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