* 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))