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