*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Z)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
Obligation:
Innermost
basic terms: {activate,first,from,s,sel}/{0,cons,n__first,n__from,n__s,nil}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
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__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),Z) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Z)) -> X
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
Obligation:
Innermost
basic terms: {activate,first,from,s,sel}/{0,cons,n__first,n__from,n__s,nil}
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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(activate) = [8] x1 + [3]
p(cons) = [1] x1 + [6]
p(first) = [1] x1 + [1] x2 + [1]
p(from) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [1]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(nil) = [1]
p(s) = [1] x1 + [0]
p(sel) = [11] x1 + [3] x2 + [1]
Following rules are strictly oriented:
activate(X) = [8] X + [3]
> [1] X + [0]
= X
activate(n__first(X1,X2)) = [8] X1 + [8] X2 + [11]
> [8] X1 + [8] X2 + [7]
= first(activate(X1),activate(X2))
first(0(),Z) = [1] Z + [2]
> [1]
= nil()
sel(0(),cons(X,Z)) = [3] X + [30]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
activate(n__from(X)) = [8] X + [3]
>= [8] X + [3]
= from(activate(X))
activate(n__s(X)) = [8] X + [3]
>= [8] X + [3]
= s(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__first(X1,X2)
from(X) = [1] X + [0]
>= [1] X + [6]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= 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 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(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:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
first(0(),Z) -> nil()
sel(0(),cons(X,Z)) -> X
Signature:
{activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
Obligation:
Innermost
basic terms: {activate,first,from,s,sel}/{0,cons,n__first,n__from,n__s,nil}
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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [12]
p(n__s) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
p(sel) = [2] x2 + [9]
Following rules are strictly oriented:
activate(n__from(X)) = [1] X + [12]
> [1] X + [0]
= from(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),Z) = [1] Z + [0]
>= [0]
= nil()
from(X) = [1] X + [0]
>= [1] X + [0]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [0]
>= [1] X + [12]
= n__from(X)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sel(0(),cons(X,Z)) = [2] X + [9]
>= [1] X + [0]
= 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__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(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:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
first(0(),Z) -> nil()
sel(0(),cons(X,Z)) -> X
Signature:
{activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
Obligation:
Innermost
basic terms: {activate,first,from,s,sel}/{0,cons,n__first,n__from,n__s,nil}
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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(activate) = [4] x1 + [2]
p(cons) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [5]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [3]
p(nil) = [0]
p(s) = [1] x1 + [6]
p(sel) = [5] x1 + [1] x2 + [3]
Following rules are strictly oriented:
activate(n__s(X)) = [4] X + [14]
> [4] X + [8]
= s(activate(X))
s(X) = [1] X + [6]
> [1] X + [3]
= n__s(X)
Following rules are (at-least) weakly oriented:
activate(X) = [4] X + [2]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [22]
>= [4] X1 + [4] X2 + [4]
= first(activate(X1),activate(X2))
activate(n__from(X)) = [4] X + [2]
>= [4] X + [2]
= from(activate(X))
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [5]
= n__first(X1,X2)
first(0(),Z) = [1] Z + [1]
>= [0]
= nil()
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)
sel(0(),cons(X,Z)) = [1] X + [8]
>= [1] X + [0]
= 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:
first(X1,X2) -> n__first(X1,X2)
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(0(),Z) -> nil()
s(X) -> n__s(X)
sel(0(),cons(X,Z)) -> X
Signature:
{activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
Obligation:
Innermost
basic terms: {activate,first,from,s,sel}/{0,cons,n__first,n__from,n__s,nil}
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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [10] x1 + [0]
p(cons) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(from) = [1] x1 + [10]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
p(sel) = [1] x2 + [0]
Following rules are strictly oriented:
from(X) = [1] X + [10]
> [1] X + [0]
= cons(X,n__from(n__s(X)))
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__first(X1,X2)) = [10] X1 + [10] X2 + [0]
>= [10] X1 + [10] X2 + [0]
= first(activate(X1),activate(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))
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),Z) = [1] Z + [0]
>= [0]
= nil()
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sel(0(),cons(X,Z)) = [1] X + [0]
>= [1] X + [0]
= 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:
first(X1,X2) -> n__first(X1,X2)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(0(),Z) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Z)) -> X
Signature:
{activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
Obligation:
Innermost
basic terms: {activate,first,from,s,sel}/{0,cons,n__first,n__from,n__s,nil}
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(first) = {1,2},
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [2] x1 + [0]
p(cons) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [4]
p(from) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [2]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
p(sel) = [11] x1 + [1] x2 + [3]
Following rules are strictly oriented:
first(X1,X2) = [1] X1 + [1] X2 + [4]
> [1] X1 + [1] X2 + [2]
= n__first(X1,X2)
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [4]
>= [2] X1 + [2] X2 + [4]
= first(activate(X1),activate(X2))
activate(n__from(X)) = [2] X + [0]
>= [2] X + [0]
= from(activate(X))
activate(n__s(X)) = [2] X + [0]
>= [2] X + [0]
= s(activate(X))
first(0(),Z) = [1] Z + [6]
>= [0]
= nil()
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 + [0]
>= [1] X + [0]
= n__s(X)
sel(0(),cons(X,Z)) = [1] X + [25]
>= [1] X + [0]
= 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:
Weak TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
first(X1,X2) -> n__first(X1,X2)
first(0(),Z) -> nil()
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Z)) -> X
Signature:
{activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0}
Obligation:
Innermost
basic terms: {activate,first,from,s,sel}/{0,cons,n__first,n__from,n__s,nil}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).