MAYBE 878.72/297.15 MAYBE 878.72/297.15 878.72/297.15 We are left with following problem, upon which TcT provides the 878.72/297.15 certificate MAYBE. 878.72/297.15 878.72/297.15 Strict Trs: 878.72/297.15 { U101(tt(), M, N) -> 878.72/297.15 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.15 , U102(tt(), M, N) -> 878.72/297.15 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.15 , isNatKind(n__0()) -> tt() 878.72/297.15 , isNatKind(n__plus(V1, V2)) -> 878.72/297.15 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.15 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.15 , isNatKind(n__x(V1, V2)) -> 878.72/297.15 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.15 , activate(X) -> X 878.72/297.15 , activate(n__0()) -> 0() 878.72/297.15 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.15 , activate(n__s(X)) -> s(activate(X)) 878.72/297.15 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.15 , U103(tt(), M, N) -> 878.72/297.15 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.15 , isNat(n__0()) -> tt() 878.72/297.15 , isNat(n__plus(V1, V2)) -> 878.72/297.15 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.15 , isNat(n__x(V1, V2)) -> 878.72/297.15 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , U104(tt(), M, N) -> 878.72/297.15 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.15 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.15 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.15 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.15 , x(X1, X2) -> n__x(X1, X2) 878.72/297.15 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.15 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.15 , U11(tt(), V1, V2) -> 878.72/297.15 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , U12(tt(), V1, V2) -> 878.72/297.15 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U13(tt(), V1, V2) -> 878.72/297.15 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.15 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.15 , U16(tt()) -> tt() 878.72/297.15 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.15 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.15 , U23(tt()) -> tt() 878.72/297.15 , U31(tt(), V1, V2) -> 878.72/297.15 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , U32(tt(), V1, V2) -> 878.72/297.15 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U33(tt(), V1, V2) -> 878.72/297.15 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.15 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.15 , U36(tt()) -> tt() 878.72/297.15 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.15 , U42(tt()) -> tt() 878.72/297.15 , U51(tt()) -> tt() 878.72/297.15 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.15 , U62(tt()) -> tt() 878.72/297.15 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.15 , U72(tt(), N) -> activate(N) 878.72/297.15 , U81(tt(), M, N) -> 878.72/297.15 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.15 , U82(tt(), M, N) -> 878.72/297.15 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.15 , U83(tt(), M, N) -> 878.72/297.15 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.15 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.15 , s(X) -> n__s(X) 878.72/297.15 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.15 , U92(tt()) -> 0() 878.72/297.15 , 0() -> n__0() } 878.72/297.15 Obligation: 878.72/297.15 runtime complexity 878.72/297.15 Answer: 878.72/297.15 MAYBE 878.72/297.15 878.72/297.15 None of the processors succeeded. 878.72/297.15 878.72/297.15 Details of failed attempt(s): 878.72/297.15 ----------------------------- 878.72/297.15 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 878.72/297.15 following reason: 878.72/297.15 878.72/297.15 Computation stopped due to timeout after 297.0 seconds. 878.72/297.15 878.72/297.15 2) 'Best' failed due to the following reason: 878.72/297.15 878.72/297.15 None of the processors succeeded. 878.72/297.15 878.72/297.15 Details of failed attempt(s): 878.72/297.15 ----------------------------- 878.72/297.15 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 878.72/297.15 seconds)' failed due to the following reason: 878.72/297.15 878.72/297.15 Computation stopped due to timeout after 148.0 seconds. 878.72/297.15 878.72/297.15 2) 'Best' failed due to the following reason: 878.72/297.15 878.72/297.15 None of the processors succeeded. 878.72/297.15 878.72/297.15 Details of failed attempt(s): 878.72/297.15 ----------------------------- 878.72/297.15 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 878.72/297.15 following reason: 878.72/297.15 878.72/297.15 The processor is inapplicable, reason: 878.72/297.15 Processor only applicable for innermost runtime complexity analysis 878.72/297.15 878.72/297.15 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 878.72/297.15 to the following reason: 878.72/297.15 878.72/297.15 The processor is inapplicable, reason: 878.72/297.15 Processor only applicable for innermost runtime complexity analysis 878.72/297.15 878.72/297.15 878.72/297.15 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 878.72/297.15 failed due to the following reason: 878.72/297.15 878.72/297.15 None of the processors succeeded. 878.72/297.15 878.72/297.15 Details of failed attempt(s): 878.72/297.15 ----------------------------- 878.72/297.15 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 878.72/297.15 failed due to the following reason: 878.72/297.15 878.72/297.15 match-boundness of the problem could not be verified. 878.72/297.15 878.72/297.15 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 878.72/297.15 failed due to the following reason: 878.72/297.15 878.72/297.15 match-boundness of the problem could not be verified. 878.72/297.15 878.72/297.15 878.72/297.15 878.72/297.15 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 878.72/297.15 the following reason: 878.72/297.15 878.72/297.15 We add the following weak dependency pairs: 878.72/297.15 878.72/297.15 Strict DPs: 878.72/297.15 { U101^#(tt(), M, N) -> 878.72/297.15 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.15 , U102^#(tt(), M, N) -> 878.72/297.15 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.15 , U103^#(tt(), M, N) -> 878.72/297.15 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.15 , isNatKind^#(n__0()) -> c_3() 878.72/297.15 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.15 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.15 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.15 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.15 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.15 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.15 , U51^#(tt()) -> c_41() 878.72/297.15 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.15 , activate^#(X) -> c_7(X) 878.72/297.15 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.15 , activate^#(n__plus(X1, X2)) -> 878.72/297.15 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.15 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.15 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.15 , 0^#() -> c_53() 878.72/297.15 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.15 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.15 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.15 , s^#(X) -> c_50(X) 878.72/297.15 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.15 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.15 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.15 , U104^#(tt(), M, N) -> 878.72/297.15 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.15 , isNat^#(n__0()) -> c_13() 878.72/297.15 , isNat^#(n__plus(V1, V2)) -> 878.72/297.15 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , isNat^#(n__s(V1)) -> 878.72/297.15 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.15 , isNat^#(n__x(V1, V2)) -> 878.72/297.15 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , U11^#(tt(), V1, V2) -> 878.72/297.15 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , U21^#(tt(), V1) -> 878.72/297.15 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.15 , U31^#(tt(), V1, V2) -> 878.72/297.15 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , U81^#(tt(), M, N) -> 878.72/297.15 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.15 , U71^#(tt(), N) -> 878.72/297.15 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.15 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.15 , U12^#(tt(), V1, V2) -> 878.72/297.15 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U13^#(tt(), V1, V2) -> 878.72/297.15 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U14^#(tt(), V1, V2) -> 878.72/297.15 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.15 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.15 , U16^#(tt()) -> c_29() 878.72/297.15 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.15 , U23^#(tt()) -> c_32() 878.72/297.15 , U32^#(tt(), V1, V2) -> 878.72/297.15 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U33^#(tt(), V1, V2) -> 878.72/297.15 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U34^#(tt(), V1, V2) -> 878.72/297.15 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.15 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.15 , U36^#(tt()) -> c_38() 878.72/297.15 , U42^#(tt()) -> c_40() 878.72/297.15 , U62^#(tt()) -> c_43() 878.72/297.15 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.15 , U82^#(tt(), M, N) -> 878.72/297.15 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.15 , U83^#(tt(), M, N) -> 878.72/297.15 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.15 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.15 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.15 878.72/297.15 and mark the set of starting terms. 878.72/297.15 878.72/297.15 We are left with following problem, upon which TcT provides the 878.72/297.15 certificate MAYBE. 878.72/297.15 878.72/297.15 Strict DPs: 878.72/297.15 { U101^#(tt(), M, N) -> 878.72/297.15 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.15 , U102^#(tt(), M, N) -> 878.72/297.15 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.15 , U103^#(tt(), M, N) -> 878.72/297.15 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.15 , isNatKind^#(n__0()) -> c_3() 878.72/297.15 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.15 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.15 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.15 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.15 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.15 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.15 , U51^#(tt()) -> c_41() 878.72/297.15 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.15 , activate^#(X) -> c_7(X) 878.72/297.15 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.15 , activate^#(n__plus(X1, X2)) -> 878.72/297.15 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.15 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.15 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.15 , 0^#() -> c_53() 878.72/297.15 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.15 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.15 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.15 , s^#(X) -> c_50(X) 878.72/297.15 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.15 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.15 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.15 , U104^#(tt(), M, N) -> 878.72/297.15 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.15 , isNat^#(n__0()) -> c_13() 878.72/297.15 , isNat^#(n__plus(V1, V2)) -> 878.72/297.15 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , isNat^#(n__s(V1)) -> 878.72/297.15 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.15 , isNat^#(n__x(V1, V2)) -> 878.72/297.15 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , U11^#(tt(), V1, V2) -> 878.72/297.15 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , U21^#(tt(), V1) -> 878.72/297.15 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.15 , U31^#(tt(), V1, V2) -> 878.72/297.15 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , U81^#(tt(), M, N) -> 878.72/297.15 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.15 , U71^#(tt(), N) -> 878.72/297.15 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.15 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.15 , U12^#(tt(), V1, V2) -> 878.72/297.15 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U13^#(tt(), V1, V2) -> 878.72/297.15 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U14^#(tt(), V1, V2) -> 878.72/297.15 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.15 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.15 , U16^#(tt()) -> c_29() 878.72/297.15 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.15 , U23^#(tt()) -> c_32() 878.72/297.15 , U32^#(tt(), V1, V2) -> 878.72/297.15 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U33^#(tt(), V1, V2) -> 878.72/297.15 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , U34^#(tt(), V1, V2) -> 878.72/297.15 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.15 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.15 , U36^#(tt()) -> c_38() 878.72/297.15 , U42^#(tt()) -> c_40() 878.72/297.15 , U62^#(tt()) -> c_43() 878.72/297.15 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.15 , U82^#(tt(), M, N) -> 878.72/297.15 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.15 , U83^#(tt(), M, N) -> 878.72/297.15 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.15 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.15 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.15 Strict Trs: 878.72/297.15 { U101(tt(), M, N) -> 878.72/297.15 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.15 , U102(tt(), M, N) -> 878.72/297.15 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.15 , isNatKind(n__0()) -> tt() 878.72/297.15 , isNatKind(n__plus(V1, V2)) -> 878.72/297.15 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.15 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.15 , isNatKind(n__x(V1, V2)) -> 878.72/297.15 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.15 , activate(X) -> X 878.72/297.15 , activate(n__0()) -> 0() 878.72/297.15 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.15 , activate(n__s(X)) -> s(activate(X)) 878.72/297.15 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.15 , U103(tt(), M, N) -> 878.72/297.15 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.15 , isNat(n__0()) -> tt() 878.72/297.15 , isNat(n__plus(V1, V2)) -> 878.72/297.15 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.15 , isNat(n__x(V1, V2)) -> 878.72/297.15 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , U104(tt(), M, N) -> 878.72/297.15 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.15 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.15 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.15 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.15 , x(X1, X2) -> n__x(X1, X2) 878.72/297.15 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.15 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.15 , U11(tt(), V1, V2) -> 878.72/297.15 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , U12(tt(), V1, V2) -> 878.72/297.15 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U13(tt(), V1, V2) -> 878.72/297.15 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.15 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.15 , U16(tt()) -> tt() 878.72/297.15 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.15 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.15 , U23(tt()) -> tt() 878.72/297.15 , U31(tt(), V1, V2) -> 878.72/297.15 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.15 , U32(tt(), V1, V2) -> 878.72/297.15 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U33(tt(), V1, V2) -> 878.72/297.15 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.15 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.15 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.15 , U36(tt()) -> tt() 878.72/297.15 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.15 , U42(tt()) -> tt() 878.72/297.15 , U51(tt()) -> tt() 878.72/297.15 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.15 , U62(tt()) -> tt() 878.72/297.15 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.15 , U72(tt(), N) -> activate(N) 878.72/297.15 , U81(tt(), M, N) -> 878.72/297.15 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.15 , U82(tt(), M, N) -> 878.72/297.15 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.15 , U83(tt(), M, N) -> 878.72/297.15 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.15 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.15 , s(X) -> n__s(X) 878.72/297.15 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.15 , U92(tt()) -> 0() 878.72/297.15 , 0() -> n__0() } 878.72/297.15 Obligation: 878.72/297.15 runtime complexity 878.72/297.15 Answer: 878.72/297.15 MAYBE 878.72/297.15 878.72/297.15 We estimate the number of application of {4,9,16,25,39,41,46,47,48} 878.72/297.15 by applications of Pre({4,9,16,25,39,41,46,47,48}) = 878.72/297.15 {6,8,10,11,12,17,20,21,38,40,45,53}. Here rules are labeled as 878.72/297.15 follows: 878.72/297.15 878.72/297.15 DPs: 878.72/297.15 { 1: U101^#(tt(), M, N) -> 878.72/297.15 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.15 , 2: U102^#(tt(), M, N) -> 878.72/297.15 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.15 , 3: U103^#(tt(), M, N) -> 878.72/297.15 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.15 , 4: isNatKind^#(n__0()) -> c_3() 878.72/297.15 , 5: isNatKind^#(n__plus(V1, V2)) -> 878.72/297.15 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.15 , 6: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.15 , 7: isNatKind^#(n__x(V1, V2)) -> 878.72/297.15 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.15 , 8: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.15 , 9: U51^#(tt()) -> c_41() 878.72/297.15 , 10: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.15 , 11: activate^#(X) -> c_7(X) 878.72/297.15 , 12: activate^#(n__0()) -> c_8(0^#()) 878.72/297.15 , 13: activate^#(n__plus(X1, X2)) -> 878.72/297.15 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.15 , 14: activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.15 , 15: activate^#(n__x(X1, X2)) -> 878.72/297.15 c_11(x^#(activate(X1), activate(X2))) 878.72/297.15 , 16: 0^#() -> c_53() 878.72/297.15 , 17: plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.15 , 18: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.15 , 19: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.15 , 20: s^#(X) -> c_50(X) 878.72/297.15 , 21: x^#(X1, X2) -> c_21(X1, X2) 878.72/297.15 , 22: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.15 , 23: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.15 , 24: U104^#(tt(), M, N) -> 878.72/297.15 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.15 , 25: isNat^#(n__0()) -> c_13() 878.72/297.15 , 26: isNat^#(n__plus(V1, V2)) -> 878.72/297.15 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , 27: isNat^#(n__s(V1)) -> 878.72/297.15 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.15 , 28: isNat^#(n__x(V1, V2)) -> 878.72/297.15 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , 29: U11^#(tt(), V1, V2) -> 878.72/297.15 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , 30: U21^#(tt(), V1) -> 878.72/297.15 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.15 , 31: U31^#(tt(), V1, V2) -> 878.72/297.15 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.15 , 32: U81^#(tt(), M, N) -> 878.72/297.15 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.15 , 33: U71^#(tt(), N) -> 878.72/297.15 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.15 , 34: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.15 , 35: U12^#(tt(), V1, V2) -> 878.72/297.15 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , 36: U13^#(tt(), V1, V2) -> 878.72/297.15 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , 37: U14^#(tt(), V1, V2) -> 878.72/297.15 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.15 , 38: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.15 , 39: U16^#(tt()) -> c_29() 878.72/297.15 , 40: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.15 , 41: U23^#(tt()) -> c_32() 878.72/297.15 , 42: U32^#(tt(), V1, V2) -> 878.72/297.15 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , 43: U33^#(tt(), V1, V2) -> 878.72/297.15 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.15 , 44: U34^#(tt(), V1, V2) -> 878.72/297.15 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.15 , 45: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.15 , 46: U36^#(tt()) -> c_38() 878.72/297.15 , 47: U42^#(tt()) -> c_40() 878.72/297.15 , 48: U62^#(tt()) -> c_43() 878.72/297.15 , 49: U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.16 , 50: U82^#(tt(), M, N) -> 878.72/297.16 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , 51: U83^#(tt(), M, N) -> 878.72/297.16 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , 52: U84^#(tt(), M, N) -> 878.72/297.16 c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.16 , 53: U92^#(tt()) -> c_52(0^#()) } 878.72/297.16 878.72/297.16 We are left with following problem, upon which TcT provides the 878.72/297.16 certificate MAYBE. 878.72/297.16 878.72/297.16 Strict DPs: 878.72/297.16 { U101^#(tt(), M, N) -> 878.72/297.16 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , U102^#(tt(), M, N) -> 878.72/297.16 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , U103^#(tt(), M, N) -> 878.72/297.16 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.16 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.16 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.16 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.16 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.16 , activate^#(X) -> c_7(X) 878.72/297.16 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.16 , activate^#(n__plus(X1, X2)) -> 878.72/297.16 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.16 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.16 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.16 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.16 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.16 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.16 , s^#(X) -> c_50(X) 878.72/297.16 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.16 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.16 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.16 , U104^#(tt(), M, N) -> 878.72/297.16 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.16 , isNat^#(n__plus(V1, V2)) -> 878.72/297.16 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , isNat^#(n__s(V1)) -> 878.72/297.16 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , isNat^#(n__x(V1, V2)) -> 878.72/297.16 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , U11^#(tt(), V1, V2) -> 878.72/297.16 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , U21^#(tt(), V1) -> 878.72/297.16 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , U31^#(tt(), V1, V2) -> 878.72/297.16 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , U81^#(tt(), M, N) -> 878.72/297.16 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , U71^#(tt(), N) -> 878.72/297.16 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.16 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.16 , U12^#(tt(), V1, V2) -> 878.72/297.16 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U13^#(tt(), V1, V2) -> 878.72/297.16 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U14^#(tt(), V1, V2) -> 878.72/297.16 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.16 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.16 , U32^#(tt(), V1, V2) -> 878.72/297.16 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U33^#(tt(), V1, V2) -> 878.72/297.16 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U34^#(tt(), V1, V2) -> 878.72/297.16 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.16 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.16 , U82^#(tt(), M, N) -> 878.72/297.16 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , U83^#(tt(), M, N) -> 878.72/297.16 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.16 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.16 Strict Trs: 878.72/297.16 { U101(tt(), M, N) -> 878.72/297.16 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.16 , U102(tt(), M, N) -> 878.72/297.16 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.16 , isNatKind(n__0()) -> tt() 878.72/297.16 , isNatKind(n__plus(V1, V2)) -> 878.72/297.16 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.16 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.16 , isNatKind(n__x(V1, V2)) -> 878.72/297.16 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.16 , activate(X) -> X 878.72/297.16 , activate(n__0()) -> 0() 878.72/297.16 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.16 , activate(n__s(X)) -> s(activate(X)) 878.72/297.16 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.16 , U103(tt(), M, N) -> 878.72/297.16 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.16 , isNat(n__0()) -> tt() 878.72/297.16 , isNat(n__plus(V1, V2)) -> 878.72/297.16 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.16 , isNat(n__x(V1, V2)) -> 878.72/297.16 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , U104(tt(), M, N) -> 878.72/297.16 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.16 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.16 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.16 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.16 , x(X1, X2) -> n__x(X1, X2) 878.72/297.16 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.16 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.16 , U11(tt(), V1, V2) -> 878.72/297.16 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , U12(tt(), V1, V2) -> 878.72/297.16 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U13(tt(), V1, V2) -> 878.72/297.16 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.16 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.16 , U16(tt()) -> tt() 878.72/297.16 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.16 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.16 , U23(tt()) -> tt() 878.72/297.16 , U31(tt(), V1, V2) -> 878.72/297.16 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , U32(tt(), V1, V2) -> 878.72/297.16 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U33(tt(), V1, V2) -> 878.72/297.16 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.16 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.16 , U36(tt()) -> tt() 878.72/297.16 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.16 , U42(tt()) -> tt() 878.72/297.16 , U51(tt()) -> tt() 878.72/297.16 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.16 , U62(tt()) -> tt() 878.72/297.16 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.16 , U72(tt(), N) -> activate(N) 878.72/297.16 , U81(tt(), M, N) -> 878.72/297.16 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.16 , U82(tt(), M, N) -> 878.72/297.16 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.16 , U83(tt(), M, N) -> 878.72/297.16 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.16 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.16 , s(X) -> n__s(X) 878.72/297.16 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.16 , U92(tt()) -> 0() 878.72/297.16 , 0() -> n__0() } 878.72/297.16 Weak DPs: 878.72/297.16 { isNatKind^#(n__0()) -> c_3() 878.72/297.16 , U51^#(tt()) -> c_41() 878.72/297.16 , 0^#() -> c_53() 878.72/297.16 , isNat^#(n__0()) -> c_13() 878.72/297.16 , U16^#(tt()) -> c_29() 878.72/297.16 , U23^#(tt()) -> c_32() 878.72/297.16 , U36^#(tt()) -> c_38() 878.72/297.16 , U42^#(tt()) -> c_40() 878.72/297.16 , U62^#(tt()) -> c_43() } 878.72/297.16 Obligation: 878.72/297.16 runtime complexity 878.72/297.16 Answer: 878.72/297.16 MAYBE 878.72/297.16 878.72/297.16 We estimate the number of application of {5,7,8,10,34,35,39,44} by 878.72/297.16 applications of Pre({5,7,8,10,34,35,39,44}) = 878.72/297.16 {4,6,9,14,17,18,26,30,33,38,40}. Here rules are labeled as follows: 878.72/297.16 878.72/297.16 DPs: 878.72/297.16 { 1: U101^#(tt(), M, N) -> 878.72/297.16 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , 2: U102^#(tt(), M, N) -> 878.72/297.16 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , 3: U103^#(tt(), M, N) -> 878.72/297.16 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , 4: isNatKind^#(n__plus(V1, V2)) -> 878.72/297.16 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , 5: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.16 , 6: isNatKind^#(n__x(V1, V2)) -> 878.72/297.16 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , 7: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.16 , 8: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.16 , 9: activate^#(X) -> c_7(X) 878.72/297.16 , 10: activate^#(n__0()) -> c_8(0^#()) 878.72/297.16 , 11: activate^#(n__plus(X1, X2)) -> 878.72/297.16 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.16 , 12: activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.16 , 13: activate^#(n__x(X1, X2)) -> 878.72/297.16 c_11(x^#(activate(X1), activate(X2))) 878.72/297.16 , 14: plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.16 , 15: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.16 , 16: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.16 , 17: s^#(X) -> c_50(X) 878.72/297.16 , 18: x^#(X1, X2) -> c_21(X1, X2) 878.72/297.16 , 19: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.16 , 20: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.16 , 21: U104^#(tt(), M, N) -> 878.72/297.16 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.16 , 22: isNat^#(n__plus(V1, V2)) -> 878.72/297.16 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 23: isNat^#(n__s(V1)) -> 878.72/297.16 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , 24: isNat^#(n__x(V1, V2)) -> 878.72/297.16 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 25: U11^#(tt(), V1, V2) -> 878.72/297.16 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 26: U21^#(tt(), V1) -> 878.72/297.16 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , 27: U31^#(tt(), V1, V2) -> 878.72/297.16 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 28: U81^#(tt(), M, N) -> 878.72/297.16 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , 29: U71^#(tt(), N) -> 878.72/297.16 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.16 , 30: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.16 , 31: U12^#(tt(), V1, V2) -> 878.72/297.16 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 32: U13^#(tt(), V1, V2) -> 878.72/297.16 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 33: U14^#(tt(), V1, V2) -> 878.72/297.16 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , 34: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.16 , 35: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.16 , 36: U32^#(tt(), V1, V2) -> 878.72/297.16 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 37: U33^#(tt(), V1, V2) -> 878.72/297.16 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 38: U34^#(tt(), V1, V2) -> 878.72/297.16 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , 39: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.16 , 40: U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.16 , 41: U82^#(tt(), M, N) -> 878.72/297.16 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , 42: U83^#(tt(), M, N) -> 878.72/297.16 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , 43: U84^#(tt(), M, N) -> 878.72/297.16 c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.16 , 44: U92^#(tt()) -> c_52(0^#()) 878.72/297.16 , 45: isNatKind^#(n__0()) -> c_3() 878.72/297.16 , 46: U51^#(tt()) -> c_41() 878.72/297.16 , 47: 0^#() -> c_53() 878.72/297.16 , 48: isNat^#(n__0()) -> c_13() 878.72/297.16 , 49: U16^#(tt()) -> c_29() 878.72/297.16 , 50: U23^#(tt()) -> c_32() 878.72/297.16 , 51: U36^#(tt()) -> c_38() 878.72/297.16 , 52: U42^#(tt()) -> c_40() 878.72/297.16 , 53: U62^#(tt()) -> c_43() } 878.72/297.16 878.72/297.16 We are left with following problem, upon which TcT provides the 878.72/297.16 certificate MAYBE. 878.72/297.16 878.72/297.16 Strict DPs: 878.72/297.16 { U101^#(tt(), M, N) -> 878.72/297.16 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , U102^#(tt(), M, N) -> 878.72/297.16 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , U103^#(tt(), M, N) -> 878.72/297.16 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.16 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.16 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , activate^#(X) -> c_7(X) 878.72/297.16 , activate^#(n__plus(X1, X2)) -> 878.72/297.16 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.16 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.16 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.16 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.16 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.16 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.16 , s^#(X) -> c_50(X) 878.72/297.16 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.16 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.16 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.16 , U104^#(tt(), M, N) -> 878.72/297.16 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.16 , isNat^#(n__plus(V1, V2)) -> 878.72/297.16 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , isNat^#(n__s(V1)) -> 878.72/297.16 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , isNat^#(n__x(V1, V2)) -> 878.72/297.16 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , U11^#(tt(), V1, V2) -> 878.72/297.16 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , U21^#(tt(), V1) -> 878.72/297.16 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , U31^#(tt(), V1, V2) -> 878.72/297.16 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , U81^#(tt(), M, N) -> 878.72/297.16 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , U71^#(tt(), N) -> 878.72/297.16 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.16 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.16 , U12^#(tt(), V1, V2) -> 878.72/297.16 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U13^#(tt(), V1, V2) -> 878.72/297.16 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U14^#(tt(), V1, V2) -> 878.72/297.16 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , U32^#(tt(), V1, V2) -> 878.72/297.16 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U33^#(tt(), V1, V2) -> 878.72/297.16 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , U34^#(tt(), V1, V2) -> 878.72/297.16 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.16 , U82^#(tt(), M, N) -> 878.72/297.16 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , U83^#(tt(), M, N) -> 878.72/297.16 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 878.72/297.16 Strict Trs: 878.72/297.16 { U101(tt(), M, N) -> 878.72/297.16 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.16 , U102(tt(), M, N) -> 878.72/297.16 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.16 , isNatKind(n__0()) -> tt() 878.72/297.16 , isNatKind(n__plus(V1, V2)) -> 878.72/297.16 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.16 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.16 , isNatKind(n__x(V1, V2)) -> 878.72/297.16 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.16 , activate(X) -> X 878.72/297.16 , activate(n__0()) -> 0() 878.72/297.16 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.16 , activate(n__s(X)) -> s(activate(X)) 878.72/297.16 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.16 , U103(tt(), M, N) -> 878.72/297.16 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.16 , isNat(n__0()) -> tt() 878.72/297.16 , isNat(n__plus(V1, V2)) -> 878.72/297.16 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.16 , isNat(n__x(V1, V2)) -> 878.72/297.16 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , U104(tt(), M, N) -> 878.72/297.16 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.16 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.16 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.16 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.16 , x(X1, X2) -> n__x(X1, X2) 878.72/297.16 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.16 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.16 , U11(tt(), V1, V2) -> 878.72/297.16 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , U12(tt(), V1, V2) -> 878.72/297.16 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U13(tt(), V1, V2) -> 878.72/297.16 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.16 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.16 , U16(tt()) -> tt() 878.72/297.16 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.16 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.16 , U23(tt()) -> tt() 878.72/297.16 , U31(tt(), V1, V2) -> 878.72/297.16 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.16 , U32(tt(), V1, V2) -> 878.72/297.16 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U33(tt(), V1, V2) -> 878.72/297.16 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.16 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.16 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.16 , U36(tt()) -> tt() 878.72/297.16 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.16 , U42(tt()) -> tt() 878.72/297.16 , U51(tt()) -> tt() 878.72/297.16 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.16 , U62(tt()) -> tt() 878.72/297.16 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.16 , U72(tt(), N) -> activate(N) 878.72/297.16 , U81(tt(), M, N) -> 878.72/297.16 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.16 , U82(tt(), M, N) -> 878.72/297.16 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.16 , U83(tt(), M, N) -> 878.72/297.16 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.16 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.16 , s(X) -> n__s(X) 878.72/297.16 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.16 , U92(tt()) -> 0() 878.72/297.16 , 0() -> n__0() } 878.72/297.16 Weak DPs: 878.72/297.16 { isNatKind^#(n__0()) -> c_3() 878.72/297.16 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.16 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.16 , U51^#(tt()) -> c_41() 878.72/297.16 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.16 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.16 , 0^#() -> c_53() 878.72/297.16 , isNat^#(n__0()) -> c_13() 878.72/297.16 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.16 , U16^#(tt()) -> c_29() 878.72/297.16 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.16 , U23^#(tt()) -> c_32() 878.72/297.16 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.16 , U36^#(tt()) -> c_38() 878.72/297.16 , U42^#(tt()) -> c_40() 878.72/297.16 , U62^#(tt()) -> c_43() 878.72/297.16 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.16 Obligation: 878.72/297.16 runtime complexity 878.72/297.16 Answer: 878.72/297.16 MAYBE 878.72/297.16 878.72/297.16 We estimate the number of application of {4,5,22,26,29,32} by 878.72/297.16 applications of Pre({4,5,22,26,29,32}) = {6,10,13,14,16,19,28,31}. 878.72/297.16 Here rules are labeled as follows: 878.72/297.16 878.72/297.16 DPs: 878.72/297.16 { 1: U101^#(tt(), M, N) -> 878.72/297.16 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , 2: U102^#(tt(), M, N) -> 878.72/297.16 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.16 , 3: U103^#(tt(), M, N) -> 878.72/297.16 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.16 , 4: isNatKind^#(n__plus(V1, V2)) -> 878.72/297.16 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , 5: isNatKind^#(n__x(V1, V2)) -> 878.72/297.16 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.16 , 6: activate^#(X) -> c_7(X) 878.72/297.16 , 7: activate^#(n__plus(X1, X2)) -> 878.72/297.16 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.16 , 8: activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.16 , 9: activate^#(n__x(X1, X2)) -> 878.72/297.16 c_11(x^#(activate(X1), activate(X2))) 878.72/297.16 , 10: plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.16 , 11: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.16 , 12: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.16 , 13: s^#(X) -> c_50(X) 878.72/297.16 , 14: x^#(X1, X2) -> c_21(X1, X2) 878.72/297.16 , 15: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.16 , 16: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.16 , 17: U104^#(tt(), M, N) -> 878.72/297.16 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.16 , 18: isNat^#(n__plus(V1, V2)) -> 878.72/297.16 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 19: isNat^#(n__s(V1)) -> 878.72/297.16 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , 20: isNat^#(n__x(V1, V2)) -> 878.72/297.16 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 21: U11^#(tt(), V1, V2) -> 878.72/297.16 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 22: U21^#(tt(), V1) -> 878.72/297.16 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.16 , 23: U31^#(tt(), V1, V2) -> 878.72/297.16 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.16 , 24: U81^#(tt(), M, N) -> 878.72/297.16 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.16 , 25: U71^#(tt(), N) -> 878.72/297.16 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.16 , 26: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.16 , 27: U12^#(tt(), V1, V2) -> 878.72/297.16 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 28: U13^#(tt(), V1, V2) -> 878.72/297.16 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 29: U14^#(tt(), V1, V2) -> 878.72/297.16 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , 30: U32^#(tt(), V1, V2) -> 878.72/297.16 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 31: U33^#(tt(), V1, V2) -> 878.72/297.16 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.16 , 32: U34^#(tt(), V1, V2) -> 878.72/297.16 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.16 , 33: U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.17 , 34: U82^#(tt(), M, N) -> 878.72/297.17 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.17 , 35: U83^#(tt(), M, N) -> 878.72/297.17 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.17 , 36: U84^#(tt(), M, N) -> 878.72/297.17 c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.17 , 37: isNatKind^#(n__0()) -> c_3() 878.72/297.17 , 38: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.17 , 39: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.17 , 40: U51^#(tt()) -> c_41() 878.72/297.17 , 41: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.17 , 42: activate^#(n__0()) -> c_8(0^#()) 878.72/297.17 , 43: 0^#() -> c_53() 878.72/297.17 , 44: isNat^#(n__0()) -> c_13() 878.72/297.17 , 45: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.17 , 46: U16^#(tt()) -> c_29() 878.72/297.17 , 47: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.17 , 48: U23^#(tt()) -> c_32() 878.72/297.17 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.17 , 50: U36^#(tt()) -> c_38() 878.72/297.17 , 51: U42^#(tt()) -> c_40() 878.72/297.17 , 52: U62^#(tt()) -> c_43() 878.72/297.17 , 53: U92^#(tt()) -> c_52(0^#()) } 878.72/297.17 878.72/297.17 We are left with following problem, upon which TcT provides the 878.72/297.17 certificate MAYBE. 878.72/297.17 878.72/297.17 Strict DPs: 878.72/297.17 { U101^#(tt(), M, N) -> 878.72/297.17 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.17 , U102^#(tt(), M, N) -> 878.72/297.17 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.17 , U103^#(tt(), M, N) -> 878.72/297.17 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.17 , activate^#(X) -> c_7(X) 878.72/297.17 , activate^#(n__plus(X1, X2)) -> 878.72/297.17 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.17 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.17 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.17 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.17 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.17 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.17 , s^#(X) -> c_50(X) 878.72/297.17 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.17 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.17 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.17 , U104^#(tt(), M, N) -> 878.72/297.17 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.17 , isNat^#(n__plus(V1, V2)) -> 878.72/297.17 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , isNat^#(n__s(V1)) -> 878.72/297.17 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.17 , isNat^#(n__x(V1, V2)) -> 878.72/297.17 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , U11^#(tt(), V1, V2) -> 878.72/297.17 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , U31^#(tt(), V1, V2) -> 878.72/297.17 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , U81^#(tt(), M, N) -> 878.72/297.17 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.17 , U71^#(tt(), N) -> 878.72/297.17 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.17 , U12^#(tt(), V1, V2) -> 878.72/297.17 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , U13^#(tt(), V1, V2) -> 878.72/297.17 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , U32^#(tt(), V1, V2) -> 878.72/297.17 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , U33^#(tt(), V1, V2) -> 878.72/297.17 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.17 , U82^#(tt(), M, N) -> 878.72/297.17 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.17 , U83^#(tt(), M, N) -> 878.72/297.17 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.17 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 878.72/297.17 Strict Trs: 878.72/297.17 { U101(tt(), M, N) -> 878.72/297.17 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.17 , U102(tt(), M, N) -> 878.72/297.17 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.17 , isNatKind(n__0()) -> tt() 878.72/297.17 , isNatKind(n__plus(V1, V2)) -> 878.72/297.17 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.17 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.17 , isNatKind(n__x(V1, V2)) -> 878.72/297.17 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.17 , activate(X) -> X 878.72/297.17 , activate(n__0()) -> 0() 878.72/297.17 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.17 , activate(n__s(X)) -> s(activate(X)) 878.72/297.17 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.17 , U103(tt(), M, N) -> 878.72/297.17 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.17 , isNat(n__0()) -> tt() 878.72/297.17 , isNat(n__plus(V1, V2)) -> 878.72/297.17 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.17 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.17 , isNat(n__x(V1, V2)) -> 878.72/297.17 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.17 , U104(tt(), M, N) -> 878.72/297.17 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.17 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.17 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.17 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.17 , x(X1, X2) -> n__x(X1, X2) 878.72/297.17 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.17 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.17 , U11(tt(), V1, V2) -> 878.72/297.17 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.17 , U12(tt(), V1, V2) -> 878.72/297.17 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.17 , U13(tt(), V1, V2) -> 878.72/297.17 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.17 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.17 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.17 , U16(tt()) -> tt() 878.72/297.17 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.17 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.17 , U23(tt()) -> tt() 878.72/297.17 , U31(tt(), V1, V2) -> 878.72/297.17 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.17 , U32(tt(), V1, V2) -> 878.72/297.17 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.17 , U33(tt(), V1, V2) -> 878.72/297.17 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.17 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.17 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.17 , U36(tt()) -> tt() 878.72/297.17 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.17 , U42(tt()) -> tt() 878.72/297.17 , U51(tt()) -> tt() 878.72/297.17 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.17 , U62(tt()) -> tt() 878.72/297.17 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.17 , U72(tt(), N) -> activate(N) 878.72/297.17 , U81(tt(), M, N) -> 878.72/297.17 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.17 , U82(tt(), M, N) -> 878.72/297.17 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.17 , U83(tt(), M, N) -> 878.72/297.17 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.17 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.17 , s(X) -> n__s(X) 878.72/297.17 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.17 , U92(tt()) -> 0() 878.72/297.17 , 0() -> n__0() } 878.72/297.17 Weak DPs: 878.72/297.17 { isNatKind^#(n__0()) -> c_3() 878.72/297.17 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.17 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.17 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.17 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.17 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.17 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.17 , U51^#(tt()) -> c_41() 878.72/297.17 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.17 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.17 , 0^#() -> c_53() 878.72/297.17 , isNat^#(n__0()) -> c_13() 878.72/297.17 , U21^#(tt(), V1) -> 878.72/297.17 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.17 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.17 , U14^#(tt(), V1, V2) -> 878.72/297.17 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.17 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.17 , U16^#(tt()) -> c_29() 878.72/297.17 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.17 , U23^#(tt()) -> c_32() 878.72/297.17 , U34^#(tt(), V1, V2) -> 878.72/297.17 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.17 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.17 , U36^#(tt()) -> c_38() 878.72/297.17 , U42^#(tt()) -> c_40() 878.72/297.17 , U62^#(tt()) -> c_43() 878.72/297.17 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.17 Obligation: 878.72/297.17 runtime complexity 878.72/297.17 Answer: 878.72/297.17 MAYBE 878.72/297.17 878.72/297.17 We estimate the number of application of {14,17,24,26} by 878.72/297.17 applications of Pre({14,17,24,26}) = {4,7,8,11,12,23,25}. Here 878.72/297.17 rules are labeled as follows: 878.72/297.17 878.72/297.17 DPs: 878.72/297.17 { 1: U101^#(tt(), M, N) -> 878.72/297.17 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.17 , 2: U102^#(tt(), M, N) -> 878.72/297.17 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.17 , 3: U103^#(tt(), M, N) -> 878.72/297.17 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.17 , 4: activate^#(X) -> c_7(X) 878.72/297.17 , 5: activate^#(n__plus(X1, X2)) -> 878.72/297.17 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.17 , 6: activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.17 , 7: activate^#(n__x(X1, X2)) -> 878.72/297.17 c_11(x^#(activate(X1), activate(X2))) 878.72/297.17 , 8: plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.17 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.17 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.17 , 11: s^#(X) -> c_50(X) 878.72/297.17 , 12: x^#(X1, X2) -> c_21(X1, X2) 878.72/297.17 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.17 , 14: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.17 , 15: U104^#(tt(), M, N) -> 878.72/297.17 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.17 , 16: isNat^#(n__plus(V1, V2)) -> 878.72/297.17 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , 17: isNat^#(n__s(V1)) -> 878.72/297.17 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.17 , 18: isNat^#(n__x(V1, V2)) -> 878.72/297.17 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , 19: U11^#(tt(), V1, V2) -> 878.72/297.17 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , 20: U31^#(tt(), V1, V2) -> 878.72/297.17 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , 21: U81^#(tt(), M, N) -> 878.72/297.17 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.17 , 22: U71^#(tt(), N) -> 878.72/297.17 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.17 , 23: U12^#(tt(), V1, V2) -> 878.72/297.17 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , 24: U13^#(tt(), V1, V2) -> 878.72/297.17 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , 25: U32^#(tt(), V1, V2) -> 878.72/297.17 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , 26: U33^#(tt(), V1, V2) -> 878.72/297.17 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , 27: U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.17 , 28: U82^#(tt(), M, N) -> 878.72/297.17 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.17 , 29: U83^#(tt(), M, N) -> 878.72/297.17 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.17 , 30: U84^#(tt(), M, N) -> 878.72/297.17 c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.17 , 31: isNatKind^#(n__0()) -> c_3() 878.72/297.17 , 32: isNatKind^#(n__plus(V1, V2)) -> 878.72/297.17 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.17 , 33: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.17 , 34: isNatKind^#(n__x(V1, V2)) -> 878.72/297.17 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.17 , 35: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.17 , 36: U51^#(tt()) -> c_41() 878.72/297.17 , 37: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.17 , 38: activate^#(n__0()) -> c_8(0^#()) 878.72/297.17 , 39: 0^#() -> c_53() 878.72/297.17 , 40: isNat^#(n__0()) -> c_13() 878.72/297.17 , 41: U21^#(tt(), V1) -> 878.72/297.17 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.17 , 42: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.17 , 43: U14^#(tt(), V1, V2) -> 878.72/297.17 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.17 , 44: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.17 , 45: U16^#(tt()) -> c_29() 878.72/297.17 , 46: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.17 , 47: U23^#(tt()) -> c_32() 878.72/297.17 , 48: U34^#(tt(), V1, V2) -> 878.72/297.17 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.17 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.17 , 50: U36^#(tt()) -> c_38() 878.72/297.17 , 51: U42^#(tt()) -> c_40() 878.72/297.17 , 52: U62^#(tt()) -> c_43() 878.72/297.17 , 53: U92^#(tt()) -> c_52(0^#()) } 878.72/297.17 878.72/297.17 We are left with following problem, upon which TcT provides the 878.72/297.17 certificate MAYBE. 878.72/297.17 878.72/297.17 Strict DPs: 878.72/297.17 { U101^#(tt(), M, N) -> 878.72/297.17 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.17 , U102^#(tt(), M, N) -> 878.72/297.17 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.17 , U103^#(tt(), M, N) -> 878.72/297.17 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.17 , activate^#(X) -> c_7(X) 878.72/297.17 , activate^#(n__plus(X1, X2)) -> 878.72/297.17 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.17 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.17 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.17 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.17 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.17 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.17 , s^#(X) -> c_50(X) 878.72/297.17 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.17 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.17 , U104^#(tt(), M, N) -> 878.72/297.17 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.17 , isNat^#(n__plus(V1, V2)) -> 878.72/297.17 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , isNat^#(n__x(V1, V2)) -> 878.72/297.17 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , U11^#(tt(), V1, V2) -> 878.72/297.17 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , U31^#(tt(), V1, V2) -> 878.72/297.17 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.17 , U81^#(tt(), M, N) -> 878.72/297.17 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.17 , U71^#(tt(), N) -> 878.72/297.17 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.17 , U12^#(tt(), V1, V2) -> 878.72/297.17 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , U32^#(tt(), V1, V2) -> 878.72/297.17 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.17 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.17 , U82^#(tt(), M, N) -> 878.72/297.17 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.17 , U83^#(tt(), M, N) -> 878.72/297.17 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.17 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 878.72/297.17 Strict Trs: 878.72/297.17 { U101(tt(), M, N) -> 878.72/297.17 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.17 , U102(tt(), M, N) -> 878.72/297.17 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.17 , isNatKind(n__0()) -> tt() 878.72/297.17 , isNatKind(n__plus(V1, V2)) -> 878.72/297.17 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.17 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.17 , isNatKind(n__x(V1, V2)) -> 878.72/297.17 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.17 , activate(X) -> X 878.72/297.17 , activate(n__0()) -> 0() 878.72/297.17 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.17 , activate(n__s(X)) -> s(activate(X)) 878.72/297.17 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.17 , U103(tt(), M, N) -> 878.72/297.17 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.17 , isNat(n__0()) -> tt() 878.72/297.17 , isNat(n__plus(V1, V2)) -> 878.72/297.17 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.17 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.17 , isNat(n__x(V1, V2)) -> 878.72/297.17 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.17 , U104(tt(), M, N) -> 878.72/297.17 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.17 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.17 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.17 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.17 , x(X1, X2) -> n__x(X1, X2) 878.72/297.17 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.17 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.17 , U11(tt(), V1, V2) -> 878.72/297.17 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.17 , U12(tt(), V1, V2) -> 878.72/297.17 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.17 , U13(tt(), V1, V2) -> 878.72/297.17 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.17 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.17 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.17 , U16(tt()) -> tt() 878.72/297.17 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.17 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.17 , U23(tt()) -> tt() 878.72/297.17 , U31(tt(), V1, V2) -> 878.72/297.17 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.18 , U32(tt(), V1, V2) -> 878.72/297.18 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.18 , U33(tt(), V1, V2) -> 878.72/297.18 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.18 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.18 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.18 , U36(tt()) -> tt() 878.72/297.18 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.18 , U42(tt()) -> tt() 878.72/297.18 , U51(tt()) -> tt() 878.72/297.18 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.18 , U62(tt()) -> tt() 878.72/297.18 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.18 , U72(tt(), N) -> activate(N) 878.72/297.18 , U81(tt(), M, N) -> 878.72/297.18 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.18 , U82(tt(), M, N) -> 878.72/297.18 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.18 , U83(tt(), M, N) -> 878.72/297.18 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.18 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.18 , s(X) -> n__s(X) 878.72/297.18 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.18 , U92(tt()) -> 0() 878.72/297.18 , 0() -> n__0() } 878.72/297.18 Weak DPs: 878.72/297.18 { isNatKind^#(n__0()) -> c_3() 878.72/297.18 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.18 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.18 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.18 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.18 , U51^#(tt()) -> c_41() 878.72/297.18 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.18 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.18 , 0^#() -> c_53() 878.72/297.18 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.18 , isNat^#(n__0()) -> c_13() 878.72/297.18 , isNat^#(n__s(V1)) -> 878.72/297.18 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.18 , U21^#(tt(), V1) -> 878.72/297.18 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.18 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.18 , U13^#(tt(), V1, V2) -> 878.72/297.18 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , U14^#(tt(), V1, V2) -> 878.72/297.18 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.18 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.18 , U16^#(tt()) -> c_29() 878.72/297.18 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.18 , U23^#(tt()) -> c_32() 878.72/297.18 , U33^#(tt(), V1, V2) -> 878.72/297.18 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , U34^#(tt(), V1, V2) -> 878.72/297.18 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.18 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.18 , U36^#(tt()) -> c_38() 878.72/297.18 , U42^#(tt()) -> c_40() 878.72/297.18 , U62^#(tt()) -> c_43() 878.72/297.18 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.18 Obligation: 878.72/297.18 runtime complexity 878.72/297.18 Answer: 878.72/297.18 MAYBE 878.72/297.18 878.72/297.18 We estimate the number of application of {21,22} by applications of 878.72/297.18 Pre({21,22}) = {4,8,11,12,17,18}. Here rules are labeled as 878.72/297.18 follows: 878.72/297.18 878.72/297.18 DPs: 878.72/297.18 { 1: U101^#(tt(), M, N) -> 878.72/297.18 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.18 , 2: U102^#(tt(), M, N) -> 878.72/297.18 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.18 , 3: U103^#(tt(), M, N) -> 878.72/297.18 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.18 , 4: activate^#(X) -> c_7(X) 878.72/297.18 , 5: activate^#(n__plus(X1, X2)) -> 878.72/297.18 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.18 , 6: activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.18 , 7: activate^#(n__x(X1, X2)) -> 878.72/297.18 c_11(x^#(activate(X1), activate(X2))) 878.72/297.18 , 8: plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.18 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.18 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.18 , 11: s^#(X) -> c_50(X) 878.72/297.18 , 12: x^#(X1, X2) -> c_21(X1, X2) 878.72/297.18 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.18 , 14: U104^#(tt(), M, N) -> 878.72/297.18 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.18 , 15: isNat^#(n__plus(V1, V2)) -> 878.72/297.18 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 16: isNat^#(n__x(V1, V2)) -> 878.72/297.18 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 17: U11^#(tt(), V1, V2) -> 878.72/297.18 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 18: U31^#(tt(), V1, V2) -> 878.72/297.18 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 19: U81^#(tt(), M, N) -> 878.72/297.18 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.18 , 20: U71^#(tt(), N) -> 878.72/297.18 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.18 , 21: U12^#(tt(), V1, V2) -> 878.72/297.18 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , 22: U32^#(tt(), V1, V2) -> 878.72/297.18 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , 23: U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.18 , 24: U82^#(tt(), M, N) -> 878.72/297.18 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.18 , 25: U83^#(tt(), M, N) -> 878.72/297.18 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.18 , 26: U84^#(tt(), M, N) -> 878.72/297.18 c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.18 , 27: isNatKind^#(n__0()) -> c_3() 878.72/297.18 , 28: isNatKind^#(n__plus(V1, V2)) -> 878.72/297.18 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , 29: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.18 , 30: isNatKind^#(n__x(V1, V2)) -> 878.72/297.18 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , 31: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.18 , 32: U51^#(tt()) -> c_41() 878.72/297.18 , 33: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.18 , 34: activate^#(n__0()) -> c_8(0^#()) 878.72/297.18 , 35: 0^#() -> c_53() 878.72/297.18 , 36: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.18 , 37: isNat^#(n__0()) -> c_13() 878.72/297.18 , 38: isNat^#(n__s(V1)) -> 878.72/297.18 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.18 , 39: U21^#(tt(), V1) -> 878.72/297.18 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.18 , 40: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.18 , 41: U13^#(tt(), V1, V2) -> 878.72/297.18 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , 42: U14^#(tt(), V1, V2) -> 878.72/297.18 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.18 , 43: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.18 , 44: U16^#(tt()) -> c_29() 878.72/297.18 , 45: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.18 , 46: U23^#(tt()) -> c_32() 878.72/297.18 , 47: U33^#(tt(), V1, V2) -> 878.72/297.18 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , 48: U34^#(tt(), V1, V2) -> 878.72/297.18 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.18 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.18 , 50: U36^#(tt()) -> c_38() 878.72/297.18 , 51: U42^#(tt()) -> c_40() 878.72/297.18 , 52: U62^#(tt()) -> c_43() 878.72/297.18 , 53: U92^#(tt()) -> c_52(0^#()) } 878.72/297.18 878.72/297.18 We are left with following problem, upon which TcT provides the 878.72/297.18 certificate MAYBE. 878.72/297.18 878.72/297.18 Strict DPs: 878.72/297.18 { U101^#(tt(), M, N) -> 878.72/297.18 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.18 , U102^#(tt(), M, N) -> 878.72/297.18 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.18 , U103^#(tt(), M, N) -> 878.72/297.18 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.18 , activate^#(X) -> c_7(X) 878.72/297.18 , activate^#(n__plus(X1, X2)) -> 878.72/297.18 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.18 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.18 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.18 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.18 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.18 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.18 , s^#(X) -> c_50(X) 878.72/297.18 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.18 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.18 , U104^#(tt(), M, N) -> 878.72/297.18 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.18 , isNat^#(n__plus(V1, V2)) -> 878.72/297.18 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , isNat^#(n__x(V1, V2)) -> 878.72/297.18 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , U11^#(tt(), V1, V2) -> 878.72/297.18 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , U31^#(tt(), V1, V2) -> 878.72/297.18 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , U81^#(tt(), M, N) -> 878.72/297.18 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.18 , U71^#(tt(), N) -> 878.72/297.18 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.18 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.18 , U82^#(tt(), M, N) -> 878.72/297.18 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.18 , U83^#(tt(), M, N) -> 878.72/297.18 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.18 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 878.72/297.18 Strict Trs: 878.72/297.18 { U101(tt(), M, N) -> 878.72/297.18 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.18 , U102(tt(), M, N) -> 878.72/297.18 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.18 , isNatKind(n__0()) -> tt() 878.72/297.18 , isNatKind(n__plus(V1, V2)) -> 878.72/297.18 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.18 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.18 , isNatKind(n__x(V1, V2)) -> 878.72/297.18 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.18 , activate(X) -> X 878.72/297.18 , activate(n__0()) -> 0() 878.72/297.18 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.18 , activate(n__s(X)) -> s(activate(X)) 878.72/297.18 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.18 , U103(tt(), M, N) -> 878.72/297.18 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.18 , isNat(n__0()) -> tt() 878.72/297.18 , isNat(n__plus(V1, V2)) -> 878.72/297.18 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.18 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.18 , isNat(n__x(V1, V2)) -> 878.72/297.18 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.18 , U104(tt(), M, N) -> 878.72/297.18 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.18 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.18 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.18 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.18 , x(X1, X2) -> n__x(X1, X2) 878.72/297.18 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.18 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.18 , U11(tt(), V1, V2) -> 878.72/297.18 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.18 , U12(tt(), V1, V2) -> 878.72/297.18 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.18 , U13(tt(), V1, V2) -> 878.72/297.18 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.18 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.18 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.18 , U16(tt()) -> tt() 878.72/297.18 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.18 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.18 , U23(tt()) -> tt() 878.72/297.18 , U31(tt(), V1, V2) -> 878.72/297.18 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.18 , U32(tt(), V1, V2) -> 878.72/297.18 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.18 , U33(tt(), V1, V2) -> 878.72/297.18 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.18 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.18 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.18 , U36(tt()) -> tt() 878.72/297.18 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.18 , U42(tt()) -> tt() 878.72/297.18 , U51(tt()) -> tt() 878.72/297.18 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.18 , U62(tt()) -> tt() 878.72/297.18 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.18 , U72(tt(), N) -> activate(N) 878.72/297.18 , U81(tt(), M, N) -> 878.72/297.18 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.18 , U82(tt(), M, N) -> 878.72/297.18 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.18 , U83(tt(), M, N) -> 878.72/297.18 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.18 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.18 , s(X) -> n__s(X) 878.72/297.18 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.18 , U92(tt()) -> 0() 878.72/297.18 , 0() -> n__0() } 878.72/297.18 Weak DPs: 878.72/297.18 { isNatKind^#(n__0()) -> c_3() 878.72/297.18 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.18 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.18 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.18 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.18 , U51^#(tt()) -> c_41() 878.72/297.18 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.18 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.18 , 0^#() -> c_53() 878.72/297.18 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.18 , isNat^#(n__0()) -> c_13() 878.72/297.18 , isNat^#(n__s(V1)) -> 878.72/297.18 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.18 , U21^#(tt(), V1) -> 878.72/297.18 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.18 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.18 , U12^#(tt(), V1, V2) -> 878.72/297.18 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , U13^#(tt(), V1, V2) -> 878.72/297.18 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , U14^#(tt(), V1, V2) -> 878.72/297.18 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.18 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.18 , U16^#(tt()) -> c_29() 878.72/297.18 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.18 , U23^#(tt()) -> c_32() 878.72/297.18 , U32^#(tt(), V1, V2) -> 878.72/297.18 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , U33^#(tt(), V1, V2) -> 878.72/297.18 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.18 , U34^#(tt(), V1, V2) -> 878.72/297.18 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.18 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.18 , U36^#(tt()) -> c_38() 878.72/297.18 , U42^#(tt()) -> c_40() 878.72/297.18 , U62^#(tt()) -> c_43() 878.72/297.18 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.18 Obligation: 878.72/297.18 runtime complexity 878.72/297.18 Answer: 878.72/297.18 MAYBE 878.72/297.18 878.72/297.18 We estimate the number of application of {17,18} by applications of 878.72/297.18 Pre({17,18}) = {4,8,11,12,15,16}. Here rules are labeled as 878.72/297.18 follows: 878.72/297.18 878.72/297.18 DPs: 878.72/297.18 { 1: U101^#(tt(), M, N) -> 878.72/297.18 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.18 , 2: U102^#(tt(), M, N) -> 878.72/297.18 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.18 , 3: U103^#(tt(), M, N) -> 878.72/297.18 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.18 , 4: activate^#(X) -> c_7(X) 878.72/297.18 , 5: activate^#(n__plus(X1, X2)) -> 878.72/297.18 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.18 , 6: activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.18 , 7: activate^#(n__x(X1, X2)) -> 878.72/297.18 c_11(x^#(activate(X1), activate(X2))) 878.72/297.18 , 8: plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.18 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.18 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.18 , 11: s^#(X) -> c_50(X) 878.72/297.18 , 12: x^#(X1, X2) -> c_21(X1, X2) 878.72/297.18 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.18 , 14: U104^#(tt(), M, N) -> 878.72/297.18 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.18 , 15: isNat^#(n__plus(V1, V2)) -> 878.72/297.18 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 16: isNat^#(n__x(V1, V2)) -> 878.72/297.18 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 17: U11^#(tt(), V1, V2) -> 878.72/297.18 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 18: U31^#(tt(), V1, V2) -> 878.72/297.18 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.18 , 19: U81^#(tt(), M, N) -> 878.72/297.18 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.18 , 20: U71^#(tt(), N) -> 878.72/297.18 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.18 , 21: U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.18 , 22: U82^#(tt(), M, N) -> 878.72/297.18 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.18 , 23: U83^#(tt(), M, N) -> 878.72/297.18 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.18 , 24: U84^#(tt(), M, N) -> 878.72/297.18 c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.18 , 25: isNatKind^#(n__0()) -> c_3() 878.72/297.18 , 26: isNatKind^#(n__plus(V1, V2)) -> 878.72/297.18 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , 27: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.18 , 28: isNatKind^#(n__x(V1, V2)) -> 878.72/297.18 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.18 , 29: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.19 , 30: U51^#(tt()) -> c_41() 878.72/297.19 , 31: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.19 , 32: activate^#(n__0()) -> c_8(0^#()) 878.72/297.19 , 33: 0^#() -> c_53() 878.72/297.19 , 34: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.19 , 35: isNat^#(n__0()) -> c_13() 878.72/297.19 , 36: isNat^#(n__s(V1)) -> 878.72/297.19 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , 37: U21^#(tt(), V1) -> 878.72/297.19 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , 38: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.19 , 39: U12^#(tt(), V1, V2) -> 878.72/297.19 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 40: U13^#(tt(), V1, V2) -> 878.72/297.19 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 41: U14^#(tt(), V1, V2) -> 878.72/297.19 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , 42: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.19 , 43: U16^#(tt()) -> c_29() 878.72/297.19 , 44: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.19 , 45: U23^#(tt()) -> c_32() 878.72/297.19 , 46: U32^#(tt(), V1, V2) -> 878.72/297.19 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 47: U33^#(tt(), V1, V2) -> 878.72/297.19 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 48: U34^#(tt(), V1, V2) -> 878.72/297.19 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.19 , 50: U36^#(tt()) -> c_38() 878.72/297.19 , 51: U42^#(tt()) -> c_40() 878.72/297.19 , 52: U62^#(tt()) -> c_43() 878.72/297.19 , 53: U92^#(tt()) -> c_52(0^#()) } 878.72/297.19 878.72/297.19 We are left with following problem, upon which TcT provides the 878.72/297.19 certificate MAYBE. 878.72/297.19 878.72/297.19 Strict DPs: 878.72/297.19 { U101^#(tt(), M, N) -> 878.72/297.19 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.19 , U102^#(tt(), M, N) -> 878.72/297.19 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.19 , U103^#(tt(), M, N) -> 878.72/297.19 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.19 , activate^#(X) -> c_7(X) 878.72/297.19 , activate^#(n__plus(X1, X2)) -> 878.72/297.19 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.19 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.19 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.19 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.19 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.19 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.19 , s^#(X) -> c_50(X) 878.72/297.19 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.19 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.19 , U104^#(tt(), M, N) -> 878.72/297.19 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.19 , isNat^#(n__plus(V1, V2)) -> 878.72/297.19 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , isNat^#(n__x(V1, V2)) -> 878.72/297.19 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , U81^#(tt(), M, N) -> 878.72/297.19 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.19 , U71^#(tt(), N) -> 878.72/297.19 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.19 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.19 , U82^#(tt(), M, N) -> 878.72/297.19 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.19 , U83^#(tt(), M, N) -> 878.72/297.19 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.19 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 878.72/297.19 Strict Trs: 878.72/297.19 { U101(tt(), M, N) -> 878.72/297.19 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.19 , U102(tt(), M, N) -> 878.72/297.19 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.19 , isNatKind(n__0()) -> tt() 878.72/297.19 , isNatKind(n__plus(V1, V2)) -> 878.72/297.19 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.19 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.19 , isNatKind(n__x(V1, V2)) -> 878.72/297.19 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.19 , activate(X) -> X 878.72/297.19 , activate(n__0()) -> 0() 878.72/297.19 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.19 , activate(n__s(X)) -> s(activate(X)) 878.72/297.19 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.19 , U103(tt(), M, N) -> 878.72/297.19 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.19 , isNat(n__0()) -> tt() 878.72/297.19 , isNat(n__plus(V1, V2)) -> 878.72/297.19 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.19 , isNat(n__x(V1, V2)) -> 878.72/297.19 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , U104(tt(), M, N) -> 878.72/297.19 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.19 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.19 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.19 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.19 , x(X1, X2) -> n__x(X1, X2) 878.72/297.19 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.19 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.19 , U11(tt(), V1, V2) -> 878.72/297.19 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , U12(tt(), V1, V2) -> 878.72/297.19 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U13(tt(), V1, V2) -> 878.72/297.19 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.19 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.19 , U16(tt()) -> tt() 878.72/297.19 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.19 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.19 , U23(tt()) -> tt() 878.72/297.19 , U31(tt(), V1, V2) -> 878.72/297.19 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , U32(tt(), V1, V2) -> 878.72/297.19 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U33(tt(), V1, V2) -> 878.72/297.19 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.19 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.19 , U36(tt()) -> tt() 878.72/297.19 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.19 , U42(tt()) -> tt() 878.72/297.19 , U51(tt()) -> tt() 878.72/297.19 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.19 , U62(tt()) -> tt() 878.72/297.19 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.19 , U72(tt(), N) -> activate(N) 878.72/297.19 , U81(tt(), M, N) -> 878.72/297.19 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.19 , U82(tt(), M, N) -> 878.72/297.19 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.19 , U83(tt(), M, N) -> 878.72/297.19 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.19 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.19 , s(X) -> n__s(X) 878.72/297.19 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.19 , U92(tt()) -> 0() 878.72/297.19 , 0() -> n__0() } 878.72/297.19 Weak DPs: 878.72/297.19 { isNatKind^#(n__0()) -> c_3() 878.72/297.19 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.19 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.19 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.19 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.19 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.19 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.19 , U51^#(tt()) -> c_41() 878.72/297.19 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.19 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.19 , 0^#() -> c_53() 878.72/297.19 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.19 , isNat^#(n__0()) -> c_13() 878.72/297.19 , isNat^#(n__s(V1)) -> 878.72/297.19 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , U11^#(tt(), V1, V2) -> 878.72/297.19 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , U21^#(tt(), V1) -> 878.72/297.19 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , U31^#(tt(), V1, V2) -> 878.72/297.19 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.19 , U12^#(tt(), V1, V2) -> 878.72/297.19 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U13^#(tt(), V1, V2) -> 878.72/297.19 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U14^#(tt(), V1, V2) -> 878.72/297.19 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.19 , U16^#(tt()) -> c_29() 878.72/297.19 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.19 , U23^#(tt()) -> c_32() 878.72/297.19 , U32^#(tt(), V1, V2) -> 878.72/297.19 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U33^#(tt(), V1, V2) -> 878.72/297.19 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U34^#(tt(), V1, V2) -> 878.72/297.19 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.19 , U36^#(tt()) -> c_38() 878.72/297.19 , U42^#(tt()) -> c_40() 878.72/297.19 , U62^#(tt()) -> c_43() 878.72/297.19 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.19 Obligation: 878.72/297.19 runtime complexity 878.72/297.19 Answer: 878.72/297.19 MAYBE 878.72/297.19 878.72/297.19 We estimate the number of application of {15,16} by applications of 878.72/297.19 Pre({15,16}) = {4,8,11,12}. Here rules are labeled as follows: 878.72/297.19 878.72/297.19 DPs: 878.72/297.19 { 1: U101^#(tt(), M, N) -> 878.72/297.19 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.19 , 2: U102^#(tt(), M, N) -> 878.72/297.19 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.19 , 3: U103^#(tt(), M, N) -> 878.72/297.19 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.19 , 4: activate^#(X) -> c_7(X) 878.72/297.19 , 5: activate^#(n__plus(X1, X2)) -> 878.72/297.19 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.19 , 6: activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.19 , 7: activate^#(n__x(X1, X2)) -> 878.72/297.19 c_11(x^#(activate(X1), activate(X2))) 878.72/297.19 , 8: plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.19 , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.19 , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.19 , 11: s^#(X) -> c_50(X) 878.72/297.19 , 12: x^#(X1, X2) -> c_21(X1, X2) 878.72/297.19 , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.19 , 14: U104^#(tt(), M, N) -> 878.72/297.19 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.19 , 15: isNat^#(n__plus(V1, V2)) -> 878.72/297.19 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , 16: isNat^#(n__x(V1, V2)) -> 878.72/297.19 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , 17: U81^#(tt(), M, N) -> 878.72/297.19 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.19 , 18: U71^#(tt(), N) -> 878.72/297.19 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.19 , 19: U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.19 , 20: U82^#(tt(), M, N) -> 878.72/297.19 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.19 , 21: U83^#(tt(), M, N) -> 878.72/297.19 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.19 , 22: U84^#(tt(), M, N) -> 878.72/297.19 c_49(s^#(plus(activate(N), activate(M)))) 878.72/297.19 , 23: isNatKind^#(n__0()) -> c_3() 878.72/297.19 , 24: isNatKind^#(n__plus(V1, V2)) -> 878.72/297.19 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.19 , 25: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.19 , 26: isNatKind^#(n__x(V1, V2)) -> 878.72/297.19 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.19 , 27: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.19 , 28: U51^#(tt()) -> c_41() 878.72/297.19 , 29: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.19 , 30: activate^#(n__0()) -> c_8(0^#()) 878.72/297.19 , 31: 0^#() -> c_53() 878.72/297.19 , 32: x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.19 , 33: isNat^#(n__0()) -> c_13() 878.72/297.19 , 34: isNat^#(n__s(V1)) -> 878.72/297.19 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , 35: U11^#(tt(), V1, V2) -> 878.72/297.19 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , 36: U21^#(tt(), V1) -> 878.72/297.19 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , 37: U31^#(tt(), V1, V2) -> 878.72/297.19 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , 38: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.19 , 39: U12^#(tt(), V1, V2) -> 878.72/297.19 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 40: U13^#(tt(), V1, V2) -> 878.72/297.19 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 41: U14^#(tt(), V1, V2) -> 878.72/297.19 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , 42: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.19 , 43: U16^#(tt()) -> c_29() 878.72/297.19 , 44: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.19 , 45: U23^#(tt()) -> c_32() 878.72/297.19 , 46: U32^#(tt(), V1, V2) -> 878.72/297.19 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 47: U33^#(tt(), V1, V2) -> 878.72/297.19 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , 48: U34^#(tt(), V1, V2) -> 878.72/297.19 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.19 , 50: U36^#(tt()) -> c_38() 878.72/297.19 , 51: U42^#(tt()) -> c_40() 878.72/297.19 , 52: U62^#(tt()) -> c_43() 878.72/297.19 , 53: U92^#(tt()) -> c_52(0^#()) } 878.72/297.19 878.72/297.19 We are left with following problem, upon which TcT provides the 878.72/297.19 certificate MAYBE. 878.72/297.19 878.72/297.19 Strict DPs: 878.72/297.19 { U101^#(tt(), M, N) -> 878.72/297.19 c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.19 , U102^#(tt(), M, N) -> 878.72/297.19 c_2(U103^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.19 , U103^#(tt(), M, N) -> 878.72/297.19 c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.19 , activate^#(X) -> c_7(X) 878.72/297.19 , activate^#(n__plus(X1, X2)) -> 878.72/297.19 c_9(plus^#(activate(X1), activate(X2))) 878.72/297.19 , activate^#(n__s(X)) -> c_10(s^#(activate(X))) 878.72/297.19 , activate^#(n__x(X1, X2)) -> c_11(x^#(activate(X1), activate(X2))) 878.72/297.19 , plus^#(X1, X2) -> c_18(X1, X2) 878.72/297.19 , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N)) 878.72/297.19 , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N)) 878.72/297.19 , s^#(X) -> c_50(X) 878.72/297.19 , x^#(X1, X2) -> c_21(X1, X2) 878.72/297.19 , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N)) 878.72/297.19 , U104^#(tt(), M, N) -> 878.72/297.19 c_17(plus^#(x(activate(N), activate(M)), activate(N))) 878.72/297.19 , U81^#(tt(), M, N) -> 878.72/297.19 c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N))) 878.72/297.19 , U71^#(tt(), N) -> 878.72/297.19 c_44(U72^#(isNatKind(activate(N)), activate(N))) 878.72/297.19 , U72^#(tt(), N) -> c_45(activate^#(N)) 878.72/297.19 , U82^#(tt(), M, N) -> 878.72/297.19 c_47(U83^#(isNat(activate(N)), activate(M), activate(N))) 878.72/297.19 , U83^#(tt(), M, N) -> 878.72/297.19 c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N))) 878.72/297.19 , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) } 878.72/297.19 Strict Trs: 878.72/297.19 { U101(tt(), M, N) -> 878.72/297.19 U102(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.19 , U102(tt(), M, N) -> 878.72/297.19 U103(isNat(activate(N)), activate(M), activate(N)) 878.72/297.19 , isNatKind(n__0()) -> tt() 878.72/297.19 , isNatKind(n__plus(V1, V2)) -> 878.72/297.19 U41(isNatKind(activate(V1)), activate(V2)) 878.72/297.19 , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 878.72/297.19 , isNatKind(n__x(V1, V2)) -> 878.72/297.19 U61(isNatKind(activate(V1)), activate(V2)) 878.72/297.19 , activate(X) -> X 878.72/297.19 , activate(n__0()) -> 0() 878.72/297.19 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 878.72/297.19 , activate(n__s(X)) -> s(activate(X)) 878.72/297.19 , activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) 878.72/297.19 , U103(tt(), M, N) -> 878.72/297.19 U104(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.19 , isNat(n__0()) -> tt() 878.72/297.19 , isNat(n__plus(V1, V2)) -> 878.72/297.19 U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 878.72/297.19 , isNat(n__x(V1, V2)) -> 878.72/297.19 U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , U104(tt(), M, N) -> 878.72/297.19 plus(x(activate(N), activate(M)), activate(N)) 878.72/297.19 , plus(X1, X2) -> n__plus(X1, X2) 878.72/297.19 , plus(N, s(M)) -> U81(isNat(M), M, N) 878.72/297.19 , plus(N, 0()) -> U71(isNat(N), N) 878.72/297.19 , x(X1, X2) -> n__x(X1, X2) 878.72/297.19 , x(N, s(M)) -> U101(isNat(M), M, N) 878.72/297.19 , x(N, 0()) -> U91(isNat(N), N) 878.72/297.19 , U11(tt(), V1, V2) -> 878.72/297.19 U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , U12(tt(), V1, V2) -> 878.72/297.19 U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U13(tt(), V1, V2) -> 878.72/297.19 U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 878.72/297.19 , U15(tt(), V2) -> U16(isNat(activate(V2))) 878.72/297.19 , U16(tt()) -> tt() 878.72/297.19 , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1)) 878.72/297.19 , U22(tt(), V1) -> U23(isNat(activate(V1))) 878.72/297.19 , U23(tt()) -> tt() 878.72/297.19 , U31(tt(), V1, V2) -> 878.72/297.19 U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 878.72/297.19 , U32(tt(), V1, V2) -> 878.72/297.19 U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U33(tt(), V1, V2) -> 878.72/297.19 U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 878.72/297.19 , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 878.72/297.19 , U35(tt(), V2) -> U36(isNat(activate(V2))) 878.72/297.19 , U36(tt()) -> tt() 878.72/297.19 , U41(tt(), V2) -> U42(isNatKind(activate(V2))) 878.72/297.19 , U42(tt()) -> tt() 878.72/297.19 , U51(tt()) -> tt() 878.72/297.19 , U61(tt(), V2) -> U62(isNatKind(activate(V2))) 878.72/297.19 , U62(tt()) -> tt() 878.72/297.19 , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N)) 878.72/297.19 , U72(tt(), N) -> activate(N) 878.72/297.19 , U81(tt(), M, N) -> 878.72/297.19 U82(isNatKind(activate(M)), activate(M), activate(N)) 878.72/297.19 , U82(tt(), M, N) -> 878.72/297.19 U83(isNat(activate(N)), activate(M), activate(N)) 878.72/297.19 , U83(tt(), M, N) -> 878.72/297.19 U84(isNatKind(activate(N)), activate(M), activate(N)) 878.72/297.19 , U84(tt(), M, N) -> s(plus(activate(N), activate(M))) 878.72/297.19 , s(X) -> n__s(X) 878.72/297.19 , U91(tt(), N) -> U92(isNatKind(activate(N))) 878.72/297.19 , U92(tt()) -> 0() 878.72/297.19 , 0() -> n__0() } 878.72/297.19 Weak DPs: 878.72/297.19 { isNatKind^#(n__0()) -> c_3() 878.72/297.19 , isNatKind^#(n__plus(V1, V2)) -> 878.72/297.19 c_4(U41^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.19 , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1)))) 878.72/297.19 , isNatKind^#(n__x(V1, V2)) -> 878.72/297.19 c_6(U61^#(isNatKind(activate(V1)), activate(V2))) 878.72/297.19 , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2)))) 878.72/297.19 , U51^#(tt()) -> c_41() 878.72/297.19 , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2)))) 878.72/297.19 , activate^#(n__0()) -> c_8(0^#()) 878.72/297.19 , 0^#() -> c_53() 878.72/297.19 , x^#(N, 0()) -> c_23(U91^#(isNat(N), N)) 878.72/297.19 , isNat^#(n__0()) -> c_13() 878.72/297.19 , isNat^#(n__plus(V1, V2)) -> 878.72/297.19 c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , isNat^#(n__s(V1)) -> 878.72/297.19 c_15(U21^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , isNat^#(n__x(V1, V2)) -> 878.72/297.19 c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , U11^#(tt(), V1, V2) -> 878.72/297.19 c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , U21^#(tt(), V1) -> 878.72/297.19 c_30(U22^#(isNatKind(activate(V1)), activate(V1))) 878.72/297.19 , U31^#(tt(), V1, V2) -> 878.72/297.19 c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2))) 878.72/297.19 , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N)))) 878.72/297.19 , U12^#(tt(), V1, V2) -> 878.72/297.19 c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U13^#(tt(), V1, V2) -> 878.72/297.19 c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U14^#(tt(), V1, V2) -> 878.72/297.19 c_27(U15^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2)))) 878.72/297.19 , U16^#(tt()) -> c_29() 878.72/297.19 , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1)))) 878.72/297.19 , U23^#(tt()) -> c_32() 878.72/297.19 , U32^#(tt(), V1, V2) -> 878.72/297.19 c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U33^#(tt(), V1, V2) -> 878.72/297.19 c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2))) 878.72/297.19 , U34^#(tt(), V1, V2) -> 878.72/297.19 c_36(U35^#(isNat(activate(V1)), activate(V2))) 878.72/297.19 , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2)))) 878.72/297.19 , U36^#(tt()) -> c_38() 878.72/297.19 , U42^#(tt()) -> c_40() 878.72/297.19 , U62^#(tt()) -> c_43() 878.72/297.19 , U92^#(tt()) -> c_52(0^#()) } 878.72/297.19 Obligation: 878.72/297.19 runtime complexity 878.72/297.19 Answer: 878.72/297.19 MAYBE 878.72/297.19 878.72/297.19 Empty strict component of the problem is NOT empty. 878.72/297.19 878.72/297.19 878.72/297.19 Arrrr.. 879.04/297.30 EOF