* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
+ Considered Problem:
- Strict TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
goal(xs) -> shuffle(xs)
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
shuffle(Cons(x,xs)) -> Cons(x,shuffle(reverse(xs)))
shuffle(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1} / {Cons/2,Nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,goal,reverse,shuffle} and constructors {Cons,Nil}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
goal(xs) -> shuffle(xs)
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
shuffle(Cons(x,xs)) -> Cons(x,shuffle(reverse(xs)))
shuffle(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1} / {Cons/2,Nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,goal,reverse,shuffle} and constructors {Cons,Nil}
+ 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){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
goal(xs) -> shuffle(xs)
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
shuffle(Cons(x,xs)) -> Cons(x,shuffle(reverse(xs)))
shuffle(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1} / {Cons/2,Nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,goal,reverse,shuffle} and constructors {Cons,Nil}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
append#(Nil(),ys) -> c_2()
goal#(xs) -> c_3(shuffle#(xs))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
reverse#(Nil()) -> c_5()
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
shuffle#(Nil()) -> c_7()
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
append#(Nil(),ys) -> c_2()
goal#(xs) -> c_3(shuffle#(xs))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
reverse#(Nil()) -> c_5()
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
shuffle#(Nil()) -> c_7()
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
goal(xs) -> shuffle(xs)
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
shuffle(Cons(x,xs)) -> Cons(x,shuffle(reverse(xs)))
shuffle(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
append#(Nil(),ys) -> c_2()
goal#(xs) -> c_3(shuffle#(xs))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
reverse#(Nil()) -> c_5()
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
shuffle#(Nil()) -> c_7()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
append#(Nil(),ys) -> c_2()
goal#(xs) -> c_3(shuffle#(xs))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
reverse#(Nil()) -> c_5()
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
shuffle#(Nil()) -> c_7()
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,5,7}
by application of
Pre({2,5,7}) = {1,3,4,6}.
Here rules are labelled as follows:
1: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
2: append#(Nil(),ys) -> c_2()
3: goal#(xs) -> c_3(shuffle#(xs))
4: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
5: reverse#(Nil()) -> c_5()
6: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
7: shuffle#(Nil()) -> c_7()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
goal#(xs) -> c_3(shuffle#(xs))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak DPs:
append#(Nil(),ys) -> c_2()
reverse#(Nil()) -> c_5()
shuffle#(Nil()) -> c_7()
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
-->_1 append#(Nil(),ys) -> c_2():5
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1
2:S:goal#(xs) -> c_3(shuffle#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4
-->_1 shuffle#(Nil()) -> c_7():7
3:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
-->_2 reverse#(Nil()) -> c_5():6
-->_1 append#(Nil(),ys) -> c_2():5
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1
4:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_1 shuffle#(Nil()) -> c_7():7
-->_2 reverse#(Nil()) -> c_5():6
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3
5:W:append#(Nil(),ys) -> c_2()
6:W:reverse#(Nil()) -> c_5()
7:W:shuffle#(Nil()) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: reverse#(Nil()) -> c_5()
7: shuffle#(Nil()) -> c_7()
5: append#(Nil(),ys) -> c_2()
** Step 1.b:5: RemoveHeads WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
goal#(xs) -> c_3(shuffle#(xs))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1
2:S:goal#(xs) -> c_3(shuffle#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4
3:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1
4:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(2,goal#(xs) -> c_3(shuffle#(xs)))]
** Step 1.b:6: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
Problem (S)
- Strict DPs:
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
*** Step 1.b:6.a:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
and a lower component
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
Further, following extension rules are added to the lower component.
shuffle#(Cons(x,xs)) -> reverse#(xs)
shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs))
**** Step 1.b:6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_6) = {1}
Following symbols are considered usable:
{append,reverse,append#,goal#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [1]
p(Nil) = [0]
p(append) = [1] x1 + [1] x2 + [0]
p(goal) = [8] x1 + [1]
p(reverse) = [1] x1 + [0]
p(shuffle) = [8] x1 + [2]
p(append#) = [1] x1 + [1] x2 + [1]
p(goal#) = [8] x1 + [0]
p(reverse#) = [1] x1 + [0]
p(shuffle#) = [4] x1 + [0]
p(c_1) = [2] x1 + [2]
p(c_2) = [1]
p(c_3) = [4] x1 + [8]
p(c_4) = [2] x1 + [1] x2 + [1]
p(c_5) = [1]
p(c_6) = [1] x1 + [3]
p(c_7) = [0]
Following rules are strictly oriented:
shuffle#(Cons(x,xs)) = [4] xs + [4]
> [4] xs + [3]
= c_6(shuffle#(reverse(xs)),reverse#(xs))
Following rules are (at-least) weakly oriented:
append(Cons(x,xs),ys) = [1] xs + [1] ys + [1]
>= [1] xs + [1] ys + [1]
= Cons(x,append(xs,ys))
append(Nil(),ys) = [1] ys + [0]
>= [1] ys + [0]
= ys
reverse(Cons(x,xs)) = [1] xs + [1]
>= [1] xs + [1]
= append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) = [0]
>= [0]
= Nil()
***** Step 1.b:6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
***** Step 1.b:6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 1.b:6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> reverse#(xs)
shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> reverse#(xs)
shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_4) = {1,2}
Following symbols are considered usable:
{append,reverse,append#,goal#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(Cons) = 1 + x1 + x2
p(Nil) = 0
p(append) = x1 + x2
p(goal) = 2*x1 + x1^2
p(reverse) = x1
p(shuffle) = 0
p(append#) = 2*x1 + 6*x1*x2 + 3*x2^2
p(goal#) = 1
p(reverse#) = 1 + 5*x1^2
p(shuffle#) = 4 + x1 + 6*x1^2
p(c_1) = x1
p(c_2) = 0
p(c_3) = x1
p(c_4) = 1 + x1 + x2
p(c_5) = 1
p(c_6) = x2
p(c_7) = 0
Following rules are strictly oriented:
append#(Cons(x,xs),ys) = 2 + 2*x + 6*x*ys + 2*xs + 6*xs*ys + 6*ys + 3*ys^2
> 2*xs + 6*xs*ys + 3*ys^2
= c_1(append#(xs,ys))
Following rules are (at-least) weakly oriented:
reverse#(Cons(x,xs)) = 6 + 10*x + 10*x*xs + 5*x^2 + 10*xs + 5*xs^2
>= 5 + 6*x + 6*x*xs + 3*x^2 + 8*xs + 5*xs^2
= c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) = 11 + 13*x + 12*x*xs + 6*x^2 + 13*xs + 6*xs^2
>= 1 + 5*xs^2
= reverse#(xs)
shuffle#(Cons(x,xs)) = 11 + 13*x + 12*x*xs + 6*x^2 + 13*xs + 6*xs^2
>= 4 + xs + 6*xs^2
= shuffle#(reverse(xs))
append(Cons(x,xs),ys) = 1 + x + xs + ys
>= 1 + x + xs + ys
= Cons(x,append(xs,ys))
append(Nil(),ys) = ys
>= ys
= ys
reverse(Cons(x,xs)) = 1 + x + xs
>= 1 + x + xs
= append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) = 0
>= 0
= Nil()
***** Step 1.b:6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> reverse#(xs)
shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> reverse#(xs)
shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1
2:W:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):2
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1
3:W:shuffle#(Cons(x,xs)) -> reverse#(xs)
-->_1 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):2
4:W:shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs))
-->_1 shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)):4
-->_1 shuffle#(Cons(x,xs)) -> reverse#(xs):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs))
3: shuffle#(Cons(x,xs)) -> reverse#(xs)
2: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
1: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
***** Step 1.b:6.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak DPs:
append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):3
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1
2:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):2
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1
3:W:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
-->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs))
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1
2:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):2
-->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
*** Step 1.b:6.b:3: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
- Weak DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
Problem (S)
- Strict DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
**** Step 1.b:6.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
- Weak DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:3.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
- Weak DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_4) = {1},
uargs(c_6) = {1,2}
Following symbols are considered usable:
{append,reverse,append#,goal#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(Cons) = 1 + x2
p(Nil) = 0
p(append) = x1 + x2
p(goal) = 4 + x1^2
p(reverse) = x1
p(shuffle) = 4*x1
p(append#) = x1 + 4*x2
p(goal#) = 1 + 4*x1
p(reverse#) = x1
p(shuffle#) = 2*x1^2
p(c_1) = 1
p(c_2) = 0
p(c_3) = 1 + x1
p(c_4) = x1
p(c_5) = 1
p(c_6) = x1 + x2
p(c_7) = 1
Following rules are strictly oriented:
reverse#(Cons(x,xs)) = 1 + xs
> xs
= c_4(reverse#(xs))
Following rules are (at-least) weakly oriented:
shuffle#(Cons(x,xs)) = 2 + 4*xs + 2*xs^2
>= xs + 2*xs^2
= c_6(shuffle#(reverse(xs)),reverse#(xs))
append(Cons(x,xs),ys) = 1 + xs + ys
>= 1 + xs + ys
= Cons(x,append(xs,ys))
append(Nil(),ys) = ys
>= ys
= ys
reverse(Cons(x,xs)) = 1 + xs
>= 1 + xs
= append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) = 0
>= 0
= Nil()
***** Step 1.b:6.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.b:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
-->_1 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):1
2:W:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):2
-->_2 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
1: reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
***** Step 1.b:6.b:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak DPs:
reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_2 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):2
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):1
2:W:reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
-->_1 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: reverse#(Cons(x,xs)) -> c_4(reverse#(xs))
**** Step 1.b:6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
**** Step 1.b:6.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_6) = {1}
Following symbols are considered usable:
{append,reverse,append#,goal#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [2]
p(Nil) = [0]
p(append) = [1] x1 + [1] x2 + [0]
p(goal) = [2] x1 + [2]
p(reverse) = [1] x1 + [0]
p(shuffle) = [1] x1 + [1]
p(append#) = [1] x2 + [0]
p(goal#) = [0]
p(reverse#) = [1] x1 + [8]
p(shuffle#) = [8] x1 + [0]
p(c_1) = [2] x1 + [0]
p(c_2) = [2]
p(c_3) = [0]
p(c_4) = [2]
p(c_5) = [1]
p(c_6) = [1] x1 + [14]
p(c_7) = [0]
Following rules are strictly oriented:
shuffle#(Cons(x,xs)) = [8] xs + [16]
> [8] xs + [14]
= c_6(shuffle#(reverse(xs)))
Following rules are (at-least) weakly oriented:
append(Cons(x,xs),ys) = [1] xs + [1] ys + [2]
>= [1] xs + [1] ys + [2]
= Cons(x,append(xs,ys))
append(Nil(),ys) = [1] ys + [0]
>= [1] ys + [0]
= ys
reverse(Cons(x,xs)) = [1] xs + [2]
>= [1] xs + [2]
= append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) = [0]
>= [0]
= Nil()
***** Step 1.b:6.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
-->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)))
***** Step 1.b:6.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
append(Nil(),ys) -> ys
reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil()))
reverse(Nil()) -> Nil()
- Signature:
{append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0
,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons
,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^3))