* 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))