* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
del(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
del(x,nil()) -> nil()
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
ge(x,0()) -> true()
ge(0(),s(x)) -> false()
ge(s(x),s(y)) -> ge(x,y)
if1(false(),x,y,xs) -> max(cons(y,xs))
if1(true(),x,y,xs) -> max(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,del(x,xs))
if2(true(),x,y,xs) -> xs
max(cons(x,cons(y,xs))) -> if1(ge(x,y),x,y,xs)
max(cons(x,nil())) -> x
max(nil()) -> 0()
sort(cons(x,xs)) -> cons(max(cons(x,xs)),sort(del(max(cons(x,xs)),cons(x,xs))))
sort(nil()) -> nil()
- Signature:
{del/2,eq/2,ge/2,if1/4,if2/4,max/1,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {del,eq,ge,if1,if2,max,sort} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
del(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
del(x,nil()) -> nil()
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
ge(x,0()) -> true()
ge(0(),s(x)) -> false()
ge(s(x),s(y)) -> ge(x,y)
if1(false(),x,y,xs) -> max(cons(y,xs))
if1(true(),x,y,xs) -> max(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,del(x,xs))
if2(true(),x,y,xs) -> xs
max(cons(x,cons(y,xs))) -> if1(ge(x,y),x,y,xs)
max(cons(x,nil())) -> x
max(nil()) -> 0()
sort(cons(x,xs)) -> cons(max(cons(x,xs)),sort(del(max(cons(x,xs)),cons(x,xs))))
sort(nil()) -> nil()
- Signature:
{del/2,eq/2,ge/2,if1/4,if2/4,max/1,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {del,eq,ge,if1,if2,max,sort} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
eq(x,y){x -> s(x),y -> s(y)} =
eq(s(x),s(y)) ->^+ eq(x,y)
= C[eq(x,y) = eq(x,y){}]
WORST_CASE(Omega(n^1),?)