* 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),?)