* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
cond1(true(),x) -> cond2(even(x),x)
cond2(false(),x) -> cond1(neq(x,0()),p(x))
cond2(true(),x) -> cond1(neq(x,0()),div2(x))
div2(0()) -> 0()
div2(s(0())) -> 0()
div2(s(s(x))) -> s(div2(x))
even(0()) -> true()
even(s(0())) -> false()
even(s(s(x))) -> even(x)
neq(0(),0()) -> false()
neq(0(),s(x)) -> true()
neq(s(x),0()) -> true()
neq(s(x),s(y())) -> neq(x,y())
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{cond1/2,cond2/2,div2/1,even/1,neq/2,p/1} / {0/0,false/0,s/1,true/0,y/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond1,cond2,div2,even,neq,p} and constructors {0,false,s
,true,y}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
cond1(true(),x) -> cond2(even(x),x)
cond2(false(),x) -> cond1(neq(x,0()),p(x))
cond2(true(),x) -> cond1(neq(x,0()),div2(x))
div2(0()) -> 0()
div2(s(0())) -> 0()
div2(s(s(x))) -> s(div2(x))
even(0()) -> true()
even(s(0())) -> false()
even(s(s(x))) -> even(x)
neq(0(),0()) -> false()
neq(0(),s(x)) -> true()
neq(s(x),0()) -> true()
neq(s(x),s(y())) -> neq(x,y())
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{cond1/2,cond2/2,div2/1,even/1,neq/2,p/1} / {0/0,false/0,s/1,true/0,y/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond1,cond2,div2,even,neq,p} and constructors {0,false,s
,true,y}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
div2(x){x -> s(s(x))} =
div2(s(s(x))) ->^+ s(div2(x))
= C[div2(x) = div2(x){}]
WORST_CASE(Omega(n^1),?)