*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
Obligation:
Innermost
basic terms: {2nd,activate,cons,from,s}/{n__cons,n__from,n__s}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
Obligation:
Innermost
basic terms: {2nd,activate,cons,from,s}/{n__cons,n__from,n__s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [1] x1 + [9]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [9]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [12]
p(n__s) = [1] x1 + [4]
p(s) = [1] x1 + [11]
Following rules are strictly oriented:
activate(X) = [1] X + [9]
> [1] X + [0]
= X
activate(n__from(X)) = [1] X + [21]
> [1] X + [18]
= from(activate(X))
from(X) = [1] X + [9]
> [1] X + [0]
= cons(X,n__from(n__s(X)))
s(X) = [1] X + [11]
> [1] X + [4]
= n__s(X)
Following rules are (at-least) weakly oriented:
activate(n__cons(X1,X2)) = [1] X1 + [9]
>= [1] X1 + [9]
= cons(activate(X1),X2)
activate(n__s(X)) = [1] X + [13]
>= [1] X + [20]
= s(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [9]
>= [1] X + [12]
= n__from(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
s(X) -> n__s(X)
Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
Obligation:
Innermost
basic terms: {2nd,activate,cons,from,s}/{n__cons,n__from,n__s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [10] x1 + [0]
p(cons) = [1] x1 + [10]
p(from) = [1] x1 + [10]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
cons(X1,X2) = [1] X1 + [10]
> [1] X1 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [10]
> [1] X + [1]
= n__from(X)
Following rules are (at-least) weakly oriented:
activate(X) = [10] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [10] X1 + [0]
>= [10] X1 + [10]
= cons(activate(X1),X2)
activate(n__from(X)) = [10] X + [10]
>= [10] X + [10]
= from(activate(X))
activate(n__s(X)) = [10] X + [0]
>= [10] X + [0]
= s(activate(X))
from(X) = [1] X + [10]
>= [1] X + [10]
= cons(X,n__from(n__s(X)))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(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:
Strict TRS Rules:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__s(X)) -> s(activate(X))
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
Obligation:
Innermost
basic terms: {2nd,activate,cons,from,s}/{n__cons,n__from,n__s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [15] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [1]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
activate(n__s(X)) = [15] X + [15]
> [15] X + [2]
= s(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [15] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [15] X1 + [0]
>= [15] X1 + [0]
= cons(activate(X1),X2)
activate(n__from(X)) = [15] X + [0]
>= [15] X + [0]
= from(activate(X))
cons(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [0]
>= [1] X + [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
s(X) = [1] X + [2]
>= [1] X + [1]
= n__s(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:
Strict TRS Rules:
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
Obligation:
Innermost
basic terms: {2nd,activate,cons,from,s}/{n__cons,n__from,n__s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, 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(cons) = {1},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(2nd) = [0]
p(activate) = [3] x1 + [0]
p(cons) = [1] x1 + [6]
p(from) = [1] x1 + [6]
p(n__cons) = [1] x1 + [5]
p(n__from) = [1] x1 + [2]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
activate(n__cons(X1,X2)) = [3] X1 + [15]
> [3] X1 + [6]
= cons(activate(X1),X2)
Following rules are (at-least) weakly oriented:
activate(X) = [3] X + [0]
>= [1] X + [0]
= X
activate(n__from(X)) = [3] X + [6]
>= [3] X + [6]
= from(activate(X))
activate(n__s(X)) = [3] X + [0]
>= [3] X + [0]
= s(activate(X))
cons(X1,X2) = [1] X1 + [6]
>= [1] X1 + [5]
= n__cons(X1,X2)
from(X) = [1] X + [6]
>= [1] X + [6]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [6]
>= [1] X + [2]
= n__from(X)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(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(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
Signature:
{2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1}
Obligation:
Innermost
basic terms: {2nd,activate,cons,from,s}/{n__cons,n__from,n__s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).