* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3))
U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3))
U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3))
U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3))
U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3))
U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3))
U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3))
U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3))
U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3))
U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3))
U15(mark(X1),X2) -> mark(U15(X1,X2))
U15(ok(X1),ok(X2)) -> ok(U15(X1,X2))
U16(mark(X)) -> mark(U16(X))
U16(ok(X)) -> ok(U16(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X1),X2) -> mark(U22(X1,X2))
U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
U23(mark(X)) -> mark(U23(X))
U23(ok(X)) -> ok(U23(X))
U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3))
U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3))
U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3))
U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3))
U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3))
U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3))
U35(mark(X1),X2) -> mark(U35(X1,X2))
U35(ok(X1),ok(X2)) -> ok(U35(X1,X2))
U36(mark(X)) -> mark(U36(X))
U36(ok(X)) -> ok(U36(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U42(mark(X)) -> mark(U42(X))
U42(ok(X)) -> ok(U42(X))
U51(mark(X)) -> mark(U51(X))
U51(ok(X)) -> ok(U51(X))
U61(mark(X1),X2) -> mark(U61(X1,X2))
U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
U62(mark(X)) -> mark(U62(X))
U62(ok(X)) -> ok(U62(X))
U71(mark(X1),X2) -> mark(U71(X1,X2))
U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
U72(mark(X1),X2) -> mark(U72(X1,X2))
U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3))
U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3))
U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3))
U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3))
U91(mark(X1),X2) -> mark(U91(X1,X2))
U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
U92(mark(X)) -> mark(U92(X))
U92(ok(X)) -> ok(U92(X))
active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N))
active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3)
active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N))
active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3)
active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N))
active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3)
active(U104(tt(),M,N)) -> mark(plus(x(N,M),N))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2))
active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2))
active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3)
active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2))
active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3)
active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2))
active(U15(X1,X2)) -> U15(active(X1),X2)
active(U15(tt(),V2)) -> mark(U16(isNat(V2)))
active(U16(X)) -> U16(active(X))
active(U16(tt())) -> mark(tt())
active(U21(X1,X2)) -> U21(active(X1),X2)
active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
active(U22(X1,X2)) -> U22(active(X1),X2)
active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
active(U23(X)) -> U23(active(X))
active(U23(tt())) -> mark(tt())
active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2))
active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3)
active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2))
active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3)
active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2))
active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3)
active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2))
active(U35(X1,X2)) -> U35(active(X1),X2)
active(U35(tt(),V2)) -> mark(U36(isNat(V2)))
active(U36(X)) -> U36(active(X))
active(U36(tt())) -> mark(tt())
active(U41(X1,X2)) -> U41(active(X1),X2)
active(U41(tt(),V2)) -> mark(U42(isNatKind(V2)))
active(U42(X)) -> U42(active(X))
active(U42(tt())) -> mark(tt())
active(U51(X)) -> U51(active(X))
active(U51(tt())) -> mark(tt())
active(U61(X1,X2)) -> U61(active(X1),X2)
active(U61(tt(),V2)) -> mark(U62(isNatKind(V2)))
active(U62(X)) -> U62(active(X))
active(U62(tt())) -> mark(tt())
active(U71(X1,X2)) -> U71(active(X1),X2)
active(U71(tt(),N)) -> mark(U72(isNatKind(N),N))
active(U72(X1,X2)) -> U72(active(X1),X2)
active(U72(tt(),N)) -> mark(N)
active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N))
active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N))
active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3)
active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N))
active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3)
active(U84(tt(),M,N)) -> mark(s(plus(N,M)))
active(U91(X1,X2)) -> U91(active(X1),X2)
active(U91(tt(),N)) -> mark(U92(isNatKind(N)))
active(U92(X)) -> U92(active(X))
active(U92(tt())) -> mark(0())
active(isNat(0())) -> mark(tt())
active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2))
active(isNatKind(0())) -> mark(tt())
active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2))
active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2))
active(plus(N,0())) -> mark(U71(isNat(N),N))
active(plus(N,s(M))) -> mark(U81(isNat(M),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
active(x(N,0())) -> mark(U91(isNat(N),N))
active(x(N,s(M))) -> mark(U101(isNat(M),M,N))
active(x(X1,X2)) -> x(X1,active(X2))
active(x(X1,X2)) -> x(active(X1),X2)
isNat(ok(X)) -> ok(isNat(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
plus(X1,mark(X2)) -> mark(plus(X1,X2))
plus(mark(X1),X2) -> mark(plus(X1,X2))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
proper(0()) -> ok(0())
proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3))
proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3))
proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3))
proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3))
proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3))
proper(U15(X1,X2)) -> U15(proper(X1),proper(X2))
proper(U16(X)) -> U16(proper(X))
proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3))
proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3))
proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3))
proper(U35(X1,X2)) -> U35(proper(X1),proper(X2))
proper(U36(X)) -> U36(proper(X))
proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X)) -> U51(proper(X))
proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3))
proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3))
proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
proper(U92(X)) -> U92(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
proper(x(X1,X2)) -> x(proper(X1),proper(X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
x(X1,mark(X2)) -> mark(x(X1,X2))
x(mark(X1),X2) -> mark(x(X1,X2))
x(ok(X1),ok(X2)) -> ok(x(X1,X2))
- Signature:
{U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3,U34/3
,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,active/1,isNat/1
,isNatKind/1,plus/2,proper/1,s/1,top/1,x/2} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U101,U102,U103,U104,U11,U12,U13,U14,U15,U16,U21,U22,U23
,U31,U32,U33,U34,U35,U36,U41,U42,U51,U61,U62,U71,U72,U81,U82,U83,U84,U91,U92,active,isNat,isNatKind,plus
,proper,s,top,x} and constructors {0,mark,ok,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3))
U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3))
U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3))
U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3))
U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3))
U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3))
U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3))
U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3))
U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3))
U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3))
U15(mark(X1),X2) -> mark(U15(X1,X2))
U15(ok(X1),ok(X2)) -> ok(U15(X1,X2))
U16(mark(X)) -> mark(U16(X))
U16(ok(X)) -> ok(U16(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X1),X2) -> mark(U22(X1,X2))
U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
U23(mark(X)) -> mark(U23(X))
U23(ok(X)) -> ok(U23(X))
U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
U32(mark(X1),X2,X3) -> mark(U32(X1,X2,X3))
U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3))
U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3))
U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3))
U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3))
U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3))
U35(mark(X1),X2) -> mark(U35(X1,X2))
U35(ok(X1),ok(X2)) -> ok(U35(X1,X2))
U36(mark(X)) -> mark(U36(X))
U36(ok(X)) -> ok(U36(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U42(mark(X)) -> mark(U42(X))
U42(ok(X)) -> ok(U42(X))
U51(mark(X)) -> mark(U51(X))
U51(ok(X)) -> ok(U51(X))
U61(mark(X1),X2) -> mark(U61(X1,X2))
U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
U62(mark(X)) -> mark(U62(X))
U62(ok(X)) -> ok(U62(X))
U71(mark(X1),X2) -> mark(U71(X1,X2))
U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
U72(mark(X1),X2) -> mark(U72(X1,X2))
U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3))
U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3))
U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3))
U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3))
U91(mark(X1),X2) -> mark(U91(X1,X2))
U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
U92(mark(X)) -> mark(U92(X))
U92(ok(X)) -> ok(U92(X))
active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
active(U101(tt(),M,N)) -> mark(U102(isNatKind(M),M,N))
active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3)
active(U102(tt(),M,N)) -> mark(U103(isNat(N),M,N))
active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3)
active(U103(tt(),M,N)) -> mark(U104(isNatKind(N),M,N))
active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3)
active(U104(tt(),M,N)) -> mark(plus(x(N,M),N))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2))
active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2))
active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3)
active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2))
active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3)
active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2))
active(U15(X1,X2)) -> U15(active(X1),X2)
active(U15(tt(),V2)) -> mark(U16(isNat(V2)))
active(U16(X)) -> U16(active(X))
active(U16(tt())) -> mark(tt())
active(U21(X1,X2)) -> U21(active(X1),X2)
active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
active(U22(X1,X2)) -> U22(active(X1),X2)
active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
active(U23(X)) -> U23(active(X))
active(U23(tt())) -> mark(tt())
active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
active(U31(tt(),V1,V2)) -> mark(U32(isNatKind(V1),V1,V2))
active(U32(X1,X2,X3)) -> U32(active(X1),X2,X3)
active(U32(tt(),V1,V2)) -> mark(U33(isNatKind(V2),V1,V2))
active(U33(X1,X2,X3)) -> U33(active(X1),X2,X3)
active(U33(tt(),V1,V2)) -> mark(U34(isNatKind(V2),V1,V2))
active(U34(X1,X2,X3)) -> U34(active(X1),X2,X3)
active(U34(tt(),V1,V2)) -> mark(U35(isNat(V1),V2))
active(U35(X1,X2)) -> U35(active(X1),X2)
active(U35(tt(),V2)) -> mark(U36(isNat(V2)))
active(U36(X)) -> U36(active(X))
active(U36(tt())) -> mark(tt())
active(U41(X1,X2)) -> U41(active(X1),X2)
active(U41(tt(),V2)) -> mark(U42(isNatKind(V2)))
active(U42(X)) -> U42(active(X))
active(U42(tt())) -> mark(tt())
active(U51(X)) -> U51(active(X))
active(U51(tt())) -> mark(tt())
active(U61(X1,X2)) -> U61(active(X1),X2)
active(U61(tt(),V2)) -> mark(U62(isNatKind(V2)))
active(U62(X)) -> U62(active(X))
active(U62(tt())) -> mark(tt())
active(U71(X1,X2)) -> U71(active(X1),X2)
active(U71(tt(),N)) -> mark(U72(isNatKind(N),N))
active(U72(X1,X2)) -> U72(active(X1),X2)
active(U72(tt(),N)) -> mark(N)
active(U81(X1,X2,X3)) -> U81(active(X1),X2,X3)
active(U81(tt(),M,N)) -> mark(U82(isNatKind(M),M,N))
active(U82(X1,X2,X3)) -> U82(active(X1),X2,X3)
active(U82(tt(),M,N)) -> mark(U83(isNat(N),M,N))
active(U83(X1,X2,X3)) -> U83(active(X1),X2,X3)
active(U83(tt(),M,N)) -> mark(U84(isNatKind(N),M,N))
active(U84(X1,X2,X3)) -> U84(active(X1),X2,X3)
active(U84(tt(),M,N)) -> mark(s(plus(N,M)))
active(U91(X1,X2)) -> U91(active(X1),X2)
active(U91(tt(),N)) -> mark(U92(isNatKind(N)))
active(U92(X)) -> U92(active(X))
active(U92(tt())) -> mark(0())
active(isNat(0())) -> mark(tt())
active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
active(isNat(x(V1,V2))) -> mark(U31(isNatKind(V1),V1,V2))
active(isNatKind(0())) -> mark(tt())
active(isNatKind(plus(V1,V2))) -> mark(U41(isNatKind(V1),V2))
active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
active(isNatKind(x(V1,V2))) -> mark(U61(isNatKind(V1),V2))
active(plus(N,0())) -> mark(U71(isNat(N),N))
active(plus(N,s(M))) -> mark(U81(isNat(M),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
active(x(N,0())) -> mark(U91(isNat(N),N))
active(x(N,s(M))) -> mark(U101(isNat(M),M,N))
active(x(X1,X2)) -> x(X1,active(X2))
active(x(X1,X2)) -> x(active(X1),X2)
isNat(ok(X)) -> ok(isNat(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
plus(X1,mark(X2)) -> mark(plus(X1,X2))
plus(mark(X1),X2) -> mark(plus(X1,X2))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
proper(0()) -> ok(0())
proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3))
proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3))
proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3))
proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3))
proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3))
proper(U15(X1,X2)) -> U15(proper(X1),proper(X2))
proper(U16(X)) -> U16(proper(X))
proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
proper(U32(X1,X2,X3)) -> U32(proper(X1),proper(X2),proper(X3))
proper(U33(X1,X2,X3)) -> U33(proper(X1),proper(X2),proper(X3))
proper(U34(X1,X2,X3)) -> U34(proper(X1),proper(X2),proper(X3))
proper(U35(X1,X2)) -> U35(proper(X1),proper(X2))
proper(U36(X)) -> U36(proper(X))
proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(U51(X)) -> U51(proper(X))
proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
proper(U72(X1,X2)) -> U72(proper(X1),proper(X2))
proper(U81(X1,X2,X3)) -> U81(proper(X1),proper(X2),proper(X3))
proper(U82(X1,X2,X3)) -> U82(proper(X1),proper(X2),proper(X3))
proper(U83(X1,X2,X3)) -> U83(proper(X1),proper(X2),proper(X3))
proper(U84(X1,X2,X3)) -> U84(proper(X1),proper(X2),proper(X3))
proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
proper(U92(X)) -> U92(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
proper(x(X1,X2)) -> x(proper(X1),proper(X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
x(X1,mark(X2)) -> mark(x(X1,X2))
x(mark(X1),X2) -> mark(x(X1,X2))
x(ok(X1),ok(X2)) -> ok(x(X1,X2))
- Signature:
{U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3,U34/3
,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,active/1,isNat/1
,isNatKind/1,plus/2,proper/1,s/1,top/1,x/2} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U101,U102,U103,U104,U11,U12,U13,U14,U15,U16,U21,U22,U23
,U31,U32,U33,U34,U35,U36,U41,U42,U51,U61,U62,U71,U72,U81,U82,U83,U84,U91,U92,active,isNat,isNatKind,plus
,proper,s,top,x} and constructors {0,mark,ok,tt}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
U101(x,y,z){x -> mark(x)} =
U101(mark(x),y,z) ->^+ mark(U101(x,y,z))
= C[U101(x,y,z) = U101(x,y,z){}]
WORST_CASE(Omega(n^1),?)