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