* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
eq(x,x) -> true()
eq(0(),0()) -> true()
eq(0(),s(x)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
f() -> g()
f() -> h()
ifPlus(false(),x,y,z) -> plus(x,z)
ifPlus(true(),x,y,z) -> y
ifTimes(false(),x,y,z,u) -> timesIter(x,y,u)
ifTimes(true(),x,y,z,u) -> z
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
minus(x,x) -> 0()
minus(x,0()) -> x
minus(0(),x) -> 0()
minus(s(x),s(y)) -> minus(x,y)
plus(x,y) -> ifPlus(eq(x,0()),minus(x,s(0())),x,inc(x))
times(x,y) -> timesIter(x,y,0())
timesIter(x,y,z) -> ifTimes(eq(x,0()),minus(x,s(0())),y,z,plus(y,z))
- Signature:
{eq/2,f/0,ifPlus/4,ifTimes/5,inc/1,minus/2,plus/2,times/2,timesIter/3} / {0/0,false/0,g/0,h/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,f,ifPlus,ifTimes,inc,minus,plus,times
,timesIter} and constructors {0,false,g,h,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
eq(x,x) -> true()
eq(0(),0()) -> true()
eq(0(),s(x)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
f() -> g()
f() -> h()
ifPlus(false(),x,y,z) -> plus(x,z)
ifPlus(true(),x,y,z) -> y
ifTimes(false(),x,y,z,u) -> timesIter(x,y,u)
ifTimes(true(),x,y,z,u) -> z
inc(0()) -> s(0())
inc(s(x)) -> s(inc(x))
minus(x,x) -> 0()
minus(x,0()) -> x
minus(0(),x) -> 0()
minus(s(x),s(y)) -> minus(x,y)
plus(x,y) -> ifPlus(eq(x,0()),minus(x,s(0())),x,inc(x))
times(x,y) -> timesIter(x,y,0())
timesIter(x,y,z) -> ifTimes(eq(x,0()),minus(x,s(0())),y,z,plus(y,z))
- Signature:
{eq/2,f/0,ifPlus/4,ifTimes/5,inc/1,minus/2,plus/2,times/2,timesIter/3} / {0/0,false/0,g/0,h/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,f,ifPlus,ifTimes,inc,minus,plus,times
,timesIter} and constructors {0,false,g,h,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),?)