* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
f() -> g()
f() -> h()
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
if(false(),b,x,y) -> logZeroError()
if(true(),false(),x,s(y)) -> y
if(true(),true(),x,y) -> logIter(x,y)
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
logarithm(x) -> logIter(x,0())
- Signature:
{f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,half,if,inc,le,logIter,logarithm} and constructors {0
,false,g,h,logZeroError,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
f() -> g()
f() -> h()
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
if(false(),b,x,y) -> logZeroError()
if(true(),false(),x,s(y)) -> y
if(true(),true(),x,y) -> logIter(x,y)
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y))
logarithm(x) -> logIter(x,0())
- Signature:
{f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,half,if,inc,le,logIter,logarithm} and constructors {0
,false,g,h,logZeroError,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),?)