*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
2nd(cons(X,XS)) -> head(activate(XS))
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
head(cons(X,XS)) -> X
s(X) -> n__s(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)))
Weak DP Rules:
Weak TRS Rules:
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0}
Obligation:
Innermost
basic terms: {2nd,activate,from,head,s,sel,take}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
2nd(cons(X,XS)) -> head(activate(XS))
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
head(cons(X,XS)) -> X
s(X) -> n__s(X)
sel(0(),cons(X,XS)) -> X
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Weak DP Rules:
Weak TRS Rules:
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0}
Obligation:
Innermost
basic terms: {2nd,activate,from,head,s,sel,take}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
DependencyPairs {dpKind_ = WIDP}
Proof:
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#(activate(X)))
activate#(n__s(X)) -> c_4(s#(activate(X)))
activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
from#(X) -> c_6()
from#(X) -> c_7()
head#(cons(X,XS)) -> c_8()
s#(X) -> c_9()
sel#(0(),cons(X,XS)) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),XS) -> c_12()
Weak DPs
and mark the set of starting terms.
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(activate(X)))
activate#(n__s(X)) -> c_4(s#(activate(X)))
activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
from#(X) -> c_6()
from#(X) -> c_7()
head#(cons(X,XS)) -> c_8()
s#(X) -> c_9()
sel#(0(),cons(X,XS)) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),XS) -> c_12()
Strict TRS Rules:
2nd(cons(X,XS)) -> head(activate(XS))
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
head(cons(X,XS)) -> X
s(X) -> n__s(X)
sel(0(),cons(X,XS)) -> X
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Weak DP Rules:
Weak TRS Rules:
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
Obligation:
Innermost
basic terms: {2nd#,activate#,from#,head#,s#,sel#,take#}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(activate(X)))
activate#(n__s(X)) -> c_4(s#(activate(X)))
activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
from#(X) -> c_6()
from#(X) -> c_7()
head#(cons(X,XS)) -> c_8()
s#(X) -> c_9()
sel#(0(),cons(X,XS)) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),XS) -> c_12()
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(activate(X)))
activate#(n__s(X)) -> c_4(s#(activate(X)))
activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
from#(X) -> c_6()
from#(X) -> c_7()
head#(cons(X,XS)) -> c_8()
s#(X) -> c_9()
sel#(0(),cons(X,XS)) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),XS) -> c_12()
Strict TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Weak DP Rules:
Weak TRS Rules:
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
Obligation:
Innermost
basic terms: {2nd#,activate#,from#,head#,s#,sel#,take#}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
Proof:
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(from) = {1},
uargs(s) = {1},
uargs(take) = {1,2},
uargs(from#) = {1},
uargs(head#) = {1},
uargs(s#) = {1},
uargs(take#) = {1,2},
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(2nd) = [1] x1 + [1]
p(activate) = [5] x1 + [1]
p(cons) = [1] x2 + [4]
p(from) = [1] x1 + [15]
p(head) = [4] x1 + [1]
p(n__from) = [1] x1 + [4]
p(n__s) = [1] x1 + [2]
p(n__take) = [1] x1 + [1] x2 + [2]
p(nil) = [1]
p(s) = [1] x1 + [9]
p(sel) = [1] x2 + [1]
p(take) = [1] x1 + [1] x2 + [7]
p(2nd#) = [7] x1 + [0]
p(activate#) = [5] x1 + [6]
p(from#) = [1] x1 + [1]
p(head#) = [1] x1 + [3]
p(s#) = [1] x1 + [0]
p(sel#) = [1] x1 + [2] x2 + [10]
p(take#) = [1] x1 + [1] x2 + [4]
p(c_1) = [1] x1 + [0]
p(c_2) = [2]
p(c_3) = [1] x1 + [8]
p(c_4) = [1] x1 + [9]
p(c_5) = [1] x1 + [12]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [1]
p(c_12) = [8]
Following rules are strictly oriented:
2nd#(cons(X,XS)) = [7] XS + [28]
> [5] XS + [4]
= c_1(head#(activate(XS)))
activate#(X) = [5] X + [6]
> [2]
= c_2()
activate#(n__from(X)) = [5] X + [26]
> [5] X + [10]
= c_3(from#(activate(X)))
activate#(n__s(X)) = [5] X + [16]
> [5] X + [10]
= c_4(s#(activate(X)))
from#(X) = [1] X + [1]
> [0]
= c_6()
from#(X) = [1] X + [1]
> [0]
= c_7()
head#(cons(X,XS)) = [1] XS + [7]
> [0]
= c_8()
sel#(0(),cons(X,XS)) = [2] XS + [20]
> [0]
= c_10()
take#(X1,X2) = [1] X1 + [1] X2 + [4]
> [1]
= c_11()
activate(X) = [5] X + [1]
> [1] X + [0]
= X
activate(n__from(X)) = [5] X + [21]
> [5] X + [16]
= from(activate(X))
activate(n__s(X)) = [5] X + [11]
> [5] X + [10]
= s(activate(X))
activate(n__take(X1,X2)) = [5] X1 + [5] X2 + [11]
> [5] X1 + [5] X2 + [9]
= take(activate(X1),activate(X2))
from(X) = [1] X + [15]
> [1] X + [10]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [15]
> [1] X + [4]
= n__from(X)
s(X) = [1] X + [9]
> [1] X + [2]
= n__s(X)
take(X1,X2) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [2]
= n__take(X1,X2)
take(0(),XS) = [1] XS + [9]
> [1]
= nil()
Following rules are (at-least) weakly oriented:
activate#(n__take(X1,X2)) = [5] X1 + [5] X2 + [16]
>= [5] X1 + [5] X2 + [18]
= c_5(take#(activate(X1)
,activate(X2)))
s#(X) = [1] X + [0]
>= [0]
= c_9()
take#(0(),XS) = [1] XS + [6]
>= [8]
= c_12()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
s#(X) -> c_9()
take#(0(),XS) -> c_12()
Strict TRS Rules:
Weak DP Rules:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(activate(X)))
activate#(n__s(X)) -> c_4(s#(activate(X)))
from#(X) -> c_6()
from#(X) -> c_7()
head#(cons(X,XS)) -> c_8()
sel#(0(),cons(X,XS)) -> c_10()
take#(X1,X2) -> c_11()
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
Obligation:
Innermost
basic terms: {2nd#,activate#,from#,head#,s#,sel#,take#}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{3}
by application of
Pre({3}) = {1}.
Here rules are labelled as follows:
1: activate#(n__take(X1,X2)) ->
c_5(take#(activate(X1)
,activate(X2)))
2: s#(X) -> c_9()
3: take#(0(),XS) -> c_12()
4: 2nd#(cons(X,XS)) ->
c_1(head#(activate(XS)))
5: activate#(X) -> c_2()
6: activate#(n__from(X)) ->
c_3(from#(activate(X)))
7: activate#(n__s(X)) ->
c_4(s#(activate(X)))
8: from#(X) -> c_6()
9: from#(X) -> c_7()
10: head#(cons(X,XS)) -> c_8()
11: sel#(0(),cons(X,XS)) -> c_10()
12: take#(X1,X2) -> c_11()
*** 1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
s#(X) -> c_9()
Strict TRS Rules:
Weak DP Rules:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(activate(X)))
activate#(n__s(X)) -> c_4(s#(activate(X)))
from#(X) -> c_6()
from#(X) -> c_7()
head#(cons(X,XS)) -> c_8()
sel#(0(),cons(X,XS)) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),XS) -> c_12()
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
Obligation:
Innermost
basic terms: {2nd#,activate#,from#,head#,s#,sel#,take#}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1}
by application of
Pre({1}) = {}.
Here rules are labelled as follows:
1: activate#(n__take(X1,X2)) ->
c_5(take#(activate(X1)
,activate(X2)))
2: s#(X) -> c_9()
3: 2nd#(cons(X,XS)) ->
c_1(head#(activate(XS)))
4: activate#(X) -> c_2()
5: activate#(n__from(X)) ->
c_3(from#(activate(X)))
6: activate#(n__s(X)) ->
c_4(s#(activate(X)))
7: from#(X) -> c_6()
8: from#(X) -> c_7()
9: head#(cons(X,XS)) -> c_8()
10: sel#(0(),cons(X,XS)) -> c_10()
11: take#(X1,X2) -> c_11()
12: take#(0(),XS) -> c_12()
*** 1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
s#(X) -> c_9()
Strict TRS Rules:
Weak DP Rules:
2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
activate#(X) -> c_2()
activate#(n__from(X)) -> c_3(from#(activate(X)))
activate#(n__s(X)) -> c_4(s#(activate(X)))
activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
from#(X) -> c_6()
from#(X) -> c_7()
head#(cons(X,XS)) -> c_8()
sel#(0(),cons(X,XS)) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),XS) -> c_12()
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
Obligation:
Innermost
basic terms: {2nd#,activate#,from#,head#,s#,sel#,take#}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:s#(X) -> c_9()
2:W:2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
-->_1 head#(cons(X,XS)) -> c_8():9
3:W:activate#(X) -> c_2()
4:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
-->_1 from#(X) -> c_7():8
-->_1 from#(X) -> c_6():7
5:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
-->_1 s#(X) -> c_9():1
6:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
-->_1 take#(0(),XS) -> c_12():12
-->_1 take#(X1,X2) -> c_11():11
7:W:from#(X) -> c_6()
8:W:from#(X) -> c_7()
9:W:head#(cons(X,XS)) -> c_8()
10:W:sel#(0(),cons(X,XS)) -> c_10()
11:W:take#(X1,X2) -> c_11()
12:W:take#(0(),XS) -> c_12()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: sel#(0(),cons(X,XS)) -> c_10()
6: activate#(n__take(X1,X2)) ->
c_5(take#(activate(X1)
,activate(X2)))
11: take#(X1,X2) -> c_11()
12: take#(0(),XS) -> c_12()
4: activate#(n__from(X)) ->
c_3(from#(activate(X)))
7: from#(X) -> c_6()
8: from#(X) -> c_7()
3: activate#(X) -> c_2()
2: 2nd#(cons(X,XS)) ->
c_1(head#(activate(XS)))
9: head#(cons(X,XS)) -> c_8()
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
s#(X) -> c_9()
Strict TRS Rules:
Weak DP Rules:
activate#(n__s(X)) -> c_4(s#(activate(X)))
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
Obligation:
Innermost
basic terms: {2nd#,activate#,from#,head#,s#,sel#,take#}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
Trivial
Proof:
Consider the dependency graph
1:S:s#(X) -> c_9()
5:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
-->_1 s#(X) -> c_9():1
The dependency graph contains no loops, we remove all dependency pairs.
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
take(X1,X2) -> n__take(X1,X2)
take(0(),XS) -> nil()
Signature:
{2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
Obligation:
Innermost
basic terms: {2nd#,activate#,from#,head#,s#,sel#,take#}/{0,cons,n__from,n__s,n__take,nil}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).