* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a() -> c()
a() -> d()
append(cons(n,l1),l2) -> cons(n,append(l1,l2))
append(nil(),l2) -> l2
if(false(),false(),l,t,n) -> lessE(l,t,s(n))
if(false(),true(),l,t,n) -> t
if(true(),b,l,t,n) -> l
le(0(),m) -> true()
le(s(n),0()) -> false()
le(s(n),s(m)) -> le(n,m)
length(cons(n,l)) -> s(length(l))
length(nil()) -> 0()
lessE(l,t,n) -> if(le(length(l),n),le(length(toList(t)),n),l,t,n)
lessElements(l,t) -> lessE(l,t,0())
toList(leaf()) -> nil()
toList(node(t1,n,t2)) -> append(toList(t1),cons(n,toList(t2)))
- Signature:
{a/0,append/2,if/5,le/2,length/1,lessE/3,lessElements/2,toList/1} / {0/0,c/0,cons/2,d/0,false/0,leaf/0,nil/0
,node/3,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,append,if,le,length,lessE,lessElements
,toList} and constructors {0,c,cons,d,false,leaf,nil,node,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a() -> c()
a() -> d()
append(cons(n,l1),l2) -> cons(n,append(l1,l2))
append(nil(),l2) -> l2
if(false(),false(),l,t,n) -> lessE(l,t,s(n))
if(false(),true(),l,t,n) -> t
if(true(),b,l,t,n) -> l
le(0(),m) -> true()
le(s(n),0()) -> false()
le(s(n),s(m)) -> le(n,m)
length(cons(n,l)) -> s(length(l))
length(nil()) -> 0()
lessE(l,t,n) -> if(le(length(l),n),le(length(toList(t)),n),l,t,n)
lessElements(l,t) -> lessE(l,t,0())
toList(leaf()) -> nil()
toList(node(t1,n,t2)) -> append(toList(t1),cons(n,toList(t2)))
- Signature:
{a/0,append/2,if/5,le/2,length/1,lessE/3,lessElements/2,toList/1} / {0/0,c/0,cons/2,d/0,false/0,leaf/0,nil/0
,node/3,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,append,if,le,length,lessE,lessElements
,toList} and constructors {0,c,cons,d,false,leaf,nil,node,s,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
append(y,z){y -> cons(x,y)} =
append(cons(x,y),z) ->^+ cons(x,append(y,z))
= C[append(y,z) = append(y,z){}]
WORST_CASE(Omega(n^1),?)