*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(X) -> length(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(inf(X)) -> a__inf(mark(X))
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(true()) -> true()
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__eq) = [0]
p(a__inf) = [1] x1 + [0]
p(a__length) = [1] x1 + [0]
p(a__take) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [0]
p(eq) = [0]
p(false) = [0]
p(inf) = [1] x1 + [0]
p(length) = [1] x1 + [1]
p(mark) = [1] x1 + [7]
p(nil) = [0]
p(s) = [0]
p(take) = [1] x1 + [1] x2 + [0]
p(true) = [0]
Following rules are strictly oriented:
mark(0()) = [7]
> [0]
= 0()
mark(cons(X1,X2)) = [1] X1 + [7]
> [1] X1 + [0]
= cons(X1,X2)
mark(eq(X1,X2)) = [7]
> [0]
= a__eq(X1,X2)
mark(false()) = [7]
> [0]
= false()
mark(length(X)) = [1] X + [8]
> [1] X + [7]
= a__length(mark(X))
mark(nil()) = [7]
> [0]
= nil()
mark(s(X)) = [7]
> [0]
= s(X)
mark(true()) = [7]
> [0]
= true()
Following rules are (at-least) weakly oriented:
a__eq(X,Y) = [0]
>= [0]
= false()
a__eq(X1,X2) = [0]
>= [0]
= eq(X1,X2)
a__eq(0(),0()) = [0]
>= [0]
= true()
a__eq(s(X),s(Y)) = [0]
>= [0]
= a__eq(X,Y)
a__inf(X) = [1] X + [0]
>= [1] X + [0]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [0]
>= [1] X + [0]
= inf(X)
a__length(X) = [1] X + [0]
>= [1] X + [1]
= length(X)
a__length(cons(X,L)) = [1] X + [0]
>= [0]
= s(length(L))
a__length(nil()) = [0]
>= [0]
= 0()
a__take(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= take(X1,X2)
a__take(0(),X) = [1] X + [0]
>= [0]
= nil()
a__take(s(X),cons(Y,L)) = [1] Y + [0]
>= [1] Y + [0]
= cons(Y,take(X,L))
mark(inf(X)) = [1] X + [7]
>= [1] X + [7]
= a__inf(mark(X))
mark(take(X1,X2)) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [14]
= a__take(mark(X1),mark(X2))
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:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(X) -> length(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(a__eq) = [1]
p(a__inf) = [1] x1 + [0]
p(a__length) = [1] x1 + [0]
p(a__take) = [1] x1 + [1] x2 + [7]
p(cons) = [1]
p(eq) = [0]
p(false) = [0]
p(inf) = [0]
p(length) = [1] x1 + [0]
p(mark) = [1]
p(nil) = [0]
p(s) = [1]
p(take) = [1] x1 + [1] x2 + [0]
p(true) = [0]
Following rules are strictly oriented:
a__eq(X,Y) = [1]
> [0]
= false()
a__eq(X1,X2) = [1]
> [0]
= eq(X1,X2)
a__eq(0(),0()) = [1]
> [0]
= true()
a__take(X1,X2) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [0]
= take(X1,X2)
a__take(0(),X) = [1] X + [8]
> [0]
= nil()
a__take(s(X),cons(Y,L)) = [9]
> [1]
= cons(Y,take(X,L))
Following rules are (at-least) weakly oriented:
a__eq(s(X),s(Y)) = [1]
>= [1]
= a__eq(X,Y)
a__inf(X) = [1] X + [0]
>= [1]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [0]
>= [0]
= inf(X)
a__length(X) = [1] X + [0]
>= [1] X + [0]
= length(X)
a__length(cons(X,L)) = [1]
>= [1]
= s(length(L))
a__length(nil()) = [0]
>= [1]
= 0()
mark(0()) = [1]
>= [1]
= 0()
mark(cons(X1,X2)) = [1]
>= [1]
= cons(X1,X2)
mark(eq(X1,X2)) = [1]
>= [1]
= a__eq(X1,X2)
mark(false()) = [1]
>= [0]
= false()
mark(inf(X)) = [1]
>= [1]
= a__inf(mark(X))
mark(length(X)) = [1]
>= [1]
= a__length(mark(X))
mark(nil()) = [1]
>= [0]
= nil()
mark(s(X)) = [1]
>= [1]
= s(X)
mark(take(X1,X2)) = [1]
>= [9]
= a__take(mark(X1),mark(X2))
mark(true()) = [1]
>= [0]
= true()
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:
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(X) -> length(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__eq) = [4]
p(a__inf) = [1] x1 + [1]
p(a__length) = [1] x1 + [0]
p(a__take) = [1] x1 + [1] x2 + [0]
p(cons) = [0]
p(eq) = [4]
p(false) = [0]
p(inf) = [0]
p(length) = [0]
p(mark) = [4]
p(nil) = [0]
p(s) = [2]
p(take) = [1] x2 + [0]
p(true) = [4]
Following rules are strictly oriented:
a__inf(X) = [1] X + [1]
> [0]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [1]
> [0]
= inf(X)
Following rules are (at-least) weakly oriented:
a__eq(X,Y) = [4]
>= [0]
= false()
a__eq(X1,X2) = [4]
>= [4]
= eq(X1,X2)
a__eq(0(),0()) = [4]
>= [4]
= true()
a__eq(s(X),s(Y)) = [4]
>= [4]
= a__eq(X,Y)
a__length(X) = [1] X + [0]
>= [0]
= length(X)
a__length(cons(X,L)) = [0]
>= [2]
= s(length(L))
a__length(nil()) = [0]
>= [0]
= 0()
a__take(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X2 + [0]
= take(X1,X2)
a__take(0(),X) = [1] X + [0]
>= [0]
= nil()
a__take(s(X),cons(Y,L)) = [2]
>= [0]
= cons(Y,take(X,L))
mark(0()) = [4]
>= [0]
= 0()
mark(cons(X1,X2)) = [4]
>= [0]
= cons(X1,X2)
mark(eq(X1,X2)) = [4]
>= [4]
= a__eq(X1,X2)
mark(false()) = [4]
>= [0]
= false()
mark(inf(X)) = [4]
>= [5]
= a__inf(mark(X))
mark(length(X)) = [4]
>= [4]
= a__length(mark(X))
mark(nil()) = [4]
>= [0]
= nil()
mark(s(X)) = [4]
>= [2]
= s(X)
mark(take(X1,X2)) = [4]
>= [8]
= a__take(mark(X1),mark(X2))
mark(true()) = [4]
>= [4]
= true()
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:
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__length(X) -> length(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(a__eq) = [0]
p(a__inf) = [1] x1 + [2]
p(a__length) = [1] x1 + [0]
p(a__take) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [2]
p(eq) = [0]
p(false) = [0]
p(inf) = [1] x1 + [2]
p(length) = [1] x1 + [0]
p(mark) = [5] x1 + [0]
p(nil) = [1]
p(s) = [0]
p(take) = [1] x1 + [1] x2 + [0]
p(true) = [0]
Following rules are strictly oriented:
a__length(cons(X,L)) = [1] X + [2]
> [0]
= s(length(L))
mark(inf(X)) = [5] X + [10]
> [5] X + [2]
= a__inf(mark(X))
Following rules are (at-least) weakly oriented:
a__eq(X,Y) = [0]
>= [0]
= false()
a__eq(X1,X2) = [0]
>= [0]
= eq(X1,X2)
a__eq(0(),0()) = [0]
>= [0]
= true()
a__eq(s(X),s(Y)) = [0]
>= [0]
= a__eq(X,Y)
a__inf(X) = [1] X + [2]
>= [1] X + [2]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [2]
>= [1] X + [2]
= inf(X)
a__length(X) = [1] X + [0]
>= [1] X + [0]
= length(X)
a__length(nil()) = [1]
>= [1]
= 0()
a__take(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= take(X1,X2)
a__take(0(),X) = [1] X + [1]
>= [1]
= nil()
a__take(s(X),cons(Y,L)) = [1] Y + [2]
>= [1] Y + [2]
= cons(Y,take(X,L))
mark(0()) = [5]
>= [1]
= 0()
mark(cons(X1,X2)) = [5] X1 + [10]
>= [1] X1 + [2]
= cons(X1,X2)
mark(eq(X1,X2)) = [0]
>= [0]
= a__eq(X1,X2)
mark(false()) = [0]
>= [0]
= false()
mark(length(X)) = [5] X + [0]
>= [5] X + [0]
= a__length(mark(X))
mark(nil()) = [5]
>= [1]
= nil()
mark(s(X)) = [0]
>= [0]
= s(X)
mark(take(X1,X2)) = [5] X1 + [5] X2 + [0]
>= [5] X1 + [5] X2 + [0]
= a__take(mark(X1),mark(X2))
mark(true()) = [0]
>= [0]
= true()
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:
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__length(X) -> length(X)
a__length(nil()) -> 0()
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(cons(X,L)) -> s(length(L))
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(inf(X)) -> a__inf(mark(X))
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(a__eq) = [1]
p(a__inf) = [1] x1 + [1]
p(a__length) = [1] x1 + [1]
p(a__take) = [1] x1 + [1] x2 + [1]
p(cons) = [1] x1 + [0]
p(eq) = [1]
p(false) = [0]
p(inf) = [1] x1 + [1]
p(length) = [1] x1 + [2]
p(mark) = [4] x1 + [0]
p(nil) = [2]
p(s) = [0]
p(take) = [1] x1 + [1] x2 + [0]
p(true) = [0]
Following rules are strictly oriented:
a__length(nil()) = [3]
> [2]
= 0()
Following rules are (at-least) weakly oriented:
a__eq(X,Y) = [1]
>= [0]
= false()
a__eq(X1,X2) = [1]
>= [1]
= eq(X1,X2)
a__eq(0(),0()) = [1]
>= [0]
= true()
a__eq(s(X),s(Y)) = [1]
>= [1]
= a__eq(X,Y)
a__inf(X) = [1] X + [1]
>= [1] X + [0]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [1]
>= [1] X + [1]
= inf(X)
a__length(X) = [1] X + [1]
>= [1] X + [2]
= length(X)
a__length(cons(X,L)) = [1] X + [1]
>= [0]
= s(length(L))
a__take(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [0]
= take(X1,X2)
a__take(0(),X) = [1] X + [3]
>= [2]
= nil()
a__take(s(X),cons(Y,L)) = [1] Y + [1]
>= [1] Y + [0]
= cons(Y,take(X,L))
mark(0()) = [8]
>= [2]
= 0()
mark(cons(X1,X2)) = [4] X1 + [0]
>= [1] X1 + [0]
= cons(X1,X2)
mark(eq(X1,X2)) = [4]
>= [1]
= a__eq(X1,X2)
mark(false()) = [0]
>= [0]
= false()
mark(inf(X)) = [4] X + [4]
>= [4] X + [1]
= a__inf(mark(X))
mark(length(X)) = [4] X + [8]
>= [4] X + [1]
= a__length(mark(X))
mark(nil()) = [8]
>= [2]
= nil()
mark(s(X)) = [0]
>= [0]
= s(X)
mark(take(X1,X2)) = [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [1]
= a__take(mark(X1),mark(X2))
mark(true()) = [0]
>= [0]
= true()
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:
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__length(X) -> length(X)
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(inf(X)) -> a__inf(mark(X))
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(a__eq) = [0]
p(a__inf) = [1] x1 + [3]
p(a__length) = [1] x1 + [0]
p(a__take) = [1] x1 + [1] x2 + [1]
p(cons) = [2]
p(eq) = [0]
p(false) = [0]
p(inf) = [1] x1 + [1]
p(length) = [1] x1 + [0]
p(mark) = [4] x1 + [0]
p(nil) = [1]
p(s) = [0]
p(take) = [1] x1 + [1] x2 + [1]
p(true) = [0]
Following rules are strictly oriented:
mark(take(X1,X2)) = [4] X1 + [4] X2 + [4]
> [4] X1 + [4] X2 + [1]
= a__take(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__eq(X,Y) = [0]
>= [0]
= false()
a__eq(X1,X2) = [0]
>= [0]
= eq(X1,X2)
a__eq(0(),0()) = [0]
>= [0]
= true()
a__eq(s(X),s(Y)) = [0]
>= [0]
= a__eq(X,Y)
a__inf(X) = [1] X + [3]
>= [2]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [3]
>= [1] X + [1]
= inf(X)
a__length(X) = [1] X + [0]
>= [1] X + [0]
= length(X)
a__length(cons(X,L)) = [2]
>= [0]
= s(length(L))
a__length(nil()) = [1]
>= [1]
= 0()
a__take(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= take(X1,X2)
a__take(0(),X) = [1] X + [2]
>= [1]
= nil()
a__take(s(X),cons(Y,L)) = [3]
>= [2]
= cons(Y,take(X,L))
mark(0()) = [4]
>= [1]
= 0()
mark(cons(X1,X2)) = [8]
>= [2]
= cons(X1,X2)
mark(eq(X1,X2)) = [0]
>= [0]
= a__eq(X1,X2)
mark(false()) = [0]
>= [0]
= false()
mark(inf(X)) = [4] X + [4]
>= [4] X + [3]
= a__inf(mark(X))
mark(length(X)) = [4] X + [0]
>= [4] X + [0]
= a__length(mark(X))
mark(nil()) = [4]
>= [1]
= nil()
mark(s(X)) = [0]
>= [0]
= s(X)
mark(true()) = [0]
>= [0]
= true()
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(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__length(X) -> length(X)
Weak DP Rules:
Weak TRS Rules:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(inf(X)) -> a__inf(mark(X))
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__eq) = [2]
p(a__inf) = [1] x1 + [0]
p(a__length) = [1] x1 + [4]
p(a__take) = [1] x1 + [1] x2 + [6]
p(cons) = [1] x1 + [0]
p(eq) = [2]
p(false) = [0]
p(inf) = [1] x1 + [0]
p(length) = [1] x1 + [1]
p(mark) = [4] x1 + [0]
p(nil) = [0]
p(s) = [0]
p(take) = [1] x1 + [1] x2 + [3]
p(true) = [0]
Following rules are strictly oriented:
a__length(X) = [1] X + [4]
> [1] X + [1]
= length(X)
Following rules are (at-least) weakly oriented:
a__eq(X,Y) = [2]
>= [0]
= false()
a__eq(X1,X2) = [2]
>= [2]
= eq(X1,X2)
a__eq(0(),0()) = [2]
>= [0]
= true()
a__eq(s(X),s(Y)) = [2]
>= [2]
= a__eq(X,Y)
a__inf(X) = [1] X + [0]
>= [1] X + [0]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [0]
>= [1] X + [0]
= inf(X)
a__length(cons(X,L)) = [1] X + [4]
>= [0]
= s(length(L))
a__length(nil()) = [4]
>= [0]
= 0()
a__take(X1,X2) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [3]
= take(X1,X2)
a__take(0(),X) = [1] X + [6]
>= [0]
= nil()
a__take(s(X),cons(Y,L)) = [1] Y + [6]
>= [1] Y + [0]
= cons(Y,take(X,L))
mark(0()) = [0]
>= [0]
= 0()
mark(cons(X1,X2)) = [4] X1 + [0]
>= [1] X1 + [0]
= cons(X1,X2)
mark(eq(X1,X2)) = [8]
>= [2]
= a__eq(X1,X2)
mark(false()) = [0]
>= [0]
= false()
mark(inf(X)) = [4] X + [0]
>= [4] X + [0]
= a__inf(mark(X))
mark(length(X)) = [4] X + [4]
>= [4] X + [4]
= a__length(mark(X))
mark(nil()) = [0]
>= [0]
= nil()
mark(s(X)) = [0]
>= [0]
= s(X)
mark(take(X1,X2)) = [4] X1 + [4] X2 + [12]
>= [4] X1 + [4] X2 + [6]
= a__take(mark(X1),mark(X2))
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__eq(s(X),s(Y)) -> a__eq(X,Y)
Weak DP Rules:
Weak TRS Rules:
a__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(X) -> length(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(inf(X)) -> a__inf(mark(X))
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
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__inf) = {1},
uargs(a__length) = {1},
uargs(a__take) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__eq) = [1] x2 + [0]
p(a__inf) = [1] x1 + [4]
p(a__length) = [1] x1 + [5]
p(a__take) = [1] x1 + [1] x2 + [4]
p(cons) = [1] x2 + [0]
p(eq) = [1] x2 + [0]
p(false) = [0]
p(inf) = [1] x1 + [2]
p(length) = [1] x1 + [4]
p(mark) = [2] x1 + [4]
p(nil) = [1]
p(s) = [1] x1 + [1]
p(take) = [1] x1 + [1] x2 + [4]
p(true) = [0]
Following rules are strictly oriented:
a__eq(s(X),s(Y)) = [1] Y + [1]
> [1] Y + [0]
= a__eq(X,Y)
Following rules are (at-least) weakly oriented:
a__eq(X,Y) = [1] Y + [0]
>= [0]
= false()
a__eq(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= eq(X1,X2)
a__eq(0(),0()) = [0]
>= [0]
= true()
a__inf(X) = [1] X + [4]
>= [1] X + [3]
= cons(X,inf(s(X)))
a__inf(X) = [1] X + [4]
>= [1] X + [2]
= inf(X)
a__length(X) = [1] X + [5]
>= [1] X + [4]
= length(X)
a__length(cons(X,L)) = [1] L + [5]
>= [1] L + [5]
= s(length(L))
a__length(nil()) = [6]
>= [0]
= 0()
a__take(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= take(X1,X2)
a__take(0(),X) = [1] X + [4]
>= [1]
= nil()
a__take(s(X),cons(Y,L)) = [1] L + [1] X + [5]
>= [1] L + [1] X + [4]
= cons(Y,take(X,L))
mark(0()) = [4]
>= [0]
= 0()
mark(cons(X1,X2)) = [2] X2 + [4]
>= [1] X2 + [0]
= cons(X1,X2)
mark(eq(X1,X2)) = [2] X2 + [4]
>= [1] X2 + [0]
= a__eq(X1,X2)
mark(false()) = [4]
>= [0]
= false()
mark(inf(X)) = [2] X + [8]
>= [2] X + [8]
= a__inf(mark(X))
mark(length(X)) = [2] X + [12]
>= [2] X + [9]
= a__length(mark(X))
mark(nil()) = [6]
>= [1]
= nil()
mark(s(X)) = [2] X + [6]
>= [1] X + [1]
= s(X)
mark(take(X1,X2)) = [2] X1 + [2] X2 + [12]
>= [2] X1 + [2] X2 + [12]
= a__take(mark(X1),mark(X2))
mark(true()) = [4]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 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__eq(X,Y) -> false()
a__eq(X1,X2) -> eq(X1,X2)
a__eq(0(),0()) -> true()
a__eq(s(X),s(Y)) -> a__eq(X,Y)
a__inf(X) -> cons(X,inf(s(X)))
a__inf(X) -> inf(X)
a__length(X) -> length(X)
a__length(cons(X,L)) -> s(length(L))
a__length(nil()) -> 0()
a__take(X1,X2) -> take(X1,X2)
a__take(0(),X) -> nil()
a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(eq(X1,X2)) -> a__eq(X1,X2)
mark(false()) -> false()
mark(inf(X)) -> a__inf(mark(X))
mark(length(X)) -> a__length(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(true()) -> true()
Signature:
{a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0}
Obligation:
Innermost
basic terms: {a__eq,a__inf,a__length,a__take,mark}/{0,cons,eq,false,inf,length,nil,s,take,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).