* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
if(false(),x,y) -> log2(half(x),y)
if(true(),x,s(y)) -> y
inc(0()) -> 0()
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(x) -> log2(x,0())
log2(x,y) -> if(le(x,s(0())),x,inc(y))
- Signature:
{half/1,if/3,inc/1,le/2,log/1,log2/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,if,inc,le,log,log2} and constructors {0,false,s
,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
if(false(),x,y) -> log2(half(x),y)
if(true(),x,s(y)) -> y
inc(0()) -> 0()
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(x) -> log2(x,0())
log2(x,y) -> if(le(x,s(0())),x,inc(y))
- Signature:
{half/1,if/3,inc/1,le/2,log/1,log2/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,if,inc,le,log,log2} and constructors {0,false,s
,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
half(x){x -> s(s(x))} =
half(s(s(x))) ->^+ s(half(x))
= C[half(x) = half(x){}]
WORST_CASE(Omega(n^1),?)