*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1}
Obligation:
Full
basic terms: {activate,from,sel}/{0,cons,n__from,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following weak dependency pairs:
Strict DPs
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Strict TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1}
Obligation:
Full
basic terms: {activate#,from#,sel#}/{0,cons,n__from,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Strict TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1}
Obligation:
Full
basic terms: {activate#,from#,sel#}/{0,cons,n__from,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [5]
p(from) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(sel) = [0]
p(activate#) = [3] x1 + [0]
p(from#) = [3] x1 + [0]
p(sel#) = [1] x2 + [0]
p(c_1) = [3] x1 + [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [3] x1 + [0]
p(c_4) = [3] x1 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
Following rules are strictly oriented:
sel#(0(),cons(X,Y)) = [1] Y + [5]
> [0]
= c_5(X)
sel#(s(X),cons(Y,Z)) = [1] Z + [5]
> [1] Z + [0]
= c_6(sel#(X,activate(Z)))
Following rules are (at-least) weakly oriented:
activate#(X) = [3] X + [0]
>= [3] X + [0]
= c_1(X)
activate#(n__from(X)) = [3] X + [0]
>= [3] X + [0]
= c_2(from#(X))
from#(X) = [3] X + [0]
>= [3] X + [0]
= c_3(X,X)
from#(X) = [3] X + [0]
>= [3] X + [0]
= c_4(X)
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__from(X)) = [1] X + [0]
>= [1] X + [0]
= from(X)
from(X) = [1] X + [0]
>= [1] X + [5]
= cons(X,n__from(s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
Strict TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Weak DP Rules:
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Weak TRS Rules:
Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1}
Obligation:
Full
basic terms: {activate#,from#,sel#}/{0,cons,n__from,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [7]
p(from) = [0]
p(n__from) = [1]
p(s) = [1] x1 + [0]
p(sel) = [4]
p(activate#) = [1] x1 + [0]
p(from#) = [0]
p(sel#) = [2] x1 + [1] x2 + [8]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [15]
p(c_6) = [1] x1 + [2]
Following rules are strictly oriented:
activate#(n__from(X)) = [1]
> [0]
= c_2(from#(X))
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__from(X)) = [3]
> [0]
= from(X)
Following rules are (at-least) weakly oriented:
activate#(X) = [1] X + [0]
>= [0]
= c_1(X)
from#(X) = [0]
>= [0]
= c_3(X,X)
from#(X) = [0]
>= [0]
= c_4(X)
sel#(0(),cons(X,Y)) = [1] Y + [15]
>= [15]
= c_5(X)
sel#(s(X),cons(Y,Z)) = [2] X + [1] Z + [15]
>= [2] X + [1] Z + [12]
= c_6(sel#(X,activate(Z)))
from(X) = [0]
>= [8]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [1]
= n__from(X)
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(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(X) -> c_1(X)
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Weak DP Rules:
activate#(n__from(X)) -> c_2(from#(X))
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1}
Obligation:
Full
basic terms: {activate#,from#,sel#}/{0,cons,n__from,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(sel) = [0]
p(activate#) = [1]
p(from#) = [1]
p(sel#) = [1] x2 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
Following rules are strictly oriented:
activate#(X) = [1]
> [0]
= c_1(X)
from#(X) = [1]
> [0]
= c_3(X,X)
from#(X) = [1]
> [0]
= c_4(X)
Following rules are (at-least) weakly oriented:
activate#(n__from(X)) = [1]
>= [1]
= c_2(from#(X))
sel#(0(),cons(X,Y)) = [1] Y + [0]
>= [0]
= c_5(X)
sel#(s(X),cons(Y,Z)) = [1] Z + [0]
>= [1] Z + [0]
= c_6(sel#(X,activate(Z)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__from(X)) = [1] X + [0]
>= [1] X + [0]
= from(X)
from(X) = [1] X + [0]
>= [1] X + [0]
= cons(X,n__from(s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Weak DP Rules:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1}
Obligation:
Full
basic terms: {activate#,from#,sel#}/{0,cons,n__from,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(sel#) = {2},
uargs(c_2) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [4]
p(cons) = [1] x2 + [0]
p(from) = [4]
p(n__from) = [2]
p(s) = [1] x1 + [1]
p(sel) = [1] x1 + [1] x2 + [1]
p(activate#) = [8]
p(from#) = [1]
p(sel#) = [4] x1 + [1] x2 + [0]
p(c_1) = [8]
p(c_2) = [1] x1 + [7]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
Following rules are strictly oriented:
from(X) = [4]
> [2]
= cons(X,n__from(s(X)))
from(X) = [4]
> [2]
= n__from(X)
Following rules are (at-least) weakly oriented:
activate#(X) = [8]
>= [8]
= c_1(X)
activate#(n__from(X)) = [8]
>= [8]
= c_2(from#(X))
from#(X) = [1]
>= [1]
= c_3(X,X)
from#(X) = [1]
>= [1]
= c_4(X)
sel#(0(),cons(X,Y)) = [1] Y + [0]
>= [0]
= c_5(X)
sel#(s(X),cons(Y,Z)) = [4] X + [1] Z + [4]
>= [4] X + [1] Z + [4]
= c_6(sel#(X,activate(Z)))
activate(X) = [1] X + [4]
>= [1] X + [0]
= X
activate(n__from(X)) = [6]
>= [4]
= from(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
activate#(X) -> c_1(X)
activate#(n__from(X)) -> c_2(from#(X))
from#(X) -> c_3(X,X)
from#(X) -> c_4(X)
sel#(0(),cons(X,Y)) -> c_5(X)
sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Signature:
{activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1}
Obligation:
Full
basic terms: {activate#,from#,sel#}/{0,cons,n__from,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).