* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: addlist(Cons(x,xs'),Cons(S(0()),xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Cons(S(0()),xs'),Cons(x,xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Nil(),ys) -> Nil() goal(xs,ys) -> addlist(xs,ys) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {addlist/2,goal/2,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {addlist,goal,notEmpty} and constructors {0,Cons,False,Nil ,S,True} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: addlist(Cons(x,xs'),Cons(S(0()),xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Cons(S(0()),xs'),Cons(x,xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Nil(),ys) -> Nil() goal(xs,ys) -> addlist(xs,ys) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {addlist/2,goal/2,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {addlist,goal,notEmpty} and constructors {0,Cons,False,Nil ,S,True} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: addlist(y,z){y -> Cons(x,y),z -> Cons(S(0()),z)} = addlist(Cons(x,y),Cons(S(0()),z)) ->^+ Cons(S(x),addlist(y,z)) = C[addlist(y,z) = addlist(y,z){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: addlist(Cons(x,xs'),Cons(S(0()),xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Cons(S(0()),xs'),Cons(x,xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Nil(),ys) -> Nil() goal(xs,ys) -> addlist(xs,ys) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {addlist/2,goal/2,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {addlist,goal,notEmpty} and constructors {0,Cons,False,Nil ,S,True} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. 0_0() -> 2 Cons_0(2,2) -> 2 Cons_1(3,4) -> 1 Cons_1(3,4) -> 4 False_0() -> 2 False_1() -> 1 Nil_0() -> 2 Nil_1() -> 1 Nil_1() -> 4 S_0(2) -> 2 S_1(2) -> 3 True_0() -> 2 True_1() -> 1 addlist_0(2,2) -> 1 addlist_1(2,2) -> 1 addlist_1(2,2) -> 4 goal_0(2,2) -> 1 notEmpty_0(2) -> 1 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: addlist(Cons(x,xs'),Cons(S(0()),xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Cons(S(0()),xs'),Cons(x,xs)) -> Cons(S(x),addlist(xs',xs)) addlist(Nil(),ys) -> Nil() goal(xs,ys) -> addlist(xs,ys) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {addlist/2,goal/2,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {addlist,goal,notEmpty} and constructors {0,Cons,False,Nil ,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))