MAYBE 880.82/297.05 MAYBE 880.82/297.05 880.82/297.05 We are left with following problem, upon which TcT provides the 880.82/297.05 certificate MAYBE. 880.82/297.05 880.82/297.05 Strict Trs: 880.82/297.05 { U101(tt(), M, N) -> 880.82/297.05 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.05 , U102(tt(), M, N) -> 880.82/297.05 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.05 , isNatKind(n__0()) -> tt() 880.82/297.05 , isNatKind(n__plus(V1, V2)) -> 880.82/297.05 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.05 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.05 , isNatKind(n__x(V1, V2)) -> 880.82/297.05 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.05 , activate(X) -> X 880.82/297.05 , activate(n__0()) -> 0() 880.82/297.05 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.05 , activate(n__s(X)) -> s(X) 880.82/297.05 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.05 , U103(tt(), M, N) -> 880.82/297.05 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.05 , isNat(n__0()) -> tt() 880.82/297.05 , isNat(n__plus(V1, V2)) -> 880.82/297.05 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.05 , isNat(n__x(V1, V2)) -> 880.82/297.05 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , U104(tt(), M, N) -> 880.82/297.05 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.05 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.05 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.05 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.05 , x(X1, X2) -> n__x(X1, X2) 880.82/297.05 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.05 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.05 , U11(tt(), V1, V2) -> 880.82/297.05 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , U12(tt(), V1, V2) -> 880.82/297.05 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.05 , U13(tt(), V1, V2) -> 880.82/297.05 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.05 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.05 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.05 , U16(tt()) -> tt() 880.82/297.05 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.05 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.05 , U23(tt()) -> tt() 880.82/297.05 , U31(tt(), V1, V2) -> 880.82/297.05 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , U32(tt(), V1, V2) -> 880.82/297.05 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.05 , U33(tt(), V1, V2) -> 880.82/297.05 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.05 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.05 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.05 , U36(tt()) -> tt() 880.82/297.05 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.05 , U42(tt()) -> tt() 880.82/297.05 , U51(tt()) -> tt() 880.82/297.05 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.05 , U62(tt()) -> tt() 880.82/297.05 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.05 , U72(tt(), N) -> activate(N) 880.82/297.05 , U81(tt(), M, N) -> 880.82/297.05 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.05 , U82(tt(), M, N) -> 880.82/297.05 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.05 , U83(tt(), M, N) -> 880.82/297.05 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.05 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.05 , s(X) -> n__s(X) 880.82/297.05 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.05 , U92(tt()) -> 0() 880.82/297.05 , 0() -> n__0() } 880.82/297.05 Obligation: 880.82/297.05 runtime complexity 880.82/297.05 Answer: 880.82/297.05 MAYBE 880.82/297.05 880.82/297.05 None of the processors succeeded. 880.82/297.05 880.82/297.05 Details of failed attempt(s): 880.82/297.05 ----------------------------- 880.82/297.05 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 880.82/297.05 following reason: 880.82/297.05 880.82/297.05 Computation stopped due to timeout after 297.0 seconds. 880.82/297.05 880.82/297.05 2) 'Best' failed due to the following reason: 880.82/297.05 880.82/297.05 None of the processors succeeded. 880.82/297.05 880.82/297.05 Details of failed attempt(s): 880.82/297.05 ----------------------------- 880.82/297.05 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 880.82/297.05 seconds)' failed due to the following reason: 880.82/297.05 880.82/297.05 Computation stopped due to timeout after 148.0 seconds. 880.82/297.05 880.82/297.05 2) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 880.82/297.05 failed due to the following reason: 880.82/297.05 880.82/297.05 None of the processors succeeded. 880.82/297.05 880.82/297.05 Details of failed attempt(s): 880.82/297.05 ----------------------------- 880.82/297.05 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 880.82/297.05 failed due to the following reason: 880.82/297.05 880.82/297.05 match-boundness of the problem could not be verified. 880.82/297.05 880.82/297.05 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 880.82/297.05 failed due to the following reason: 880.82/297.05 880.82/297.05 match-boundness of the problem could not be verified. 880.82/297.05 880.82/297.05 880.82/297.05 3) 'Best' failed due to the following reason: 880.82/297.05 880.82/297.05 None of the processors succeeded. 880.82/297.05 880.82/297.05 Details of failed attempt(s): 880.82/297.05 ----------------------------- 880.82/297.05 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 880.82/297.05 following reason: 880.82/297.05 880.82/297.05 The processor is inapplicable, reason: 880.82/297.05 Processor only applicable for innermost runtime complexity analysis 880.82/297.05 880.82/297.05 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 880.82/297.05 to the following reason: 880.82/297.05 880.82/297.05 The processor is inapplicable, reason: 880.82/297.05 Processor only applicable for innermost runtime complexity analysis 880.82/297.05 880.82/297.05 880.82/297.05 880.82/297.05 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 880.82/297.05 the following reason: 880.82/297.05 880.82/297.05 We add the following weak dependency pairs: 880.82/297.05 880.82/297.05 Strict DPs: 880.82/297.05 { U101^#(tt(), M, N) -> 880.82/297.05 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.05 , U102^#(tt(), M, N) -> 880.82/297.05 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.05 , U103^#(tt(), M, N) -> 880.82/297.05 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.05 , isNatKind^#(n__0()) -> c_3() 880.82/297.05 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.05 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.05 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.05 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.05 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.05 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.05 , U51^#(tt()) -> c_41() 880.82/297.05 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.05 , activate^#(X) -> c_7(X) 880.82/297.05 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.05 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.05 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.05 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.05 , 0^#() -> c_53() 880.82/297.05 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.05 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.05 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.05 , s^#(X) -> c_50(X) 880.82/297.05 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.05 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.05 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.05 , U104^#(tt(), M, N) -> 880.82/297.05 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.05 , isNat^#(n__0()) -> c_13() 880.82/297.05 , isNat^#(n__plus(V1, V2)) -> 880.82/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , isNat^#(n__s(V1)) -> 880.82/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.05 , isNat^#(n__x(V1, V2)) -> 880.82/297.05 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , U11^#(tt(), V1, V2) -> 880.82/297.05 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , U21^#(tt(), V1) -> 880.82/297.05 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.05 , U31^#(tt(), V1, V2) -> 880.82/297.05 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , U81^#(tt(), M, N) -> 880.82/297.05 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.05 , U71^#(tt(), N) -> 880.82/297.05 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.05 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.05 , U12^#(tt(), V1, V2) -> 880.82/297.05 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U13^#(tt(), V1, V2) -> 880.82/297.05 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U14^#(tt(), V1, V2) -> 880.82/297.05 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.05 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.05 , U16^#(tt()) -> c_29() 880.82/297.05 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.05 , U23^#(tt()) -> c_32() 880.82/297.05 , U32^#(tt(), V1, V2) -> 880.82/297.05 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U33^#(tt(), V1, V2) -> 880.82/297.05 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U34^#(tt(), V1, V2) -> 880.82/297.05 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.05 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.05 , U36^#(tt()) -> c_38() 880.82/297.05 , U42^#(tt()) -> c_40() 880.82/297.05 , U62^#(tt()) -> c_43() 880.82/297.05 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.05 , U82^#(tt(), M, N) -> 880.82/297.05 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.05 , U83^#(tt(), M, N) -> 880.82/297.05 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.05 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.05 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.05 880.82/297.05 and mark the set of starting terms. 880.82/297.05 880.82/297.05 We are left with following problem, upon which TcT provides the 880.82/297.05 certificate MAYBE. 880.82/297.05 880.82/297.05 Strict DPs: 880.82/297.05 { U101^#(tt(), M, N) -> 880.82/297.05 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.05 , U102^#(tt(), M, N) -> 880.82/297.05 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.05 , U103^#(tt(), M, N) -> 880.82/297.05 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.05 , isNatKind^#(n__0()) -> c_3() 880.82/297.05 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.05 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.05 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.05 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.05 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.05 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.05 , U51^#(tt()) -> c_41() 880.82/297.05 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.05 , activate^#(X) -> c_7(X) 880.82/297.05 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.05 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.05 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.05 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.05 , 0^#() -> c_53() 880.82/297.05 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.05 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.05 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.05 , s^#(X) -> c_50(X) 880.82/297.05 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.05 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.05 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.05 , U104^#(tt(), M, N) -> 880.82/297.05 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.05 , isNat^#(n__0()) -> c_13() 880.82/297.05 , isNat^#(n__plus(V1, V2)) -> 880.82/297.05 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , isNat^#(n__s(V1)) -> 880.82/297.05 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.05 , isNat^#(n__x(V1, V2)) -> 880.82/297.05 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , U11^#(tt(), V1, V2) -> 880.82/297.05 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , U21^#(tt(), V1) -> 880.82/297.05 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.05 , U31^#(tt(), V1, V2) -> 880.82/297.05 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.05 , U81^#(tt(), M, N) -> 880.82/297.05 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.05 , U71^#(tt(), N) -> 880.82/297.05 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.05 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.05 , U12^#(tt(), V1, V2) -> 880.82/297.05 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U13^#(tt(), V1, V2) -> 880.82/297.05 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U14^#(tt(), V1, V2) -> 880.82/297.05 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.05 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.05 , U16^#(tt()) -> c_29() 880.82/297.05 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.05 , U23^#(tt()) -> c_32() 880.82/297.05 , U32^#(tt(), V1, V2) -> 880.82/297.05 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U33^#(tt(), V1, V2) -> 880.82/297.05 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.05 , U34^#(tt(), V1, V2) -> 880.82/297.05 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.05 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.05 , U36^#(tt()) -> c_38() 880.82/297.05 , U42^#(tt()) -> c_40() 880.82/297.05 , U62^#(tt()) -> c_43() 880.82/297.05 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.05 , U82^#(tt(), M, N) -> 880.82/297.05 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.05 , U83^#(tt(), M, N) -> 880.82/297.05 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.05 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.05 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.05 Strict Trs: 880.82/297.05 { U101(tt(), M, N) -> 880.82/297.05 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.05 , U102(tt(), M, N) -> 880.82/297.05 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.05 , isNatKind(n__0()) -> tt() 880.82/297.05 , isNatKind(n__plus(V1, V2)) -> 880.82/297.05 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.05 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.05 , isNatKind(n__x(V1, V2)) -> 880.82/297.05 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.05 , activate(X) -> X 880.82/297.05 , activate(n__0()) -> 0() 880.82/297.05 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.05 , activate(n__s(X)) -> s(X) 880.82/297.05 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.05 , U103(tt(), M, N) -> 880.82/297.05 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.05 , isNat(n__0()) -> tt() 880.82/297.05 , isNat(n__plus(V1, V2)) -> 880.82/297.05 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.05 , isNat(n__x(V1, V2)) -> 880.82/297.05 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , U104(tt(), M, N) -> 880.82/297.05 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.05 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.05 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.05 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.05 , x(X1, X2) -> n__x(X1, X2) 880.82/297.05 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.05 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.05 , U11(tt(), V1, V2) -> 880.82/297.05 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , U12(tt(), V1, V2) -> 880.82/297.05 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.05 , U13(tt(), V1, V2) -> 880.82/297.05 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.05 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.05 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.05 , U16(tt()) -> tt() 880.82/297.05 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.05 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.05 , U23(tt()) -> tt() 880.82/297.05 , U31(tt(), V1, V2) -> 880.82/297.05 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.05 , U32(tt(), V1, V2) -> 880.82/297.05 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.05 , U33(tt(), V1, V2) -> 880.82/297.05 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.06 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.06 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.06 , U36(tt()) -> tt() 880.82/297.06 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.06 , U42(tt()) -> tt() 880.82/297.06 , U51(tt()) -> tt() 880.82/297.06 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.06 , U62(tt()) -> tt() 880.82/297.06 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.06 , U72(tt(), N) -> activate(N) 880.82/297.06 , U81(tt(), M, N) -> 880.82/297.06 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.06 , U82(tt(), M, N) -> 880.82/297.06 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.06 , U83(tt(), M, N) -> 880.82/297.06 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.06 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.06 , s(X) -> n__s(X) 880.82/297.06 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.06 , U92(tt()) -> 0() 880.82/297.06 , 0() -> n__0() } 880.82/297.06 Obligation: 880.82/297.06 runtime complexity 880.82/297.06 Answer: 880.82/297.06 MAYBE 880.82/297.06 880.82/297.06 We estimate the number of application of {4,9,16,25,39,41,46,47,48} 880.82/297.06 by applications of Pre({4,9,16,25,39,41,46,47,48}) = 880.82/297.06 {6,8,10,11,12,17,20,21,38,40,45,53}. Here rules are labeled as 880.82/297.06 follows: 880.82/297.06 880.82/297.06 DPs: 880.82/297.06 { 1: U101^#(tt(), M, N) -> 880.82/297.06 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , 2: U102^#(tt(), M, N) -> 880.82/297.06 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.06 , 3: U103^#(tt(), M, N) -> 880.82/297.06 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.06 , 4: isNatKind^#(n__0()) -> c_3() 880.82/297.06 , 5: isNatKind^#(n__plus(V1, V2)) -> 880.82/297.06 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , 6: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.06 , 7: isNatKind^#(n__x(V1, V2)) -> 880.82/297.06 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , 8: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.06 , 9: U51^#(tt()) -> c_41() 880.82/297.06 , 10: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.06 , 11: activate^#(X) -> c_7(X) 880.82/297.06 , 12: activate^#(n__0()) -> c_8(0^#()) 880.82/297.06 , 13: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.06 , 14: activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.06 , 15: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.06 , 16: 0^#() -> c_53() 880.82/297.06 , 17: plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.06 , 18: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.06 , 19: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.06 , 20: s^#(X) -> c_50(X) 880.82/297.06 , 21: x^#(X1, X2) -> c_21(X1, X2) 880.82/297.06 , 22: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.06 , 23: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.06 , 24: U104^#(tt(), M, N) -> 880.82/297.06 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.06 , 25: isNat^#(n__0()) -> c_13() 880.82/297.06 , 26: isNat^#(n__plus(V1, V2)) -> 880.82/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 27: isNat^#(n__s(V1)) -> 880.82/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , 28: isNat^#(n__x(V1, V2)) -> 880.82/297.06 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 29: U11^#(tt(), V1, V2) -> 880.82/297.06 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 30: U21^#(tt(), V1) -> 880.82/297.06 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , 31: U31^#(tt(), V1, V2) -> 880.82/297.06 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 32: U81^#(tt(), M, N) -> 880.82/297.06 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , 33: U71^#(tt(), N) -> 880.82/297.06 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.06 , 34: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.06 , 35: U12^#(tt(), V1, V2) -> 880.82/297.06 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 36: U13^#(tt(), V1, V2) -> 880.82/297.06 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 37: U14^#(tt(), V1, V2) -> 880.82/297.06 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.06 , 38: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.06 , 39: U16^#(tt()) -> c_29() 880.82/297.06 , 40: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.06 , 41: U23^#(tt()) -> c_32() 880.82/297.06 , 42: U32^#(tt(), V1, V2) -> 880.82/297.06 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 43: U33^#(tt(), V1, V2) -> 880.82/297.06 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 44: U34^#(tt(), V1, V2) -> 880.82/297.06 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.06 , 45: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.06 , 46: U36^#(tt()) -> c_38() 880.82/297.06 , 47: U42^#(tt()) -> c_40() 880.82/297.06 , 48: U62^#(tt()) -> c_43() 880.82/297.06 , 49: U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.06 , 50: U82^#(tt(), M, N) -> 880.82/297.06 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.06 , 51: U83^#(tt(), M, N) -> 880.82/297.06 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.06 , 52: U84^#(tt(), M, N) -> 880.82/297.06 c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.06 , 53: U92^#(tt()) -> c_52(0^#()) } 880.82/297.06 880.82/297.06 We are left with following problem, upon which TcT provides the 880.82/297.06 certificate MAYBE. 880.82/297.06 880.82/297.06 Strict DPs: 880.82/297.06 { U101^#(tt(), M, N) -> 880.82/297.06 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , U102^#(tt(), M, N) -> 880.82/297.06 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.06 , U103^#(tt(), M, N) -> 880.82/297.06 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.06 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.06 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.06 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.06 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.06 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.06 , activate^#(X) -> c_7(X) 880.82/297.06 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.06 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.06 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.06 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.06 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.06 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.06 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.06 , s^#(X) -> c_50(X) 880.82/297.06 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.06 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.06 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.06 , U104^#(tt(), M, N) -> 880.82/297.06 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.06 , isNat^#(n__plus(V1, V2)) -> 880.82/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , isNat^#(n__s(V1)) -> 880.82/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , isNat^#(n__x(V1, V2)) -> 880.82/297.06 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , U11^#(tt(), V1, V2) -> 880.82/297.06 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , U21^#(tt(), V1) -> 880.82/297.06 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , U31^#(tt(), V1, V2) -> 880.82/297.06 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , U81^#(tt(), M, N) -> 880.82/297.06 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , U71^#(tt(), N) -> 880.82/297.06 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.06 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.06 , U12^#(tt(), V1, V2) -> 880.82/297.06 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , U13^#(tt(), V1, V2) -> 880.82/297.06 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , U14^#(tt(), V1, V2) -> 880.82/297.06 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.06 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.06 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.06 , U32^#(tt(), V1, V2) -> 880.82/297.06 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , U33^#(tt(), V1, V2) -> 880.82/297.06 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , U34^#(tt(), V1, V2) -> 880.82/297.06 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.06 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.06 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.06 , U82^#(tt(), M, N) -> 880.82/297.06 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.06 , U83^#(tt(), M, N) -> 880.82/297.06 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.06 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.06 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.06 Strict Trs: 880.82/297.06 { U101(tt(), M, N) -> 880.82/297.06 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.06 , U102(tt(), M, N) -> 880.82/297.06 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.06 , isNatKind(n__0()) -> tt() 880.82/297.06 , isNatKind(n__plus(V1, V2)) -> 880.82/297.06 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.06 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.06 , isNatKind(n__x(V1, V2)) -> 880.82/297.06 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.06 , activate(X) -> X 880.82/297.06 , activate(n__0()) -> 0() 880.82/297.06 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.06 , activate(n__s(X)) -> s(X) 880.82/297.06 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.06 , U103(tt(), M, N) -> 880.82/297.06 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.06 , isNat(n__0()) -> tt() 880.82/297.06 , isNat(n__plus(V1, V2)) -> 880.82/297.06 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.06 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.06 , isNat(n__x(V1, V2)) -> 880.82/297.06 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.06 , U104(tt(), M, N) -> 880.82/297.06 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.06 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.06 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.06 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.06 , x(X1, X2) -> n__x(X1, X2) 880.82/297.06 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.06 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.06 , U11(tt(), V1, V2) -> 880.82/297.06 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.06 , U12(tt(), V1, V2) -> 880.82/297.06 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.06 , U13(tt(), V1, V2) -> 880.82/297.06 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.06 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.06 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.06 , U16(tt()) -> tt() 880.82/297.06 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.06 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.06 , U23(tt()) -> tt() 880.82/297.06 , U31(tt(), V1, V2) -> 880.82/297.06 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.06 , U32(tt(), V1, V2) -> 880.82/297.06 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.06 , U33(tt(), V1, V2) -> 880.82/297.06 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.06 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.06 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.06 , U36(tt()) -> tt() 880.82/297.06 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.06 , U42(tt()) -> tt() 880.82/297.06 , U51(tt()) -> tt() 880.82/297.06 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.06 , U62(tt()) -> tt() 880.82/297.06 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.06 , U72(tt(), N) -> activate(N) 880.82/297.06 , U81(tt(), M, N) -> 880.82/297.06 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.06 , U82(tt(), M, N) -> 880.82/297.06 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.06 , U83(tt(), M, N) -> 880.82/297.06 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.06 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.06 , s(X) -> n__s(X) 880.82/297.06 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.06 , U92(tt()) -> 0() 880.82/297.06 , 0() -> n__0() } 880.82/297.06 Weak DPs: 880.82/297.06 { isNatKind^#(n__0()) -> c_3() 880.82/297.06 , U51^#(tt()) -> c_41() 880.82/297.06 , 0^#() -> c_53() 880.82/297.06 , isNat^#(n__0()) -> c_13() 880.82/297.06 , U16^#(tt()) -> c_29() 880.82/297.06 , U23^#(tt()) -> c_32() 880.82/297.06 , U36^#(tt()) -> c_38() 880.82/297.06 , U42^#(tt()) -> c_40() 880.82/297.06 , U62^#(tt()) -> c_43() } 880.82/297.06 Obligation: 880.82/297.06 runtime complexity 880.82/297.06 Answer: 880.82/297.06 MAYBE 880.82/297.06 880.82/297.06 We estimate the number of application of {5,7,8,10,34,35,39,44} by 880.82/297.06 applications of Pre({5,7,8,10,34,35,39,44}) = 880.82/297.06 {4,6,9,14,17,18,26,30,33,38,40}. Here rules are labeled as follows: 880.82/297.06 880.82/297.06 DPs: 880.82/297.06 { 1: U101^#(tt(), M, N) -> 880.82/297.06 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , 2: U102^#(tt(), M, N) -> 880.82/297.06 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.06 , 3: U103^#(tt(), M, N) -> 880.82/297.06 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.06 , 4: isNatKind^#(n__plus(V1, V2)) -> 880.82/297.06 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , 5: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.06 , 6: isNatKind^#(n__x(V1, V2)) -> 880.82/297.06 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , 7: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.06 , 8: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.06 , 9: activate^#(X) -> c_7(X) 880.82/297.06 , 10: activate^#(n__0()) -> c_8(0^#()) 880.82/297.06 , 11: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.06 , 12: activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.06 , 13: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.06 , 14: plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.06 , 15: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.06 , 16: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.06 , 17: s^#(X) -> c_50(X) 880.82/297.06 , 18: x^#(X1, X2) -> c_21(X1, X2) 880.82/297.06 , 19: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.06 , 20: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.06 , 21: U104^#(tt(), M, N) -> 880.82/297.06 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.06 , 22: isNat^#(n__plus(V1, V2)) -> 880.82/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 23: isNat^#(n__s(V1)) -> 880.82/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , 24: isNat^#(n__x(V1, V2)) -> 880.82/297.06 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 25: U11^#(tt(), V1, V2) -> 880.82/297.06 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 26: U21^#(tt(), V1) -> 880.82/297.06 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , 27: U31^#(tt(), V1, V2) -> 880.82/297.06 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , 28: U81^#(tt(), M, N) -> 880.82/297.06 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , 29: U71^#(tt(), N) -> 880.82/297.06 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.06 , 30: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.06 , 31: U12^#(tt(), V1, V2) -> 880.82/297.06 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 32: U13^#(tt(), V1, V2) -> 880.82/297.06 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 33: U14^#(tt(), V1, V2) -> 880.82/297.06 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.06 , 34: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.06 , 35: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.06 , 36: U32^#(tt(), V1, V2) -> 880.82/297.06 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 37: U33^#(tt(), V1, V2) -> 880.82/297.06 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , 38: U34^#(tt(), V1, V2) -> 880.82/297.06 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.06 , 39: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.06 , 40: U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.06 , 41: U82^#(tt(), M, N) -> 880.82/297.06 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.06 , 42: U83^#(tt(), M, N) -> 880.82/297.06 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.06 , 43: U84^#(tt(), M, N) -> 880.82/297.06 c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.06 , 44: U92^#(tt()) -> c_52(0^#()) 880.82/297.06 , 45: isNatKind^#(n__0()) -> c_3() 880.82/297.06 , 46: U51^#(tt()) -> c_41() 880.82/297.06 , 47: 0^#() -> c_53() 880.82/297.06 , 48: isNat^#(n__0()) -> c_13() 880.82/297.06 , 49: U16^#(tt()) -> c_29() 880.82/297.06 , 50: U23^#(tt()) -> c_32() 880.82/297.06 , 51: U36^#(tt()) -> c_38() 880.82/297.06 , 52: U42^#(tt()) -> c_40() 880.82/297.06 , 53: U62^#(tt()) -> c_43() } 880.82/297.06 880.82/297.06 We are left with following problem, upon which TcT provides the 880.82/297.06 certificate MAYBE. 880.82/297.06 880.82/297.06 Strict DPs: 880.82/297.06 { U101^#(tt(), M, N) -> 880.82/297.06 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , U102^#(tt(), M, N) -> 880.82/297.06 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.06 , U103^#(tt(), M, N) -> 880.82/297.06 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.06 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.06 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.06 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.06 , activate^#(X) -> c_7(X) 880.82/297.06 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.06 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.06 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.06 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.06 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.06 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.06 , s^#(X) -> c_50(X) 880.82/297.06 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.06 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.06 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.06 , U104^#(tt(), M, N) -> 880.82/297.06 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.06 , isNat^#(n__plus(V1, V2)) -> 880.82/297.06 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , isNat^#(n__s(V1)) -> 880.82/297.06 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , isNat^#(n__x(V1, V2)) -> 880.82/297.06 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , U11^#(tt(), V1, V2) -> 880.82/297.06 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , U21^#(tt(), V1) -> 880.82/297.06 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.06 , U31^#(tt(), V1, V2) -> 880.82/297.06 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.06 , U81^#(tt(), M, N) -> 880.82/297.06 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.06 , U71^#(tt(), N) -> 880.82/297.06 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.06 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.06 , U12^#(tt(), V1, V2) -> 880.82/297.06 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , U13^#(tt(), V1, V2) -> 880.82/297.06 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.06 , U14^#(tt(), V1, V2) -> 880.82/297.06 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.07 , U32^#(tt(), V1, V2) -> 880.82/297.07 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , U33^#(tt(), V1, V2) -> 880.82/297.07 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , U34^#(tt(), V1, V2) -> 880.82/297.07 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.07 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.07 , U82^#(tt(), M, N) -> 880.82/297.07 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.07 , U83^#(tt(), M, N) -> 880.82/297.07 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.07 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 880.82/297.07 Strict Trs: 880.82/297.07 { U101(tt(), M, N) -> 880.82/297.07 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.07 , U102(tt(), M, N) -> 880.82/297.07 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.07 , isNatKind(n__0()) -> tt() 880.82/297.07 , isNatKind(n__plus(V1, V2)) -> 880.82/297.07 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.07 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.07 , isNatKind(n__x(V1, V2)) -> 880.82/297.07 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.07 , activate(X) -> X 880.82/297.07 , activate(n__0()) -> 0() 880.82/297.07 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.07 , activate(n__s(X)) -> s(X) 880.82/297.07 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.07 , U103(tt(), M, N) -> 880.82/297.07 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.07 , isNat(n__0()) -> tt() 880.82/297.07 , isNat(n__plus(V1, V2)) -> 880.82/297.07 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.07 , isNat(n__x(V1, V2)) -> 880.82/297.07 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , U104(tt(), M, N) -> 880.82/297.07 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.07 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.07 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.07 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.07 , x(X1, X2) -> n__x(X1, X2) 880.82/297.07 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.07 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.07 , U11(tt(), V1, V2) -> 880.82/297.07 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , U12(tt(), V1, V2) -> 880.82/297.07 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U13(tt(), V1, V2) -> 880.82/297.07 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.07 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.07 , U16(tt()) -> tt() 880.82/297.07 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.07 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.07 , U23(tt()) -> tt() 880.82/297.07 , U31(tt(), V1, V2) -> 880.82/297.07 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , U32(tt(), V1, V2) -> 880.82/297.07 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U33(tt(), V1, V2) -> 880.82/297.07 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.07 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.07 , U36(tt()) -> tt() 880.82/297.07 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.07 , U42(tt()) -> tt() 880.82/297.07 , U51(tt()) -> tt() 880.82/297.07 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.07 , U62(tt()) -> tt() 880.82/297.07 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.07 , U72(tt(), N) -> activate(N) 880.82/297.07 , U81(tt(), M, N) -> 880.82/297.07 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.07 , U82(tt(), M, N) -> 880.82/297.07 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.07 , U83(tt(), M, N) -> 880.82/297.07 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.07 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.07 , s(X) -> n__s(X) 880.82/297.07 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.07 , U92(tt()) -> 0() 880.82/297.07 , 0() -> n__0() } 880.82/297.07 Weak DPs: 880.82/297.07 { isNatKind^#(n__0()) -> c_3() 880.82/297.07 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.07 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.07 , U51^#(tt()) -> c_41() 880.82/297.07 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.07 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.07 , 0^#() -> c_53() 880.82/297.07 , isNat^#(n__0()) -> c_13() 880.82/297.07 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.07 , U16^#(tt()) -> c_29() 880.82/297.07 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.07 , U23^#(tt()) -> c_32() 880.82/297.07 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.07 , U36^#(tt()) -> c_38() 880.82/297.07 , U42^#(tt()) -> c_40() 880.82/297.07 , U62^#(tt()) -> c_43() 880.82/297.07 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.07 Obligation: 880.82/297.07 runtime complexity 880.82/297.07 Answer: 880.82/297.07 MAYBE 880.82/297.07 880.82/297.07 We estimate the number of application of {4,5,22,26,29,32} by 880.82/297.07 applications of Pre({4,5,22,26,29,32}) = {6,10,13,14,16,19,28,31}. 880.82/297.07 Here rules are labeled as follows: 880.82/297.07 880.82/297.07 DPs: 880.82/297.07 { 1: U101^#(tt(), M, N) -> 880.82/297.07 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.07 , 2: U102^#(tt(), M, N) -> 880.82/297.07 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.07 , 3: U103^#(tt(), M, N) -> 880.82/297.07 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.07 , 4: isNatKind^#(n__plus(V1, V2)) -> 880.82/297.07 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.07 , 5: isNatKind^#(n__x(V1, V2)) -> 880.82/297.07 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.07 , 6: activate^#(X) -> c_7(X) 880.82/297.07 , 7: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.07 , 8: activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.07 , 9: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.07 , 10: plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.07 , 11: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.07 , 12: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.07 , 13: s^#(X) -> c_50(X) 880.82/297.07 , 14: x^#(X1, X2) -> c_21(X1, X2) 880.82/297.07 , 15: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.07 , 16: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.07 , 17: U104^#(tt(), M, N) -> 880.82/297.07 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.07 , 18: isNat^#(n__plus(V1, V2)) -> 880.82/297.07 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , 19: isNat^#(n__s(V1)) -> 880.82/297.07 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.07 , 20: isNat^#(n__x(V1, V2)) -> 880.82/297.07 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , 21: U11^#(tt(), V1, V2) -> 880.82/297.07 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , 22: U21^#(tt(), V1) -> 880.82/297.07 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.07 , 23: U31^#(tt(), V1, V2) -> 880.82/297.07 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , 24: U81^#(tt(), M, N) -> 880.82/297.07 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.07 , 25: U71^#(tt(), N) -> 880.82/297.07 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.07 , 26: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.07 , 27: U12^#(tt(), V1, V2) -> 880.82/297.07 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , 28: U13^#(tt(), V1, V2) -> 880.82/297.07 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , 29: U14^#(tt(), V1, V2) -> 880.82/297.07 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.07 , 30: U32^#(tt(), V1, V2) -> 880.82/297.07 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , 31: U33^#(tt(), V1, V2) -> 880.82/297.07 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , 32: U34^#(tt(), V1, V2) -> 880.82/297.07 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.07 , 33: U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.07 , 34: U82^#(tt(), M, N) -> 880.82/297.07 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.07 , 35: U83^#(tt(), M, N) -> 880.82/297.07 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.07 , 36: U84^#(tt(), M, N) -> 880.82/297.07 c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.07 , 37: isNatKind^#(n__0()) -> c_3() 880.82/297.07 , 38: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.07 , 39: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.07 , 40: U51^#(tt()) -> c_41() 880.82/297.07 , 41: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.07 , 42: activate^#(n__0()) -> c_8(0^#()) 880.82/297.07 , 43: 0^#() -> c_53() 880.82/297.07 , 44: isNat^#(n__0()) -> c_13() 880.82/297.07 , 45: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.07 , 46: U16^#(tt()) -> c_29() 880.82/297.07 , 47: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.07 , 48: U23^#(tt()) -> c_32() 880.82/297.07 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.07 , 50: U36^#(tt()) -> c_38() 880.82/297.07 , 51: U42^#(tt()) -> c_40() 880.82/297.07 , 52: U62^#(tt()) -> c_43() 880.82/297.07 , 53: U92^#(tt()) -> c_52(0^#()) } 880.82/297.07 880.82/297.07 We are left with following problem, upon which TcT provides the 880.82/297.07 certificate MAYBE. 880.82/297.07 880.82/297.07 Strict DPs: 880.82/297.07 { U101^#(tt(), M, N) -> 880.82/297.07 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.07 , U102^#(tt(), M, N) -> 880.82/297.07 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.07 , U103^#(tt(), M, N) -> 880.82/297.07 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.07 , activate^#(X) -> c_7(X) 880.82/297.07 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.07 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.07 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.07 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.07 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.07 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.07 , s^#(X) -> c_50(X) 880.82/297.07 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.07 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.07 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.07 , U104^#(tt(), M, N) -> 880.82/297.07 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.07 , isNat^#(n__plus(V1, V2)) -> 880.82/297.07 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , isNat^#(n__s(V1)) -> 880.82/297.07 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.07 , isNat^#(n__x(V1, V2)) -> 880.82/297.07 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , U11^#(tt(), V1, V2) -> 880.82/297.07 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , U31^#(tt(), V1, V2) -> 880.82/297.07 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.07 , U81^#(tt(), M, N) -> 880.82/297.07 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.07 , U71^#(tt(), N) -> 880.82/297.07 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.07 , U12^#(tt(), V1, V2) -> 880.82/297.07 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , U13^#(tt(), V1, V2) -> 880.82/297.07 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , U32^#(tt(), V1, V2) -> 880.82/297.07 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , U33^#(tt(), V1, V2) -> 880.82/297.07 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.07 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.07 , U82^#(tt(), M, N) -> 880.82/297.07 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.07 , U83^#(tt(), M, N) -> 880.82/297.07 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.07 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 880.82/297.07 Strict Trs: 880.82/297.07 { U101(tt(), M, N) -> 880.82/297.07 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.07 , U102(tt(), M, N) -> 880.82/297.07 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.07 , isNatKind(n__0()) -> tt() 880.82/297.07 , isNatKind(n__plus(V1, V2)) -> 880.82/297.07 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.07 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.07 , isNatKind(n__x(V1, V2)) -> 880.82/297.07 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.07 , activate(X) -> X 880.82/297.07 , activate(n__0()) -> 0() 880.82/297.07 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.07 , activate(n__s(X)) -> s(X) 880.82/297.07 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.07 , U103(tt(), M, N) -> 880.82/297.07 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.07 , isNat(n__0()) -> tt() 880.82/297.07 , isNat(n__plus(V1, V2)) -> 880.82/297.07 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.07 , isNat(n__x(V1, V2)) -> 880.82/297.07 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , U104(tt(), M, N) -> 880.82/297.07 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.07 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.07 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.07 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.07 , x(X1, X2) -> n__x(X1, X2) 880.82/297.07 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.07 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.07 , U11(tt(), V1, V2) -> 880.82/297.07 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , U12(tt(), V1, V2) -> 880.82/297.07 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U13(tt(), V1, V2) -> 880.82/297.07 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.07 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.07 , U16(tt()) -> tt() 880.82/297.07 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.07 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.07 , U23(tt()) -> tt() 880.82/297.07 , U31(tt(), V1, V2) -> 880.82/297.07 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.07 , U32(tt(), V1, V2) -> 880.82/297.07 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U33(tt(), V1, V2) -> 880.82/297.07 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.07 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.07 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.07 , U36(tt()) -> tt() 880.82/297.07 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.07 , U42(tt()) -> tt() 880.82/297.07 , U51(tt()) -> tt() 880.82/297.07 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.07 , U62(tt()) -> tt() 880.82/297.07 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.07 , U72(tt(), N) -> activate(N) 880.82/297.07 , U81(tt(), M, N) -> 880.82/297.07 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.07 , U82(tt(), M, N) -> 880.82/297.07 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.07 , U83(tt(), M, N) -> 880.82/297.07 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.07 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.07 , s(X) -> n__s(X) 880.82/297.07 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.07 , U92(tt()) -> 0() 880.82/297.07 , 0() -> n__0() } 880.82/297.07 Weak DPs: 880.82/297.07 { isNatKind^#(n__0()) -> c_3() 880.82/297.07 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.07 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.07 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.07 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.07 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.07 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.07 , U51^#(tt()) -> c_41() 880.82/297.07 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.07 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.07 , 0^#() -> c_53() 880.82/297.07 , isNat^#(n__0()) -> c_13() 880.82/297.07 , U21^#(tt(), V1) -> 880.82/297.07 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.07 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.07 , U14^#(tt(), V1, V2) -> 880.82/297.07 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.07 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.07 , U16^#(tt()) -> c_29() 880.82/297.07 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.07 , U23^#(tt()) -> c_32() 880.82/297.07 , U34^#(tt(), V1, V2) -> 880.82/297.07 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.07 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.07 , U36^#(tt()) -> c_38() 880.82/297.07 , U42^#(tt()) -> c_40() 880.82/297.07 , U62^#(tt()) -> c_43() 880.82/297.07 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.07 Obligation: 880.82/297.07 runtime complexity 880.82/297.07 Answer: 880.82/297.07 MAYBE 880.82/297.07 880.82/297.07 We estimate the number of application of {14,17,24,26} by 880.82/297.07 applications of Pre({14,17,24,26}) = {4,7,8,11,12,23,25}. Here 880.82/297.07 rules are labeled as follows: 880.82/297.07 880.82/297.07 DPs: 880.82/297.07 { 1: U101^#(tt(), M, N) -> 880.82/297.07 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.07 , 2: U102^#(tt(), M, N) -> 880.82/297.07 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.07 , 3: U103^#(tt(), M, N) -> 880.82/297.07 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.07 , 4: activate^#(X) -> c_7(X) 880.82/297.07 , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.07 , 6: activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.07 , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.07 , 8: plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.07 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.07 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.07 , 11: s^#(X) -> c_50(X) 880.82/297.07 , 12: x^#(X1, X2) -> c_21(X1, X2) 880.82/297.07 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.08 , 14: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.08 , 15: U104^#(tt(), M, N) -> 880.82/297.08 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.08 , 16: isNat^#(n__plus(V1, V2)) -> 880.82/297.08 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 17: isNat^#(n__s(V1)) -> 880.82/297.08 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.08 , 18: isNat^#(n__x(V1, V2)) -> 880.82/297.08 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 19: U11^#(tt(), V1, V2) -> 880.82/297.08 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 20: U31^#(tt(), V1, V2) -> 880.82/297.08 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 21: U81^#(tt(), M, N) -> 880.82/297.08 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.08 , 22: U71^#(tt(), N) -> 880.82/297.08 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.08 , 23: U12^#(tt(), V1, V2) -> 880.82/297.08 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 24: U13^#(tt(), V1, V2) -> 880.82/297.08 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 25: U32^#(tt(), V1, V2) -> 880.82/297.08 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 26: U33^#(tt(), V1, V2) -> 880.82/297.08 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 27: U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.08 , 28: U82^#(tt(), M, N) -> 880.82/297.08 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.08 , 29: U83^#(tt(), M, N) -> 880.82/297.08 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.08 , 30: U84^#(tt(), M, N) -> 880.82/297.08 c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.08 , 31: isNatKind^#(n__0()) -> c_3() 880.82/297.08 , 32: isNatKind^#(n__plus(V1, V2)) -> 880.82/297.08 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.08 , 33: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.08 , 34: isNatKind^#(n__x(V1, V2)) -> 880.82/297.08 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.08 , 35: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.08 , 36: U51^#(tt()) -> c_41() 880.82/297.08 , 37: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.08 , 38: activate^#(n__0()) -> c_8(0^#()) 880.82/297.08 , 39: 0^#() -> c_53() 880.82/297.08 , 40: isNat^#(n__0()) -> c_13() 880.82/297.08 , 41: U21^#(tt(), V1) -> 880.82/297.08 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.08 , 42: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.08 , 43: U14^#(tt(), V1, V2) -> 880.82/297.08 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.08 , 44: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.08 , 45: U16^#(tt()) -> c_29() 880.82/297.08 , 46: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.08 , 47: U23^#(tt()) -> c_32() 880.82/297.08 , 48: U34^#(tt(), V1, V2) -> 880.82/297.08 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.08 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.08 , 50: U36^#(tt()) -> c_38() 880.82/297.08 , 51: U42^#(tt()) -> c_40() 880.82/297.08 , 52: U62^#(tt()) -> c_43() 880.82/297.08 , 53: U92^#(tt()) -> c_52(0^#()) } 880.82/297.08 880.82/297.08 We are left with following problem, upon which TcT provides the 880.82/297.08 certificate MAYBE. 880.82/297.08 880.82/297.08 Strict DPs: 880.82/297.08 { U101^#(tt(), M, N) -> 880.82/297.08 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.08 , U102^#(tt(), M, N) -> 880.82/297.08 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.08 , U103^#(tt(), M, N) -> 880.82/297.08 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.08 , activate^#(X) -> c_7(X) 880.82/297.08 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.08 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.08 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.08 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.08 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.08 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.08 , s^#(X) -> c_50(X) 880.82/297.08 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.08 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.08 , U104^#(tt(), M, N) -> 880.82/297.08 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.08 , isNat^#(n__plus(V1, V2)) -> 880.82/297.08 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , isNat^#(n__x(V1, V2)) -> 880.82/297.08 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , U11^#(tt(), V1, V2) -> 880.82/297.08 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , U31^#(tt(), V1, V2) -> 880.82/297.08 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , U81^#(tt(), M, N) -> 880.82/297.08 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.08 , U71^#(tt(), N) -> 880.82/297.08 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.08 , U12^#(tt(), V1, V2) -> 880.82/297.08 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , U32^#(tt(), V1, V2) -> 880.82/297.08 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.08 , U82^#(tt(), M, N) -> 880.82/297.08 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.08 , U83^#(tt(), M, N) -> 880.82/297.08 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.08 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 880.82/297.08 Strict Trs: 880.82/297.08 { U101(tt(), M, N) -> 880.82/297.08 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.08 , U102(tt(), M, N) -> 880.82/297.08 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.08 , isNatKind(n__0()) -> tt() 880.82/297.08 , isNatKind(n__plus(V1, V2)) -> 880.82/297.08 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.08 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.08 , isNatKind(n__x(V1, V2)) -> 880.82/297.08 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.08 , activate(X) -> X 880.82/297.08 , activate(n__0()) -> 0() 880.82/297.08 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.08 , activate(n__s(X)) -> s(X) 880.82/297.08 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.08 , U103(tt(), M, N) -> 880.82/297.08 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.08 , isNat(n__0()) -> tt() 880.82/297.08 , isNat(n__plus(V1, V2)) -> 880.82/297.08 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.08 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.08 , isNat(n__x(V1, V2)) -> 880.82/297.08 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.08 , U104(tt(), M, N) -> 880.82/297.08 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.08 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.08 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.08 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.08 , x(X1, X2) -> n__x(X1, X2) 880.82/297.08 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.08 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.08 , U11(tt(), V1, V2) -> 880.82/297.08 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.08 , U12(tt(), V1, V2) -> 880.82/297.08 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.08 , U13(tt(), V1, V2) -> 880.82/297.08 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.08 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.08 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.08 , U16(tt()) -> tt() 880.82/297.08 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.08 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.08 , U23(tt()) -> tt() 880.82/297.08 , U31(tt(), V1, V2) -> 880.82/297.08 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.08 , U32(tt(), V1, V2) -> 880.82/297.08 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.08 , U33(tt(), V1, V2) -> 880.82/297.08 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.08 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.08 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.08 , U36(tt()) -> tt() 880.82/297.08 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.08 , U42(tt()) -> tt() 880.82/297.08 , U51(tt()) -> tt() 880.82/297.08 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.08 , U62(tt()) -> tt() 880.82/297.08 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.08 , U72(tt(), N) -> activate(N) 880.82/297.08 , U81(tt(), M, N) -> 880.82/297.08 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.08 , U82(tt(), M, N) -> 880.82/297.08 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.08 , U83(tt(), M, N) -> 880.82/297.08 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.08 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.08 , s(X) -> n__s(X) 880.82/297.08 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.08 , U92(tt()) -> 0() 880.82/297.08 , 0() -> n__0() } 880.82/297.08 Weak DPs: 880.82/297.08 { isNatKind^#(n__0()) -> c_3() 880.82/297.08 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.08 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.08 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.08 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.08 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.08 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.08 , U51^#(tt()) -> c_41() 880.82/297.08 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.08 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.08 , 0^#() -> c_53() 880.82/297.08 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.08 , isNat^#(n__0()) -> c_13() 880.82/297.08 , isNat^#(n__s(V1)) -> 880.82/297.08 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.08 , U21^#(tt(), V1) -> 880.82/297.08 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.08 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.08 , U13^#(tt(), V1, V2) -> 880.82/297.08 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , U14^#(tt(), V1, V2) -> 880.82/297.08 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.08 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.08 , U16^#(tt()) -> c_29() 880.82/297.08 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.08 , U23^#(tt()) -> c_32() 880.82/297.08 , U33^#(tt(), V1, V2) -> 880.82/297.08 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , U34^#(tt(), V1, V2) -> 880.82/297.08 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.08 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.08 , U36^#(tt()) -> c_38() 880.82/297.08 , U42^#(tt()) -> c_40() 880.82/297.08 , U62^#(tt()) -> c_43() 880.82/297.08 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.08 Obligation: 880.82/297.08 runtime complexity 880.82/297.08 Answer: 880.82/297.08 MAYBE 880.82/297.08 880.82/297.08 We estimate the number of application of {21,22} by applications of 880.82/297.08 Pre({21,22}) = {4,8,11,12,17,18}. Here rules are labeled as 880.82/297.08 follows: 880.82/297.08 880.82/297.08 DPs: 880.82/297.08 { 1: U101^#(tt(), M, N) -> 880.82/297.08 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.08 , 2: U102^#(tt(), M, N) -> 880.82/297.08 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.08 , 3: U103^#(tt(), M, N) -> 880.82/297.08 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.08 , 4: activate^#(X) -> c_7(X) 880.82/297.08 , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.08 , 6: activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.08 , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.08 , 8: plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.08 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.08 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.08 , 11: s^#(X) -> c_50(X) 880.82/297.08 , 12: x^#(X1, X2) -> c_21(X1, X2) 880.82/297.08 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.08 , 14: U104^#(tt(), M, N) -> 880.82/297.08 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.08 , 15: isNat^#(n__plus(V1, V2)) -> 880.82/297.08 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 16: isNat^#(n__x(V1, V2)) -> 880.82/297.08 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 17: U11^#(tt(), V1, V2) -> 880.82/297.08 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 18: U31^#(tt(), V1, V2) -> 880.82/297.08 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , 19: U81^#(tt(), M, N) -> 880.82/297.08 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.08 , 20: U71^#(tt(), N) -> 880.82/297.08 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.08 , 21: U12^#(tt(), V1, V2) -> 880.82/297.08 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 22: U32^#(tt(), V1, V2) -> 880.82/297.08 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 23: U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.08 , 24: U82^#(tt(), M, N) -> 880.82/297.08 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.08 , 25: U83^#(tt(), M, N) -> 880.82/297.08 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.08 , 26: U84^#(tt(), M, N) -> 880.82/297.08 c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.08 , 27: isNatKind^#(n__0()) -> c_3() 880.82/297.08 , 28: isNatKind^#(n__plus(V1, V2)) -> 880.82/297.08 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.08 , 29: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.08 , 30: isNatKind^#(n__x(V1, V2)) -> 880.82/297.08 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.08 , 31: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.08 , 32: U51^#(tt()) -> c_41() 880.82/297.08 , 33: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.08 , 34: activate^#(n__0()) -> c_8(0^#()) 880.82/297.08 , 35: 0^#() -> c_53() 880.82/297.08 , 36: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.08 , 37: isNat^#(n__0()) -> c_13() 880.82/297.08 , 38: isNat^#(n__s(V1)) -> 880.82/297.08 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.08 , 39: U21^#(tt(), V1) -> 880.82/297.08 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.08 , 40: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.08 , 41: U13^#(tt(), V1, V2) -> 880.82/297.08 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 42: U14^#(tt(), V1, V2) -> 880.82/297.08 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.08 , 43: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.08 , 44: U16^#(tt()) -> c_29() 880.82/297.08 , 45: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.08 , 46: U23^#(tt()) -> c_32() 880.82/297.08 , 47: U33^#(tt(), V1, V2) -> 880.82/297.08 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.08 , 48: U34^#(tt(), V1, V2) -> 880.82/297.08 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.08 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.08 , 50: U36^#(tt()) -> c_38() 880.82/297.08 , 51: U42^#(tt()) -> c_40() 880.82/297.08 , 52: U62^#(tt()) -> c_43() 880.82/297.08 , 53: U92^#(tt()) -> c_52(0^#()) } 880.82/297.08 880.82/297.08 We are left with following problem, upon which TcT provides the 880.82/297.08 certificate MAYBE. 880.82/297.08 880.82/297.08 Strict DPs: 880.82/297.08 { U101^#(tt(), M, N) -> 880.82/297.08 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.08 , U102^#(tt(), M, N) -> 880.82/297.08 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.08 , U103^#(tt(), M, N) -> 880.82/297.08 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.08 , activate^#(X) -> c_7(X) 880.82/297.08 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.08 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.08 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.08 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.08 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.08 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.08 , s^#(X) -> c_50(X) 880.82/297.08 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.08 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.08 , U104^#(tt(), M, N) -> 880.82/297.08 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.08 , isNat^#(n__plus(V1, V2)) -> 880.82/297.08 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , isNat^#(n__x(V1, V2)) -> 880.82/297.08 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , U11^#(tt(), V1, V2) -> 880.82/297.08 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , U31^#(tt(), V1, V2) -> 880.82/297.08 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.08 , U81^#(tt(), M, N) -> 880.82/297.08 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.08 , U71^#(tt(), N) -> 880.82/297.08 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.08 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.08 , U82^#(tt(), M, N) -> 880.82/297.08 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.08 , U83^#(tt(), M, N) -> 880.82/297.08 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.08 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 880.82/297.08 Strict Trs: 880.82/297.08 { U101(tt(), M, N) -> 880.82/297.08 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.08 , U102(tt(), M, N) -> 880.82/297.08 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.08 , isNatKind(n__0()) -> tt() 880.82/297.08 , isNatKind(n__plus(V1, V2)) -> 880.82/297.08 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.08 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.08 , isNatKind(n__x(V1, V2)) -> 880.82/297.08 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.09 , activate(X) -> X 880.82/297.09 , activate(n__0()) -> 0() 880.82/297.09 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.09 , activate(n__s(X)) -> s(X) 880.82/297.09 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.09 , U103(tt(), M, N) -> 880.82/297.09 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.09 , isNat(n__0()) -> tt() 880.82/297.09 , isNat(n__plus(V1, V2)) -> 880.82/297.09 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.09 , isNat(n__x(V1, V2)) -> 880.82/297.09 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , U104(tt(), M, N) -> 880.82/297.09 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.09 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.09 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.09 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.09 , x(X1, X2) -> n__x(X1, X2) 880.82/297.09 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.09 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.09 , U11(tt(), V1, V2) -> 880.82/297.09 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , U12(tt(), V1, V2) -> 880.82/297.09 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U13(tt(), V1, V2) -> 880.82/297.09 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.09 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.09 , U16(tt()) -> tt() 880.82/297.09 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.09 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.09 , U23(tt()) -> tt() 880.82/297.09 , U31(tt(), V1, V2) -> 880.82/297.09 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , U32(tt(), V1, V2) -> 880.82/297.09 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U33(tt(), V1, V2) -> 880.82/297.09 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.09 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.09 , U36(tt()) -> tt() 880.82/297.09 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.09 , U42(tt()) -> tt() 880.82/297.09 , U51(tt()) -> tt() 880.82/297.09 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.09 , U62(tt()) -> tt() 880.82/297.09 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.09 , U72(tt(), N) -> activate(N) 880.82/297.09 , U81(tt(), M, N) -> 880.82/297.09 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.09 , U82(tt(), M, N) -> 880.82/297.09 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.09 , U83(tt(), M, N) -> 880.82/297.09 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.09 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.09 , s(X) -> n__s(X) 880.82/297.09 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.09 , U92(tt()) -> 0() 880.82/297.09 , 0() -> n__0() } 880.82/297.09 Weak DPs: 880.82/297.09 { isNatKind^#(n__0()) -> c_3() 880.82/297.09 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.09 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.09 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.09 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.09 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.09 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.09 , U51^#(tt()) -> c_41() 880.82/297.09 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.09 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.09 , 0^#() -> c_53() 880.82/297.09 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.09 , isNat^#(n__0()) -> c_13() 880.82/297.09 , isNat^#(n__s(V1)) -> 880.82/297.09 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.09 , U21^#(tt(), V1) -> 880.82/297.09 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.09 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.09 , U12^#(tt(), V1, V2) -> 880.82/297.09 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U13^#(tt(), V1, V2) -> 880.82/297.09 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U14^#(tt(), V1, V2) -> 880.82/297.09 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.09 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.09 , U16^#(tt()) -> c_29() 880.82/297.09 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.09 , U23^#(tt()) -> c_32() 880.82/297.09 , U32^#(tt(), V1, V2) -> 880.82/297.09 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U33^#(tt(), V1, V2) -> 880.82/297.09 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U34^#(tt(), V1, V2) -> 880.82/297.09 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.09 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.09 , U36^#(tt()) -> c_38() 880.82/297.09 , U42^#(tt()) -> c_40() 880.82/297.09 , U62^#(tt()) -> c_43() 880.82/297.09 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.09 Obligation: 880.82/297.09 runtime complexity 880.82/297.09 Answer: 880.82/297.09 MAYBE 880.82/297.09 880.82/297.09 We estimate the number of application of {17,18} by applications of 880.82/297.09 Pre({17,18}) = {4,8,11,12,15,16}. Here rules are labeled as 880.82/297.09 follows: 880.82/297.09 880.82/297.09 DPs: 880.82/297.09 { 1: U101^#(tt(), M, N) -> 880.82/297.09 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.09 , 2: U102^#(tt(), M, N) -> 880.82/297.09 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.09 , 3: U103^#(tt(), M, N) -> 880.82/297.09 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.09 , 4: activate^#(X) -> c_7(X) 880.82/297.09 , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.09 , 6: activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.09 , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.09 , 8: plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.09 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.09 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.09 , 11: s^#(X) -> c_50(X) 880.82/297.09 , 12: x^#(X1, X2) -> c_21(X1, X2) 880.82/297.09 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.09 , 14: U104^#(tt(), M, N) -> 880.82/297.09 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.09 , 15: isNat^#(n__plus(V1, V2)) -> 880.82/297.09 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , 16: isNat^#(n__x(V1, V2)) -> 880.82/297.09 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , 17: U11^#(tt(), V1, V2) -> 880.82/297.09 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , 18: U31^#(tt(), V1, V2) -> 880.82/297.09 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , 19: U81^#(tt(), M, N) -> 880.82/297.09 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.09 , 20: U71^#(tt(), N) -> 880.82/297.09 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.09 , 21: U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.09 , 22: U82^#(tt(), M, N) -> 880.82/297.09 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.09 , 23: U83^#(tt(), M, N) -> 880.82/297.09 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.09 , 24: U84^#(tt(), M, N) -> 880.82/297.09 c_49(s^#(plus(activate(N), activate(M)))) 880.82/297.09 , 25: isNatKind^#(n__0()) -> c_3() 880.82/297.09 , 26: isNatKind^#(n__plus(V1, V2)) -> 880.82/297.09 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.09 , 27: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.09 , 28: isNatKind^#(n__x(V1, V2)) -> 880.82/297.09 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.09 , 29: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.09 , 30: U51^#(tt()) -> c_41() 880.82/297.09 , 31: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.09 , 32: activate^#(n__0()) -> c_8(0^#()) 880.82/297.09 , 33: 0^#() -> c_53() 880.82/297.09 , 34: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.09 , 35: isNat^#(n__0()) -> c_13() 880.82/297.09 , 36: isNat^#(n__s(V1)) -> 880.82/297.09 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.09 , 37: U21^#(tt(), V1) -> 880.82/297.09 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.09 , 38: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.09 , 39: U12^#(tt(), V1, V2) -> 880.82/297.09 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , 40: U13^#(tt(), V1, V2) -> 880.82/297.09 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , 41: U14^#(tt(), V1, V2) -> 880.82/297.09 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.09 , 42: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.09 , 43: U16^#(tt()) -> c_29() 880.82/297.09 , 44: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.09 , 45: U23^#(tt()) -> c_32() 880.82/297.09 , 46: U32^#(tt(), V1, V2) -> 880.82/297.09 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , 47: U33^#(tt(), V1, V2) -> 880.82/297.09 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , 48: U34^#(tt(), V1, V2) -> 880.82/297.09 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.09 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.09 , 50: U36^#(tt()) -> c_38() 880.82/297.09 , 51: U42^#(tt()) -> c_40() 880.82/297.09 , 52: U62^#(tt()) -> c_43() 880.82/297.09 , 53: U92^#(tt()) -> c_52(0^#()) } 880.82/297.09 880.82/297.09 We are left with following problem, upon which TcT provides the 880.82/297.09 certificate MAYBE. 880.82/297.09 880.82/297.09 Strict DPs: 880.82/297.09 { U101^#(tt(), M, N) -> 880.82/297.09 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.09 , U102^#(tt(), M, N) -> 880.82/297.09 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.09 , U103^#(tt(), M, N) -> 880.82/297.09 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.09 , activate^#(X) -> c_7(X) 880.82/297.09 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.09 , activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.09 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.09 , plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.09 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.09 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.09 , s^#(X) -> c_50(X) 880.82/297.09 , x^#(X1, X2) -> c_21(X1, X2) 880.82/297.09 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.09 , U104^#(tt(), M, N) -> 880.82/297.09 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 880.82/297.09 , isNat^#(n__plus(V1, V2)) -> 880.82/297.09 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , isNat^#(n__x(V1, V2)) -> 880.82/297.09 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , U81^#(tt(), M, N) -> 880.82/297.09 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.09 , U71^#(tt(), N) -> 880.82/297.09 c_44(U72^#(isNatKind(activate(N)), activate(N))) 880.82/297.09 , U72^#(tt(), N) -> c_45(activate^#(N)) 880.82/297.09 , U82^#(tt(), M, N) -> 880.82/297.09 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.09 , U83^#(tt(), M, N) -> 880.82/297.09 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.09 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 880.82/297.09 Strict Trs: 880.82/297.09 { U101(tt(), M, N) -> 880.82/297.09 U102(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.09 , U102(tt(), M, N) -> 880.82/297.09 U103(isNat(activate(N)), activate(M), activate(N)) 880.82/297.09 , isNatKind(n__0()) -> tt() 880.82/297.09 , isNatKind(n__plus(V1, V2)) -> 880.82/297.09 U41(isNatKind(activate(V1)), activate(V2)) 880.82/297.09 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 880.82/297.09 , isNatKind(n__x(V1, V2)) -> 880.82/297.09 U61(isNatKind(activate(V1)), activate(V2)) 880.82/297.09 , activate(X) -> X 880.82/297.09 , activate(n__0()) -> 0() 880.82/297.09 , activate(n__plus(X1, X2)) -> plus(X1, X2) 880.82/297.09 , activate(n__s(X)) -> s(X) 880.82/297.09 , activate(n__x(X1, X2)) -> x(X1, X2) 880.82/297.09 , U103(tt(), M, N) -> 880.82/297.09 U104(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.09 , isNat(n__0()) -> tt() 880.82/297.09 , isNat(n__plus(V1, V2)) -> 880.82/297.09 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 880.82/297.09 , isNat(n__x(V1, V2)) -> 880.82/297.09 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , U104(tt(), M, N) -> 880.82/297.09 plus(x(activate(N), activate(M)), activate(N)) 880.82/297.09 , plus(X1, X2) -> n__plus(X1, X2) 880.82/297.09 , plus(N, s(M)) -> U81(isNat(M), M, N) 880.82/297.09 , plus(N, 0()) -> U71(isNat(N), N) 880.82/297.09 , x(X1, X2) -> n__x(X1, X2) 880.82/297.09 , x(N, s(M)) -> U101(isNat(M), M, N) 880.82/297.09 , x(N, 0()) -> U91(isNat(N), N) 880.82/297.09 , U11(tt(), V1, V2) -> 880.82/297.09 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , U12(tt(), V1, V2) -> 880.82/297.09 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U13(tt(), V1, V2) -> 880.82/297.09 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 880.82/297.09 , U15(tt(), V2) -> U16(isNat(activate(V2))) 880.82/297.09 , U16(tt()) -> tt() 880.82/297.09 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 880.82/297.09 , U22(tt(), V1) -> U23(isNat(activate(V1))) 880.82/297.09 , U23(tt()) -> tt() 880.82/297.09 , U31(tt(), V1, V2) -> 880.82/297.09 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 880.82/297.09 , U32(tt(), V1, V2) -> 880.82/297.09 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U33(tt(), V1, V2) -> 880.82/297.09 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 880.82/297.09 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 880.82/297.09 , U35(tt(), V2) -> U36(isNat(activate(V2))) 880.82/297.09 , U36(tt()) -> tt() 880.82/297.09 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 880.82/297.09 , U42(tt()) -> tt() 880.82/297.09 , U51(tt()) -> tt() 880.82/297.09 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 880.82/297.09 , U62(tt()) -> tt() 880.82/297.09 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 880.82/297.09 , U72(tt(), N) -> activate(N) 880.82/297.09 , U81(tt(), M, N) -> 880.82/297.09 U82(isNatKind(activate(M)), activate(M), activate(N)) 880.82/297.09 , U82(tt(), M, N) -> 880.82/297.09 U83(isNat(activate(N)), activate(M), activate(N)) 880.82/297.09 , U83(tt(), M, N) -> 880.82/297.09 U84(isNatKind(activate(N)), activate(M), activate(N)) 880.82/297.09 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 880.82/297.09 , s(X) -> n__s(X) 880.82/297.09 , U91(tt(), N) -> U92(isNatKind(activate(N))) 880.82/297.09 , U92(tt()) -> 0() 880.82/297.09 , 0() -> n__0() } 880.82/297.09 Weak DPs: 880.82/297.09 { isNatKind^#(n__0()) -> c_3() 880.82/297.09 , isNatKind^#(n__plus(V1, V2)) -> 880.82/297.09 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.09 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 880.82/297.09 , isNatKind^#(n__x(V1, V2)) -> 880.82/297.09 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 880.82/297.09 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 880.82/297.09 , U51^#(tt()) -> c_41() 880.82/297.09 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 880.82/297.09 , activate^#(n__0()) -> c_8(0^#()) 880.82/297.09 , 0^#() -> c_53() 880.82/297.09 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 880.82/297.09 , isNat^#(n__0()) -> c_13() 880.82/297.09 , isNat^#(n__s(V1)) -> 880.82/297.09 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.09 , U11^#(tt(), V1, V2) -> 880.82/297.09 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , U21^#(tt(), V1) -> 880.82/297.09 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 880.82/297.09 , U31^#(tt(), V1, V2) -> 880.82/297.09 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 880.82/297.09 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 880.82/297.09 , U12^#(tt(), V1, V2) -> 880.82/297.09 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U13^#(tt(), V1, V2) -> 880.82/297.09 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U14^#(tt(), V1, V2) -> 880.82/297.09 c_27(U15^#(isNat(activate(V1)), activate(V2))) 880.82/297.09 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 880.82/297.09 , U16^#(tt()) -> c_29() 880.82/297.09 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 880.82/297.09 , U23^#(tt()) -> c_32() 880.82/297.09 , U32^#(tt(), V1, V2) -> 880.82/297.09 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U33^#(tt(), V1, V2) -> 880.82/297.09 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 880.82/297.09 , U34^#(tt(), V1, V2) -> 880.82/297.09 c_36(U35^#(isNat(activate(V1)), activate(V2))) 880.82/297.09 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 880.82/297.09 , U36^#(tt()) -> c_38() 880.82/297.09 , U42^#(tt()) -> c_40() 880.82/297.09 , U62^#(tt()) -> c_43() 880.82/297.09 , U92^#(tt()) -> c_52(0^#()) } 880.82/297.09 Obligation: 880.82/297.09 runtime complexity 880.82/297.09 Answer: 880.82/297.09 MAYBE 880.82/297.09 880.82/297.09 We estimate the number of application of {15,16} by applications of 880.82/297.09 Pre({15,16}) = {4,8,11,12}. Here rules are labeled as follows: 880.82/297.09 880.82/297.09 DPs: 880.82/297.09 { 1: U101^#(tt(), M, N) -> 880.82/297.09 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 880.82/297.09 , 2: U102^#(tt(), M, N) -> 880.82/297.09 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 880.82/297.09 , 3: U103^#(tt(), M, N) -> 880.82/297.09 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 880.82/297.09 , 4: activate^#(X) -> c_7(X) 880.82/297.09 , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 880.82/297.09 , 6: activate^#(n__s(X)) -> c_10(s^#(X)) 880.82/297.09 , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 880.82/297.09 , 8: plus^#(X1, X2) -> c_18(X1, X2) 880.82/297.09 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 880.82/297.09 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 880.82/297.09 , 11: s^#(X) -> c_50(X) 880.82/297.09 , 12: x^#(X1, X2) -> c_21(X1, X2) 880.82/297.09 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 880.82/297.09 , 14: U104^#(tt(), M, N) -> 881.03/297.10 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 881.03/297.10 , 15: isNat^#(n__plus(V1, V2)) -> 881.03/297.10 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , 16: isNat^#(n__x(V1, V2)) -> 881.03/297.10 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , 17: U81^#(tt(), M, N) -> 881.03/297.10 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 881.03/297.10 , 18: U71^#(tt(), N) -> 881.03/297.10 c_44(U72^#(isNatKind(activate(N)), activate(N))) 881.03/297.10 , 19: U72^#(tt(), N) -> c_45(activate^#(N)) 881.03/297.10 , 20: U82^#(tt(), M, N) -> 881.03/297.10 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 881.03/297.10 , 21: U83^#(tt(), M, N) -> 881.03/297.10 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 881.03/297.10 , 22: U84^#(tt(), M, N) -> 881.03/297.10 c_49(s^#(plus(activate(N), activate(M)))) 881.03/297.10 , 23: isNatKind^#(n__0()) -> c_3() 881.03/297.10 , 24: isNatKind^#(n__plus(V1, V2)) -> 881.03/297.10 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 881.03/297.10 , 25: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 881.03/297.10 , 26: isNatKind^#(n__x(V1, V2)) -> 881.03/297.10 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 881.03/297.10 , 27: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 881.03/297.10 , 28: U51^#(tt()) -> c_41() 881.03/297.10 , 29: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 881.03/297.10 , 30: activate^#(n__0()) -> c_8(0^#()) 881.03/297.10 , 31: 0^#() -> c_53() 881.03/297.10 , 32: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 881.03/297.10 , 33: isNat^#(n__0()) -> c_13() 881.03/297.10 , 34: isNat^#(n__s(V1)) -> 881.03/297.10 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 881.03/297.10 , 35: U11^#(tt(), V1, V2) -> 881.03/297.10 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , 36: U21^#(tt(), V1) -> 881.03/297.10 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 881.03/297.10 , 37: U31^#(tt(), V1, V2) -> 881.03/297.10 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , 38: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 881.03/297.10 , 39: U12^#(tt(), V1, V2) -> 881.03/297.10 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , 40: U13^#(tt(), V1, V2) -> 881.03/297.10 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , 41: U14^#(tt(), V1, V2) -> 881.03/297.10 c_27(U15^#(isNat(activate(V1)), activate(V2))) 881.03/297.10 , 42: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 881.03/297.10 , 43: U16^#(tt()) -> c_29() 881.03/297.10 , 44: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 881.03/297.10 , 45: U23^#(tt()) -> c_32() 881.03/297.10 , 46: U32^#(tt(), V1, V2) -> 881.03/297.10 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , 47: U33^#(tt(), V1, V2) -> 881.03/297.10 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , 48: U34^#(tt(), V1, V2) -> 881.03/297.10 c_36(U35^#(isNat(activate(V1)), activate(V2))) 881.03/297.10 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 881.03/297.10 , 50: U36^#(tt()) -> c_38() 881.03/297.10 , 51: U42^#(tt()) -> c_40() 881.03/297.10 , 52: U62^#(tt()) -> c_43() 881.03/297.10 , 53: U92^#(tt()) -> c_52(0^#()) } 881.03/297.10 881.03/297.10 We are left with following problem, upon which TcT provides the 881.03/297.10 certificate MAYBE. 881.03/297.10 881.03/297.10 Strict DPs: 881.03/297.10 { U101^#(tt(), M, N) -> 881.03/297.10 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 881.03/297.10 , U102^#(tt(), M, N) -> 881.03/297.10 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 881.03/297.10 , U103^#(tt(), M, N) -> 881.03/297.10 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 881.03/297.10 , activate^#(X) -> c_7(X) 881.03/297.10 , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2)) 881.03/297.10 , activate^#(n__s(X)) -> c_10(s^#(X)) 881.03/297.10 , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2)) 881.03/297.10 , plus^#(X1, X2) -> c_18(X1, X2) 881.03/297.10 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 881.03/297.10 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 881.03/297.10 , s^#(X) -> c_50(X) 881.03/297.10 , x^#(X1, X2) -> c_21(X1, X2) 881.03/297.10 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 881.03/297.10 , U104^#(tt(), M, N) -> 881.03/297.10 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 881.03/297.10 , U81^#(tt(), M, N) -> 881.03/297.10 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 881.03/297.10 , U71^#(tt(), N) -> 881.03/297.10 c_44(U72^#(isNatKind(activate(N)), activate(N))) 881.03/297.10 , U72^#(tt(), N) -> c_45(activate^#(N)) 881.03/297.10 , U82^#(tt(), M, N) -> 881.03/297.10 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 881.03/297.10 , U83^#(tt(), M, N) -> 881.03/297.10 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 881.03/297.10 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 881.03/297.10 Strict Trs: 881.03/297.10 { U101(tt(), M, N) -> 881.03/297.10 U102(isNatKind(activate(M)), activate(M), activate(N)) 881.03/297.10 , U102(tt(), M, N) -> 881.03/297.10 U103(isNat(activate(N)), activate(M), activate(N)) 881.03/297.10 , isNatKind(n__0()) -> tt() 881.03/297.10 , isNatKind(n__plus(V1, V2)) -> 881.03/297.10 U41(isNatKind(activate(V1)), activate(V2)) 881.03/297.10 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 881.03/297.10 , isNatKind(n__x(V1, V2)) -> 881.03/297.10 U61(isNatKind(activate(V1)), activate(V2)) 881.03/297.10 , activate(X) -> X 881.03/297.10 , activate(n__0()) -> 0() 881.03/297.10 , activate(n__plus(X1, X2)) -> plus(X1, X2) 881.03/297.10 , activate(n__s(X)) -> s(X) 881.03/297.10 , activate(n__x(X1, X2)) -> x(X1, X2) 881.03/297.10 , U103(tt(), M, N) -> 881.03/297.10 U104(isNatKind(activate(N)), activate(M), activate(N)) 881.03/297.10 , isNat(n__0()) -> tt() 881.03/297.10 , isNat(n__plus(V1, V2)) -> 881.03/297.10 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 881.03/297.10 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 881.03/297.10 , isNat(n__x(V1, V2)) -> 881.03/297.10 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 881.03/297.10 , U104(tt(), M, N) -> 881.03/297.10 plus(x(activate(N), activate(M)), activate(N)) 881.03/297.10 , plus(X1, X2) -> n__plus(X1, X2) 881.03/297.10 , plus(N, s(M)) -> U81(isNat(M), M, N) 881.03/297.10 , plus(N, 0()) -> U71(isNat(N), N) 881.03/297.10 , x(X1, X2) -> n__x(X1, X2) 881.03/297.10 , x(N, s(M)) -> U101(isNat(M), M, N) 881.03/297.10 , x(N, 0()) -> U91(isNat(N), N) 881.03/297.10 , U11(tt(), V1, V2) -> 881.03/297.10 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 881.03/297.10 , U12(tt(), V1, V2) -> 881.03/297.10 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 881.03/297.10 , U13(tt(), V1, V2) -> 881.03/297.10 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 881.03/297.10 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 881.03/297.10 , U15(tt(), V2) -> U16(isNat(activate(V2))) 881.03/297.10 , U16(tt()) -> tt() 881.03/297.10 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 881.03/297.10 , U22(tt(), V1) -> U23(isNat(activate(V1))) 881.03/297.10 , U23(tt()) -> tt() 881.03/297.10 , U31(tt(), V1, V2) -> 881.03/297.10 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 881.03/297.10 , U32(tt(), V1, V2) -> 881.03/297.10 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 881.03/297.10 , U33(tt(), V1, V2) -> 881.03/297.10 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 881.03/297.10 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 881.03/297.10 , U35(tt(), V2) -> U36(isNat(activate(V2))) 881.03/297.10 , U36(tt()) -> tt() 881.03/297.10 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 881.03/297.10 , U42(tt()) -> tt() 881.03/297.10 , U51(tt()) -> tt() 881.03/297.10 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 881.03/297.10 , U62(tt()) -> tt() 881.03/297.10 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 881.03/297.10 , U72(tt(), N) -> activate(N) 881.03/297.10 , U81(tt(), M, N) -> 881.03/297.10 U82(isNatKind(activate(M)), activate(M), activate(N)) 881.03/297.10 , U82(tt(), M, N) -> 881.03/297.10 U83(isNat(activate(N)), activate(M), activate(N)) 881.03/297.10 , U83(tt(), M, N) -> 881.03/297.10 U84(isNatKind(activate(N)), activate(M), activate(N)) 881.03/297.10 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 881.03/297.10 , s(X) -> n__s(X) 881.03/297.10 , U91(tt(), N) -> U92(isNatKind(activate(N))) 881.03/297.10 , U92(tt()) -> 0() 881.03/297.10 , 0() -> n__0() } 881.03/297.10 Weak DPs: 881.03/297.10 { isNatKind^#(n__0()) -> c_3() 881.03/297.10 , isNatKind^#(n__plus(V1, V2)) -> 881.03/297.10 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 881.03/297.10 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 881.03/297.10 , isNatKind^#(n__x(V1, V2)) -> 881.03/297.10 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 881.03/297.10 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 881.03/297.10 , U51^#(tt()) -> c_41() 881.03/297.10 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 881.03/297.10 , activate^#(n__0()) -> c_8(0^#()) 881.03/297.10 , 0^#() -> c_53() 881.03/297.10 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 881.03/297.10 , isNat^#(n__0()) -> c_13() 881.03/297.10 , isNat^#(n__plus(V1, V2)) -> 881.03/297.10 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , isNat^#(n__s(V1)) -> 881.03/297.10 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 881.03/297.10 , isNat^#(n__x(V1, V2)) -> 881.03/297.10 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , U11^#(tt(), V1, V2) -> 881.03/297.10 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , U21^#(tt(), V1) -> 881.03/297.10 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 881.03/297.10 , U31^#(tt(), V1, V2) -> 881.03/297.10 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 881.03/297.10 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 881.03/297.10 , U12^#(tt(), V1, V2) -> 881.03/297.10 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , U13^#(tt(), V1, V2) -> 881.03/297.10 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , U14^#(tt(), V1, V2) -> 881.03/297.10 c_27(U15^#(isNat(activate(V1)), activate(V2))) 881.03/297.10 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 881.03/297.10 , U16^#(tt()) -> c_29() 881.03/297.10 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 881.03/297.10 , U23^#(tt()) -> c_32() 881.03/297.10 , U32^#(tt(), V1, V2) -> 881.03/297.10 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , U33^#(tt(), V1, V2) -> 881.03/297.10 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 881.03/297.10 , U34^#(tt(), V1, V2) -> 881.03/297.10 c_36(U35^#(isNat(activate(V1)), activate(V2))) 881.03/297.10 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 881.03/297.10 , U36^#(tt()) -> c_38() 881.03/297.10 , U42^#(tt()) -> c_40() 881.03/297.10 , U62^#(tt()) -> c_43() 881.03/297.10 , U92^#(tt()) -> c_52(0^#()) } 881.03/297.10 Obligation: 881.03/297.10 runtime complexity 881.03/297.10 Answer: 881.03/297.10 MAYBE 881.03/297.10 881.03/297.10 Empty strict component of the problem is NOT empty. 881.03/297.10 881.03/297.10 881.03/297.10 Arrrr.. 881.13/297.20 EOF