* Step 1: Sum WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
a__c() -> c()
a__c() -> d()
a__g(X) -> a__h(X)
a__g(X) -> g(X)
a__h(X) -> h(X)
a__h(d()) -> a__g(c())
mark(c()) -> a__c()
mark(d()) -> d()
mark(g(X)) -> a__g(X)
mark(h(X)) -> a__h(X)
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1} / {c/0,d/0,g/1,h/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__g,a__h,mark} and constructors {c,d,g,h}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
a__c() -> c()
a__c() -> d()
a__g(X) -> a__h(X)
a__g(X) -> g(X)
a__h(X) -> h(X)
a__h(d()) -> a__g(c())
mark(c()) -> a__c()
mark(d()) -> d()
mark(g(X)) -> a__g(X)
mark(h(X)) -> a__h(X)
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1} / {c/0,d/0,g/1,h/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__g,a__h,mark} and constructors {c,d,g,h}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
a__c#() -> c_1()
a__c#() -> c_2()
a__g#(X) -> c_3(a__h#(X))
a__g#(X) -> c_4()
a__h#(X) -> c_5()
a__h#(d()) -> c_6(a__g#(c()))
mark#(c()) -> c_7(a__c#())
mark#(d()) -> c_8()
mark#(g(X)) -> c_9(a__g#(X))
mark#(h(X)) -> c_10(a__h#(X))
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__c#() -> c_1()
a__c#() -> c_2()
a__g#(X) -> c_3(a__h#(X))
a__g#(X) -> c_4()
a__h#(X) -> c_5()
a__h#(d()) -> c_6(a__g#(c()))
mark#(c()) -> c_7(a__c#())
mark#(d()) -> c_8()
mark#(g(X)) -> c_9(a__g#(X))
mark#(h(X)) -> c_10(a__h#(X))
- Weak TRS:
a__c() -> c()
a__c() -> d()
a__g(X) -> a__h(X)
a__g(X) -> g(X)
a__h(X) -> h(X)
a__h(d()) -> a__g(c())
mark(c()) -> a__c()
mark(d()) -> d()
mark(g(X)) -> a__g(X)
mark(h(X)) -> a__h(X)
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
a__c#() -> c_1()
a__c#() -> c_2()
a__g#(X) -> c_3(a__h#(X))
a__g#(X) -> c_4()
a__h#(X) -> c_5()
a__h#(d()) -> c_6(a__g#(c()))
mark#(c()) -> c_7(a__c#())
mark#(d()) -> c_8()
mark#(g(X)) -> c_9(a__g#(X))
mark#(h(X)) -> c_10(a__h#(X))
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__c#() -> c_1()
a__c#() -> c_2()
a__g#(X) -> c_3(a__h#(X))
a__g#(X) -> c_4()
a__h#(X) -> c_5()
a__h#(d()) -> c_6(a__g#(c()))
mark#(c()) -> c_7(a__c#())
mark#(d()) -> c_8()
mark#(g(X)) -> c_9(a__g#(X))
mark#(h(X)) -> c_10(a__h#(X))
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,4,5,8}
by application of
Pre({1,2,4,5,8}) = {3,6,7,9,10}.
Here rules are labelled as follows:
1: a__c#() -> c_1()
2: a__c#() -> c_2()
3: a__g#(X) -> c_3(a__h#(X))
4: a__g#(X) -> c_4()
5: a__h#(X) -> c_5()
6: a__h#(d()) -> c_6(a__g#(c()))
7: mark#(c()) -> c_7(a__c#())
8: mark#(d()) -> c_8()
9: mark#(g(X)) -> c_9(a__g#(X))
10: mark#(h(X)) -> c_10(a__h#(X))
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__g#(X) -> c_3(a__h#(X))
a__h#(d()) -> c_6(a__g#(c()))
mark#(c()) -> c_7(a__c#())
mark#(g(X)) -> c_9(a__g#(X))
mark#(h(X)) -> c_10(a__h#(X))
- Weak DPs:
a__c#() -> c_1()
a__c#() -> c_2()
a__g#(X) -> c_4()
a__h#(X) -> c_5()
mark#(d()) -> c_8()
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3}
by application of
Pre({3}) = {}.
Here rules are labelled as follows:
1: a__g#(X) -> c_3(a__h#(X))
2: a__h#(d()) -> c_6(a__g#(c()))
3: mark#(c()) -> c_7(a__c#())
4: mark#(g(X)) -> c_9(a__g#(X))
5: mark#(h(X)) -> c_10(a__h#(X))
6: a__c#() -> c_1()
7: a__c#() -> c_2()
8: a__g#(X) -> c_4()
9: a__h#(X) -> c_5()
10: mark#(d()) -> c_8()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__g#(X) -> c_3(a__h#(X))
a__h#(d()) -> c_6(a__g#(c()))
mark#(g(X)) -> c_9(a__g#(X))
mark#(h(X)) -> c_10(a__h#(X))
- Weak DPs:
a__c#() -> c_1()
a__c#() -> c_2()
a__g#(X) -> c_4()
a__h#(X) -> c_5()
mark#(c()) -> c_7(a__c#())
mark#(d()) -> c_8()
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:a__g#(X) -> c_3(a__h#(X))
-->_1 a__h#(d()) -> c_6(a__g#(c())):2
-->_1 a__h#(X) -> c_5():8
2:S:a__h#(d()) -> c_6(a__g#(c()))
-->_1 a__g#(X) -> c_4():7
-->_1 a__g#(X) -> c_3(a__h#(X)):1
3:S:mark#(g(X)) -> c_9(a__g#(X))
-->_1 a__g#(X) -> c_4():7
-->_1 a__g#(X) -> c_3(a__h#(X)):1
4:S:mark#(h(X)) -> c_10(a__h#(X))
-->_1 a__h#(X) -> c_5():8
-->_1 a__h#(d()) -> c_6(a__g#(c())):2
5:W:a__c#() -> c_1()
6:W:a__c#() -> c_2()
7:W:a__g#(X) -> c_4()
8:W:a__h#(X) -> c_5()
9:W:mark#(c()) -> c_7(a__c#())
-->_1 a__c#() -> c_2():6
-->_1 a__c#() -> c_1():5
10:W:mark#(d()) -> c_8()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: mark#(d()) -> c_8()
9: mark#(c()) -> c_7(a__c#())
6: a__c#() -> c_2()
5: a__c#() -> c_1()
8: a__h#(X) -> c_5()
7: a__g#(X) -> c_4()
* Step 7: RemoveHeads WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__g#(X) -> c_3(a__h#(X))
a__h#(d()) -> c_6(a__g#(c()))
mark#(g(X)) -> c_9(a__g#(X))
mark#(h(X)) -> c_10(a__h#(X))
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:a__g#(X) -> c_3(a__h#(X))
-->_1 a__h#(d()) -> c_6(a__g#(c())):2
2:S:a__h#(d()) -> c_6(a__g#(c()))
-->_1 a__g#(X) -> c_3(a__h#(X)):1
3:S:mark#(g(X)) -> c_9(a__g#(X))
-->_1 a__g#(X) -> c_3(a__h#(X)):1
4:S:mark#(h(X)) -> c_10(a__h#(X))
-->_1 a__h#(d()) -> c_6(a__g#(c())):2
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).
[(3,mark#(g(X)) -> c_9(a__g#(X))),(4,mark#(h(X)) -> c_10(a__h#(X)))]
* Step 8: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__g#(X) -> c_3(a__h#(X))
a__h#(d()) -> c_6(a__g#(c()))
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
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_3) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__c) = [0]
p(a__g) = [0]
p(a__h) = [0]
p(c) = [0]
p(d) = [0]
p(g) = [0]
p(h) = [0]
p(mark) = [0]
p(a__c#) = [0]
p(a__g#) = [0]
p(a__h#) = [11]
p(mark#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [0]
Following rules are strictly oriented:
a__h#(d()) = [11]
> [0]
= c_6(a__g#(c()))
Following rules are (at-least) weakly oriented:
a__g#(X) = [0]
>= [11]
= c_3(a__h#(X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 9: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__g#(X) -> c_3(a__h#(X))
- Weak DPs:
a__h#(d()) -> c_6(a__g#(c()))
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
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_3) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__c) = [0]
p(a__g) = [0]
p(a__h) = [2] x1 + [0]
p(c) = [0]
p(d) = [7]
p(g) = [0]
p(h) = [0]
p(mark) = [2] x1 + [1]
p(a__c#) = [1]
p(a__g#) = [8] x1 + [8]
p(a__h#) = [3] x1 + [2]
p(mark#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [2]
p(c_4) = [8]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [2] x1 + [4]
p(c_8) = [0]
p(c_9) = [8] x1 + [1]
p(c_10) = [1]
Following rules are strictly oriented:
a__g#(X) = [8] X + [8]
> [3] X + [4]
= c_3(a__h#(X))
Following rules are (at-least) weakly oriented:
a__h#(d()) = [23]
>= [8]
= c_6(a__g#(c()))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 10: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
a__g#(X) -> c_3(a__h#(X))
a__h#(d()) -> c_6(a__g#(c()))
- Signature:
{a__c/0,a__g/1,a__h/1,mark/1,a__c#/0,a__g#/1,a__h#/1,mark#/1} / {c/0,d/0,g/1,h/1,c_1/0,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__g#,a__h#,mark#} and constructors {c,d,g,h}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(1))