WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2))
- Signature:
{app_xs#1/2,main/2} / {Cons/2,Nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1,main} and constructors {Cons,Nil}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Strict TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2))
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Strict TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(app_xs#1#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [1]
p(Nil) = [0]
p(app_xs#1) = [4] x1 + [1] x2 + [3]
p(main) = [0]
p(app_xs#1#) = [9] x1 + [1] x2 + [0]
p(main#) = [14] x1 + [1] x2 + [2]
p(c_1) = [1] x1 + [2]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
Following rules are strictly oriented:
app_xs#1#(Cons(x7,x8),x10) = [1] x10 + [9] x8 + [9]
> [1] x10 + [9] x8 + [2]
= c_1(app_xs#1#(x8,x10))
app_xs#1(Cons(x7,x8),x10) = [1] x10 + [4] x8 + [7]
> [1] x10 + [4] x8 + [4]
= Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) = [1] x6 + [3]
> [1] x6 + [0]
= x6
Following rules are (at-least) weakly oriented:
app_xs#1#(Nil(),x6) = [1] x6 + [0]
>= [0]
= c_2()
main#(x4,x2) = [1] x2 + [14] x4 + [2]
>= [1] x2 + [13] x4 + [3]
= c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: Decompose WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} 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:
app_xs#1#(Nil(),x6) -> c_2()
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
Problem (S)
- Strict DPs:
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
** Step 4.a:1: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
app_xs#1#(Nil(),x6) -> c_2()
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: app_xs#1#(Nil(),x6) -> c_2()
Consider the set of all dependency pairs
1: app_xs#1#(Nil(),x6) -> c_2()
2: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
3: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
Processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** Step 4.a:1.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
app_xs#1#(Nil(),x6) -> c_2()
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
{app_xs#1,app_xs#1#,main#}
TcT has computed the following interpretation:
p(Cons) = [2]
p(Nil) = [0]
p(app_xs#1) = [2] x1 + [1] x2 + [0]
p(main) = [1] x1 + [1]
p(app_xs#1#) = [1] x2 + [1]
p(main#) = [8] x1 + [4] x2 + [4]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [4] x1 + [0]
Following rules are strictly oriented:
app_xs#1#(Nil(),x6) = [1] x6 + [1]
> [0]
= c_2()
Following rules are (at-least) weakly oriented:
app_xs#1#(Cons(x7,x8),x10) = [1] x10 + [1]
>= [1] x10 + [1]
= c_1(app_xs#1#(x8,x10))
main#(x4,x2) = [4] x2 + [8] x4 + [4]
>= [4] x2 + [8] x4 + [4]
= c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
app_xs#1(Cons(x7,x8),x10) = [1] x10 + [4]
>= [2]
= Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) = [1] x6 + [0]
>= [1] x6 + [0]
= x6
*** Step 4.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
-->_1 app_xs#1#(Nil(),x6) -> c_2():2
-->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1
2:W:app_xs#1#(Nil(),x6) -> c_2()
3:W:main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
-->_1 app_xs#1#(Nil(),x6) -> c_2():2
-->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
1: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
2: app_xs#1#(Nil(),x6) -> c_2()
*** Step 4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 4.b:1: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {}.
Here rules are labelled as follows:
1: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
2: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
3: app_xs#1#(Nil(),x6) -> c_2()
** Step 4.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
app_xs#1#(Nil(),x6) -> c_2()
main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
-->_1 app_xs#1#(Nil(),x6) -> c_2():2
-->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1
2:W:app_xs#1#(Nil(),x6) -> c_2()
3:W:main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
-->_1 app_xs#1#(Nil(),x6) -> c_2():2
-->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2)))
1: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10))
2: app_xs#1#(Nil(),x6) -> c_2()
** Step 4.b:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
app_xs#1(Nil(),x6) -> x6
- Signature:
{app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))