* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: goal(x) -> list(x) list(Cons(x,xs)) -> list(xs) list(Nil()) -> True() list(Nil()) -> isEmpty[Match](Nil()) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1} - Obligation: innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True ,isEmpty[Match]} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: goal(x) -> list(x) list(Cons(x,xs)) -> list(xs) list(Nil()) -> True() list(Nil()) -> isEmpty[Match](Nil()) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1} - Obligation: innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True ,isEmpty[Match]} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: list(y){y -> Cons(x,y)} = list(Cons(x,y)) ->^+ list(y) = C[list(y) = list(y){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: goal(x) -> list(x) list(Cons(x,xs)) -> list(xs) list(Nil()) -> True() list(Nil()) -> isEmpty[Match](Nil()) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1} - Obligation: innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True ,isEmpty[Match]} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. Cons_0(2,2) -> 2 False_0() -> 2 False_1() -> 1 Nil_0() -> 2 Nil_1() -> 3 True_0() -> 2 True_1() -> 1 goal_0(2) -> 1 isEmpty[Match]_0(2) -> 2 isEmpty[Match]_1(3) -> 1 list_0(2) -> 1 list_1(2) -> 1 notEmpty_0(2) -> 1 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: goal(x) -> list(x) list(Cons(x,xs)) -> list(xs) list(Nil()) -> True() list(Nil()) -> isEmpty[Match](Nil()) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1} - Obligation: innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True ,isEmpty[Match]} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))