*** 1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,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(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [11]
p(a__from) = [1] x1 + [1]
p(a__length) = [0]
p(a__length1) = [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(length) = [0]
p(length1) = [0]
p(mark) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
a__from(X) = [1] X + [1]
> [1] X + [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [1]
> [1] X + [0]
= from(X)
Following rules are (at-least) weakly oriented:
a__length(X) = [0]
>= [0]
= length(X)
a__length(cons(X,Y)) = [0]
>= [0]
= s(a__length1(Y))
a__length(nil()) = [0]
>= [11]
= 0()
a__length1(X) = [0]
>= [0]
= a__length(X)
a__length1(X) = [0]
>= [0]
= length1(X)
mark(0()) = [11]
>= [11]
= 0()
mark(cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(mark(X1),X2)
mark(from(X)) = [1] X + [0]
>= [1] X + [1]
= a__from(mark(X))
mark(length(X)) = [0]
>= [0]
= a__length(X)
mark(length1(X)) = [0]
>= [0]
= a__length1(X)
mark(nil()) = [0]
>= [0]
= nil()
mark(s(X)) = [1] X + [0]
>= [1] X + [0]
= s(mark(X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,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(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__from) = [1] x1 + [1]
p(a__length) = [7]
p(a__length1) = [0]
p(cons) = [1] x1 + [1]
p(from) = [1] x1 + [1]
p(length) = [9]
p(length1) = [0]
p(mark) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [9]
Following rules are strictly oriented:
a__length(nil()) = [7]
> [0]
= 0()
mark(length(X)) = [9]
> [7]
= a__length(X)
Following rules are (at-least) weakly oriented:
a__from(X) = [1] X + [1]
>= [1] X + [1]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [1]
>= [1] X + [1]
= from(X)
a__length(X) = [7]
>= [9]
= length(X)
a__length(cons(X,Y)) = [7]
>= [9]
= s(a__length1(Y))
a__length1(X) = [0]
>= [7]
= a__length(X)
a__length1(X) = [0]
>= [0]
= length1(X)
mark(0()) = [0]
>= [0]
= 0()
mark(cons(X1,X2)) = [1] X1 + [1]
>= [1] X1 + [1]
= cons(mark(X1),X2)
mark(from(X)) = [1] X + [1]
>= [1] X + [1]
= a__from(mark(X))
mark(length1(X)) = [0]
>= [0]
= a__length1(X)
mark(nil()) = [0]
>= [0]
= nil()
mark(s(X)) = [1] X + [9]
>= [1] X + [9]
= s(mark(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^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(nil()) -> 0()
mark(length(X)) -> a__length(X)
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__from,a__length,a__length1,mark}
TcT has computed the following interpretation:
p(0) = [3]
p(a__from) = [1] x1 + [5]
p(a__length) = [3]
p(a__length1) = [3]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [5]
p(length) = [0]
p(length1) = [3]
p(mark) = [1] x1 + [4]
p(nil) = [12]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
a__length(X) = [3]
> [0]
= length(X)
mark(0()) = [7]
> [3]
= 0()
mark(length1(X)) = [7]
> [3]
= a__length1(X)
mark(nil()) = [16]
> [12]
= nil()
Following rules are (at-least) weakly oriented:
a__from(X) = [1] X + [5]
>= [1] X + [4]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [5]
>= [1] X + [5]
= from(X)
a__length(cons(X,Y)) = [3]
>= [3]
= s(a__length1(Y))
a__length(nil()) = [3]
>= [3]
= 0()
a__length1(X) = [3]
>= [3]
= a__length(X)
a__length1(X) = [3]
>= [3]
= length1(X)
mark(cons(X1,X2)) = [1] X1 + [4]
>= [1] X1 + [4]
= cons(mark(X1),X2)
mark(from(X)) = [1] X + [9]
>= [1] X + [9]
= a__from(mark(X))
mark(length(X)) = [4]
>= [3]
= a__length(X)
mark(s(X)) = [1] X + [4]
>= [1] X + [4]
= s(mark(X))
*** 1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(nil()) -> 0()
mark(0()) -> 0()
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,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(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(a__from) = [1] x1 + [6]
p(a__length) = [2]
p(a__length1) = [0]
p(cons) = [1] x1 + [6]
p(from) = [1] x1 + [6]
p(length) = [2]
p(length1) = [0]
p(mark) = [1] x1 + [0]
p(nil) = [2]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
a__length(cons(X,Y)) = [2]
> [0]
= s(a__length1(Y))
Following rules are (at-least) weakly oriented:
a__from(X) = [1] X + [6]
>= [1] X + [6]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [6]
>= [1] X + [6]
= from(X)
a__length(X) = [2]
>= [2]
= length(X)
a__length(nil()) = [2]
>= [1]
= 0()
a__length1(X) = [0]
>= [2]
= a__length(X)
a__length1(X) = [0]
>= [0]
= length1(X)
mark(0()) = [1]
>= [1]
= 0()
mark(cons(X1,X2)) = [1] X1 + [6]
>= [1] X1 + [6]
= cons(mark(X1),X2)
mark(from(X)) = [1] X + [6]
>= [1] X + [6]
= a__from(mark(X))
mark(length(X)) = [2]
>= [2]
= a__length(X)
mark(length1(X)) = [0]
>= [0]
= a__length1(X)
mark(nil()) = [2]
>= [2]
= nil()
mark(s(X)) = [1] X + [0]
>= [1] X + [0]
= s(mark(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^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
mark(0()) -> 0()
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,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(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [8]
p(a__from) = [1] x1 + [1]
p(a__length) = [8]
p(a__length1) = [8]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [1]
p(length) = [8]
p(length1) = [7]
p(mark) = [1] x1 + [1]
p(nil) = [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
a__length1(X) = [8]
> [7]
= length1(X)
Following rules are (at-least) weakly oriented:
a__from(X) = [1] X + [1]
>= [1] X + [1]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [1]
>= [1] X + [1]
= from(X)
a__length(X) = [8]
>= [8]
= length(X)
a__length(cons(X,Y)) = [8]
>= [8]
= s(a__length1(Y))
a__length(nil()) = [8]
>= [8]
= 0()
a__length1(X) = [8]
>= [8]
= a__length(X)
mark(0()) = [9]
>= [8]
= 0()
mark(cons(X1,X2)) = [1] X1 + [1]
>= [1] X1 + [1]
= cons(mark(X1),X2)
mark(from(X)) = [1] X + [2]
>= [1] X + [2]
= a__from(mark(X))
mark(length(X)) = [9]
>= [8]
= a__length(X)
mark(length1(X)) = [8]
>= [8]
= a__length1(X)
mark(nil()) = [1]
>= [0]
= nil()
mark(s(X)) = [1] X + [1]
>= [1] X + [1]
= s(mark(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^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__length1(X) -> a__length(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,nil,s}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__from,a__length,a__length1,mark}
TcT has computed the following interpretation:
p(0) = [1]
[0]
p(a__from) = [1 0] x1 + [0]
[0 0] [7]
p(a__length) = [0 1] x1 + [0]
[0 0] [0]
p(a__length1) = [0 1] x1 + [4]
[0 0] [0]
p(cons) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [4]
p(from) = [1 0] x1 + [0]
[0 0] [2]
p(length) = [0 1] x1 + [0]
[0 0] [0]
p(length1) = [0 1] x1 + [4]
[0 0] [0]
p(mark) = [1 0] x1 + [0]
[2 2] [5]
p(nil) = [0]
[2]
p(s) = [1 0] x1 + [0]
[0 1] [0]
Following rules are strictly oriented:
a__length1(X) = [0 1] X + [4]
[0 0] [0]
> [0 1] X + [0]
[0 0] [0]
= a__length(X)
Following rules are (at-least) weakly oriented:
a__from(X) = [1 0] X + [0]
[0 0] [7]
>= [1 0] X + [0]
[0 0] [6]
= cons(mark(X),from(s(X)))
a__from(X) = [1 0] X + [0]
[0 0] [7]
>= [1 0] X + [0]
[0 0] [2]
= from(X)
a__length(X) = [0 1] X + [0]
[0 0] [0]
>= [0 1] X + [0]
[0 0] [0]
= length(X)
a__length(cons(X,Y)) = [0 1] Y + [4]
[0 0] [0]
>= [0 1] Y + [4]
[0 0] [0]
= s(a__length1(Y))
a__length(nil()) = [2]
[0]
>= [1]
[0]
= 0()
a__length1(X) = [0 1] X + [4]
[0 0] [0]
>= [0 1] X + [4]
[0 0] [0]
= length1(X)
mark(0()) = [1]
[7]
>= [1]
[0]
= 0()
mark(cons(X1,X2)) = [1 0] X1 + [0 0] X2 + [0]
[2 0] [0 2] [13]
>= [1 0] X1 + [0 0] X2 + [0]
[0 0] [0 1] [4]
= cons(mark(X1),X2)
mark(from(X)) = [1 0] X + [0]
[2 0] [9]
>= [1 0] X + [0]
[0 0] [7]
= a__from(mark(X))
mark(length(X)) = [0 1] X + [0]
[0 2] [5]
>= [0 1] X + [0]
[0 0] [0]
= a__length(X)
mark(length1(X)) = [0 1] X + [4]
[0 2] [13]
>= [0 1] X + [4]
[0 0] [0]
= a__length1(X)
mark(nil()) = [0]
[9]
>= [0]
[2]
= nil()
mark(s(X)) = [1 0] X + [0]
[2 2] [5]
>= [1 0] X + [0]
[2 2] [5]
= s(mark(X))
*** 1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,nil,s}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__from,a__length,a__length1,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__from) = [1 4] x1 + [5]
[0 1] [3]
p(a__length) = [0]
[0]
p(a__length1) = [0]
[0]
p(cons) = [1 0] x1 + [0 1] x2 + [0]
[0 1] [0 0] [0]
p(from) = [1 4] x1 + [0]
[0 1] [3]
p(length) = [0]
[0]
p(length1) = [0]
[0]
p(mark) = [1 3] x1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(s) = [1 0] x1 + [0]
[0 1] [0]
Following rules are strictly oriented:
mark(from(X)) = [1 7] X + [9]
[0 1] [3]
> [1 7] X + [5]
[0 1] [3]
= a__from(mark(X))
Following rules are (at-least) weakly oriented:
a__from(X) = [1 4] X + [5]
[0 1] [3]
>= [1 4] X + [3]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [5]
[0 1] [3]
>= [1 4] X + [0]
[0 1] [3]
= from(X)
a__length(X) = [0]
[0]
>= [0]
[0]
= length(X)
a__length(cons(X,Y)) = [0]
[0]
>= [0]
[0]
= s(a__length1(Y))
a__length(nil()) = [0]
[0]
>= [0]
[0]
= 0()
a__length1(X) = [0]
[0]
>= [0]
[0]
= a__length(X)
a__length1(X) = [0]
[0]
>= [0]
[0]
= length1(X)
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(cons(X1,X2)) = [1 3] X1 + [0 1] X2 + [0]
[0 1] [0 0] [0]
>= [1 3] X1 + [0 1] X2 + [0]
[0 1] [0 0] [0]
= cons(mark(X1),X2)
mark(length(X)) = [0]
[0]
>= [0]
[0]
= a__length(X)
mark(length1(X)) = [0]
[0]
>= [0]
[0]
= a__length1(X)
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(s(X)) = [1 3] X + [0]
[0 1] [0]
>= [1 3] X + [0]
[0 1] [0]
= s(mark(X))
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,nil,s}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__from,a__length,a__length1,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__from) = [1 4] x1 + [5]
[0 1] [4]
p(a__length) = [0]
[0]
p(a__length1) = [0]
[0]
p(cons) = [1 0] x1 + [0]
[0 1] [2]
p(from) = [1 4] x1 + [1]
[0 1] [4]
p(length) = [0]
[0]
p(length1) = [0]
[0]
p(mark) = [1 1] x1 + [0]
[0 1] [0]
p(nil) = [4]
[4]
p(s) = [1 0] x1 + [0]
[0 1] [0]
Following rules are strictly oriented:
mark(cons(X1,X2)) = [1 1] X1 + [2]
[0 1] [2]
> [1 1] X1 + [0]
[0 1] [2]
= cons(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__from(X) = [1 4] X + [5]
[0 1] [4]
>= [1 1] X + [0]
[0 1] [2]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [5]
[0 1] [4]
>= [1 4] X + [1]
[0 1] [4]
= from(X)
a__length(X) = [0]
[0]
>= [0]
[0]
= length(X)
a__length(cons(X,Y)) = [0]
[0]
>= [0]
[0]
= s(a__length1(Y))
a__length(nil()) = [0]
[0]
>= [0]
[0]
= 0()
a__length1(X) = [0]
[0]
>= [0]
[0]
= a__length(X)
a__length1(X) = [0]
[0]
>= [0]
[0]
= length1(X)
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(from(X)) = [1 5] X + [5]
[0 1] [4]
>= [1 5] X + [5]
[0 1] [4]
= a__from(mark(X))
mark(length(X)) = [0]
[0]
>= [0]
[0]
= a__length(X)
mark(length1(X)) = [0]
[0]
>= [0]
[0]
= a__length1(X)
mark(nil()) = [8]
[4]
>= [4]
[4]
= nil()
mark(s(X)) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= s(mark(X))
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,nil,s}
Applied Processor:
NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(a__from) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__from,a__length,a__length1,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
[0]
p(a__from) = [1 1 0 0] [1]
[0 1 0 0] x1 + [1]
[0 0 0 0] [0]
[0 0 1 0] [1]
p(a__length) = [0 0 0 0] [1]
[0 0 0 1] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(a__length1) = [0 0 0 0] [1]
[0 0 0 1] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(cons) = [1 0 0 0] [0 0 0
1] [1]
[0 1 0 0] x1 + [0 0 0
0] x2 + [1]
[0 0 0 0] [0 0 0
0] [0]
[0 0 0 0] [0 0 0
1] [1]
p(from) = [1 1 0 0] [1]
[0 1 0 0] x1 + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(length) = [0 0 0 0] [1]
[0 0 0 1] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(length1) = [0 0 0 0] [1]
[0 0 0 1] x1 + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
p(mark) = [1 1 0 0] [0]
[0 1 0 0] x1 + [0]
[0 0 0 0] [0]
[1 0 0 0] [0]
p(nil) = [0]
[0]
[0]
[0]
p(s) = [1 0 0 0] [0]
[0 1 0 0] x1 + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
Following rules are strictly oriented:
mark(s(X)) = [1 1 0 0] [1]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[1 0 0 0] [0]
> [1 1 0 0] [0]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
= s(mark(X))
Following rules are (at-least) weakly oriented:
a__from(X) = [1 1 0 0] [1]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[0 0 1 0] [1]
>= [1 1 0 0] [1]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[0 0 0 0] [1]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0 0] [1]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[0 0 1 0] [1]
>= [1 1 0 0] [1]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
= from(X)
a__length(X) = [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
>= [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
= length(X)
a__length(cons(X,Y)) = [0 0 0 0] [1]
[0 0 0 1] Y + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
>= [0 0 0 0] [1]
[0 0 0 1] Y + [1]
[0 0 0 0] [0]
[0 0 0 0] [0]
= s(a__length1(Y))
a__length(nil()) = [1]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
= 0()
a__length1(X) = [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
>= [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
= a__length(X)
a__length1(X) = [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
>= [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
= length1(X)
mark(0()) = [0]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
= 0()
mark(cons(X1,X2)) = [1 1 0 0] [0 0 0
1] [2]
[0 1 0 0] X1 + [0 0 0
0] X2 + [1]
[0 0 0 0] [0 0 0
0] [0]
[1 0 0 0] [0 0 0
1] [1]
>= [1 1 0 0] [0 0 0
1] [1]
[0 1 0 0] X1 + [0 0 0
0] X2 + [1]
[0 0 0 0] [0 0 0
0] [0]
[0 0 0 0] [0 0 0
1] [1]
= cons(mark(X1),X2)
mark(from(X)) = [1 2 0 0] [2]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[1 1 0 0] [1]
>= [1 2 0 0] [1]
[0 1 0 0] X + [1]
[0 0 0 0] [0]
[0 0 0 0] [1]
= a__from(mark(X))
mark(length(X)) = [0 0 0 1] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [1]
>= [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
= a__length(X)
mark(length1(X)) = [0 0 0 1] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [1]
>= [0 0 0 0] [1]
[0 0 0 1] X + [0]
[0 0 0 0] [0]
[0 0 0 0] [0]
= a__length1(X)
mark(nil()) = [0]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
= nil()
*** 1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length(cons(X,Y)) -> s(a__length1(Y))
a__length(nil()) -> 0()
a__length1(X) -> a__length(X)
a__length1(X) -> length1(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(nil()) -> nil()
mark(s(X)) -> s(mark(X))
Signature:
{a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1}
Obligation:
Innermost
basic terms: {a__from,a__length,a__length1,mark}/{0,cons,from,length,length1,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).