*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,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) = {2},
uargs(n__first) = {2}
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(first) = [1] x2 + [0]
p(from) = [0]
p(n__first) = [1] x2 + [0]
p(n__from) = [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
activate(X) = [1] X + [4]
> [1] X + [0]
= X
activate(n__first(X1,X2)) = [1] X2 + [4]
> [1] X2 + [0]
= first(X1,X2)
activate(n__from(X)) = [4]
> [0]
= from(X)
Following rules are (at-least) weakly oriented:
first(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
first(s(X),cons(Y,Z)) = [1] Z + [0]
>= [1] Z + [4]
= cons(Y,n__first(X,activate(Z)))
from(X) = [0]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,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) = {2},
uargs(n__first) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [10] x1 + [8]
p(cons) = [1] x2 + [0]
p(first) = [8] x1 + [10] x2 + [0]
p(from) = [2]
p(n__first) = [1] x1 + [1] x2 + [1]
p(n__from) = [0]
p(nil) = [0]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
first(s(X),cons(Y,Z)) = [8] X + [10] Z + [16]
> [1] X + [10] Z + [9]
= cons(Y,n__first(X,activate(Z)))
from(X) = [2]
> [0]
= cons(X,n__from(s(X)))
from(X) = [2]
> [0]
= n__from(X)
Following rules are (at-least) weakly oriented:
activate(X) = [10] X + [8]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [10] X1 + [10] X2 + [18]
>= [8] X1 + [10] X2 + [0]
= first(X1,X2)
activate(n__from(X)) = [8]
>= [2]
= from(X)
first(X1,X2) = [8] X1 + [10] X2 + [0]
>= [1] X1 + [1] X2 + [1]
= n__first(X1,X2)
first(0(),X) = [10] X + [0]
>= [0]
= nil()
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:
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,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) = {2},
uargs(n__first) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [2] x1 + [8]
p(cons) = [1] x2 + [1]
p(first) = [2] x2 + [7]
p(from) = [4]
p(n__first) = [1] x2 + [0]
p(n__from) = [0]
p(nil) = [1]
p(s) = [1] x1 + [8]
Following rules are strictly oriented:
first(X1,X2) = [2] X2 + [7]
> [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [2] X + [7]
> [1]
= nil()
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [8]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [2] X2 + [8]
>= [2] X2 + [7]
= first(X1,X2)
activate(n__from(X)) = [8]
>= [4]
= from(X)
first(s(X),cons(Y,Z)) = [2] Z + [9]
>= [2] Z + [9]
= cons(Y,n__first(X,activate(Z)))
from(X) = [4]
>= [1]
= cons(X,n__from(s(X)))
from(X) = [4]
>= [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(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__from(X)) -> from(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
Signature:
{activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).