* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__U11(X1,X2) -> U11(X1,X2)
a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
a__U12(X1,X2) -> U12(X1,X2)
a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
a__U13(X) -> U13(X)
a__U13(tt()) -> tt()
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(X1,X2) -> U22(X1,X2)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(X) -> U23(X)
a__U23(tt()) -> tt()
a__U31(X1,X2) -> U31(X1,X2)
a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
a__U32(X1,X2) -> U32(X1,X2)
a__U32(tt(),V) -> a__U33(a__isNatList(V))
a__U33(X) -> U33(X)
a__U33(tt()) -> tt()
a__U41(X1,X2,X3) -> U41(X1,X2,X3)
a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
a__U42(X1,X2,X3) -> U42(X1,X2,X3)
a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
a__U43(X1,X2,X3) -> U43(X1,X2,X3)
a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
a__U44(X1,X2,X3) -> U44(X1,X2,X3)
a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
a__U45(X1,X2) -> U45(X1,X2)
a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
a__U46(X) -> U46(X)
a__U46(tt()) -> tt()
a__U51(X1,X2) -> U51(X1,X2)
a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
a__U52(X) -> U52(X)
a__U52(tt()) -> tt()
a__U61(X) -> U61(X)
a__U61(tt()) -> tt()
a__U71(X) -> U71(X)
a__U71(tt()) -> tt()
a__U81(X1,X2,X3) -> U81(X1,X2,X3)
a__U81(tt(),V1,V2) -> a__U82(a__isNatKind(V1),V1,V2)
a__U82(X1,X2,X3) -> U82(X1,X2,X3)
a__U82(tt(),V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2)
a__U83(X1,X2,X3) -> U83(X1,X2,X3)
a__U83(tt(),V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2)
a__U84(X1,X2,X3) -> U84(X1,X2,X3)
a__U84(tt(),V1,V2) -> a__U85(a__isNat(V1),V2)
a__U85(X1,X2) -> U85(X1,X2)
a__U85(tt(),V2) -> a__U86(a__isNatList(V2))
a__U86(X) -> U86(X)
a__U86(tt()) -> tt()
a__U91(X1,X2,X3) -> U91(X1,X2,X3)
a__U91(tt(),L,N) -> a__U92(a__isNatIListKind(L),L,N)
a__U92(X1,X2,X3) -> U92(X1,X2,X3)
a__U92(tt(),L,N) -> a__U93(a__isNat(N),L,N)
a__U93(X1,X2,X3) -> U93(X1,X2,X3)
a__U93(tt(),L,N) -> a__U94(a__isNatKind(N),L)
a__U94(X1,X2) -> U94(X1,X2)
a__U94(tt(),L) -> s(a__length(mark(L)))
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
a__isNatIList(X) -> isNatIList(X)
a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
a__isNatIList(zeros()) -> tt()
a__isNatIListKind(X) -> isNatIListKind(X)
a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
a__isNatIListKind(nil()) -> tt()
a__isNatIListKind(zeros()) -> tt()
a__isNatKind(X) -> isNatKind(X)
a__isNatKind(0()) -> tt()
a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1))
a__isNatList(X) -> isNatList(X)
a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2)
a__isNatList(nil()) -> tt()
a__length(X) -> length(X)
a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N)
a__length(nil()) -> 0()
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
mark(U13(X)) -> a__U13(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
mark(U33(X)) -> a__U33(mark(X))
mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
mark(U46(X)) -> a__U46(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X)) -> a__U52(mark(X))
mark(U61(X)) -> a__U61(mark(X))
mark(U71(X)) -> a__U71(mark(X))
mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3)
mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3)
mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3)
mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3)
mark(U85(X1,X2)) -> a__U85(mark(X1),X2)
mark(U86(X)) -> a__U86(mark(X))
mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
mark(U94(X1,X2)) -> a__U94(mark(X1),X2)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(isNatIList(X)) -> a__isNatIList(X)
mark(isNatIListKind(X)) -> a__isNatIListKind(X)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(isNatList(X)) -> a__isNatList(X)
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
mark(zeros()) -> a__zeros()
- Signature:
{a__U11/2,a__U12/2,a__U13/1,a__U21/2,a__U22/2,a__U23/1,a__U31/2,a__U32/2,a__U33/1,a__U41/3,a__U42/3,a__U43/3
,a__U44/3,a__U45/2,a__U46/1,a__U51/2,a__U52/1,a__U61/1,a__U71/1,a__U81/3,a__U82/3,a__U83/3,a__U84/3,a__U85/2
,a__U86/1,a__U91/3,a__U92/3,a__U93/3,a__U94/2,a__isNat/1,a__isNatIList/1,a__isNatIListKind/1,a__isNatKind/1
,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2,U12/2,U13/1,U21/2,U22/2,U23/1,U31/2,U32/2,U33/1
,U41/3,U42/3,U43/3,U44/3,U45/2,U46/1,U51/2,U52/1,U61/1,U71/1,U81/3,U82/3,U83/3,U84/3,U85/2,U86/1,U91/3,U92/3
,U93/3,U94/2,cons/2,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,nil/0,s/1,tt/0
,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__U13,a__U21,a__U22,a__U23,a__U31,a__U32
,a__U33,a__U41,a__U42,a__U43,a__U44,a__U45,a__U46,a__U51,a__U52,a__U61,a__U71,a__U81,a__U82,a__U83,a__U84
,a__U85,a__U86,a__U91,a__U92,a__U93,a__U94,a__isNat,a__isNatIList,a__isNatIListKind,a__isNatKind
,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U12,U13,U21,U22,U23,U31,U32,U33,U41,U42,U43
,U44,U45,U46,U51,U52,U61,U71,U81,U82,U83,U84,U85,U86,U91,U92,U93,U94,cons,isNat,isNatIList,isNatIListKind
,isNatKind,isNatList,length,nil,s,tt,zeros}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__U11(X1,X2) -> U11(X1,X2)
a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
a__U12(X1,X2) -> U12(X1,X2)
a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
a__U13(X) -> U13(X)
a__U13(tt()) -> tt()
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(X1,X2) -> U22(X1,X2)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(X) -> U23(X)
a__U23(tt()) -> tt()
a__U31(X1,X2) -> U31(X1,X2)
a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
a__U32(X1,X2) -> U32(X1,X2)
a__U32(tt(),V) -> a__U33(a__isNatList(V))
a__U33(X) -> U33(X)
a__U33(tt()) -> tt()
a__U41(X1,X2,X3) -> U41(X1,X2,X3)
a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
a__U42(X1,X2,X3) -> U42(X1,X2,X3)
a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
a__U43(X1,X2,X3) -> U43(X1,X2,X3)
a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
a__U44(X1,X2,X3) -> U44(X1,X2,X3)
a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
a__U45(X1,X2) -> U45(X1,X2)
a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
a__U46(X) -> U46(X)
a__U46(tt()) -> tt()
a__U51(X1,X2) -> U51(X1,X2)
a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
a__U52(X) -> U52(X)
a__U52(tt()) -> tt()
a__U61(X) -> U61(X)
a__U61(tt()) -> tt()
a__U71(X) -> U71(X)
a__U71(tt()) -> tt()
a__U81(X1,X2,X3) -> U81(X1,X2,X3)
a__U81(tt(),V1,V2) -> a__U82(a__isNatKind(V1),V1,V2)
a__U82(X1,X2,X3) -> U82(X1,X2,X3)
a__U82(tt(),V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2)
a__U83(X1,X2,X3) -> U83(X1,X2,X3)
a__U83(tt(),V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2)
a__U84(X1,X2,X3) -> U84(X1,X2,X3)
a__U84(tt(),V1,V2) -> a__U85(a__isNat(V1),V2)
a__U85(X1,X2) -> U85(X1,X2)
a__U85(tt(),V2) -> a__U86(a__isNatList(V2))
a__U86(X) -> U86(X)
a__U86(tt()) -> tt()
a__U91(X1,X2,X3) -> U91(X1,X2,X3)
a__U91(tt(),L,N) -> a__U92(a__isNatIListKind(L),L,N)
a__U92(X1,X2,X3) -> U92(X1,X2,X3)
a__U92(tt(),L,N) -> a__U93(a__isNat(N),L,N)
a__U93(X1,X2,X3) -> U93(X1,X2,X3)
a__U93(tt(),L,N) -> a__U94(a__isNatKind(N),L)
a__U94(X1,X2) -> U94(X1,X2)
a__U94(tt(),L) -> s(a__length(mark(L)))
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
a__isNatIList(X) -> isNatIList(X)
a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
a__isNatIList(zeros()) -> tt()
a__isNatIListKind(X) -> isNatIListKind(X)
a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
a__isNatIListKind(nil()) -> tt()
a__isNatIListKind(zeros()) -> tt()
a__isNatKind(X) -> isNatKind(X)
a__isNatKind(0()) -> tt()
a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1))
a__isNatList(X) -> isNatList(X)
a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2)
a__isNatList(nil()) -> tt()
a__length(X) -> length(X)
a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N)
a__length(nil()) -> 0()
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
mark(U13(X)) -> a__U13(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
mark(U33(X)) -> a__U33(mark(X))
mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
mark(U46(X)) -> a__U46(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X)) -> a__U52(mark(X))
mark(U61(X)) -> a__U61(mark(X))
mark(U71(X)) -> a__U71(mark(X))
mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3)
mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3)
mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3)
mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3)
mark(U85(X1,X2)) -> a__U85(mark(X1),X2)
mark(U86(X)) -> a__U86(mark(X))
mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
mark(U94(X1,X2)) -> a__U94(mark(X1),X2)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(isNatIList(X)) -> a__isNatIList(X)
mark(isNatIListKind(X)) -> a__isNatIListKind(X)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(isNatList(X)) -> a__isNatList(X)
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
mark(zeros()) -> a__zeros()
- Signature:
{a__U11/2,a__U12/2,a__U13/1,a__U21/2,a__U22/2,a__U23/1,a__U31/2,a__U32/2,a__U33/1,a__U41/3,a__U42/3,a__U43/3
,a__U44/3,a__U45/2,a__U46/1,a__U51/2,a__U52/1,a__U61/1,a__U71/1,a__U81/3,a__U82/3,a__U83/3,a__U84/3,a__U85/2
,a__U86/1,a__U91/3,a__U92/3,a__U93/3,a__U94/2,a__isNat/1,a__isNatIList/1,a__isNatIListKind/1,a__isNatKind/1
,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2,U12/2,U13/1,U21/2,U22/2,U23/1,U31/2,U32/2,U33/1
,U41/3,U42/3,U43/3,U44/3,U45/2,U46/1,U51/2,U52/1,U61/1,U71/1,U81/3,U82/3,U83/3,U84/3,U85/2,U86/1,U91/3,U92/3
,U93/3,U94/2,cons/2,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,nil/0,s/1,tt/0
,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__U13,a__U21,a__U22,a__U23,a__U31,a__U32
,a__U33,a__U41,a__U42,a__U43,a__U44,a__U45,a__U46,a__U51,a__U52,a__U61,a__U71,a__U81,a__U82,a__U83,a__U84
,a__U85,a__U86,a__U91,a__U92,a__U93,a__U94,a__isNat,a__isNatIList,a__isNatIListKind,a__isNatKind
,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U12,U13,U21,U22,U23,U31,U32,U33,U41,U42,U43
,U44,U45,U46,U51,U52,U61,U71,U81,U82,U83,U84,U85,U86,U91,U92,U93,U94,cons,isNat,isNatIList,isNatIListKind
,isNatKind,isNatList,length,nil,s,tt,zeros}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
a__isNatKind(x){x -> s(x)} =
a__isNatKind(s(x)) ->^+ a__U71(a__isNatKind(x))
= C[a__isNatKind(x) = a__isNatKind(x){}]
WORST_CASE(Omega(n^1),?)