* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> b()
a__b() -> c()
a__f(X,g(X),Y) -> a__f(Y,Y,Y)
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__g(X) -> g(X)
a__g(b()) -> c()
mark(b()) -> a__b()
mark(c()) -> c()
mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
mark(g(X)) -> a__g(mark(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1} / {b/0,c/0,f/3,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b,a__f,a__g,mark} and constructors {b,c,f,g}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
a__f#(X1,X2,X3) -> c_4()
a__g#(X) -> c_5()
a__g#(b()) -> c_6()
mark#(b()) -> c_7(a__b#())
mark#(c()) -> c_8()
mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
a__f#(X1,X2,X3) -> c_4()
a__g#(X) -> c_5()
a__g#(b()) -> c_6()
mark#(b()) -> c_7(a__b#())
mark#(c()) -> c_8()
mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
- Weak TRS:
a__b() -> b()
a__b() -> c()
a__f(X,g(X),Y) -> a__f(Y,Y,Y)
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__g(X) -> g(X)
a__g(b()) -> c()
mark(b()) -> a__b()
mark(c()) -> c()
mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
mark(g(X)) -> a__g(mark(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,4,5,6,8}
by application of
Pre({1,2,4,5,6,8}) = {3,7,9,10}.
Here rules are labelled as follows:
1: a__b#() -> c_1()
2: a__b#() -> c_2()
3: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
4: a__f#(X1,X2,X3) -> c_4()
5: a__g#(X) -> c_5()
6: a__g#(b()) -> c_6()
7: mark#(b()) -> c_7(a__b#())
8: mark#(c()) -> c_8()
9: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
10: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
mark#(b()) -> c_7(a__b#())
mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
- Weak DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X1,X2,X3) -> c_4()
a__g#(X) -> c_5()
a__g#(b()) -> c_6()
mark#(c()) -> c_8()
- Weak TRS:
a__b() -> b()
a__b() -> c()
a__f(X,g(X),Y) -> a__f(Y,Y,Y)
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__g(X) -> g(X)
a__g(b()) -> c()
mark(b()) -> a__b()
mark(c()) -> c()
mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
mark(g(X)) -> a__g(mark(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2}
by application of
Pre({1,2}) = {3,4}.
Here rules are labelled as follows:
1: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
2: mark#(b()) -> c_7(a__b#())
3: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
4: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
5: a__b#() -> c_1()
6: a__b#() -> c_2()
7: a__f#(X1,X2,X3) -> c_4()
8: a__g#(X) -> c_5()
9: a__g#(b()) -> c_6()
10: mark#(c()) -> c_8()
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
- Weak DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
a__f#(X1,X2,X3) -> c_4()
a__g#(X) -> c_5()
a__g#(b()) -> c_6()
mark#(b()) -> c_7(a__b#())
mark#(c()) -> c_8()
- Weak TRS:
a__b() -> b()
a__b() -> c()
a__f(X,g(X),Y) -> a__f(Y,Y,Y)
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__g(X) -> g(X)
a__g(b()) -> c()
mark(b()) -> a__b()
mark(c()) -> c()
mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
mark(g(X)) -> a__g(mark(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {2}.
Here rules are labelled as follows:
1: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
2: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
3: a__b#() -> c_1()
4: a__b#() -> c_2()
5: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
6: a__f#(X1,X2,X3) -> c_4()
7: a__g#(X) -> c_5()
8: a__g#(b()) -> c_6()
9: mark#(b()) -> c_7(a__b#())
10: mark#(c()) -> c_8()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
- Weak DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
a__f#(X1,X2,X3) -> c_4()
a__g#(X) -> c_5()
a__g#(b()) -> c_6()
mark#(b()) -> c_7(a__b#())
mark#(c()) -> c_8()
mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
- Weak TRS:
a__b() -> b()
a__b() -> c()
a__f(X,g(X),Y) -> a__f(Y,Y,Y)
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__g(X) -> g(X)
a__g(b()) -> c()
mark(b()) -> a__b()
mark(c()) -> c()
mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
mark(g(X)) -> a__g(mark(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
-->_2 mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)):10
-->_2 mark#(b()) -> c_7(a__b#()):8
-->_2 mark#(c()) -> c_8():9
-->_1 a__g#(b()) -> c_6():7
-->_1 a__g#(X) -> c_5():6
-->_2 mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)):1
2:W:a__b#() -> c_1()
3:W:a__b#() -> c_2()
4:W:a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
-->_1 a__f#(X1,X2,X3) -> c_4():5
5:W:a__f#(X1,X2,X3) -> c_4()
6:W:a__g#(X) -> c_5()
7:W:a__g#(b()) -> c_6()
8:W:mark#(b()) -> c_7(a__b#())
-->_1 a__b#() -> c_2():3
-->_1 a__b#() -> c_1():2
9:W:mark#(c()) -> c_8()
10:W:mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
-->_1 a__f#(X1,X2,X3) -> c_4():5
-->_1 a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: a__g#(X) -> c_5()
7: a__g#(b()) -> c_6()
9: mark#(c()) -> c_8()
8: mark#(b()) -> c_7(a__b#())
2: a__b#() -> c_1()
3: a__b#() -> c_2()
10: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
4: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
5: a__f#(X1,X2,X3) -> c_4()
* Step 6: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
- Weak TRS:
a__b() -> b()
a__b() -> c()
a__f(X,g(X),Y) -> a__f(Y,Y,Y)
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__g(X) -> g(X)
a__g(b()) -> c()
mark(b()) -> a__b()
mark(c()) -> c()
mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
mark(g(X)) -> a__g(mark(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
-->_2 mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mark#(g(X)) -> c_10(mark#(X))
* Step 7: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(g(X)) -> c_10(mark#(X))
- Weak TRS:
a__b() -> b()
a__b() -> c()
a__f(X,g(X),Y) -> a__f(Y,Y,Y)
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__g(X) -> g(X)
a__g(b()) -> c()
mark(b()) -> a__b()
mark(c()) -> c()
mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
mark(g(X)) -> a__g(mark(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
mark#(g(X)) -> c_10(mark#(X))
* Step 8: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(g(X)) -> c_10(mark#(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ 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: mark#(g(X)) -> c_10(mark#(X))
The strictly oriented rules are moved into the weak component.
** Step 8.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(g(X)) -> c_10(mark#(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ 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_10) = {1}
Following symbols are considered usable:
{a__b#,a__f#,a__g#,mark#}
TcT has computed the following interpretation:
p(a__b) = [0]
p(a__f) = [1] x2 + [1] x3 + [0]
p(a__g) = [1] x1 + [1]
p(b) = [0]
p(c) = [1]
p(f) = [1]
p(g) = [1] x1 + [6]
p(mark) = [0]
p(a__b#) = [1]
p(a__f#) = [1] x3 + [1]
p(a__g#) = [1] x1 + [1]
p(mark#) = [4] x1 + [0]
p(c_1) = [8]
p(c_2) = [0]
p(c_3) = [2] x1 + [1]
p(c_4) = [1]
p(c_5) = [8]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1]
p(c_9) = [1]
p(c_10) = [1] x1 + [15]
Following rules are strictly oriented:
mark#(g(X)) = [4] X + [24]
> [4] X + [15]
= c_10(mark#(X))
Following rules are (at-least) weakly oriented:
** Step 8.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mark#(g(X)) -> c_10(mark#(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mark#(g(X)) -> c_10(mark#(X))
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:mark#(g(X)) -> c_10(mark#(X))
-->_1 mark#(g(X)) -> c_10(mark#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mark#(g(X)) -> c_10(mark#(X))
** Step 8.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))