* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X1,X2,X3) -> c_3()
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(a()) -> c_5()
mark#(b()) -> c_6(a__b#())
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
Weak DPs
and mark the set of starting terms.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X1,X2,X3) -> c_3()
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(a()) -> c_5()
mark#(b()) -> c_6(a__b#())
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Strict TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ 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(a__f) = {2},
uargs(a__f#) = {2},
uargs(c_4) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [3]
p(a__b) = [8]
p(a__f) = [3] x1 + [1] x2 + [4] x3 + [2]
p(b) = [0]
p(f) = [1] x1 + [1] x2 + [1] x3 + [1]
p(mark) = [4] x1 + [10]
p(a__b#) = [1]
p(a__f#) = [1] x2 + [4] x3 + [2]
p(mark#) = [4] x1 + [4]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [1] x1 + [8]
p(c_5) = [1]
p(c_6) = [1] x1 + [1]
p(c_7) = [1] x1 + [4]
Following rules are strictly oriented:
a__f#(X1,X2,X3) = [1] X2 + [4] X3 + [2]
> [0]
= c_3()
mark#(a()) = [16]
> [1]
= c_5()
mark#(b()) = [4]
> [2]
= c_6(a__b#())
a__b() = [8]
> [3]
= a()
a__b() = [8]
> [0]
= b()
a__f(X1,X2,X3) = [3] X1 + [1] X2 + [4] X3 + [2]
> [1] X1 + [1] X2 + [1] X3 + [1]
= f(X1,X2,X3)
a__f(a(),X,X) = [5] X + [11]
> [3] X + [10]
= a__f(X,a__b(),b())
mark(a()) = [22]
> [3]
= a()
mark(b()) = [10]
> [8]
= a__b()
mark(f(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [14]
> [3] X1 + [4] X2 + [4] X3 + [12]
= a__f(X1,mark(X2),X3)
Following rules are (at-least) weakly oriented:
a__b#() = [1]
>= [1]
= c_1()
a__b#() = [1]
>= [1]
= c_2()
a__f#(a(),X,X) = [5] X + [2]
>= [18]
= c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [8]
>= [4] X2 + [4] X3 + [16]
= c_7(a__f#(X1,mark(X2),X3))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak DPs:
a__f#(X1,X2,X3) -> c_3()
mark#(a()) -> c_5()
mark#(b()) -> c_6(a__b#())
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:a__b#() -> c_1()
2:S:a__b#() -> c_2()
3:S:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
-->_1 a__f#(X1,X2,X3) -> c_3():5
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
4:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
-->_1 a__f#(X1,X2,X3) -> c_3():5
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
5:W:a__f#(X1,X2,X3) -> c_3()
6:W:mark#(a()) -> c_5()
7:W:mark#(b()) -> c_6(a__b#())
-->_1 a__b#() -> c_2():2
-->_1 a__b#() -> c_1():1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: mark#(a()) -> c_5()
5: a__f#(X1,X2,X3) -> c_3()
* Step 4: RemoveHeads WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak DPs:
mark#(b()) -> c_6(a__b#())
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:a__b#() -> c_1()
2:S:a__b#() -> c_2()
3:S:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
4:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
7:W:mark#(b()) -> c_6(a__b#())
-->_1 a__b#() -> c_2():2
-->_1 a__b#() -> c_1():1
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).
[(7,mark#(b()) -> c_6(a__b#()))]
* Step 5: RemoveHeads WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:a__b#() -> c_1()
2:S:a__b#() -> c_2()
3:S:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
4:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):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).
[(1,a__b#() -> c_1()),(2,a__b#() -> c_2())]
* Step 6: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ 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:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
- Weak DPs:
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
Problem (S)
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
- Weak DPs:
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ 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:
3: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
Consider the set of all dependency pairs
3: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
4: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{3}
These cover all (indirect) predecessors of dependency pairs
{3,4}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** Step 6.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
- Weak DPs:
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ 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_4) = {1},
uargs(c_7) = {1}
Following symbols are considered usable:
{a__b#,a__f#,mark#}
TcT has computed the following interpretation:
p(a) = [2]
p(a__b) = [0]
p(a__f) = [5] x1 + [2] x2 + [6]
p(b) = [0]
p(f) = [1] x1 + [1] x3 + [0]
p(mark) = [5] x1 + [9]
p(a__b#) = [1]
p(a__f#) = [2] x1 + [8] x3 + [0]
p(mark#) = [8] x1 + [1]
p(c_1) = [0]
p(c_2) = [2]
p(c_3) = [4]
p(c_4) = [2] x1 + [0]
p(c_5) = [1]
p(c_6) = [1] x1 + [1]
p(c_7) = [1] x1 + [0]
Following rules are strictly oriented:
a__f#(a(),X,X) = [8] X + [4]
> [4] X + [0]
= c_4(a__f#(X,a__b(),b()))
Following rules are (at-least) weakly oriented:
mark#(f(X1,X2,X3)) = [8] X1 + [8] X3 + [1]
>= [2] X1 + [8] X3 + [0]
= c_7(a__f#(X1,mark(X2),X3))
*** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
2:W:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
1: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
*** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 6.b:1: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ 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: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
2: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
** Step 6.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
2:W:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
-->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
1: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))