* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),V1) -> U12(isNatList(activate(V1)))
U12(tt()) -> tt()
U21(tt(),V1) -> U22(isNat(activate(V1)))
U22(tt()) -> tt()
U31(tt(),V) -> U32(isNatList(activate(V)))
U32(tt()) -> tt()
U41(tt(),V1,V2) -> U42(isNat(activate(V1)),activate(V2))
U42(tt(),V2) -> U43(isNatIList(activate(V2)))
U43(tt()) -> tt()
U51(tt(),V1,V2) -> U52(isNat(activate(V1)),activate(V2))
U52(tt(),V2) -> U53(isNatList(activate(V2)))
U53(tt()) -> tt()
U61(tt(),L) -> s(length(activate(L)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__and(X1,X2)) -> and(activate(X1),X2)
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__isNat(X)) -> isNat(X)
activate(n__isNatIListKind(X)) -> isNatIListKind(X)
activate(n__isNatKind(X)) -> isNatKind(X)
activate(n__length(X)) -> length(activate(X))
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(activate(X))
activate(n__zeros()) -> zeros()
and(X1,X2) -> n__and(X1,X2)
and(tt(),X) -> activate(X)
cons(X1,X2) -> n__cons(X1,X2)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)),activate(V1))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatIList(V) -> U31(isNatIListKind(activate(V)),activate(V))
isNatIList(n__cons(V1,V2)) -> U41(and(isNatKind(activate(V1)),n__isNatIListKind(activate(V2)))
,activate(V1)
,activate(V2))
isNatIList(n__zeros()) -> tt()
isNatIListKind(X) -> n__isNatIListKind(X)
isNatIListKind(n__cons(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatIListKind(activate(V2)))
isNatIListKind(n__nil()) -> tt()
isNatIListKind(n__zeros()) -> tt()
isNatKind(X) -> n__isNatKind(X)
isNatKind(n__0()) -> tt()
isNatKind(n__length(V1)) -> isNatIListKind(activate(V1))
isNatKind(n__s(V1)) -> isNatKind(activate(V1))
isNatList(n__cons(V1,V2)) -> U51(and(isNatKind(activate(V1)),n__isNatIListKind(activate(V2)))
,activate(V1)
,activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
length(cons(N,L)) -> U61(and(and(isNatList(activate(L)),n__isNatIListKind(activate(L)))
,n__and(n__isNat(N),n__isNatKind(N)))
,activate(L))
length(nil()) -> 0()
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/2,U12/1,U21/2,U22/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/2,activate/1,and/2,cons/2
,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,nil/0,s/1,zeros/0} / {n__0/0
,n__and/2,n__cons/2,n__isNat/1,n__isNatIListKind/1,n__isNatKind/1,n__length/1,n__nil/0,n__s/1,n__zeros/0
,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U22,U31,U32,U41,U42,U43,U51,U52,U53,U61
,activate,and,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length,nil,s
,zeros} and constructors {n__0,n__and,n__cons,n__isNat,n__isNatIListKind,n__isNatKind,n__length,n__nil,n__s
,n__zeros,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),V1) -> U12(isNatList(activate(V1)))
U12(tt()) -> tt()
U21(tt(),V1) -> U22(isNat(activate(V1)))
U22(tt()) -> tt()
U31(tt(),V) -> U32(isNatList(activate(V)))
U32(tt()) -> tt()
U41(tt(),V1,V2) -> U42(isNat(activate(V1)),activate(V2))
U42(tt(),V2) -> U43(isNatIList(activate(V2)))
U43(tt()) -> tt()
U51(tt(),V1,V2) -> U52(isNat(activate(V1)),activate(V2))
U52(tt(),V2) -> U53(isNatList(activate(V2)))
U53(tt()) -> tt()
U61(tt(),L) -> s(length(activate(L)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__and(X1,X2)) -> and(activate(X1),X2)
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__isNat(X)) -> isNat(X)
activate(n__isNatIListKind(X)) -> isNatIListKind(X)
activate(n__isNatKind(X)) -> isNatKind(X)
activate(n__length(X)) -> length(activate(X))
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(activate(X))
activate(n__zeros()) -> zeros()
and(X1,X2) -> n__and(X1,X2)
and(tt(),X) -> activate(X)
cons(X1,X2) -> n__cons(X1,X2)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)),activate(V1))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatIList(V) -> U31(isNatIListKind(activate(V)),activate(V))
isNatIList(n__cons(V1,V2)) -> U41(and(isNatKind(activate(V1)),n__isNatIListKind(activate(V2)))
,activate(V1)
,activate(V2))
isNatIList(n__zeros()) -> tt()
isNatIListKind(X) -> n__isNatIListKind(X)
isNatIListKind(n__cons(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatIListKind(activate(V2)))
isNatIListKind(n__nil()) -> tt()
isNatIListKind(n__zeros()) -> tt()
isNatKind(X) -> n__isNatKind(X)
isNatKind(n__0()) -> tt()
isNatKind(n__length(V1)) -> isNatIListKind(activate(V1))
isNatKind(n__s(V1)) -> isNatKind(activate(V1))
isNatList(n__cons(V1,V2)) -> U51(and(isNatKind(activate(V1)),n__isNatIListKind(activate(V2)))
,activate(V1)
,activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
length(cons(N,L)) -> U61(and(and(isNatList(activate(L)),n__isNatIListKind(activate(L)))
,n__and(n__isNat(N),n__isNatKind(N)))
,activate(L))
length(nil()) -> 0()
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/2,U12/1,U21/2,U22/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/2,activate/1,and/2,cons/2
,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,nil/0,s/1,zeros/0} / {n__0/0
,n__and/2,n__cons/2,n__isNat/1,n__isNatIListKind/1,n__isNatKind/1,n__length/1,n__nil/0,n__s/1,n__zeros/0
,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U22,U31,U32,U41,U42,U43,U51,U52,U53,U61
,activate,and,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length,nil,s
,zeros} and constructors {n__0,n__and,n__cons,n__isNat,n__isNatIListKind,n__isNatKind,n__length,n__nil,n__s
,n__zeros,tt}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__and(x,y)} =
activate(n__and(x,y)) ->^+ and(activate(x),y)
= C[activate(x) = activate(x){}]
WORST_CASE(Omega(n^1),?)