* Step 1: Sum WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
2nd(cons(X,XS)) -> head(activate(XS))
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
head(cons(X,XS)) -> X
sel(0(),cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons
,n__from,n__take,nil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
2nd(cons(X,XS)) -> head(activate(XS))
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
head(cons(X,XS)) -> X
sel(0(),cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons
,n__from,n__take,nil,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5()
from#(X) -> c_6()
head#(cons(X,XS)) -> c_7()
sel#(0(),cons(X,XS)) -> c_8()
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10()
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5()
from#(X) -> c_6()
head#(cons(X,XS)) -> c_7()
sel#(0(),cons(X,XS)) -> c_8()
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10()
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Strict TRS:
2nd(cons(X,XS)) -> head(activate(XS))
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
head(cons(X,XS)) -> X
sel(0(),cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5()
from#(X) -> c_6()
head#(cons(X,XS)) -> c_7()
sel#(0(),cons(X,XS)) -> c_8()
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10()
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5()
from#(X) -> c_6()
head#(cons(X,XS)) -> c_7()
sel#(0(),cons(X,XS)) -> c_8()
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10()
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ 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(n__take) = {2},
uargs(head#) = {1},
uargs(sel#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_9) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(2nd) = [0]
p(activate) = [1] x1 + [3]
p(cons) = [1] x2 + [1]
p(from) = [7]
p(head) = [1] x1 + [1]
p(n__from) = [5]
p(n__take) = [1] x1 + [1] x2 + [1]
p(nil) = [0]
p(s) = [1] x1 + [2]
p(sel) = [0]
p(take) = [1] x1 + [1] x2 + [3]
p(2nd#) = [2] x1 + [0]
p(activate#) = [0]
p(from#) = [0]
p(head#) = [1] x1 + [0]
p(sel#) = [5] x1 + [1] x2 + [0]
p(take#) = [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
Following rules are strictly oriented:
head#(cons(X,XS)) = [1] XS + [1]
> [0]
= c_7()
sel#(0(),cons(X,XS)) = [1] XS + [6]
> [0]
= c_8()
sel#(s(N),cons(X,XS)) = [5] N + [1] XS + [11]
> [5] N + [1] XS + [3]
= c_9(sel#(N,activate(XS)))
activate(X) = [1] X + [3]
> [1] X + [0]
= X
activate(n__from(X)) = [8]
> [7]
= from(X)
activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [4]
> [1] X1 + [1] X2 + [3]
= take(X1,X2)
from(X) = [7]
> [6]
= cons(X,n__from(s(X)))
from(X) = [7]
> [5]
= n__from(X)
take(X1,X2) = [1] X1 + [1] X2 + [3]
> [1] X1 + [1] X2 + [1]
= n__take(X1,X2)
take(0(),XS) = [1] XS + [4]
> [0]
= nil()
take(s(N),cons(X,XS)) = [1] N + [1] XS + [6]
> [1] N + [1] XS + [5]
= cons(X,n__take(N,activate(XS)))
Following rules are (at-least) weakly oriented:
2nd#(cons(X,XS)) = [2] XS + [2]
>= [1] XS + [3]
= c_1(head#(activate(XS)))
activate#(X) = [0]
>= [0]
= c_2()
activate#(n__from(X)) = [0]
>= [0]
= c_3(from#(X))
activate#(n__take(X1,X2)) = [0]
>= [0]
= c_4(take#(X1,X2))
from#(X) = [0]
>= [0]
= c_5()
from#(X) = [0]
>= [0]
= c_6()
take#(X1,X2) = [0]
>= [0]
= c_10()
take#(0(),XS) = [0]
>= [0]
= c_11()
take#(s(N),cons(X,XS)) = [0]
>= [0]
= c_12(activate#(XS))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
from#(X) -> c_5()
from#(X) -> c_6()
take#(X1,X2) -> c_10()
take#(0(),XS) -> c_11()
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Weak DPs:
head#(cons(X,XS)) -> c_7()
sel#(0(),cons(X,XS)) -> c_8()
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,5,6,7,8}
by application of
Pre({1,2,5,6,7,8}) = {3,4,9}.
Here rules are labelled as follows:
1: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
2: activate#(X) -> c_2()
3: activate#(n__from(X)) -> c_3(from#(X))
4: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
5: from#(X) -> c_5()
6: from#(X) -> c_6()
7: take#(X1,X2) -> c_10()
8: take#(0(),XS) -> c_11()
9: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
10: head#(cons(X,XS)) -> c_7()
11: sel#(0(),cons(X,XS)) -> c_8()
12: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
* Step 6: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__from(X)) -> c_3(from#(X))
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Weak DPs:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
from#(X) -> c_5()
from#(X) -> c_6()
head#(cons(X,XS)) -> c_7()
sel#(0(),cons(X,XS)) -> c_8()
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10()
take#(0(),XS) -> c_11()
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {3}.
Here rules are labelled as follows:
1: activate#(n__from(X)) -> c_3(from#(X))
2: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
3: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
4: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
5: activate#(X) -> c_2()
6: from#(X) -> c_5()
7: from#(X) -> c_6()
8: head#(cons(X,XS)) -> c_7()
9: sel#(0(),cons(X,XS)) -> c_8()
10: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
11: take#(X1,X2) -> c_10()
12: take#(0(),XS) -> c_11()
* Step 7: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Weak DPs:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(X))
from#(X) -> c_5()
from#(X) -> c_6()
head#(cons(X,XS)) -> c_7()
sel#(0(),cons(X,XS)) -> c_8()
sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
take#(X1,X2) -> c_10()
take#(0(),XS) -> c_11()
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
-->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):2
-->_1 take#(0(),XS) -> c_11():12
-->_1 take#(X1,X2) -> c_10():11
2:S:take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
-->_1 activate#(n__from(X)) -> c_3(from#(X)):5
-->_1 activate#(X) -> c_2():4
-->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):1
3:W:2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
-->_1 head#(cons(X,XS)) -> c_7():8
4:W:activate#(X) -> c_2()
5:W:activate#(n__from(X)) -> c_3(from#(X))
-->_1 from#(X) -> c_6():7
-->_1 from#(X) -> c_5():6
6:W:from#(X) -> c_5()
7:W:from#(X) -> c_6()
8:W:head#(cons(X,XS)) -> c_7()
9:W:sel#(0(),cons(X,XS)) -> c_8()
10:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
-->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):10
-->_1 sel#(0(),cons(X,XS)) -> c_8():9
11:W:take#(X1,X2) -> c_10()
12:W:take#(0(),XS) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
9: sel#(0(),cons(X,XS)) -> c_8()
3: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
8: head#(cons(X,XS)) -> c_7()
11: take#(X1,X2) -> c_10()
12: take#(0(),XS) -> c_11()
4: activate#(X) -> c_2()
5: activate#(n__from(X)) -> c_3(from#(X))
6: from#(X) -> c_5()
7: from#(X) -> c_6()
* Step 8: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__take(X1,X2)) -> take(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
* Step 9: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ 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:
2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
Consider the set of all dependency pairs
1: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
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
{2}
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 9.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ 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_12) = {1}
Following symbols are considered usable:
{2nd#,activate#,from#,head#,sel#,take#}
TcT has computed the following interpretation:
p(0) = [0]
p(2nd) = [2] x1 + [1]
p(activate) = [8] x1 + [1]
p(cons) = [1] x2 + [5]
p(from) = [1] x1 + [0]
p(head) = [8] x1 + [1]
p(n__from) = [1]
p(n__take) = [1] x1 + [1] x2 + [10]
p(nil) = [8]
p(s) = [1] x1 + [8]
p(sel) = [0]
p(take) = [1]
p(2nd#) = [1]
p(activate#) = [1] x1 + [2]
p(from#) = [1] x1 + [1]
p(head#) = [1]
p(sel#) = [1]
p(take#) = [1] x1 + [1] x2 + [8]
p(c_1) = [8] x1 + [1]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [1] x1 + [4]
p(c_5) = [0]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [1]
p(c_9) = [1]
p(c_10) = [2]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
Following rules are strictly oriented:
take#(s(N),cons(X,XS)) = [1] N + [1] XS + [21]
> [1] XS + [2]
= c_12(activate#(XS))
Following rules are (at-least) weakly oriented:
activate#(n__take(X1,X2)) = [1] X1 + [1] X2 + [12]
>= [1] X1 + [1] X2 + [12]
= c_4(take#(X1,X2))
** Step 9.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
- Weak DPs:
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 9.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
-->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):2
2:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
-->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
** Step 9.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
,take#} and constructors {0,cons,n__from,n__take,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))