* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U12(mark(X)) -> mark(U12(X))
U12(ok(X)) -> ok(U12(X))
U21(mark(X1),X2,X3) -> mark(U21(X1,X2,X3))
U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3))
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) -> mark(U31(X1,X2))
U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
U32(mark(X)) -> mark(U32(X))
U32(ok(X)) -> ok(U32(X))
U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
U42(mark(X1),X2) -> mark(U42(X1,X2))
U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
U43(mark(X)) -> mark(U43(X))
U43(ok(X)) -> ok(U43(X))
U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
U52(mark(X1),X2) -> mark(U52(X1,X2))
U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
U53(mark(X)) -> mark(U53(X))
U53(ok(X)) -> ok(U53(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(X)) -> mark(U72(X))
U72(ok(X)) -> ok(U72(X))
__(X1,mark(X2)) -> mark(__(X1,X2))
__(mark(X1),X2) -> mark(__(X1,X2))
__(ok(X1),ok(X2)) -> ok(__(X1,X2))
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),V)) -> mark(U12(isNeList(V)))
active(U12(X)) -> U12(active(X))
active(U12(tt())) -> mark(tt())
active(U21(X1,X2,X3)) -> U21(active(X1),X2,X3)
active(U21(tt(),V1,V2)) -> mark(U22(isList(V1),V2))
active(U22(X1,X2)) -> U22(active(X1),X2)
active(U22(tt(),V2)) -> mark(U23(isList(V2)))
active(U23(X)) -> U23(active(X))
active(U23(tt())) -> mark(tt())
active(U31(X1,X2)) -> U31(active(X1),X2)
active(U31(tt(),V)) -> mark(U32(isQid(V)))
active(U32(X)) -> U32(active(X))
active(U32(tt())) -> mark(tt())
active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
active(U41(tt(),V1,V2)) -> mark(U42(isList(V1),V2))
active(U42(X1,X2)) -> U42(active(X1),X2)
active(U42(tt(),V2)) -> mark(U43(isNeList(V2)))
active(U43(X)) -> U43(active(X))
active(U43(tt())) -> mark(tt())
active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
active(U51(tt(),V1,V2)) -> mark(U52(isNeList(V1),V2))
active(U52(X1,X2)) -> U52(active(X1),X2)
active(U52(tt(),V2)) -> mark(U53(isList(V2)))
active(U53(X)) -> U53(active(X))
active(U53(tt())) -> mark(tt())
active(U61(X1,X2)) -> U61(active(X1),X2)
active(U61(tt(),V)) -> mark(U62(isQid(V)))
active(U62(X)) -> U62(active(X))
active(U62(tt())) -> mark(tt())
active(U71(X1,X2)) -> U71(active(X1),X2)
active(U71(tt(),V)) -> mark(U72(isNePal(V)))
active(U72(X)) -> U72(active(X))
active(U72(tt())) -> mark(tt())
active(__(X,nil())) -> mark(X)
active(__(X1,X2)) -> __(X1,active(X2))
active(__(X1,X2)) -> __(active(X1),X2)
active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z)))
active(__(nil(),X)) -> mark(X)
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isList(V)) -> mark(U11(isPalListKind(V),V))
active(isList(__(V1,V2))) -> mark(U21(and(isPalListKind(V1),isPalListKind(V2)),V1,V2))
active(isList(nil())) -> mark(tt())
active(isNeList(V)) -> mark(U31(isPalListKind(V),V))
active(isNeList(__(V1,V2))) -> mark(U41(and(isPalListKind(V1),isPalListKind(V2)),V1,V2))
active(isNeList(__(V1,V2))) -> mark(U51(and(isPalListKind(V1),isPalListKind(V2)),V1,V2))
active(isNePal(V)) -> mark(U61(isPalListKind(V),V))
active(isNePal(__(I,__(P,I)))) -> mark(and(and(isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P))))
active(isPal(V)) -> mark(U71(isPalListKind(V),V))
active(isPal(nil())) -> mark(tt())
active(isPalListKind(__(V1,V2))) -> mark(and(isPalListKind(V1),isPalListKind(V2)))
active(isPalListKind(a())) -> mark(tt())
active(isPalListKind(e())) -> mark(tt())
active(isPalListKind(i())) -> mark(tt())
active(isPalListKind(nil())) -> mark(tt())
active(isPalListKind(o())) -> mark(tt())
active(isPalListKind(u())) -> mark(tt())
active(isQid(a())) -> mark(tt())
active(isQid(e())) -> mark(tt())
active(isQid(i())) -> mark(tt())
active(isQid(o())) -> mark(tt())
active(isQid(u())) -> mark(tt())
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
isPalListKind(ok(X)) -> ok(isPalListKind(X))
isQid(ok(X)) -> ok(isQid(X))
proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U12(X)) -> U12(proper(X))
proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3))
proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
proper(U53(X)) -> U53(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(X)) -> U72(proper(X))
proper(__(X1,X2)) -> __(proper(X1),proper(X2))
proper(a()) -> ok(a())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(e()) -> ok(e())
proper(i()) -> ok(i())
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isNePal(X)) -> isNePal(proper(X))
proper(isPal(X)) -> isPal(proper(X))
proper(isPalListKind(X)) -> isPalListKind(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(nil()) -> ok(nil())
proper(o()) -> ok(o())
proper(tt()) -> ok(tt())
proper(u()) -> ok(u())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{U11/2,U12/1,U21/3,U22/2,U23/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/2,U62/1,U71/2,U72/1,__/2
,active/1,and/2,isList/1,isNeList/1,isNePal/1,isPal/1,isPalListKind/1,isQid/1,proper/1,top/1} / {a/0,e/0,i/0
,mark/1,nil/0,o/0,ok/1,tt/0,u/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,U21,U22,U23,U31,U32,U41,U42,U43,U51,U52,U53,U61
,U62,U71,U72,__,active,and,isList,isNeList,isNePal,isPal,isPalListKind,isQid,proper,top} and constructors {a
,e,i,mark,nil,o,ok,tt,u}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U12(mark(X)) -> mark(U12(X))
U12(ok(X)) -> ok(U12(X))
U21(mark(X1),X2,X3) -> mark(U21(X1,X2,X3))
U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3))
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) -> mark(U31(X1,X2))
U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
U32(mark(X)) -> mark(U32(X))
U32(ok(X)) -> ok(U32(X))
U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
U42(mark(X1),X2) -> mark(U42(X1,X2))
U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
U43(mark(X)) -> mark(U43(X))
U43(ok(X)) -> ok(U43(X))
U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
U52(mark(X1),X2) -> mark(U52(X1,X2))
U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
U53(mark(X)) -> mark(U53(X))
U53(ok(X)) -> ok(U53(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(X)) -> mark(U72(X))
U72(ok(X)) -> ok(U72(X))
__(X1,mark(X2)) -> mark(__(X1,X2))
__(mark(X1),X2) -> mark(__(X1,X2))
__(ok(X1),ok(X2)) -> ok(__(X1,X2))
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),V)) -> mark(U12(isNeList(V)))
active(U12(X)) -> U12(active(X))
active(U12(tt())) -> mark(tt())
active(U21(X1,X2,X3)) -> U21(active(X1),X2,X3)
active(U21(tt(),V1,V2)) -> mark(U22(isList(V1),V2))
active(U22(X1,X2)) -> U22(active(X1),X2)
active(U22(tt(),V2)) -> mark(U23(isList(V2)))
active(U23(X)) -> U23(active(X))
active(U23(tt())) -> mark(tt())
active(U31(X1,X2)) -> U31(active(X1),X2)
active(U31(tt(),V)) -> mark(U32(isQid(V)))
active(U32(X)) -> U32(active(X))
active(U32(tt())) -> mark(tt())
active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
active(U41(tt(),V1,V2)) -> mark(U42(isList(V1),V2))
active(U42(X1,X2)) -> U42(active(X1),X2)
active(U42(tt(),V2)) -> mark(U43(isNeList(V2)))
active(U43(X)) -> U43(active(X))
active(U43(tt())) -> mark(tt())
active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
active(U51(tt(),V1,V2)) -> mark(U52(isNeList(V1),V2))
active(U52(X1,X2)) -> U52(active(X1),X2)
active(U52(tt(),V2)) -> mark(U53(isList(V2)))
active(U53(X)) -> U53(active(X))
active(U53(tt())) -> mark(tt())
active(U61(X1,X2)) -> U61(active(X1),X2)
active(U61(tt(),V)) -> mark(U62(isQid(V)))
active(U62(X)) -> U62(active(X))
active(U62(tt())) -> mark(tt())
active(U71(X1,X2)) -> U71(active(X1),X2)
active(U71(tt(),V)) -> mark(U72(isNePal(V)))
active(U72(X)) -> U72(active(X))
active(U72(tt())) -> mark(tt())
active(__(X,nil())) -> mark(X)
active(__(X1,X2)) -> __(X1,active(X2))
active(__(X1,X2)) -> __(active(X1),X2)
active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z)))
active(__(nil(),X)) -> mark(X)
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isList(V)) -> mark(U11(isPalListKind(V),V))
active(isList(__(V1,V2))) -> mark(U21(and(isPalListKind(V1),isPalListKind(V2)),V1,V2))
active(isList(nil())) -> mark(tt())
active(isNeList(V)) -> mark(U31(isPalListKind(V),V))
active(isNeList(__(V1,V2))) -> mark(U41(and(isPalListKind(V1),isPalListKind(V2)),V1,V2))
active(isNeList(__(V1,V2))) -> mark(U51(and(isPalListKind(V1),isPalListKind(V2)),V1,V2))
active(isNePal(V)) -> mark(U61(isPalListKind(V),V))
active(isNePal(__(I,__(P,I)))) -> mark(and(and(isQid(I),isPalListKind(I)),and(isPal(P),isPalListKind(P))))
active(isPal(V)) -> mark(U71(isPalListKind(V),V))
active(isPal(nil())) -> mark(tt())
active(isPalListKind(__(V1,V2))) -> mark(and(isPalListKind(V1),isPalListKind(V2)))
active(isPalListKind(a())) -> mark(tt())
active(isPalListKind(e())) -> mark(tt())
active(isPalListKind(i())) -> mark(tt())
active(isPalListKind(nil())) -> mark(tt())
active(isPalListKind(o())) -> mark(tt())
active(isPalListKind(u())) -> mark(tt())
active(isQid(a())) -> mark(tt())
active(isQid(e())) -> mark(tt())
active(isQid(i())) -> mark(tt())
active(isQid(o())) -> mark(tt())
active(isQid(u())) -> mark(tt())
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
isPalListKind(ok(X)) -> ok(isPalListKind(X))
isQid(ok(X)) -> ok(isQid(X))
proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U12(X)) -> U12(proper(X))
proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3))
proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
proper(U53(X)) -> U53(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(X)) -> U72(proper(X))
proper(__(X1,X2)) -> __(proper(X1),proper(X2))
proper(a()) -> ok(a())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(e()) -> ok(e())
proper(i()) -> ok(i())
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isNePal(X)) -> isNePal(proper(X))
proper(isPal(X)) -> isPal(proper(X))
proper(isPalListKind(X)) -> isPalListKind(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(nil()) -> ok(nil())
proper(o()) -> ok(o())
proper(tt()) -> ok(tt())
proper(u()) -> ok(u())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{U11/2,U12/1,U21/3,U22/2,U23/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/2,U62/1,U71/2,U72/1,__/2
,active/1,and/2,isList/1,isNeList/1,isNePal/1,isPal/1,isPalListKind/1,isQid/1,proper/1,top/1} / {a/0,e/0,i/0
,mark/1,nil/0,o/0,ok/1,tt/0,u/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,U21,U22,U23,U31,U32,U41,U42,U43,U51,U52,U53,U61
,U62,U71,U72,__,active,and,isList,isNeList,isNePal,isPal,isPalListKind,isQid,proper,top} and constructors {a
,e,i,mark,nil,o,ok,tt,u}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
U11(x,y){x -> mark(x)} =
U11(mark(x),y) ->^+ mark(U11(x,y))
= C[U11(x,y) = U11(x,y){}]
WORST_CASE(Omega(n^1),?)