* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
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) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
U13(mark(X)) -> mark(U13(X))
U13(ok(X)) -> ok(U13(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X)) -> mark(U22(X))
U22(ok(X)) -> ok(U22(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) -> mark(U32(X1,X2))
U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
U33(mark(X)) -> mark(U33(X))
U33(ok(X)) -> ok(U33(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
U61(mark(X)) -> mark(U61(X))
U61(ok(X)) -> ok(U61(X))
U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
active(U12(X1,X2)) -> U12(active(X1),X2)
active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
active(U13(X)) -> U13(active(X))
active(U13(tt())) -> mark(tt())
active(U21(X1,X2)) -> U21(active(X1),X2)
active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
active(U22(X)) -> U22(active(X))
active(U22(tt())) -> mark(tt())
active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
active(U32(X1,X2)) -> U32(active(X1),X2)
active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
active(U33(X)) -> U33(active(X))
active(U33(tt())) -> mark(tt())
active(U41(X1,X2)) -> U41(active(X1),X2)
active(U41(tt(),N)) -> mark(N)
active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
active(U61(X)) -> U61(active(X))
active(U61(tt())) -> mark(0())
active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isNat(0())) -> mark(tt())
active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
active(isNatKind(0())) -> mark(tt())
active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),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(U61(and(isNat(N),isNatKind(N))))
active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
active(x(X1,X2)) -> x(X1,active(X2))
active(x(X1,X2)) -> x(active(X1),X2)
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(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(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
proper(U13(X)) -> U13(proper(X))
proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
proper(U61(X)) -> U61(proper(X))
proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
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:
{U11/3,U12/2,U13/1,U21/2,U22/1,U31/3,U32/2,U33/1,U41/2,U51/3,U61/1,U71/3,active/1,and/2,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 {U11,U12,U13,U21,U22,U31,U32,U33,U41,U51,U61,U71,active
,and,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:
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) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
U13(mark(X)) -> mark(U13(X))
U13(ok(X)) -> ok(U13(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X)) -> mark(U22(X))
U22(ok(X)) -> ok(U22(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) -> mark(U32(X1,X2))
U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
U33(mark(X)) -> mark(U33(X))
U33(ok(X)) -> ok(U33(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
U61(mark(X)) -> mark(U61(X))
U61(ok(X)) -> ok(U61(X))
U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
active(U12(X1,X2)) -> U12(active(X1),X2)
active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
active(U13(X)) -> U13(active(X))
active(U13(tt())) -> mark(tt())
active(U21(X1,X2)) -> U21(active(X1),X2)
active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
active(U22(X)) -> U22(active(X))
active(U22(tt())) -> mark(tt())
active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
active(U32(X1,X2)) -> U32(active(X1),X2)
active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
active(U33(X)) -> U33(active(X))
active(U33(tt())) -> mark(tt())
active(U41(X1,X2)) -> U41(active(X1),X2)
active(U41(tt(),N)) -> mark(N)
active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
active(U61(X)) -> U61(active(X))
active(U61(tt())) -> mark(0())
active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isNat(0())) -> mark(tt())
active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
active(isNatKind(0())) -> mark(tt())
active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),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(U61(and(isNat(N),isNatKind(N))))
active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
active(x(X1,X2)) -> x(X1,active(X2))
active(x(X1,X2)) -> x(active(X1),X2)
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(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(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
proper(U13(X)) -> U13(proper(X))
proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
proper(U61(X)) -> U61(proper(X))
proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
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:
{U11/3,U12/2,U13/1,U21/2,U22/1,U31/3,U32/2,U33/1,U41/2,U51/3,U61/1,U71/3,active/1,and/2,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 {U11,U12,U13,U21,U22,U31,U32,U33,U41,U51,U61,U71,active
,and,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:
U11(x,y,z){x -> mark(x)} =
U11(mark(x),y,z) ->^+ mark(U11(x,y,z))
= C[U11(x,y,z) = U11(x,y,z){}]
WORST_CASE(Omega(n^1),?)