*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(nil()) -> nil()
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a__app) = [1] x1 + [1] x2 + [2]
p(a__from) = [1] x1 + [0]
p(a__prefix) = [1] x1 + [0]
p(a__zWadr) = [1] x1 + [1] x2 + [0]
p(app) = [0]
p(cons) = [1] x1 + [1]
p(from) = [0]
p(mark) = [5]
p(nil) = [0]
p(prefix) = [0]
p(s) = [1] x1 + [0]
p(zWadr) = [0]
Following rules are strictly oriented:
a__app(X1,X2) = [1] X1 + [1] X2 + [2]
> [0]
= app(X1,X2)
mark(nil()) = [5]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
a__app(cons(X,XS),YS) = [1] X + [1] YS + [3]
>= [6]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1] YS + [2]
>= [5]
= mark(YS)
a__from(X) = [1] X + [0]
>= [6]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [0]
>= [0]
= from(X)
a__prefix(L) = [1] L + [0]
>= [1]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1] X + [0]
>= [0]
= prefix(X)
a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1] XS + [0]
>= [0]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [2]
>= [14]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1] YS + [0]
>= [0]
= nil()
mark(app(X1,X2)) = [5]
>= [12]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [5]
>= [6]
= cons(mark(X1),X2)
mark(from(X)) = [5]
>= [5]
= a__from(mark(X))
mark(prefix(X)) = [5]
>= [5]
= a__prefix(mark(X))
mark(s(X)) = [5]
>= [5]
= s(mark(X))
mark(zWadr(X1,X2)) = [5]
>= [10]
= a__zWadr(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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
mark(nil()) -> nil()
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a__app) = [1] x1 + [1] x2 + [4]
p(a__from) = [1] x1 + [0]
p(a__prefix) = [1] x1 + [0]
p(a__zWadr) = [1] x1 + [1] x2 + [0]
p(app) = [0]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(mark) = [3]
p(nil) = [3]
p(prefix) = [0]
p(s) = [1] x1 + [0]
p(zWadr) = [0]
Following rules are strictly oriented:
a__app(cons(X,XS),YS) = [1] X + [1] YS + [4]
> [3]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1] YS + [7]
> [3]
= mark(YS)
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1] X1 + [1] X2 + [4]
>= [0]
= app(X1,X2)
a__from(X) = [1] X + [0]
>= [3]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [0]
>= [0]
= from(X)
a__prefix(L) = [1] L + [0]
>= [3]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1] X + [0]
>= [0]
= prefix(X)
a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1] XS + [3]
>= [3]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [0]
>= [10]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1] YS + [3]
>= [3]
= nil()
mark(app(X1,X2)) = [3]
>= [10]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [3]
>= [3]
= cons(mark(X1),X2)
mark(from(X)) = [3]
>= [3]
= a__from(mark(X))
mark(nil()) = [3]
>= [3]
= nil()
mark(prefix(X)) = [3]
>= [3]
= a__prefix(mark(X))
mark(s(X)) = [3]
>= [3]
= s(mark(X))
mark(zWadr(X1,X2)) = [3]
>= [6]
= a__zWadr(mark(X1),mark(X2))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
mark(nil()) -> nil()
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a__app) = [1] x1 + [1] x2 + [4]
p(a__from) = [1] x1 + [1]
p(a__prefix) = [1] x1 + [0]
p(a__zWadr) = [1] x1 + [1] x2 + [0]
p(app) = [0]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(mark) = [3]
p(nil) = [3]
p(prefix) = [0]
p(s) = [1] x1 + [0]
p(zWadr) = [0]
Following rules are strictly oriented:
a__from(X) = [1] X + [1]
> [0]
= from(X)
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1] X1 + [1] X2 + [4]
>= [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1] X + [1] YS + [4]
>= [3]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1] YS + [7]
>= [3]
= mark(YS)
a__from(X) = [1] X + [1]
>= [3]
= cons(mark(X),from(s(X)))
a__prefix(L) = [1] L + [0]
>= [3]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1] X + [0]
>= [0]
= prefix(X)
a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1] XS + [3]
>= [3]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [0]
>= [10]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1] YS + [3]
>= [3]
= nil()
mark(app(X1,X2)) = [3]
>= [10]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [3]
>= [3]
= cons(mark(X1),X2)
mark(from(X)) = [3]
>= [4]
= a__from(mark(X))
mark(nil()) = [3]
>= [3]
= nil()
mark(prefix(X)) = [3]
>= [3]
= a__prefix(mark(X))
mark(s(X)) = [3]
>= [3]
= s(mark(X))
mark(zWadr(X1,X2)) = [3]
>= [6]
= a__zWadr(mark(X1),mark(X2))
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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> from(X)
mark(nil()) -> nil()
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a__app) = [1] x1 + [1] x2 + [2]
p(a__from) = [1] x1 + [1]
p(a__prefix) = [1] x1 + [7]
p(a__zWadr) = [1] x1 + [1] x2 + [0]
p(app) = [1] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(mark) = [1]
p(nil) = [1]
p(prefix) = [0]
p(s) = [1] x1 + [0]
p(zWadr) = [0]
Following rules are strictly oriented:
a__prefix(L) = [1] L + [7]
> [1]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1] X + [7]
> [0]
= prefix(X)
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1] X + [1] YS + [2]
>= [1]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1] YS + [3]
>= [1]
= mark(YS)
a__from(X) = [1] X + [1]
>= [1]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [1]
>= [0]
= from(X)
a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1] XS + [1]
>= [1]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [0]
>= [4]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1] YS + [1]
>= [1]
= nil()
mark(app(X1,X2)) = [1]
>= [4]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1]
>= [1]
= cons(mark(X1),X2)
mark(from(X)) = [1]
>= [2]
= a__from(mark(X))
mark(nil()) = [1]
>= [1]
= nil()
mark(prefix(X)) = [1]
>= [8]
= a__prefix(mark(X))
mark(s(X)) = [1]
>= [1]
= s(mark(X))
mark(zWadr(X1,X2)) = [1]
>= [2]
= a__zWadr(mark(X1),mark(X2))
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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
mark(nil()) -> nil()
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a__app) = [1] x1 + [1] x2 + [2]
p(a__from) = [1] x1 + [1]
p(a__prefix) = [1] x1 + [1]
p(a__zWadr) = [1] x1 + [1] x2 + [4]
p(app) = [1] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(mark) = [1]
p(nil) = [1]
p(prefix) = [0]
p(s) = [1] x1 + [0]
p(zWadr) = [0]
Following rules are strictly oriented:
a__zWadr(X1,X2) = [1] X1 + [1] X2 + [4]
> [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1] XS + [5]
> [1]
= nil()
a__zWadr(nil(),YS) = [1] YS + [5]
> [1]
= nil()
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1] X + [1] YS + [2]
>= [1]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1] YS + [3]
>= [1]
= mark(YS)
a__from(X) = [1] X + [1]
>= [1]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [1]
>= [0]
= from(X)
a__prefix(L) = [1] L + [1]
>= [1]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1] X + [1]
>= [0]
= prefix(X)
a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [4]
>= [4]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
mark(app(X1,X2)) = [1]
>= [4]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1]
>= [1]
= cons(mark(X1),X2)
mark(from(X)) = [1]
>= [2]
= a__from(mark(X))
mark(nil()) = [1]
>= [1]
= nil()
mark(prefix(X)) = [1]
>= [2]
= a__prefix(mark(X))
mark(s(X)) = [1]
>= [1]
= s(mark(X))
mark(zWadr(X1,X2)) = [1]
>= [6]
= a__zWadr(mark(X1),mark(X2))
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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__from(X) -> cons(mark(X),from(s(X)))
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(nil(),YS) -> nil()
mark(nil()) -> nil()
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a__app) = [1] x1 + [1] x2 + [4]
p(a__from) = [1] x1 + [6]
p(a__prefix) = [1] x1 + [4]
p(a__zWadr) = [1] x1 + [1] x2 + [5]
p(app) = [1] x1 + [0]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(mark) = [3]
p(nil) = [3]
p(prefix) = [0]
p(s) = [1] x1 + [0]
p(zWadr) = [0]
Following rules are strictly oriented:
a__from(X) = [1] X + [6]
> [3]
= cons(mark(X),from(s(X)))
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1] X + [1] YS + [4]
>= [3]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1] YS + [7]
>= [3]
= mark(YS)
a__from(X) = [1] X + [6]
>= [0]
= from(X)
a__prefix(L) = [1] L + [4]
>= [3]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1] X + [4]
>= [0]
= prefix(X)
a__zWadr(X1,X2) = [1] X1 + [1] X2 + [5]
>= [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1] XS + [8]
>= [3]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [5]
>= [10]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1] YS + [8]
>= [3]
= nil()
mark(app(X1,X2)) = [3]
>= [10]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [3]
>= [3]
= cons(mark(X1),X2)
mark(from(X)) = [3]
>= [9]
= a__from(mark(X))
mark(nil()) = [3]
>= [3]
= nil()
mark(prefix(X)) = [3]
>= [7]
= a__prefix(mark(X))
mark(s(X)) = [3]
>= [3]
= s(mark(X))
mark(zWadr(X1,X2)) = [3]
>= [11]
= a__zWadr(mark(X1),mark(X2))
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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(nil(),YS) -> nil()
mark(nil()) -> nil()
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a__app) = [1] x1 + [1] x2 + [0]
p(a__from) = [1] x1 + [0]
p(a__prefix) = [1] x1 + [0]
p(a__zWadr) = [1] x1 + [1] x2 + [1]
p(app) = [0]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(mark) = [0]
p(nil) = [0]
p(prefix) = [0]
p(s) = [1] x1 + [0]
p(zWadr) = [1] x2 + [1]
Following rules are strictly oriented:
a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [1]
> [0]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1] X + [1] YS + [0]
>= [0]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1] YS + [0]
>= [0]
= mark(YS)
a__from(X) = [1] X + [0]
>= [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [0]
>= [0]
= from(X)
a__prefix(L) = [1] L + [0]
>= [0]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1] X + [0]
>= [0]
= prefix(X)
a__zWadr(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X2 + [1]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1] XS + [1]
>= [0]
= nil()
a__zWadr(nil(),YS) = [1] YS + [1]
>= [0]
= nil()
mark(app(X1,X2)) = [0]
>= [0]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [0]
>= [0]
= cons(mark(X1),X2)
mark(from(X)) = [0]
>= [0]
= a__from(mark(X))
mark(nil()) = [0]
>= [0]
= nil()
mark(prefix(X)) = [0]
>= [0]
= a__prefix(mark(X))
mark(s(X)) = [0]
>= [0]
= s(mark(X))
mark(zWadr(X1,X2)) = [0]
>= [1]
= a__zWadr(mark(X1),mark(X2))
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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(nil()) -> nil()
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__app,a__from,a__prefix,a__zWadr,mark}
TcT has computed the following interpretation:
p(a__app) = [1 1] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(a__from) = [1 1] x1 + [0]
[0 1] [0]
p(a__prefix) = [1 4] x1 + [0]
[0 1] [2]
p(a__zWadr) = [1 5] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(app) = [1 1] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x1 + [0]
[0 1] [0]
p(from) = [1 1] x1 + [0]
[0 1] [0]
p(mark) = [1 1] x1 + [0]
[0 1] [0]
p(nil) = [0]
[2]
p(prefix) = [1 4] x1 + [0]
[0 1] [2]
p(s) = [1 0] x1 + [0]
[0 1] [0]
p(zWadr) = [1 5] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
Following rules are strictly oriented:
mark(prefix(X)) = [1 5] X + [2]
[0 1] [2]
> [1 5] X + [0]
[0 1] [2]
= a__prefix(mark(X))
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1 1] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 1] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1 1] X + [1 4] YS + [0]
[0 1] [0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1 4] YS + [2]
[0 1] [2]
>= [1 1] YS + [0]
[0 1] [0]
= mark(YS)
a__from(X) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= from(X)
a__prefix(L) = [1 4] L + [0]
[0 1] [2]
>= [0]
[2]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1 4] X + [0]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [2]
= prefix(X)
a__zWadr(X1,X2) = [1 5] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 5] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1 5] XS + [8]
[0 1] [2]
>= [0]
[2]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1 5] X + [1 4] Y + [0]
[0 1] [0 1] [0]
>= [1 5] X + [1 2] Y + [0]
[0 1] [0 1] [0]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1 4] YS + [10]
[0 1] [2]
>= [0]
[2]
= nil()
mark(app(X1,X2)) = [1 2] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
>= [1 2] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1] X1 + [0]
[0 1] [0]
>= [1 1] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 2] X + [0]
[0 1] [0]
>= [1 2] X + [0]
[0 1] [0]
= a__from(mark(X))
mark(nil()) = [2]
[2]
>= [0]
[2]
= nil()
mark(s(X)) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= s(mark(X))
mark(zWadr(X1,X2)) = [1 6] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
>= [1 6] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
= a__zWadr(mark(X1),mark(X2))
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(nil()) -> nil()
mark(prefix(X)) -> a__prefix(mark(X))
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__app,a__from,a__prefix,a__zWadr,mark}
TcT has computed the following interpretation:
p(a__app) = [1 4] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
p(a__from) = [1 4] x1 + [4]
[0 1] [2]
p(a__prefix) = [1 4] x1 + [4]
[0 1] [0]
p(a__zWadr) = [1 4] x1 + [1 6] x2 + [4]
[0 1] [0 1] [4]
p(app) = [1 4] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x1 + [0 1] x2 + [0]
[0 1] [0 0] [0]
p(from) = [1 4] x1 + [0]
[0 1] [2]
p(mark) = [1 2] x1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(prefix) = [1 4] x1 + [4]
[0 1] [0]
p(s) = [1 0] x1 + [0]
[0 1] [2]
p(zWadr) = [1 4] x1 + [1 6] x2 + [0]
[0 1] [0 1] [4]
Following rules are strictly oriented:
mark(s(X)) = [1 2] X + [4]
[0 1] [2]
> [1 2] X + [0]
[0 1] [2]
= s(mark(X))
mark(zWadr(X1,X2)) = [1 6] X1 + [1 8] X2 + [8]
[0 1] [0 1] [4]
> [1 6] X1 + [1 8] X2 + [4]
[0 1] [0 1] [4]
= a__zWadr(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1 4] X1 + [1 2] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 2] X2 + [0]
[0 1] [0 1] [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1 4] X + [0 1] XS + [1
2] YS + [0]
[0 1] [0 0] [0
1] [0]
>= [1 2] X + [0 1] XS + [0
1] YS + [0]
[0 1] [0 0] [0
0] [0]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1 2] YS + [0]
[0 1] [0]
>= [1 2] YS + [0]
[0 1] [0]
= mark(YS)
a__from(X) = [1 4] X + [4]
[0 1] [2]
>= [1 3] X + [4]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [4]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [2]
= from(X)
a__prefix(L) = [1 4] L + [4]
[0 1] [0]
>= [0 2] L + [4]
[0 0] [0]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1 4] X + [4]
[0 1] [0]
>= [1 4] X + [4]
[0 1] [0]
= prefix(X)
a__zWadr(X1,X2) = [1 4] X1 + [1 6] X2 + [4]
[0 1] [0 1] [4]
>= [1 4] X1 + [1 6] X2 + [0]
[0 1] [0 1] [4]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1 4] XS + [4]
[0 1] [4]
>= [0]
[0]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1 4] X + [0 1] XS + [1
6] Y + [0 1] YS + [4]
[0 1] [0 0] [0
1] [0 0] [4]
>= [1 4] X + [0 1] XS + [1
6] Y + [0 1] YS + [4]
[0 1] [0 0] [0
1] [0 0] [0]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1 6] YS + [4]
[0 1] [4]
>= [0]
[0]
= nil()
mark(app(X1,X2)) = [1 6] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 6] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 2] X1 + [0 1] X2 + [0]
[0 1] [0 0] [0]
>= [1 2] X1 + [0 1] X2 + [0]
[0 1] [0 0] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 6] X + [4]
[0 1] [2]
>= [1 6] X + [4]
[0 1] [2]
= a__from(mark(X))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(prefix(X)) = [1 6] X + [4]
[0 1] [0]
>= [1 6] X + [4]
[0 1] [0]
= a__prefix(mark(X))
*** 1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(nil()) -> nil()
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__app,a__from,a__prefix,a__zWadr,mark}
TcT has computed the following interpretation:
p(a__app) = [1 2] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
p(a__from) = [1 4] x1 + [6]
[0 1] [1]
p(a__prefix) = [1 4] x1 + [6]
[0 1] [3]
p(a__zWadr) = [1 4] x1 + [1 5] x2 + [0]
[0 1] [0 1] [0]
p(app) = [1 2] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x1 + [0 2] x2 + [0]
[0 1] [0 0] [0]
p(from) = [1 4] x1 + [5]
[0 1] [1]
p(mark) = [1 2] x1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(prefix) = [1 4] x1 + [0]
[0 1] [3]
p(s) = [1 0] x1 + [1]
[0 1] [2]
p(zWadr) = [1 4] x1 + [1 5] x2 + [0]
[0 1] [0 1] [0]
Following rules are strictly oriented:
mark(from(X)) = [1 6] X + [7]
[0 1] [1]
> [1 6] X + [6]
[0 1] [1]
= a__from(mark(X))
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1 2] X1 + [1 2] X2 + [0]
[0 1] [0 1] [0]
>= [1 2] X1 + [1 2] X2 + [0]
[0 1] [0 1] [0]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1 2] X + [0 2] XS + [1
2] YS + [0]
[0 1] [0 0] [0
1] [0]
>= [1 2] X + [0 2] XS + [0
2] YS + [0]
[0 1] [0 0] [0
0] [0]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1 2] YS + [0]
[0 1] [0]
>= [1 2] YS + [0]
[0 1] [0]
= mark(YS)
a__from(X) = [1 4] X + [6]
[0 1] [1]
>= [1 4] X + [6]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [6]
[0 1] [1]
>= [1 4] X + [5]
[0 1] [1]
= from(X)
a__prefix(L) = [1 4] L + [6]
[0 1] [3]
>= [0 4] L + [6]
[0 0] [0]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1 4] X + [6]
[0 1] [3]
>= [1 4] X + [0]
[0 1] [3]
= prefix(X)
a__zWadr(X1,X2) = [1 4] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1 4] XS + [0]
[0 1] [0]
>= [0]
[0]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1 4] X + [0 2] XS + [1
5] Y + [0 2] YS + [0]
[0 1] [0 0] [0
1] [0 0] [0]
>= [1 4] X + [0 2] XS + [1
4] Y + [0 2] YS + [0]
[0 1] [0 0] [0
1] [0 0] [0]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1 5] YS + [0]
[0 1] [0]
>= [0]
[0]
= nil()
mark(app(X1,X2)) = [1 4] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
= a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 2] X1 + [0 2] X2 + [0]
[0 1] [0 0] [0]
>= [1 2] X1 + [0 2] X2 + [0]
[0 1] [0 0] [0]
= cons(mark(X1),X2)
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(prefix(X)) = [1 6] X + [6]
[0 1] [3]
>= [1 6] X + [6]
[0 1] [3]
= a__prefix(mark(X))
mark(s(X)) = [1 2] X + [5]
[0 1] [2]
>= [1 2] X + [1]
[0 1] [2]
= s(mark(X))
mark(zWadr(X1,X2)) = [1 6] X1 + [1 7] X2 + [0]
[0 1] [0 1] [0]
>= [1 6] X1 + [1 7] X2 + [0]
[0 1] [0 1] [0]
= a__zWadr(mark(X1),mark(X2))
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(from(X)) -> a__from(mark(X))
mark(nil()) -> nil()
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__app,a__from,a__prefix,a__zWadr,mark}
TcT has computed the following interpretation:
p(a__app) = [1 1] x1 + [1 4] x2 + [0]
[0 1] [0 1] [1]
p(a__from) = [1 1] x1 + [2]
[0 1] [0]
p(a__prefix) = [1 0] x1 + [2]
[0 1] [0]
p(a__zWadr) = [1 6] x1 + [1 2] x2 + [7]
[0 1] [0 1] [1]
p(app) = [1 1] x1 + [1 4] x2 + [0]
[0 1] [0 1] [1]
p(cons) = [1 0] x1 + [2]
[0 1] [0]
p(from) = [1 1] x1 + [2]
[0 1] [0]
p(mark) = [1 1] x1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(prefix) = [1 0] x1 + [2]
[0 1] [0]
p(s) = [1 5] x1 + [6]
[0 1] [3]
p(zWadr) = [1 6] x1 + [1 2] x2 + [7]
[0 1] [0 1] [1]
Following rules are strictly oriented:
mark(app(X1,X2)) = [1 2] X1 + [1 5] X2 + [1]
[0 1] [0 1] [1]
> [1 2] X1 + [1 5] X2 + [0]
[0 1] [0 1] [1]
= a__app(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1 1] X1 + [1 4] X2 + [0]
[0 1] [0 1] [1]
>= [1 1] X1 + [1 4] X2 + [0]
[0 1] [0 1] [1]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1 1] X + [1 4] YS + [2]
[0 1] [0 1] [1]
>= [1 1] X + [2]
[0 1] [0]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1 4] YS + [0]
[0 1] [1]
>= [1 1] YS + [0]
[0 1] [0]
= mark(YS)
a__from(X) = [1 1] X + [2]
[0 1] [0]
>= [1 1] X + [2]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1] X + [2]
[0 1] [0]
>= [1 1] X + [2]
[0 1] [0]
= from(X)
a__prefix(L) = [1 0] L + [2]
[0 1] [0]
>= [2]
[0]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1 0] X + [2]
[0 1] [0]
>= [1 0] X + [2]
[0 1] [0]
= prefix(X)
a__zWadr(X1,X2) = [1 6] X1 + [1 2] X2 + [7]
[0 1] [0 1] [1]
>= [1 6] X1 + [1 2] X2 + [7]
[0 1] [0 1] [1]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1 6] XS + [7]
[0 1] [1]
>= [0]
[0]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1 6] X + [1 2] Y + [11]
[0 1] [0 1] [1]
>= [1 5] X + [1 2] Y + [4]
[0 1] [0 1] [1]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1 2] YS + [7]
[0 1] [1]
>= [0]
[0]
= nil()
mark(cons(X1,X2)) = [1 1] X1 + [2]
[0 1] [0]
>= [1 1] X1 + [2]
[0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 2] X + [2]
[0 1] [0]
>= [1 2] X + [2]
[0 1] [0]
= a__from(mark(X))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(prefix(X)) = [1 1] X + [2]
[0 1] [0]
>= [1 1] X + [2]
[0 1] [0]
= a__prefix(mark(X))
mark(s(X)) = [1 6] X + [9]
[0 1] [3]
>= [1 6] X + [6]
[0 1] [3]
= s(mark(X))
mark(zWadr(X1,X2)) = [1 7] X1 + [1 3] X2 + [8]
[0 1] [0 1] [1]
>= [1 7] X1 + [1 3] X2 + [7]
[0 1] [0 1] [1]
= a__zWadr(mark(X1),mark(X2))
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(cons(X1,X2)) -> cons(mark(X1),X2)
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(nil()) -> nil()
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
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__app) = {1,2},
uargs(a__from) = {1},
uargs(a__prefix) = {1},
uargs(a__zWadr) = {1,2},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__app,a__from,a__prefix,a__zWadr,mark}
TcT has computed the following interpretation:
p(a__app) = [1 2] x1 + [1 2] x2 + [4]
[0 1] [0 1] [1]
p(a__from) = [1 4] x1 + [4]
[0 1] [1]
p(a__prefix) = [1 0] x1 + [5]
[0 1] [1]
p(a__zWadr) = [1 4] x1 + [1 4] x2 + [1]
[0 1] [0 1] [6]
p(app) = [1 2] x1 + [1 2] x2 + [3]
[0 1] [0 1] [1]
p(cons) = [1 0] x1 + [3]
[0 1] [1]
p(from) = [1 4] x1 + [2]
[0 1] [1]
p(mark) = [1 2] x1 + [1]
[0 1] [0]
p(nil) = [0]
[0]
p(prefix) = [1 0] x1 + [3]
[0 1] [1]
p(s) = [1 0] x1 + [0]
[0 1] [0]
p(zWadr) = [1 4] x1 + [1 4] x2 + [0]
[0 1] [0 1] [6]
Following rules are strictly oriented:
mark(cons(X1,X2)) = [1 2] X1 + [6]
[0 1] [1]
> [1 2] X1 + [4]
[0 1] [1]
= cons(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__app(X1,X2) = [1 2] X1 + [1 2] X2 + [4]
[0 1] [0 1] [1]
>= [1 2] X1 + [1 2] X2 + [3]
[0 1] [0 1] [1]
= app(X1,X2)
a__app(cons(X,XS),YS) = [1 2] X + [1 2] YS + [9]
[0 1] [0 1] [2]
>= [1 2] X + [4]
[0 1] [1]
= cons(mark(X),app(XS,YS))
a__app(nil(),YS) = [1 2] YS + [4]
[0 1] [1]
>= [1 2] YS + [1]
[0 1] [0]
= mark(YS)
a__from(X) = [1 4] X + [4]
[0 1] [1]
>= [1 2] X + [4]
[0 1] [1]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [4]
[0 1] [1]
>= [1 4] X + [2]
[0 1] [1]
= from(X)
a__prefix(L) = [1 0] L + [5]
[0 1] [1]
>= [3]
[1]
= cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) = [1 0] X + [5]
[0 1] [1]
>= [1 0] X + [3]
[0 1] [1]
= prefix(X)
a__zWadr(X1,X2) = [1 4] X1 + [1 4] X2 + [1]
[0 1] [0 1] [6]
>= [1 4] X1 + [1 4] X2 + [0]
[0 1] [0 1] [6]
= zWadr(X1,X2)
a__zWadr(XS,nil()) = [1 4] XS + [1]
[0 1] [6]
>= [0]
[0]
= nil()
a__zWadr(cons(X,XS),cons(Y,YS)) = [1 4] X + [1 4] Y + [15]
[0 1] [0 1] [8]
>= [1 4] X + [1 4] Y + [14]
[0 1] [0 1] [3]
= cons(a__app(mark(Y)
,cons(mark(X),nil()))
,zWadr(XS,YS))
a__zWadr(nil(),YS) = [1 4] YS + [1]
[0 1] [6]
>= [0]
[0]
= nil()
mark(app(X1,X2)) = [1 4] X1 + [1 4] X2 + [6]
[0 1] [0 1] [1]
>= [1 4] X1 + [1 4] X2 + [6]
[0 1] [0 1] [1]
= a__app(mark(X1),mark(X2))
mark(from(X)) = [1 6] X + [5]
[0 1] [1]
>= [1 6] X + [5]
[0 1] [1]
= a__from(mark(X))
mark(nil()) = [1]
[0]
>= [0]
[0]
= nil()
mark(prefix(X)) = [1 2] X + [6]
[0 1] [1]
>= [1 2] X + [6]
[0 1] [1]
= a__prefix(mark(X))
mark(s(X)) = [1 2] X + [1]
[0 1] [0]
>= [1 2] X + [1]
[0 1] [0]
= s(mark(X))
mark(zWadr(X1,X2)) = [1 6] X1 + [1 6] X2 + [13]
[0 1] [0 1] [6]
>= [1 6] X1 + [1 6] X2 + [3]
[0 1] [0 1] [6]
= a__zWadr(mark(X1),mark(X2))
*** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
a__app(X1,X2) -> app(X1,X2)
a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS))
a__app(nil(),YS) -> mark(YS)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__prefix(L) -> cons(nil(),zWadr(L,prefix(L)))
a__prefix(X) -> prefix(X)
a__zWadr(X1,X2) -> zWadr(X1,X2)
a__zWadr(XS,nil()) -> nil()
a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS))
a__zWadr(nil(),YS) -> nil()
mark(app(X1,X2)) -> a__app(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(nil()) -> nil()
mark(prefix(X)) -> a__prefix(mark(X))
mark(s(X)) -> s(mark(X))
mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2))
Signature:
{a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2}
Obligation:
Innermost
basic terms: {a__app,a__from,a__prefix,a__zWadr,mark}/{app,cons,from,nil,prefix,s,zWadr}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).