tt(), V1) -> 839.39/297.05 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.05 , U23^#(tt()) -> c_19() 839.39/297.05 , U32^#(tt()) -> c_21() 839.39/297.05 , U51^#(tt(), N) -> 839.39/297.05 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.05 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.05 , U61^#(tt(), M, N) -> 839.39/297.05 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.05 , U62^#(tt(), M, N) -> 839.39/297.05 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.05 , U63^#(tt(), M, N) -> 839.39/297.05 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.05 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.05 839.39/297.05 and mark the set of starting terms. 839.39/297.05 839.39/297.05 We are left with following problem, upon which TcT provides the 839.39/297.05 certificate MAYBE. 839.39/297.05 839.39/297.05 Strict DPs: 839.39/297.05 { U11^#(tt(), V1, V2) -> 839.39/297.05 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , U12^#(tt(), V1, V2) -> 839.39/297.05 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , U13^#(tt(), V1, V2) -> 839.39/297.05 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , isNatKind^#(n__0()) -> c_3() 839.39/297.05 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.05 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.05 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.05 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.05 , U41^#(tt()) -> c_22() 839.39/297.05 , activate^#(X) -> c_6(X) 839.39/297.05 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.05 , activate^#(n__plus(X1, X2)) -> 839.39/297.05 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.05 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.05 , 0^#() -> c_33() 839.39/297.05 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.05 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.05 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.05 , s^#(X) -> c_29(X) 839.39/297.05 , U14^#(tt(), V1, V2) -> 839.39/297.05 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.05 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.05 , U16^#(tt()) -> c_16() 839.39/297.05 , isNat^#(n__0()) -> c_13() 839.39/297.05 , isNat^#(n__plus(V1, V2)) -> 839.39/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , isNat^#(n__s(V1)) -> 839.39/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U21^#(tt(), V1) -> 839.39/297.05 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.05 , U23^#(tt()) -> c_19() 839.39/297.05 , U32^#(tt()) -> c_21() 839.39/297.05 , U51^#(tt(), N) -> 839.39/297.05 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.05 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.05 , U61^#(tt(), M, N) -> 839.39/297.05 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.05 , U62^#(tt(), M, N) -> 839.39/297.05 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.05 , U63^#(tt(), M, N) -> 839.39/297.05 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.05 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.05 Strict Trs: 839.39/297.05 { U11(tt(), V1, V2) -> 839.39/297.05 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.05 , U12(tt(), V1, V2) -> 839.39/297.05 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.05 , isNatKind(n__0()) -> tt() 839.39/297.05 , isNatKind(n__plus(V1, V2)) -> 839.39/297.05 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.05 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.05 , activate(X) -> X 839.39/297.05 , activate(n__0()) -> 0() 839.39/297.05 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.05 , activate(n__s(X)) -> s(activate(X)) 839.39/297.05 , U13(tt(), V1, V2) -> 839.39/297.05 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.05 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.05 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.05 , isNat(n__0()) -> tt() 839.39/297.05 , isNat(n__plus(V1, V2)) -> 839.39/297.05 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.05 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.05 , U16(tt()) -> tt() 839.39/297.05 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.05 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.05 , U23(tt()) -> tt() 839.39/297.05 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.05 , U32(tt()) -> tt() 839.39/297.05 , U41(tt()) -> tt() 839.39/297.05 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.05 , U52(tt(), N) -> activate(N) 839.39/297.05 , U61(tt(), M, N) -> 839.39/297.05 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.05 , U62(tt(), M, N) -> 839.39/297.05 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.05 , U63(tt(), M, N) -> 839.39/297.05 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.05 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.05 , s(X) -> n__s(X) 839.39/297.05 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.05 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.05 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.05 , 0() -> n__0() } 839.39/297.05 Obligation: 839.39/297.05 runtime complexity 839.39/297.05 Answer: 839.39/297.05 MAYBE 839.39/297.05 839.39/297.05 We estimate the number of application of {4,8,13,20,21,26,27} by 839.39/297.05 applications of Pre({4,8,13,20,21,26,27}) = {6,7,9,10,14,17,19,25}. 839.39/297.05 Here rules are labeled as follows: 839.39/297.05 839.39/297.05 DPs: 839.39/297.05 { 1: U11^#(tt(), V1, V2) -> 839.39/297.05 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , 2: U12^#(tt(), V1, V2) -> 839.39/297.05 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , 3: U13^#(tt(), V1, V2) -> 839.39/297.05 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , 4: isNatKind^#(n__0()) -> c_3() 839.39/297.05 , 5: isNatKind^#(n__plus(V1, V2)) -> 839.39/297.05 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.05 , 6: isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.05 , 7: U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.05 , 8: U41^#(tt()) -> c_22() 839.39/297.05 , 9: activate^#(X) -> c_6(X) 839.39/297.05 , 10: activate^#(n__0()) -> c_7(0^#()) 839.39/297.05 , 11: activate^#(n__plus(X1, X2)) -> 839.39/297.05 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.05 , 12: activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.05 , 13: 0^#() -> c_33() 839.39/297.05 , 14: plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.05 , 15: plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.05 , 16: plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.05 , 17: s^#(X) -> c_29(X) 839.39/297.05 , 18: U14^#(tt(), V1, V2) -> 839.39/297.05 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.05 , 19: U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.05 , 20: U16^#(tt()) -> c_16() 839.39/297.05 , 21: isNat^#(n__0()) -> c_13() 839.39/297.05 , 22: isNat^#(n__plus(V1, V2)) -> 839.39/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , 23: isNat^#(n__s(V1)) -> 839.39/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , 24: U21^#(tt(), V1) -> 839.39/297.05 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , 25: U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.05 , 26: U23^#(tt()) -> c_19() 839.39/297.05 , 27: U32^#(tt()) -> c_21() 839.39/297.05 , 28: U51^#(tt(), N) -> 839.39/297.05 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.05 , 29: U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.05 , 30: U61^#(tt(), M, N) -> 839.39/297.05 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.05 , 31: U62^#(tt(), M, N) -> 839.39/297.05 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.05 , 32: U63^#(tt(), M, N) -> 839.39/297.05 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.05 , 33: U64^#(tt(), M, N) -> 839.39/297.05 c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.05 839.39/297.05 We are left with following problem, upon which TcT provides the 839.39/297.05 certificate MAYBE. 839.39/297.05 839.39/297.05 Strict DPs: 839.39/297.05 { U11^#(tt(), V1, V2) -> 839.39/297.05 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , U12^#(tt(), V1, V2) -> 839.39/297.05 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , U13^#(tt(), V1, V2) -> 839.39/297.05 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.05 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.05 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.05 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.05 , activate^#(X) -> c_6(X) 839.39/297.05 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.05 , activate^#(n__plus(X1, X2)) -> 839.39/297.05 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.05 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.05 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.05 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.05 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.05 , s^#(X) -> c_29(X) 839.39/297.05 , U14^#(tt(), V1, V2) -> 839.39/297.05 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.05 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.05 , isNat^#(n__plus(V1, V2)) -> 839.39/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , isNat^#(n__s(V1)) -> 839.39/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U21^#(tt(), V1) -> 839.39/297.05 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.05 , U51^#(tt(), N) -> 839.39/297.05 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.05 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.05 , U61^#(tt(), M, N) -> 839.39/297.05 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.05 , U62^#(tt(), M, N) -> 839.39/297.05 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.05 , U63^#(tt(), M, N) -> 839.39/297.05 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.05 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.05 Strict Trs: 839.39/297.05 { U11(tt(), V1, V2) -> 839.39/297.05 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.05 , U12(tt(), V1, V2) -> 839.39/297.05 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.05 , isNatKind(n__0()) -> tt() 839.39/297.05 , isNatKind(n__plus(V1, V2)) -> 839.39/297.05 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.05 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.05 , activate(X) -> X 839.39/297.05 , activate(n__0()) -> 0() 839.39/297.05 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.05 , activate(n__s(X)) -> s(activate(X)) 839.39/297.05 , U13(tt(), V1, V2) -> 839.39/297.05 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.05 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.05 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.05 , isNat(n__0()) -> tt() 839.39/297.05 , isNat(n__plus(V1, V2)) -> 839.39/297.05 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.05 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.05 , U16(tt()) -> tt() 839.39/297.05 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.05 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.05 , U23(tt()) -> tt() 839.39/297.05 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.05 , U32(tt()) -> tt() 839.39/297.05 , U41(tt()) -> tt() 839.39/297.05 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.05 , U52(tt(), N) -> activate(N) 839.39/297.05 , U61(tt(), M, N) -> 839.39/297.05 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.05 , U62(tt(), M, N) -> 839.39/297.05 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.05 , U63(tt(), M, N) -> 839.39/297.05 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.05 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.05 , s(X) -> n__s(X) 839.39/297.05 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.05 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.05 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.05 , 0() -> n__0() } 839.39/297.05 Weak DPs: 839.39/297.05 { isNatKind^#(n__0()) -> c_3() 839.39/297.05 , U41^#(tt()) -> c_22() 839.39/297.05 , 0^#() -> c_33() 839.39/297.05 , U16^#(tt()) -> c_16() 839.39/297.05 , isNat^#(n__0()) -> c_13() 839.39/297.05 , U23^#(tt()) -> c_19() 839.39/297.05 , U32^#(tt()) -> c_21() } 839.39/297.05 Obligation: 839.39/297.05 runtime complexity 839.39/297.05 Answer: 839.39/297.05 MAYBE 839.39/297.05 839.39/297.05 We estimate the number of application of {5,6,8,16,20} by 839.39/297.05 applications of Pre({5,6,8,16,20}) = {4,7,11,14,15,19,22}. Here 839.39/297.05 rules are labeled as follows: 839.39/297.05 839.39/297.05 DPs: 839.39/297.05 { 1: U11^#(tt(), V1, V2) -> 839.39/297.05 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , 2: U12^#(tt(), V1, V2) -> 839.39/297.05 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , 3: U13^#(tt(), V1, V2) -> 839.39/297.05 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , 4: isNatKind^#(n__plus(V1, V2)) -> 839.39/297.05 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.05 , 5: isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.05 , 6: U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.05 , 7: activate^#(X) -> c_6(X) 839.39/297.05 , 8: activate^#(n__0()) -> c_7(0^#()) 839.39/297.05 , 9: activate^#(n__plus(X1, X2)) -> 839.39/297.05 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.05 , 10: activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.05 , 11: plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.05 , 12: plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.05 , 13: plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.05 , 14: s^#(X) -> c_29(X) 839.39/297.05 , 15: U14^#(tt(), V1, V2) -> 839.39/297.05 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.05 , 16: U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.05 , 17: isNat^#(n__plus(V1, V2)) -> 839.39/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , 18: isNat^#(n__s(V1)) -> 839.39/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , 19: U21^#(tt(), V1) -> 839.39/297.05 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , 20: U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.05 , 21: U51^#(tt(), N) -> 839.39/297.05 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.05 , 22: U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.05 , 23: U61^#(tt(), M, N) -> 839.39/297.05 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.05 , 24: U62^#(tt(), M, N) -> 839.39/297.05 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.05 , 25: U63^#(tt(), M, N) -> 839.39/297.05 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.05 , 26: U64^#(tt(), M, N) -> 839.39/297.05 c_28(s^#(plus(activate(N), activate(M)))) 839.39/297.05 , 27: isNatKind^#(n__0()) -> c_3() 839.39/297.05 , 28: U41^#(tt()) -> c_22() 839.39/297.05 , 29: 0^#() -> c_33() 839.39/297.05 , 30: U16^#(tt()) -> c_16() 839.39/297.05 , 31: isNat^#(n__0()) -> c_13() 839.39/297.05 , 32: U23^#(tt()) -> c_19() 839.39/297.05 , 33: U32^#(tt()) -> c_21() } 839.39/297.05 839.39/297.05 We are left with following problem, upon which TcT provides the 839.39/297.05 certificate MAYBE. 839.39/297.05 839.39/297.05 Strict DPs: 839.39/297.05 { U11^#(tt(), V1, V2) -> 839.39/297.05 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , U12^#(tt(), V1, V2) -> 839.39/297.05 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , U13^#(tt(), V1, V2) -> 839.39/297.05 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.05 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.05 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.05 , activate^#(X) -> c_6(X) 839.39/297.05 , activate^#(n__plus(X1, X2)) -> 839.39/297.05 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.05 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.05 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.05 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.05 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.05 , s^#(X) -> c_29(X) 839.39/297.05 , U14^#(tt(), V1, V2) -> 839.39/297.05 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.05 , isNat^#(n__plus(V1, V2)) -> 839.39/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.05 , isNat^#(n__s(V1)) -> 839.39/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U21^#(tt(), V1) -> 839.39/297.05 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.05 , U51^#(tt(), N) -> 839.39/297.05 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.05 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.05 , U61^#(tt(), M, N) -> 839.39/297.05 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.05 , U62^#(tt(), M, N) -> 839.39/297.05 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.05 , U63^#(tt(), M, N) -> 839.39/297.05 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.05 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.05 Strict Trs: 839.39/297.05 { U11(tt(), V1, V2) -> 839.39/297.05 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.05 , U12(tt(), V1, V2) -> 839.39/297.05 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.05 , isNatKind(n__0()) -> tt() 839.39/297.05 , isNatKind(n__plus(V1, V2)) -> 839.39/297.05 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.05 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.05 , activate(X) -> X 839.39/297.05 , activate(n__0()) -> 0() 839.39/297.06 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.06 , activate(n__s(X)) -> s(activate(X)) 839.39/297.06 , U13(tt(), V1, V2) -> 839.39/297.06 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.06 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.06 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.06 , isNat(n__0()) -> tt() 839.39/297.06 , isNat(n__plus(V1, V2)) -> 839.39/297.06 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.06 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.06 , U16(tt()) -> tt() 839.39/297.06 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.06 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.06 , U23(tt()) -> tt() 839.39/297.06 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.06 , U32(tt()) -> tt() 839.39/297.06 , U41(tt()) -> tt() 839.39/297.06 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.06 , U52(tt(), N) -> activate(N) 839.39/297.06 , U61(tt(), M, N) -> 839.39/297.06 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.06 , U62(tt(), M, N) -> 839.39/297.06 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.06 , U63(tt(), M, N) -> 839.39/297.06 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.06 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.06 , s(X) -> n__s(X) 839.39/297.06 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.06 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.06 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.06 , 0() -> n__0() } 839.39/297.06 Weak DPs: 839.39/297.06 { isNatKind^#(n__0()) -> c_3() 839.39/297.06 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.06 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.06 , U41^#(tt()) -> c_22() 839.39/297.06 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.06 , 0^#() -> c_33() 839.39/297.06 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.06 , U16^#(tt()) -> c_16() 839.39/297.06 , isNat^#(n__0()) -> c_13() 839.39/297.06 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.06 , U23^#(tt()) -> c_19() 839.39/297.06 , U32^#(tt()) -> c_21() } 839.39/297.06 Obligation: 839.39/297.06 runtime complexity 839.39/297.06 Answer: 839.39/297.06 MAYBE 839.39/297.06 839.39/297.06 We estimate the number of application of {4,12,15} by applications 839.39/297.06 of Pre({4,12,15}) = {3,5,8,11,14}. Here rules are labeled as 839.39/297.06 follows: 839.39/297.06 839.39/297.06 DPs: 839.39/297.06 { 1: U11^#(tt(), V1, V2) -> 839.39/297.06 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , 2: U12^#(tt(), V1, V2) -> 839.39/297.06 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , 3: U13^#(tt(), V1, V2) -> 839.39/297.06 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , 4: isNatKind^#(n__plus(V1, V2)) -> 839.39/297.06 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.06 , 5: activate^#(X) -> c_6(X) 839.39/297.06 , 6: activate^#(n__plus(X1, X2)) -> 839.39/297.06 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.06 , 7: activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.06 , 8: plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.06 , 9: plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.06 , 10: plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.06 , 11: s^#(X) -> c_29(X) 839.39/297.06 , 12: U14^#(tt(), V1, V2) -> 839.39/297.06 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.06 , 13: isNat^#(n__plus(V1, V2)) -> 839.39/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , 14: isNat^#(n__s(V1)) -> 839.39/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , 15: U21^#(tt(), V1) -> 839.39/297.06 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , 16: U51^#(tt(), N) -> 839.39/297.06 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.06 , 17: U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.06 , 18: U61^#(tt(), M, N) -> 839.39/297.06 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.06 , 19: U62^#(tt(), M, N) -> 839.39/297.06 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.06 , 20: U63^#(tt(), M, N) -> 839.39/297.06 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.06 , 21: U64^#(tt(), M, N) -> 839.39/297.06 c_28(s^#(plus(activate(N), activate(M)))) 839.39/297.06 , 22: isNatKind^#(n__0()) -> c_3() 839.39/297.06 , 23: isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.06 , 24: U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.06 , 25: U41^#(tt()) -> c_22() 839.39/297.06 , 26: activate^#(n__0()) -> c_7(0^#()) 839.39/297.06 , 27: 0^#() -> c_33() 839.39/297.06 , 28: U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.06 , 29: U16^#(tt()) -> c_16() 839.39/297.06 , 30: isNat^#(n__0()) -> c_13() 839.39/297.06 , 31: U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.06 , 32: U23^#(tt()) -> c_19() 839.39/297.06 , 33: U32^#(tt()) -> c_21() } 839.39/297.06 839.39/297.06 We are left with following problem, upon which TcT provides the 839.39/297.06 certificate MAYBE. 839.39/297.06 839.39/297.06 Strict DPs: 839.39/297.06 { U11^#(tt(), V1, V2) -> 839.39/297.06 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , U12^#(tt(), V1, V2) -> 839.39/297.06 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , U13^#(tt(), V1, V2) -> 839.39/297.06 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , activate^#(X) -> c_6(X) 839.39/297.06 , activate^#(n__plus(X1, X2)) -> 839.39/297.06 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.06 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.06 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.06 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.06 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.06 , s^#(X) -> c_29(X) 839.39/297.06 , isNat^#(n__plus(V1, V2)) -> 839.39/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , isNat^#(n__s(V1)) -> 839.39/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , U51^#(tt(), N) -> 839.39/297.06 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.06 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.06 , U61^#(tt(), M, N) -> 839.39/297.06 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.06 , U62^#(tt(), M, N) -> 839.39/297.06 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.06 , U63^#(tt(), M, N) -> 839.39/297.06 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.06 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.06 Strict Trs: 839.39/297.06 { U11(tt(), V1, V2) -> 839.39/297.06 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.06 , U12(tt(), V1, V2) -> 839.39/297.06 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.06 , isNatKind(n__0()) -> tt() 839.39/297.06 , isNatKind(n__plus(V1, V2)) -> 839.39/297.06 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.06 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.06 , activate(X) -> X 839.39/297.06 , activate(n__0()) -> 0() 839.39/297.06 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.06 , activate(n__s(X)) -> s(activate(X)) 839.39/297.06 , U13(tt(), V1, V2) -> 839.39/297.06 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.06 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.06 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.06 , isNat(n__0()) -> tt() 839.39/297.06 , isNat(n__plus(V1, V2)) -> 839.39/297.06 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.06 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.06 , U16(tt()) -> tt() 839.39/297.06 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.06 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.06 , U23(tt()) -> tt() 839.39/297.06 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.06 , U32(tt()) -> tt() 839.39/297.06 , U41(tt()) -> tt() 839.39/297.06 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.06 , U52(tt(), N) -> activate(N) 839.39/297.06 , U61(tt(), M, N) -> 839.39/297.06 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.06 , U62(tt(), M, N) -> 839.39/297.06 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.06 , U63(tt(), M, N) -> 839.39/297.06 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.06 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.06 , s(X) -> n__s(X) 839.39/297.06 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.06 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.06 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.06 , 0() -> n__0() } 839.39/297.06 Weak DPs: 839.39/297.06 { isNatKind^#(n__0()) -> c_3() 839.39/297.06 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.06 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.06 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.06 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.06 , U41^#(tt()) -> c_22() 839.39/297.06 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.06 , 0^#() -> c_33() 839.39/297.06 , U14^#(tt(), V1, V2) -> 839.39/297.06 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.06 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.06 , U16^#(tt()) -> c_16() 839.39/297.06 , isNat^#(n__0()) -> c_13() 839.39/297.06 , U21^#(tt(), V1) -> 839.39/297.06 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.06 , U23^#(tt()) -> c_19() 839.39/297.06 , U32^#(tt()) -> c_21() } 839.39/297.06 Obligation: 839.39/297.06 runtime complexity 839.39/297.06 Answer: 839.39/297.06 MAYBE 839.39/297.06 839.39/297.06 We estimate the number of application of {3,12} by applications of 839.39/297.06 Pre({3,12}) = {2,4,7,10}. Here rules are labeled as follows: 839.39/297.06 839.39/297.06 DPs: 839.39/297.06 { 1: U11^#(tt(), V1, V2) -> 839.39/297.06 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , 2: U12^#(tt(), V1, V2) -> 839.39/297.06 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , 3: U13^#(tt(), V1, V2) -> 839.39/297.06 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , 4: activate^#(X) -> c_6(X) 839.39/297.06 , 5: activate^#(n__plus(X1, X2)) -> 839.39/297.06 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.06 , 6: activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.06 , 7: plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.06 , 8: plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.06 , 9: plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.06 , 10: s^#(X) -> c_29(X) 839.39/297.06 , 11: isNat^#(n__plus(V1, V2)) -> 839.39/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , 12: isNat^#(n__s(V1)) -> 839.39/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , 13: U51^#(tt(), N) -> 839.39/297.06 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.06 , 14: U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.06 , 15: U61^#(tt(), M, N) -> 839.39/297.06 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.06 , 16: U62^#(tt(), M, N) -> 839.39/297.06 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.06 , 17: U63^#(tt(), M, N) -> 839.39/297.06 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.06 , 18: U64^#(tt(), M, N) -> 839.39/297.06 c_28(s^#(plus(activate(N), activate(M)))) 839.39/297.06 , 19: isNatKind^#(n__0()) -> c_3() 839.39/297.06 , 20: isNatKind^#(n__plus(V1, V2)) -> 839.39/297.06 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.06 , 21: isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.06 , 22: U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.06 , 23: U41^#(tt()) -> c_22() 839.39/297.06 , 24: activate^#(n__0()) -> c_7(0^#()) 839.39/297.06 , 25: 0^#() -> c_33() 839.39/297.06 , 26: U14^#(tt(), V1, V2) -> 839.39/297.06 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.06 , 27: U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.06 , 28: U16^#(tt()) -> c_16() 839.39/297.06 , 29: isNat^#(n__0()) -> c_13() 839.39/297.06 , 30: U21^#(tt(), V1) -> 839.39/297.06 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , 31: U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.06 , 32: U23^#(tt()) -> c_19() 839.39/297.06 , 33: U32^#(tt()) -> c_21() } 839.39/297.06 839.39/297.06 We are left with following problem, upon which TcT provides the 839.39/297.06 certificate MAYBE. 839.39/297.06 839.39/297.06 Strict DPs: 839.39/297.06 { U11^#(tt(), V1, V2) -> 839.39/297.06 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , U12^#(tt(), V1, V2) -> 839.39/297.06 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , activate^#(X) -> c_6(X) 839.39/297.06 , activate^#(n__plus(X1, X2)) -> 839.39/297.06 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.06 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.06 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.06 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.06 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.06 , s^#(X) -> c_29(X) 839.39/297.06 , isNat^#(n__plus(V1, V2)) -> 839.39/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , U51^#(tt(), N) -> 839.39/297.06 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.06 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.06 , U61^#(tt(), M, N) -> 839.39/297.06 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.06 , U62^#(tt(), M, N) -> 839.39/297.06 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.06 , U63^#(tt(), M, N) -> 839.39/297.06 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.06 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.06 Strict Trs: 839.39/297.06 { U11(tt(), V1, V2) -> 839.39/297.06 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.06 , U12(tt(), V1, V2) -> 839.39/297.06 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.06 , isNatKind(n__0()) -> tt() 839.39/297.06 , isNatKind(n__plus(V1, V2)) -> 839.39/297.06 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.06 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.06 , activate(X) -> X 839.39/297.06 , activate(n__0()) -> 0() 839.39/297.06 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.06 , activate(n__s(X)) -> s(activate(X)) 839.39/297.06 , U13(tt(), V1, V2) -> 839.39/297.06 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.06 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.06 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.06 , isNat(n__0()) -> tt() 839.39/297.06 , isNat(n__plus(V1, V2)) -> 839.39/297.06 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.06 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.06 , U16(tt()) -> tt() 839.39/297.06 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.06 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.06 , U23(tt()) -> tt() 839.39/297.06 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.06 , U32(tt()) -> tt() 839.39/297.06 , U41(tt()) -> tt() 839.39/297.06 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.06 , U52(tt(), N) -> activate(N) 839.39/297.06 , U61(tt(), M, N) -> 839.39/297.06 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.06 , U62(tt(), M, N) -> 839.39/297.06 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.06 , U63(tt(), M, N) -> 839.39/297.06 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.06 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.06 , s(X) -> n__s(X) 839.39/297.06 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.06 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.06 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.06 , 0() -> n__0() } 839.39/297.06 Weak DPs: 839.39/297.06 { U13^#(tt(), V1, V2) -> 839.39/297.06 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , isNatKind^#(n__0()) -> c_3() 839.39/297.06 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.06 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.06 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.06 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.06 , U41^#(tt()) -> c_22() 839.39/297.06 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.06 , 0^#() -> c_33() 839.39/297.06 , U14^#(tt(), V1, V2) -> 839.39/297.06 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.06 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.06 , U16^#(tt()) -> c_16() 839.39/297.06 , isNat^#(n__0()) -> c_13() 839.39/297.06 , isNat^#(n__s(V1)) -> 839.39/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , U21^#(tt(), V1) -> 839.39/297.06 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.06 , U23^#(tt()) -> c_19() 839.39/297.06 , U32^#(tt()) -> c_21() } 839.39/297.06 Obligation: 839.39/297.06 runtime complexity 839.39/297.06 Answer: 839.39/297.06 MAYBE 839.39/297.06 839.39/297.06 We estimate the number of application of {2} by applications of 839.39/297.06 Pre({2}) = {1,3,6,9}. Here rules are labeled as follows: 839.39/297.06 839.39/297.06 DPs: 839.39/297.06 { 1: U11^#(tt(), V1, V2) -> 839.39/297.06 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , 2: U12^#(tt(), V1, V2) -> 839.39/297.06 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , 3: activate^#(X) -> c_6(X) 839.39/297.06 , 4: activate^#(n__plus(X1, X2)) -> 839.39/297.06 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.06 , 5: activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.06 , 6: plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.06 , 7: plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.06 , 8: plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.06 , 9: s^#(X) -> c_29(X) 839.39/297.06 , 10: isNat^#(n__plus(V1, V2)) -> 839.39/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , 11: U51^#(tt(), N) -> 839.39/297.06 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.06 , 12: U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.06 , 13: U61^#(tt(), M, N) -> 839.39/297.06 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.06 , 14: U62^#(tt(), M, N) -> 839.39/297.06 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.06 , 15: U63^#(tt(), M, N) -> 839.39/297.06 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.06 , 16: U64^#(tt(), M, N) -> 839.39/297.06 c_28(s^#(plus(activate(N), activate(M)))) 839.39/297.06 , 17: U13^#(tt(), V1, V2) -> 839.39/297.06 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.06 , 18: isNatKind^#(n__0()) -> c_3() 839.39/297.06 , 19: isNatKind^#(n__plus(V1, V2)) -> 839.39/297.06 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.06 , 20: isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.06 , 21: U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.06 , 22: U41^#(tt()) -> c_22() 839.39/297.06 , 23: activate^#(n__0()) -> c_7(0^#()) 839.39/297.06 , 24: 0^#() -> c_33() 839.39/297.06 , 25: U14^#(tt(), V1, V2) -> 839.39/297.06 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.06 , 26: U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.06 , 27: U16^#(tt()) -> c_16() 839.39/297.06 , 28: isNat^#(n__0()) -> c_13() 839.39/297.06 , 29: isNat^#(n__s(V1)) -> 839.39/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , 30: U21^#(tt(), V1) -> 839.39/297.06 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.06 , 31: U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.06 , 32: U23^#(tt()) -> c_19() 839.39/297.06 , 33: U32^#(tt()) -> c_21() } 839.39/297.06 839.39/297.06 We are left with following problem, upon which TcT provides the 839.39/297.06 certificate MAYBE. 839.39/297.06 839.39/297.06 Strict DPs: 839.39/297.06 { U11^#(tt(), V1, V2) -> 839.39/297.06 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , activate^#(X) -> c_6(X) 839.39/297.06 , activate^#(n__plus(X1, X2)) -> 839.39/297.06 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.06 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.06 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.06 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.06 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.06 , s^#(X) -> c_29(X) 839.39/297.06 , isNat^#(n__plus(V1, V2)) -> 839.39/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.06 , U51^#(tt(), N) -> 839.39/297.06 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.06 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.06 , U61^#(tt(), M, N) -> 839.39/297.06 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.06 , U62^#(tt(), M, N) -> 839.39/297.06 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.06 , U63^#(tt(), M, N) -> 839.39/297.06 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.06 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.06 Strict Trs: 839.39/297.06 { U11(tt(), V1, V2) -> 839.39/297.06 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.06 , U12(tt(), V1, V2) -> 839.39/297.06 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.06 , isNatKind(n__0()) -> tt() 839.39/297.06 , isNatKind(n__plus(V1, V2)) -> 839.39/297.06 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.06 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.06 , activate(X) -> X 839.39/297.06 , activate(n__0()) -> 0() 839.39/297.07 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.07 , activate(n__s(X)) -> s(activate(X)) 839.39/297.07 , U13(tt(), V1, V2) -> 839.39/297.07 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.07 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.07 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.07 , isNat(n__0()) -> tt() 839.39/297.07 , isNat(n__plus(V1, V2)) -> 839.39/297.07 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.07 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.07 , U16(tt()) -> tt() 839.39/297.07 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.07 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.07 , U23(tt()) -> tt() 839.39/297.07 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.07 , U32(tt()) -> tt() 839.39/297.07 , U41(tt()) -> tt() 839.39/297.07 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.07 , U52(tt(), N) -> activate(N) 839.39/297.07 , U61(tt(), M, N) -> 839.39/297.07 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.07 , U62(tt(), M, N) -> 839.39/297.07 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.07 , U63(tt(), M, N) -> 839.39/297.07 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.07 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.07 , s(X) -> n__s(X) 839.39/297.07 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.07 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.07 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.07 , 0() -> n__0() } 839.39/297.07 Weak DPs: 839.39/297.07 { U12^#(tt(), V1, V2) -> 839.39/297.07 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , U13^#(tt(), V1, V2) -> 839.39/297.07 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , isNatKind^#(n__0()) -> c_3() 839.39/297.07 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.07 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.07 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.07 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.07 , U41^#(tt()) -> c_22() 839.39/297.07 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.07 , 0^#() -> c_33() 839.39/297.07 , U14^#(tt(), V1, V2) -> 839.39/297.07 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.07 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.07 , U16^#(tt()) -> c_16() 839.39/297.07 , isNat^#(n__0()) -> c_13() 839.39/297.07 , isNat^#(n__s(V1)) -> 839.39/297.07 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , U21^#(tt(), V1) -> 839.39/297.07 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.07 , U23^#(tt()) -> c_19() 839.39/297.07 , U32^#(tt()) -> c_21() } 839.39/297.07 Obligation: 839.39/297.07 runtime complexity 839.39/297.07 Answer: 839.39/297.07 MAYBE 839.39/297.07 839.39/297.07 We estimate the number of application of {1} by applications of 839.39/297.07 Pre({1}) = {2,5,8,9}. Here rules are labeled as follows: 839.39/297.07 839.39/297.07 DPs: 839.39/297.07 { 1: U11^#(tt(), V1, V2) -> 839.39/297.07 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , 2: activate^#(X) -> c_6(X) 839.39/297.07 , 3: activate^#(n__plus(X1, X2)) -> 839.39/297.07 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.07 , 4: activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.07 , 5: plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.07 , 6: plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.07 , 7: plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.07 , 8: s^#(X) -> c_29(X) 839.39/297.07 , 9: isNat^#(n__plus(V1, V2)) -> 839.39/297.07 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , 10: U51^#(tt(), N) -> 839.39/297.07 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.07 , 11: U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.07 , 12: U61^#(tt(), M, N) -> 839.39/297.07 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.07 , 13: U62^#(tt(), M, N) -> 839.39/297.07 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.07 , 14: U63^#(tt(), M, N) -> 839.39/297.07 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.07 , 15: U64^#(tt(), M, N) -> 839.39/297.07 c_28(s^#(plus(activate(N), activate(M)))) 839.39/297.07 , 16: U12^#(tt(), V1, V2) -> 839.39/297.07 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , 17: U13^#(tt(), V1, V2) -> 839.39/297.07 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , 18: isNatKind^#(n__0()) -> c_3() 839.39/297.07 , 19: isNatKind^#(n__plus(V1, V2)) -> 839.39/297.07 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.07 , 20: isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.07 , 21: U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.07 , 22: U41^#(tt()) -> c_22() 839.39/297.07 , 23: activate^#(n__0()) -> c_7(0^#()) 839.39/297.07 , 24: 0^#() -> c_33() 839.39/297.07 , 25: U14^#(tt(), V1, V2) -> 839.39/297.07 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.07 , 26: U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.07 , 27: U16^#(tt()) -> c_16() 839.39/297.07 , 28: isNat^#(n__0()) -> c_13() 839.39/297.07 , 29: isNat^#(n__s(V1)) -> 839.39/297.07 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , 30: U21^#(tt(), V1) -> 839.39/297.07 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , 31: U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.07 , 32: U23^#(tt()) -> c_19() 839.39/297.07 , 33: U32^#(tt()) -> c_21() } 839.39/297.07 839.39/297.07 We are left with following problem, upon which TcT provides the 839.39/297.07 certificate MAYBE. 839.39/297.07 839.39/297.07 Strict DPs: 839.39/297.07 { activate^#(X) -> c_6(X) 839.39/297.07 , activate^#(n__plus(X1, X2)) -> 839.39/297.07 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.07 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.07 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.07 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.07 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.07 , s^#(X) -> c_29(X) 839.39/297.07 , isNat^#(n__plus(V1, V2)) -> 839.39/297.07 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , U51^#(tt(), N) -> 839.39/297.07 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.07 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.07 , U61^#(tt(), M, N) -> 839.39/297.07 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.07 , U62^#(tt(), M, N) -> 839.39/297.07 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.07 , U63^#(tt(), M, N) -> 839.39/297.07 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.07 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.07 Strict Trs: 839.39/297.07 { U11(tt(), V1, V2) -> 839.39/297.07 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.07 , U12(tt(), V1, V2) -> 839.39/297.07 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.07 , isNatKind(n__0()) -> tt() 839.39/297.07 , isNatKind(n__plus(V1, V2)) -> 839.39/297.07 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.07 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.07 , activate(X) -> X 839.39/297.07 , activate(n__0()) -> 0() 839.39/297.07 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.07 , activate(n__s(X)) -> s(activate(X)) 839.39/297.07 , U13(tt(), V1, V2) -> 839.39/297.07 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.07 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.07 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.07 , isNat(n__0()) -> tt() 839.39/297.07 , isNat(n__plus(V1, V2)) -> 839.39/297.07 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.07 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.07 , U16(tt()) -> tt() 839.39/297.07 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.07 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.07 , U23(tt()) -> tt() 839.39/297.07 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.07 , U32(tt()) -> tt() 839.39/297.07 , U41(tt()) -> tt() 839.39/297.07 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.07 , U52(tt(), N) -> activate(N) 839.39/297.07 , U61(tt(), M, N) -> 839.39/297.07 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.07 , U62(tt(), M, N) -> 839.39/297.07 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.07 , U63(tt(), M, N) -> 839.39/297.07 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.07 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.07 , s(X) -> n__s(X) 839.39/297.07 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.07 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.07 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.07 , 0() -> n__0() } 839.39/297.07 Weak DPs: 839.39/297.07 { U11^#(tt(), V1, V2) -> 839.39/297.07 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , U12^#(tt(), V1, V2) -> 839.39/297.07 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , U13^#(tt(), V1, V2) -> 839.39/297.07 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , isNatKind^#(n__0()) -> c_3() 839.39/297.07 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.07 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.07 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.07 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.07 , U41^#(tt()) -> c_22() 839.39/297.07 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.07 , 0^#() -> c_33() 839.39/297.07 , U14^#(tt(), V1, V2) -> 839.39/297.07 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.07 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.07 , U16^#(tt()) -> c_16() 839.39/297.07 , isNat^#(n__0()) -> c_13() 839.39/297.07 , isNat^#(n__s(V1)) -> 839.39/297.07 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , U21^#(tt(), V1) -> 839.39/297.07 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.07 , U23^#(tt()) -> c_19() 839.39/297.07 , U32^#(tt()) -> c_21() } 839.39/297.07 Obligation: 839.39/297.07 runtime complexity 839.39/297.07 Answer: 839.39/297.07 MAYBE 839.39/297.07 839.39/297.07 We estimate the number of application of {8} by applications of 839.39/297.07 Pre({8}) = {1,4,7}. Here rules are labeled as follows: 839.39/297.07 839.39/297.07 DPs: 839.39/297.07 { 1: activate^#(X) -> c_6(X) 839.39/297.07 , 2: activate^#(n__plus(X1, X2)) -> 839.39/297.07 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.07 , 3: activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.07 , 4: plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.07 , 5: plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.07 , 6: plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.07 , 7: s^#(X) -> c_29(X) 839.39/297.07 , 8: isNat^#(n__plus(V1, V2)) -> 839.39/297.07 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , 9: U51^#(tt(), N) -> 839.39/297.07 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.07 , 10: U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.07 , 11: U61^#(tt(), M, N) -> 839.39/297.07 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.07 , 12: U62^#(tt(), M, N) -> 839.39/297.07 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.07 , 13: U63^#(tt(), M, N) -> 839.39/297.07 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.07 , 14: U64^#(tt(), M, N) -> 839.39/297.07 c_28(s^#(plus(activate(N), activate(M)))) 839.39/297.07 , 15: U11^#(tt(), V1, V2) -> 839.39/297.07 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , 16: U12^#(tt(), V1, V2) -> 839.39/297.07 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , 17: U13^#(tt(), V1, V2) -> 839.39/297.07 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , 18: isNatKind^#(n__0()) -> c_3() 839.39/297.07 , 19: isNatKind^#(n__plus(V1, V2)) -> 839.39/297.07 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.07 , 20: isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.07 , 21: U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.07 , 22: U41^#(tt()) -> c_22() 839.39/297.07 , 23: activate^#(n__0()) -> c_7(0^#()) 839.39/297.07 , 24: 0^#() -> c_33() 839.39/297.07 , 25: U14^#(tt(), V1, V2) -> 839.39/297.07 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.07 , 26: U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.07 , 27: U16^#(tt()) -> c_16() 839.39/297.07 , 28: isNat^#(n__0()) -> c_13() 839.39/297.07 , 29: isNat^#(n__s(V1)) -> 839.39/297.07 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , 30: U21^#(tt(), V1) -> 839.39/297.07 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , 31: U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.07 , 32: U23^#(tt()) -> c_19() 839.39/297.07 , 33: U32^#(tt()) -> c_21() } 839.39/297.07 839.39/297.07 We are left with following problem, upon which TcT provides the 839.39/297.07 certificate MAYBE. 839.39/297.07 839.39/297.07 Strict DPs: 839.39/297.07 { activate^#(X) -> c_6(X) 839.39/297.07 , activate^#(n__plus(X1, X2)) -> 839.39/297.07 c_8(plus^#(activate(X1), activate(X2))) 839.39/297.07 , activate^#(n__s(X)) -> c_9(s^#(activate(X))) 839.39/297.07 , plus^#(X1, X2) -> c_30(X1, X2) 839.39/297.07 , plus^#(N, s(M)) -> c_31(U61^#(isNat(M), M, N)) 839.39/297.07 , plus^#(N, 0()) -> c_32(U51^#(isNat(N), N)) 839.39/297.07 , s^#(X) -> c_29(X) 839.39/297.07 , U51^#(tt(), N) -> 839.39/297.07 c_23(U52^#(isNatKind(activate(N)), activate(N))) 839.39/297.07 , U52^#(tt(), N) -> c_24(activate^#(N)) 839.39/297.07 , U61^#(tt(), M, N) -> 839.39/297.07 c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N))) 839.39/297.07 , U62^#(tt(), M, N) -> 839.39/297.07 c_26(U63^#(isNat(activate(N)), activate(M), activate(N))) 839.39/297.07 , U63^#(tt(), M, N) -> 839.39/297.07 c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N))) 839.39/297.07 , U64^#(tt(), M, N) -> c_28(s^#(plus(activate(N), activate(M)))) } 839.39/297.07 Strict Trs: 839.39/297.07 { U11(tt(), V1, V2) -> 839.39/297.07 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.07 , U12(tt(), V1, V2) -> 839.39/297.07 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.07 , isNatKind(n__0()) -> tt() 839.39/297.07 , isNatKind(n__plus(V1, V2)) -> 839.39/297.07 U31(isNatKind(activate(V1)), activate(V2)) 839.39/297.07 , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 839.39/297.07 , activate(X) -> X 839.39/297.07 , activate(n__0()) -> 0() 839.39/297.07 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 839.39/297.07 , activate(n__s(X)) -> s(activate(X)) 839.39/297.07 , U13(tt(), V1, V2) -> 839.39/297.07 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 839.39/297.07 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 839.39/297.07 , U15(tt(), V2) -> U16(isNat(activate(V2))) 839.39/297.07 , isNat(n__0()) -> tt() 839.39/297.07 , isNat(n__plus(V1, V2)) -> 839.39/297.07 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 839.39/297.07 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 839.39/297.07 , U16(tt()) -> tt() 839.39/297.07 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 839.39/297.07 , U22(tt(), V1) -> U23(isNat(activate(V1))) 839.39/297.07 , U23(tt()) -> tt() 839.39/297.07 , U31(tt(), V2) -> U32(isNatKind(activate(V2))) 839.39/297.07 , U32(tt()) -> tt() 839.39/297.07 , U41(tt()) -> tt() 839.39/297.07 , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N)) 839.39/297.07 , U52(tt(), N) -> activate(N) 839.39/297.07 , U61(tt(), M, N) -> 839.39/297.07 U62(isNatKind(activate(M)), activate(M), activate(N)) 839.39/297.07 , U62(tt(), M, N) -> 839.39/297.07 U63(isNat(activate(N)), activate(M), activate(N)) 839.39/297.07 , U63(tt(), M, N) -> 839.39/297.07 U64(isNatKind(activate(N)), activate(M), activate(N)) 839.39/297.07 , U64(tt(), M, N) -> s(plus(activate(N), activate(M))) 839.39/297.07 , s(X) -> n__s(X) 839.39/297.07 , plus(X1, X2) -> n__plus(X1, X2) 839.39/297.07 , plus(N, s(M)) -> U61(isNat(M), M, N) 839.39/297.07 , plus(N, 0()) -> U51(isNat(N), N) 839.39/297.07 , 0() -> n__0() } 839.39/297.07 Weak DPs: 839.39/297.07 { U11^#(tt(), V1, V2) -> 839.39/297.07 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , U12^#(tt(), V1, V2) -> 839.39/297.07 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , U13^#(tt(), V1, V2) -> 839.39/297.07 c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 839.39/297.07 , isNatKind^#(n__0()) -> c_3() 839.39/297.07 , isNatKind^#(n__plus(V1, V2)) -> 839.39/297.07 c_4(U31^#(isNatKind(activate(V1)), activate(V2))) 839.39/297.07 , isNatKind^#(n__s(V1)) -> c_5(U41^#(isNatKind(activate(V1)))) 839.39/297.07 , U31^#(tt(), V2) -> c_20(U32^#(isNatKind(activate(V2)))) 839.39/297.07 , U41^#(tt()) -> c_22() 839.39/297.07 , activate^#(n__0()) -> c_7(0^#()) 839.39/297.07 , 0^#() -> c_33() 839.39/297.07 , U14^#(tt(), V1, V2) -> 839.39/297.07 c_11(U15^#(isNat(activate(V1)), activate(V2))) 839.39/297.07 , U15^#(tt(), V2) -> c_12(U16^#(isNat(activate(V2)))) 839.39/297.07 , U16^#(tt()) -> c_16() 839.39/297.07 , isNat^#(n__0()) -> c_13() 839.39/297.07 , isNat^#(n__plus(V1, V2)) -> 839.39/297.07 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 839.39/297.07 , isNat^#(n__s(V1)) -> 839.39/297.07 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , U21^#(tt(), V1) -> 839.39/297.07 c_17(U22^#(isNatKind(activate(V1)), activate(V1))) 839.39/297.07 , U22^#(tt(), V1) -> c_18(U23^#(isNat(activate(V1)))) 839.39/297.07 , U23^#(tt()) -> c_19() 839.39/297.07 , U32^#(tt()) -> c_21() } 839.39/297.07 Obligation: 839.39/297.07 runtime complexity 839.39/297.07 Answer: 839.39/297.07 MAYBE 839.39/297.07 839.39/297.07 Empty strict component of the problem is NOT empty. 839.39/297.07 839.39/297.07 839.39/297.07 Arrrr.. 839.67/297.26 EOF