* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
+(x,h()) -> x
+(+(x,y),z) -> +(x,+(y,z))
+(h(),x) -> x
+(s(x),s(y)) -> s(s(+(x,y)))
a(l,x,s(y),h()) -> a(l,x,y,s(h()))
a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z))
a(l,s(x),h(),z) -> a(l,x,z,z)
a(h(),h(),h(),x) -> s(x)
a(s(l),h(),h(),z) -> a(l,z,h(),z)
app(l,nil()) -> l
app(cons(x,l),k) -> cons(x,app(l,k))
app(nil(),k) -> k
s(h()) -> 1()
sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h(),h()),l))
sum(cons(x,nil())) -> cons(x,nil())
- Signature:
{+/2,a/4,app/2,s/1,sum/1} / {1/0,cons/2,h/0,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,a,app,s,sum} and constructors {1,cons,h,nil}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
+(x,h()) -> x
+(+(x,y),z) -> +(x,+(y,z))
+(h(),x) -> x
+(s(x),s(y)) -> s(s(+(x,y)))
a(l,x,s(y),h()) -> a(l,x,y,s(h()))
a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z))
a(l,s(x),h(),z) -> a(l,x,z,z)
a(h(),h(),h(),x) -> s(x)
a(s(l),h(),h(),z) -> a(l,z,h(),z)
app(l,nil()) -> l
app(cons(x,l),k) -> cons(x,app(l,k))
app(nil(),k) -> k
s(h()) -> 1()
sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h(),h()),l))
sum(cons(x,nil())) -> cons(x,nil())
- Signature:
{+/2,a/4,app/2,s/1,sum/1} / {1/0,cons/2,h/0,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,a,app,s,sum} and constructors {1,cons,h,nil}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
app(y,z){y -> cons(x,y)} =
app(cons(x,y),z) ->^+ cons(x,app(y,z))
= C[app(y,z) = app(y,z){}]
WORST_CASE(Omega(n^1),?)