* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(a__add(mark(X),mark(Y)))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(a__dbl(mark(X))))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)),a__dbl(mark(X))))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
- Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1
,s/1,sqr/1,terms/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(a__add(mark(X),mark(Y)))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(a__dbl(mark(X))))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)),a__dbl(mark(X))))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
- Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1
,s/1,sqr/1,terms/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(x){x -> add(x,y)} =
mark(add(x,y)) ->^+ a__add(mark(x),mark(y))
= C[mark(x) = mark(x){}]
WORST_CASE(Omega(n^1),?)