* Step 1: ToInnermost WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
shuffle(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1} / {add/2,nil/0}
- Obligation:
runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
+ Applied Processor:
ToInnermost
+ Details:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
shuffle(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1} / {add/2,nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_4()
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_6()
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_4()
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_6()
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
shuffle(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_4()
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_6()
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_4()
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_6()
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4,6}
by application of
Pre({2,4,6}) = {1,3,5}.
Here rules are labelled as follows:
1: app#(add(n,x),y) -> c_1(app#(x,y))
2: app#(nil(),y) -> c_2()
3: reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
4: reverse#(nil()) -> c_4()
5: shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
6: shuffle#(nil()) -> c_6()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
app#(nil(),y) -> c_2()
reverse#(nil()) -> c_4()
shuffle#(nil()) -> c_6()
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:app#(add(n,x),y) -> c_1(app#(x,y))
-->_1 app#(nil(),y) -> c_2():4
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
2:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
-->_2 reverse#(nil()) -> c_4():5
-->_1 app#(nil(),y) -> c_2():4
-->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
3:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(nil()) -> c_6():6
-->_2 reverse#(nil()) -> c_4():5
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):3
-->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
4:W:app#(nil(),y) -> c_2()
5:W:reverse#(nil()) -> c_4()
6:W:shuffle#(nil()) -> c_6()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: shuffle#(nil()) -> c_6()
5: reverse#(nil()) -> c_4()
4: app#(nil(),y) -> c_2()
* Step 6: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2
,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
Problem (S)
- Strict DPs:
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2
,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
** Step 6.a:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
and a lower component
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
Further, following extension rules are added to the lower component.
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
*** Step 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
The strictly oriented rules are moved into the weak component.
**** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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_5) = {1}
Following symbols are considered usable:
{app,reverse,app#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(add) = [1] x2 + [1]
p(app) = [1] x1 + [1] x2 + [0]
p(nil) = [0]
p(reverse) = [1] x1 + [0]
p(shuffle) = [1] x1 + [1]
p(app#) = [1] x1 + [1] x2 + [1]
p(reverse#) = [0]
p(shuffle#) = [8] x1 + [0]
p(c_1) = [1]
p(c_2) = [0]
p(c_3) = [4] x1 + [1] x2 + [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [4] x2 + [4]
p(c_6) = [8]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [8] x + [8]
> [8] x + [4]
= c_5(shuffle#(reverse(x)),reverse#(x))
Following rules are (at-least) weakly oriented:
app(add(n,x),y) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] x + [1]
>= [1] x + [1]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [0]
>= [0]
= nil()
**** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
**** Step 6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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: app#(add(n,x),y) -> c_1(app#(x,y))
The strictly oriented rules are moved into the weak component.
**** Step 6.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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_3) = {1,2}
Following symbols are considered usable:
{app,reverse,app#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(add) = 1 + x2
p(app) = x1 + x2
p(nil) = 0
p(reverse) = x1
p(shuffle) = 1 + x1
p(app#) = 4*x1 + x2^2
p(reverse#) = 1 + x1 + 2*x1^2
p(shuffle#) = 3 + 2*x1 + 3*x1^2
p(c_1) = x1
p(c_2) = 0
p(c_3) = x1 + x2
p(c_4) = 0
p(c_5) = 1 + x2
p(c_6) = 1
Following rules are strictly oriented:
app#(add(n,x),y) = 4 + 4*x + y^2
> 4*x + y^2
= c_1(app#(x,y))
Following rules are (at-least) weakly oriented:
reverse#(add(n,x)) = 4 + 5*x + 2*x^2
>= 2 + 5*x + 2*x^2
= c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) = 8 + 8*x + 3*x^2
>= 1 + x + 2*x^2
= reverse#(x)
shuffle#(add(n,x)) = 8 + 8*x + 3*x^2
>= 3 + 2*x + 3*x^2
= shuffle#(reverse(x))
app(add(n,x),y) = 1 + x + y
>= 1 + x + y
= add(n,app(x,y))
app(nil(),y) = y
>= y
= y
reverse(add(n,x)) = 1 + x
>= 1 + x
= app(reverse(x),add(n,nil()))
reverse(nil()) = 0
>= 0
= nil()
**** Step 6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 6.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:app#(add(n,x),y) -> c_1(app#(x,y))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
2:W:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
3:W:shuffle#(add(n,x)) -> reverse#(x)
-->_1 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
4:W:shuffle#(add(n,x)) -> shuffle#(reverse(x))
-->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):4
-->_1 shuffle#(add(n,x)) -> reverse#(x):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: shuffle#(add(n,x)) -> shuffle#(reverse(x))
3: shuffle#(add(n,x)) -> reverse#(x)
2: reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
1: app#(add(n,x),y) -> c_1(app#(x,y))
**** Step 6.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):3
-->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
2:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
-->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
3:W:app#(add(n,x),y) -> c_1(app#(x,y))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: app#(add(n,x),y) -> c_1(app#(x,y))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
2:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
-->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
reverse#(add(n,x)) -> c_3(reverse#(x))
** Step 6.b:3: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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#(add(n,x)) -> c_3(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2
,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
Problem (S)
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2
,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
*** Step 6.b:3.a:1: DecomposeDG WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
and a lower component
reverse#(add(n,x)) -> c_3(reverse#(x))
Further, following extension rules are added to the lower component.
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
**** Step 6.b:3.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
The strictly oriented rules are moved into the weak component.
***** Step 6.b:3.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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_5) = {1}
Following symbols are considered usable:
{app,reverse,app#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(add) = [1] x2 + [3]
p(app) = [1] x1 + [1] x2 + [0]
p(nil) = [0]
p(reverse) = [1] x1 + [0]
p(shuffle) = [2] x1 + [1]
p(app#) = [1] x1 + [1]
p(reverse#) = [2]
p(shuffle#) = [8] x1 + [4]
p(c_1) = [1] x1 + [1]
p(c_2) = [2]
p(c_3) = [1] x1 + [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [1] x2 + [14]
p(c_6) = [1]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [8] x + [28]
> [8] x + [20]
= c_5(shuffle#(reverse(x)),reverse#(x))
Following rules are (at-least) weakly oriented:
app(add(n,x),y) = [1] x + [1] y + [3]
>= [1] x + [1] y + [3]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] x + [3]
>= [1] x + [3]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [0]
>= [0]
= nil()
***** Step 6.b:3.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 6.b:3.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
***** Step 6.b:3.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 6.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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: reverse#(add(n,x)) -> c_3(reverse#(x))
The strictly oriented rules are moved into the weak component.
***** Step 6.b:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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_3) = {1}
Following symbols are considered usable:
{app,reverse,app#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(add) = [1] x1 + [1] x2 + [2]
p(app) = [1] x1 + [1] x2 + [0]
p(nil) = [0]
p(reverse) = [1] x1 + [0]
p(shuffle) = [1] x1 + [2]
p(app#) = [1] x1 + [0]
p(reverse#) = [8] x1 + [0]
p(shuffle#) = [8] x1 + [0]
p(c_1) = [1] x1 + [1]
p(c_2) = [0]
p(c_3) = [1] x1 + [8]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
Following rules are strictly oriented:
reverse#(add(n,x)) = [8] n + [8] x + [16]
> [8] x + [8]
= c_3(reverse#(x))
Following rules are (at-least) weakly oriented:
shuffle#(add(n,x)) = [8] n + [8] x + [16]
>= [8] x + [0]
= reverse#(x)
shuffle#(add(n,x)) = [8] n + [8] x + [16]
>= [8] x + [0]
= shuffle#(reverse(x))
app(add(n,x),y) = [1] n + [1] x + [1] y + [2]
>= [1] n + [1] x + [1] y + [2]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] n + [1] x + [2]
>= [1] n + [1] x + [2]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [0]
>= [0]
= nil()
***** Step 6.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 6.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:reverse#(add(n,x)) -> c_3(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_3(reverse#(x)):1
2:W:shuffle#(add(n,x)) -> reverse#(x)
-->_1 reverse#(add(n,x)) -> c_3(reverse#(x)):1
3:W:shuffle#(add(n,x)) -> shuffle#(reverse(x))
-->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):3
-->_1 shuffle#(add(n,x)) -> reverse#(x):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: shuffle#(add(n,x)) -> shuffle#(reverse(x))
2: shuffle#(add(n,x)) -> reverse#(x)
1: reverse#(add(n,x)) -> c_3(reverse#(x))
***** Step 6.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
reverse#(add(n,x)) -> c_3(reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_3(reverse#(x)):2
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
2:W:reverse#(add(n,x)) -> c_3(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_3(reverse#(x)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: reverse#(add(n,x)) -> c_3(reverse#(x))
*** Step 6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
*** Step 6.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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#(add(n,x)) -> c_5(shuffle#(reverse(x)))
The strictly oriented rules are moved into the weak component.
**** Step 6.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,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_5) = {1}
Following symbols are considered usable:
{app,reverse,app#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(add) = [1] x2 + [4]
p(app) = [1] x1 + [1] x2 + [0]
p(nil) = [0]
p(reverse) = [1] x1 + [0]
p(shuffle) = [1]
p(app#) = [8] x1 + [1] x2 + [1]
p(reverse#) = [8]
p(shuffle#) = [4] x1 + [4]
p(c_1) = [0]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [2]
p(c_5) = [1] x1 + [14]
p(c_6) = [8]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [4] x + [20]
> [4] x + [18]
= c_5(shuffle#(reverse(x)))
Following rules are (at-least) weakly oriented:
app(add(n,x),y) = [1] x + [1] y + [4]
>= [1] x + [1] y + [4]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] x + [4]
>= [1] x + [4]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [0]
>= [0]
= nil()
**** Step 6.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
-->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
**** Step 6.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^3))