* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
append(cons(y(),ys),x) -> cons(y(),append(ys,x))
append(nil(),x) -> cons(x,nil())
elem(node(l,x,r)) -> x
if(false(),false(),n,m,xs,ys) -> listify(m,xs)
if(false(),true(),n,m,xs,ys) -> listify(n,ys)
if(true(),b,n,m,xs,ys) -> xs
isEmpty(empty()) -> true()
isEmpty(node(l,x,r)) -> false()
left(empty()) -> empty()
left(node(l,x,r)) -> l
listify(n,xs) -> if(isEmpty(n)
,isEmpty(left(n))
,right(n)
,node(left(left(n)),elem(left(n)),node(right(left(n)),elem(n),right(n)))
,xs
,append(xs,n))
right(empty()) -> empty()
right(node(l,x,r)) -> r
toList(n) -> listify(n,nil())
- Signature:
{append/2,elem/1,if/6,isEmpty/1,left/1,listify/2,right/1,toList/1} / {cons/2,empty/0,false/0,nil/0,node/3
,true/0,y/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,elem,if,isEmpty,left,listify,right
,toList} and constructors {cons,empty,false,nil,node,true,y}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
append(cons(y(),ys),x) -> cons(y(),append(ys,x))
append(nil(),x) -> cons(x,nil())
elem(node(l,x,r)) -> x
if(false(),false(),n,m,xs,ys) -> listify(m,xs)
if(false(),true(),n,m,xs,ys) -> listify(n,ys)
if(true(),b,n,m,xs,ys) -> xs
isEmpty(empty()) -> true()
isEmpty(node(l,x,r)) -> false()
left(empty()) -> empty()
left(node(l,x,r)) -> l
listify(n,xs) -> if(isEmpty(n)
,isEmpty(left(n))
,right(n)
,node(left(left(n)),elem(left(n)),node(right(left(n)),elem(n),right(n)))
,xs
,append(xs,n))
right(empty()) -> empty()
right(node(l,x,r)) -> r
toList(n) -> listify(n,nil())
- Signature:
{append/2,elem/1,if/6,isEmpty/1,left/1,listify/2,right/1,toList/1} / {cons/2,empty/0,false/0,nil/0,node/3
,true/0,y/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,elem,if,isEmpty,left,listify,right
,toList} and constructors {cons,empty,false,nil,node,true,y}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
append(x,y){x -> cons(y(),x)} =
append(cons(y(),x),y) ->^+ cons(y(),append(x,y))
= C[append(x,y) = append(x,y){}]
WORST_CASE(Omega(n^1),?)