* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
fcons(X,Z) -> cons(X,Z)
first(0(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
first1(0(),Z) -> nil1()
first1(s(X),cons(Y,Z)) -> cons1(quote(Y),first1(X,Z))
from(X) -> cons(X,from(s(X)))
quote(0()) -> 01()
quote(s(X)) -> s1(quote(X))
quote(sel(X,Z)) -> sel1(X,Z)
quote1(cons(X,Z)) -> cons1(quote(X),quote1(Z))
quote1(first(X,Z)) -> first1(X,Z)
quote1(nil()) -> nil1()
sel(0(),cons(X,Z)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,Z)
sel1(0(),cons(X,Z)) -> quote(X)
sel1(s(X),cons(Y,Z)) -> sel1(X,Z)
unquote(01()) -> 0()
unquote(s1(X)) -> s(unquote(X))
unquote1(cons1(X,Z)) -> fcons(unquote(X),unquote1(Z))
unquote1(nil1()) -> nil()
- Signature:
{fcons/2,first/2,first1/2,from/1,quote/1,quote1/1,sel/2,sel1/2,unquote/1,unquote1/1} / {0/0,01/0,cons/2
,cons1/2,nil/0,nil1/0,s/1,s1/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fcons,first,first1,from,quote,quote1,sel,sel1,unquote
,unquote1} and constructors {0,01,cons,cons1,nil,nil1,s,s1}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
fcons(X,Z) -> cons(X,Z)
first(0(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
first1(0(),Z) -> nil1()
first1(s(X),cons(Y,Z)) -> cons1(quote(Y),first1(X,Z))
from(X) -> cons(X,from(s(X)))
quote(0()) -> 01()
quote(s(X)) -> s1(quote(X))
quote(sel(X,Z)) -> sel1(X,Z)
quote1(cons(X,Z)) -> cons1(quote(X),quote1(Z))
quote1(first(X,Z)) -> first1(X,Z)
quote1(nil()) -> nil1()
sel(0(),cons(X,Z)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,Z)
sel1(0(),cons(X,Z)) -> quote(X)
sel1(s(X),cons(Y,Z)) -> sel1(X,Z)
unquote(01()) -> 0()
unquote(s1(X)) -> s(unquote(X))
unquote1(cons1(X,Z)) -> fcons(unquote(X),unquote1(Z))
unquote1(nil1()) -> nil()
- Signature:
{fcons/2,first/2,first1/2,from/1,quote/1,quote1/1,sel/2,sel1/2,unquote/1,unquote1/1} / {0/0,01/0,cons/2
,cons1/2,nil/0,nil1/0,s/1,s1/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {fcons,first,first1,from,quote,quote1,sel,sel1,unquote
,unquote1} and constructors {0,01,cons,cons1,nil,nil1,s,s1}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
first(x,z){x -> s(x),z -> cons(y,z)} =
first(s(x),cons(y,z)) ->^+ cons(y,first(x,z))
= C[first(x,z) = first(x,z){}]
WORST_CASE(Omega(n^1),?)