WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
after(0(),XS) -> XS
after(s(N),cons(X,XS)) -> after(N,activate(XS))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,after/2,from/1,s/1} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,after,from,s} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
activate#(X) -> c_1()
activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
after#(0(),XS) -> c_4()
after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
from#(X) -> c_6()
from#(X) -> c_7()
s#(X) -> c_8()
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
after#(0(),XS) -> c_4()
after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
from#(X) -> c_6()
from#(X) -> c_7()
s#(X) -> c_8()
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
after(0(),XS) -> XS
after(s(N),cons(X,XS)) -> after(N,activate(XS))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
,c_3/2,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
,n__from,n__s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,4,5,6,7,8}
by application of
Pre({1,4,5,6,7,8}) = {2,3}.
Here rules are labelled as follows:
1: activate#(X) -> c_1()
2: activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
3: activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
4: after#(0(),XS) -> c_4()
5: after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
6: from#(X) -> c_6()
7: from#(X) -> c_7()
8: s#(X) -> c_8()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
- Weak DPs:
activate#(X) -> c_1()
after#(0(),XS) -> c_4()
after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
from#(X) -> c_6()
from#(X) -> c_7()
s#(X) -> c_8()
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
after(0(),XS) -> XS
after(s(N),cons(X,XS)) -> after(N,activate(XS))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
,c_3/2,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
,n__from,n__s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
-->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
-->_1 from#(X) -> c_7():7
-->_1 from#(X) -> c_6():6
-->_2 activate#(X) -> c_1():3
-->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
-->_1 s#(X) -> c_8():8
-->_2 activate#(X) -> c_1():3
-->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
-->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
3:W:activate#(X) -> c_1()
4:W:after#(0(),XS) -> c_4()
5:W:after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
6:W:from#(X) -> c_6()
7:W:from#(X) -> c_7()
8:W:s#(X) -> c_8()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
4: after#(0(),XS) -> c_4()
6: from#(X) -> c_6()
7: from#(X) -> c_7()
3: activate#(X) -> c_1()
8: s#(X) -> c_8()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
after(0(),XS) -> XS
after(s(N),cons(X,XS)) -> after(N,activate(XS))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
,c_3/2,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
,n__from,n__s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
-->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
-->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
-->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
-->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
activate#(n__from(X)) -> c_2(activate#(X))
activate#(n__s(X)) -> c_3(activate#(X))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__from(X)) -> c_2(activate#(X))
activate#(n__s(X)) -> c_3(activate#(X))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
after(0(),XS) -> XS
after(s(N),cons(X,XS)) -> after(N,activate(XS))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
,c_3/1,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
,n__from,n__s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate#(n__from(X)) -> c_2(activate#(X))
activate#(n__s(X)) -> c_3(activate#(X))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__from(X)) -> c_2(activate#(X))
activate#(n__s(X)) -> c_3(activate#(X))
- Signature:
{activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
,c_3/1,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
,n__from,n__s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [0]
p(after) = [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(from) = [0]
p(n__from) = [1] x1 + [8]
p(n__s) = [1] x1 + [10]
p(s) = [0]
p(activate#) = [2] x1 + [10]
p(after#) = [8] x2 + [2]
p(from#) = [1]
p(s#) = [2] x1 + [4]
p(c_1) = [4]
p(c_2) = [1] x1 + [10]
p(c_3) = [1] x1 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
Following rules are strictly oriented:
activate#(n__from(X)) = [2] X + [26]
> [2] X + [20]
= c_2(activate#(X))
activate#(n__s(X)) = [2] X + [30]
> [2] X + [10]
= c_3(activate#(X))
Following rules are (at-least) weakly oriented:
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(n__from(X)) -> c_2(activate#(X))
activate#(n__s(X)) -> c_3(activate#(X))
- Signature:
{activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
,c_3/1,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
,n__from,n__s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))