* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__head(X) -> head(X)
a__head(cons(X,XS)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
a__nats() -> cons(0(),incr(nats()))
a__nats() -> nats()
a__odds() -> a__incr(a__pairs())
a__odds() -> odds()
a__pairs() -> cons(0(),incr(odds()))
a__pairs() -> pairs()
a__tail(X) -> tail(X)
a__tail(cons(X,XS)) -> mark(XS)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(head(X)) -> a__head(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(odds()) -> a__odds()
mark(pairs()) -> a__pairs()
mark(s(X)) -> s(mark(X))
mark(tail(X)) -> a__tail(mark(X))
- Signature:
{a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0
,odds/0,pairs/0,s/1,tail/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail
,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__head(X) -> head(X)
a__head(cons(X,XS)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
a__nats() -> cons(0(),incr(nats()))
a__nats() -> nats()
a__odds() -> a__incr(a__pairs())
a__odds() -> odds()
a__pairs() -> cons(0(),incr(odds()))
a__pairs() -> pairs()
a__tail(X) -> tail(X)
a__tail(cons(X,XS)) -> mark(XS)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(head(X)) -> a__head(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(odds()) -> a__odds()
mark(pairs()) -> a__pairs()
mark(s(X)) -> s(mark(X))
mark(tail(X)) -> a__tail(mark(X))
- Signature:
{a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0
,odds/0,pairs/0,s/1,tail/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail
,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(x){x -> cons(x,y)} =
mark(cons(x,y)) ->^+ cons(mark(x),y)
= C[mark(x) = mark(x){}]
WORST_CASE(Omega(n^1),?)