*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(a__add) = [1] x1 + [1] x2 + [0]
p(a__dbl) = [1] x1 + [0]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__sqr) = [1] x1 + [1]
p(a__terms) = [1] x1 + [0]
p(add) = [0]
p(cons) = [1] x1 + [0]
p(dbl) = [4]
p(first) = [0]
p(mark) = [0]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [1]
p(sqr) = [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
a__add(0(),X) = [1] X + [1]
> [0]
= mark(X)
a__first(0(),X) = [1] X + [1]
> [0]
= nil()
a__first(s(X),cons(Y,Z)) = [1] Y + [1]
> [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1] X + [1]
> [0]
= sqr(X)
a__sqr(0()) = [2]
> [1]
= 0()
a__sqr(s(X)) = [2]
> [1]
= s(add(sqr(X),dbl(X)))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= add(X1,X2)
a__add(s(X),Y) = [1] Y + [1]
>= [1]
= s(add(X,Y))
a__dbl(X) = [1] X + [0]
>= [4]
= dbl(X)
a__dbl(0()) = [1]
>= [1]
= 0()
a__dbl(s(X)) = [1]
>= [1]
= s(s(dbl(X)))
a__first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= first(X1,X2)
a__terms(N) = [1] N + [0]
>= [1]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1] X + [0]
>= [1] X + [0]
= terms(X)
mark(0()) = [0]
>= [1]
= 0()
mark(add(X1,X2)) = [0]
>= [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [0]
>= [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [0]
>= [0]
= a__dbl(mark(X))
mark(first(X1,X2)) = [0]
>= [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [0]
>= [0]
= nil()
mark(recip(X)) = [0]
>= [0]
= recip(mark(X))
mark(s(X)) = [0]
>= [1]
= s(X)
mark(sqr(X)) = [0]
>= [1]
= a__sqr(mark(X))
mark(terms(X)) = [0]
>= [0]
= a__terms(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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(0(),X) -> mark(X)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [6]
p(a__add) = [1] x1 + [1] x2 + [4]
p(a__dbl) = [1] x1 + [0]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__sqr) = [1] x1 + [0]
p(a__terms) = [1] x1 + [0]
p(add) = [0]
p(cons) = [1] x1 + [5]
p(dbl) = [4]
p(first) = [1] x1 + [1] x2 + [0]
p(mark) = [4]
p(nil) = [6]
p(recip) = [1] x1 + [5]
p(s) = [4]
p(sqr) = [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [4]
> [0]
= add(X1,X2)
a__add(s(X),Y) = [1] Y + [8]
> [4]
= s(add(X,Y))
Following rules are (at-least) weakly oriented:
a__add(0(),X) = [1] X + [10]
>= [4]
= mark(X)
a__dbl(X) = [1] X + [0]
>= [4]
= dbl(X)
a__dbl(0()) = [6]
>= [6]
= 0()
a__dbl(s(X)) = [4]
>= [4]
= s(s(dbl(X)))
a__first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= first(X1,X2)
a__first(0(),X) = [1] X + [6]
>= [6]
= nil()
a__first(s(X),cons(Y,Z)) = [1] Y + [9]
>= [9]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1] X + [0]
>= [0]
= sqr(X)
a__sqr(0()) = [6]
>= [6]
= 0()
a__sqr(s(X)) = [4]
>= [4]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1] N + [0]
>= [14]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1] X + [0]
>= [1] X + [0]
= terms(X)
mark(0()) = [4]
>= [6]
= 0()
mark(add(X1,X2)) = [4]
>= [12]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [4]
>= [9]
= cons(mark(X1),X2)
mark(dbl(X)) = [4]
>= [4]
= a__dbl(mark(X))
mark(first(X1,X2)) = [4]
>= [8]
= a__first(mark(X1),mark(X2))
mark(nil()) = [4]
>= [6]
= nil()
mark(recip(X)) = [4]
>= [9]
= recip(mark(X))
mark(s(X)) = [4]
>= [4]
= s(X)
mark(sqr(X)) = [4]
>= [4]
= a__sqr(mark(X))
mark(terms(X)) = [4]
>= [4]
= a__terms(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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [1] x2 + [5]
p(a__dbl) = [1] x1 + [0]
p(a__first) = [1] x1 + [1] x2 + [3]
p(a__sqr) = [1] x1 + [0]
p(a__terms) = [1] x1 + [0]
p(add) = [0]
p(cons) = [1] x1 + [0]
p(dbl) = [0]
p(first) = [1] x1 + [1] x2 + [0]
p(mark) = [5]
p(nil) = [3]
p(recip) = [1] x1 + [2]
p(s) = [2]
p(sqr) = [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
a__first(X1,X2) = [1] X1 + [1] X2 + [3]
> [1] X1 + [1] X2 + [0]
= first(X1,X2)
mark(0()) = [5]
> [0]
= 0()
mark(nil()) = [5]
> [3]
= nil()
mark(s(X)) = [5]
> [2]
= s(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [5]
>= [0]
= add(X1,X2)
a__add(0(),X) = [1] X + [5]
>= [5]
= mark(X)
a__add(s(X),Y) = [1] Y + [7]
>= [2]
= s(add(X,Y))
a__dbl(X) = [1] X + [0]
>= [0]
= dbl(X)
a__dbl(0()) = [0]
>= [0]
= 0()
a__dbl(s(X)) = [2]
>= [2]
= s(s(dbl(X)))
a__first(0(),X) = [1] X + [3]
>= [3]
= nil()
a__first(s(X),cons(Y,Z)) = [1] Y + [5]
>= [5]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1] X + [0]
>= [0]
= sqr(X)
a__sqr(0()) = [0]
>= [0]
= 0()
a__sqr(s(X)) = [2]
>= [2]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1] N + [0]
>= [7]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1] X + [0]
>= [1] X + [0]
= terms(X)
mark(add(X1,X2)) = [5]
>= [15]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [5]
>= [5]
= cons(mark(X1),X2)
mark(dbl(X)) = [5]
>= [5]
= a__dbl(mark(X))
mark(first(X1,X2)) = [5]
>= [13]
= a__first(mark(X1),mark(X2))
mark(recip(X)) = [5]
>= [7]
= recip(mark(X))
mark(sqr(X)) = [5]
>= [5]
= a__sqr(mark(X))
mark(terms(X)) = [5]
>= [5]
= a__terms(mark(X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
mark(0()) -> 0()
mark(nil()) -> nil()
mark(s(X)) -> s(X)
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [1] x2 + [1]
p(a__dbl) = [1] x1 + [0]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__sqr) = [1] x1 + [0]
p(a__terms) = [1] x1 + [4]
p(add) = [1] x2 + [0]
p(cons) = [1] x1 + [2]
p(dbl) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(mark) = [0]
p(nil) = [0]
p(recip) = [1] x1 + [1]
p(s) = [0]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
a__terms(N) = [1] N + [4]
> [3]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1] X + [4]
> [1] X + [0]
= terms(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X2 + [0]
= add(X1,X2)
a__add(0(),X) = [1] X + [1]
>= [0]
= mark(X)
a__add(s(X),Y) = [1] Y + [1]
>= [0]
= s(add(X,Y))
a__dbl(X) = [1] X + [0]
>= [1] X + [0]
= dbl(X)
a__dbl(0()) = [0]
>= [0]
= 0()
a__dbl(s(X)) = [0]
>= [0]
= s(s(dbl(X)))
a__first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= first(X1,X2)
a__first(0(),X) = [1] X + [0]
>= [0]
= nil()
a__first(s(X),cons(Y,Z)) = [1] Y + [2]
>= [2]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1] X + [0]
>= [1] X + [0]
= sqr(X)
a__sqr(0()) = [0]
>= [0]
= 0()
a__sqr(s(X)) = [0]
>= [0]
= s(add(sqr(X),dbl(X)))
mark(0()) = [0]
>= [0]
= 0()
mark(add(X1,X2)) = [0]
>= [1]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [0]
>= [2]
= cons(mark(X1),X2)
mark(dbl(X)) = [0]
>= [0]
= a__dbl(mark(X))
mark(first(X1,X2)) = [0]
>= [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [0]
>= [0]
= nil()
mark(recip(X)) = [0]
>= [1]
= recip(mark(X))
mark(s(X)) = [0]
>= [0]
= s(X)
mark(sqr(X)) = [0]
>= [0]
= a__sqr(mark(X))
mark(terms(X)) = [0]
>= [4]
= a__terms(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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(nil()) -> nil()
mark(s(X)) -> s(X)
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [1] x2 + [4]
p(a__dbl) = [1] x1 + [0]
p(a__first) = [1] x1 + [1] x2 + [0]
p(a__sqr) = [1] x1 + [1]
p(a__terms) = [1] x1 + [3]
p(add) = [1] x1 + [1] x2 + [4]
p(cons) = [1] x1 + [0]
p(dbl) = [1] x1 + [2]
p(first) = [1] x1 + [1] x2 + [0]
p(mark) = [1] x1 + [0]
p(nil) = [0]
p(recip) = [1] x1 + [2]
p(s) = [1]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
mark(dbl(X)) = [1] X + [2]
> [1] X + [0]
= a__dbl(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= add(X1,X2)
a__add(0(),X) = [1] X + [4]
>= [1] X + [0]
= mark(X)
a__add(s(X),Y) = [1] Y + [5]
>= [1]
= s(add(X,Y))
a__dbl(X) = [1] X + [0]
>= [1] X + [2]
= dbl(X)
a__dbl(0()) = [0]
>= [0]
= 0()
a__dbl(s(X)) = [1]
>= [1]
= s(s(dbl(X)))
a__first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= first(X1,X2)
a__first(0(),X) = [1] X + [0]
>= [0]
= nil()
a__first(s(X),cons(Y,Z)) = [1] Y + [1]
>= [1] Y + [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1] X + [1]
>= [1] X + [0]
= sqr(X)
a__sqr(0()) = [1]
>= [0]
= 0()
a__sqr(s(X)) = [2]
>= [1]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1] N + [3]
>= [1] N + [3]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1] X + [3]
>= [1] X + [0]
= terms(X)
mark(0()) = [0]
>= [0]
= 0()
mark(add(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(mark(X1),X2)
mark(first(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [0]
>= [0]
= nil()
mark(recip(X)) = [1] X + [2]
>= [1] X + [2]
= recip(mark(X))
mark(s(X)) = [1]
>= [1]
= s(X)
mark(sqr(X)) = [1] X + [0]
>= [1] X + [1]
= a__sqr(mark(X))
mark(terms(X)) = [1] X + [0]
>= [1] X + [3]
= a__terms(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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(dbl(X)) -> a__dbl(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(a__add) = [1] x1 + [1] x2 + [5]
p(a__dbl) = [1] x1 + [1]
p(a__first) = [1] x1 + [1] x2 + [5]
p(a__sqr) = [1] x1 + [4]
p(a__terms) = [1] x1 + [6]
p(add) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [0]
p(dbl) = [1] x1 + [4]
p(first) = [1] x1 + [1] x2 + [0]
p(mark) = [1] x1 + [0]
p(nil) = [1]
p(recip) = [1] x1 + [1]
p(s) = [0]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
a__dbl(0()) = [2]
> [1]
= 0()
a__dbl(s(X)) = [1]
> [0]
= s(s(dbl(X)))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [0]
= add(X1,X2)
a__add(0(),X) = [1] X + [6]
>= [1] X + [0]
= mark(X)
a__add(s(X),Y) = [1] Y + [5]
>= [0]
= s(add(X,Y))
a__dbl(X) = [1] X + [1]
>= [1] X + [4]
= dbl(X)
a__first(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [0]
= first(X1,X2)
a__first(0(),X) = [1] X + [6]
>= [1]
= nil()
a__first(s(X),cons(Y,Z)) = [1] Y + [5]
>= [1] Y + [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1] X + [4]
>= [1] X + [0]
= sqr(X)
a__sqr(0()) = [5]
>= [1]
= 0()
a__sqr(s(X)) = [4]
>= [0]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1] N + [6]
>= [1] N + [5]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1] X + [6]
>= [1] X + [0]
= terms(X)
mark(0()) = [1]
>= [1]
= 0()
mark(add(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [5]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [1] X + [4]
>= [1] X + [1]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [5]
= a__first(mark(X1),mark(X2))
mark(nil()) = [1]
>= [1]
= nil()
mark(recip(X)) = [1] X + [1]
>= [1] X + [1]
= recip(mark(X))
mark(s(X)) = [0]
>= [0]
= s(X)
mark(sqr(X)) = [1] X + [0]
>= [1] X + [4]
= a__sqr(mark(X))
mark(terms(X)) = [1] X + [0]
>= [1] X + [6]
= a__terms(mark(X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__dbl(X) -> dbl(X)
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(dbl(X)) -> a__dbl(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{a__add,a__dbl,a__first,a__sqr,a__terms,mark}
TcT has computed the following interpretation:
p(0) = [1]
[0]
p(a__add) = [1 0] x1 + [1 4] x2 + [5]
[0 1] [0 1] [0]
p(a__dbl) = [1 0] x1 + [2]
[0 1] [2]
p(a__first) = [1 6] x1 + [1 7] x2 + [4]
[0 1] [0 1] [0]
p(a__sqr) = [1 0] x1 + [0]
[0 1] [0]
p(a__terms) = [1 4] x1 + [4]
[0 1] [0]
p(add) = [1 0] x1 + [1 4] x2 + [5]
[0 1] [0 1] [0]
p(cons) = [1 0] x1 + [2]
[0 1] [0]
p(dbl) = [1 0] x1 + [0]
[0 1] [2]
p(first) = [1 6] x1 + [1 7] x2 + [4]
[0 1] [0 1] [0]
p(mark) = [1 4] x1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(recip) = [1 0] x1 + [2]
[0 1] [0]
p(s) = [4]
[0]
p(sqr) = [1 0] x1 + [0]
[0 1] [0]
p(terms) = [1 4] x1 + [4]
[0 1] [0]
Following rules are strictly oriented:
a__dbl(X) = [1 0] X + [2]
[0 1] [2]
> [1 0] X + [0]
[0 1] [2]
= dbl(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [5]
[0 1] [0 1] [0]
>= [1 0] X1 + [1 4] X2 + [5]
[0 1] [0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [6]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [9]
[0 1] [0]
>= [4]
[0]
= s(add(X,Y))
a__dbl(0()) = [3]
[2]
>= [1]
[0]
= 0()
a__dbl(s(X)) = [6]
[2]
>= [4]
[0]
= s(s(dbl(X)))
a__first(X1,X2) = [1 6] X1 + [1 7] X2 + [4]
[0 1] [0 1] [0]
>= [1 6] X1 + [1 7] X2 + [4]
[0 1] [0 1] [0]
= first(X1,X2)
a__first(0(),X) = [1 7] X + [5]
[0 1] [0]
>= [0]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 7] Y + [10]
[0 1] [0]
>= [1 4] Y + [2]
[0 1] [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= sqr(X)
a__sqr(0()) = [1]
[0]
>= [1]
[0]
= 0()
a__sqr(s(X)) = [4]
[0]
>= [4]
[0]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [4]
[0 1] [0]
>= [1 4] N + [4]
[0 1] [0]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1 4] X + [4]
[0 1] [0]
>= [1 4] X + [4]
[0 1] [0]
= terms(X)
mark(0()) = [1]
[0]
>= [1]
[0]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [5]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 8] X2 + [5]
[0 1] [0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 4] X1 + [2]
[0 1] [0]
>= [1 4] X1 + [2]
[0 1] [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [1 4] X + [8]
[0 1] [2]
>= [1 4] X + [2]
[0 1] [2]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 10] X1 + [1 11] X2 + [4]
[0 1] [0 1] [0]
>= [1 10] X1 + [1 11] X2 + [4]
[0 1] [0 1] [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(recip(X)) = [1 4] X + [2]
[0 1] [0]
>= [1 4] X + [2]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [4]
[0]
>= [4]
[0]
= s(X)
mark(sqr(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= a__sqr(mark(X))
mark(terms(X)) = [1 8] X + [4]
[0 1] [0]
>= [1 8] X + [4]
[0 1] [0]
= a__terms(mark(X))
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(dbl(X)) -> a__dbl(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{a__add,a__dbl,a__first,a__sqr,a__terms,mark}
TcT has computed the following interpretation:
p(0) = [0]
[2]
p(a__add) = [1 4] x1 + [1 4] x2 + [5]
[0 1] [0 1] [1]
p(a__dbl) = [1 0] x1 + [1]
[0 1] [1]
p(a__first) = [1 2] x1 + [1 5] x2 + [0]
[0 1] [0 1] [0]
p(a__sqr) = [1 0] x1 + [0]
[0 1] [0]
p(a__terms) = [1 7] x1 + [6]
[0 1] [1]
p(add) = [1 4] x1 + [1 4] x2 + [3]
[0 1] [0 1] [1]
p(cons) = [1 2] x1 + [4]
[0 1] [0]
p(dbl) = [1 0] x1 + [1]
[0 1] [1]
p(first) = [1 2] x1 + [1 5] x2 + [0]
[0 1] [0 1] [0]
p(mark) = [1 4] x1 + [0]
[0 1] [0]
p(nil) = [3]
[1]
p(recip) = [1 1] x1 + [0]
[0 1] [0]
p(s) = [7]
[0]
p(sqr) = [1 0] x1 + [0]
[0 1] [0]
p(terms) = [1 7] x1 + [4]
[0 1] [1]
Following rules are strictly oriented:
mark(add(X1,X2)) = [1 8] X1 + [1 8] X2 + [7]
[0 1] [0 1] [1]
> [1 8] X1 + [1 8] X2 + [5]
[0 1] [0 1] [1]
= a__add(mark(X1),mark(X2))
mark(terms(X)) = [1 11] X + [8]
[0 1] [1]
> [1 11] X + [6]
[0 1] [1]
= a__terms(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 4] X1 + [1 4] X2 + [5]
[0 1] [0 1] [1]
>= [1 4] X1 + [1 4] X2 + [3]
[0 1] [0 1] [1]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [13]
[0 1] [3]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [12]
[0 1] [1]
>= [7]
[0]
= s(add(X,Y))
a__dbl(X) = [1 0] X + [1]
[0 1] [1]
>= [1 0] X + [1]
[0 1] [1]
= dbl(X)
a__dbl(0()) = [1]
[3]
>= [0]
[2]
= 0()
a__dbl(s(X)) = [8]
[1]
>= [7]
[0]
= s(s(dbl(X)))
a__first(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]
= first(X1,X2)
a__first(0(),X) = [1 5] X + [4]
[0 1] [2]
>= [3]
[1]
= nil()
a__first(s(X),cons(Y,Z)) = [1 7] Y + [11]
[0 1] [0]
>= [1 6] Y + [4]
[0 1] [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= sqr(X)
a__sqr(0()) = [0]
[2]
>= [0]
[2]
= 0()
a__sqr(s(X)) = [7]
[0]
>= [7]
[0]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 7] N + [6]
[0 1] [1]
>= [1 7] N + [4]
[0 1] [0]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1 7] X + [6]
[0 1] [1]
>= [1 7] X + [4]
[0 1] [1]
= terms(X)
mark(0()) = [8]
[2]
>= [0]
[2]
= 0()
mark(cons(X1,X2)) = [1 6] X1 + [4]
[0 1] [0]
>= [1 6] X1 + [4]
[0 1] [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [1 4] X + [5]
[0 1] [1]
>= [1 4] X + [1]
[0 1] [1]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 6] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
>= [1 6] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [7]
[1]
>= [3]
[1]
= nil()
mark(recip(X)) = [1 5] X + [0]
[0 1] [0]
>= [1 5] X + [0]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [7]
[0]
>= [7]
[0]
= s(X)
mark(sqr(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= a__sqr(mark(X))
*** 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)
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(dbl(X)) -> a__dbl(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(terms(X)) -> a__terms(mark(X))
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{a__add,a__dbl,a__first,a__sqr,a__terms,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__add) = [1 0] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(a__dbl) = [1 4] x1 + [3]
[0 1] [0]
p(a__first) = [1 0] x1 + [1 5] x2 + [0]
[0 1] [0 1] [0]
p(a__sqr) = [1 0] x1 + [1]
[0 1] [1]
p(a__terms) = [1 4] x1 + [1]
[0 1] [1]
p(add) = [1 0] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x1 + [0]
[0 1] [0]
p(dbl) = [1 4] x1 + [3]
[0 1] [0]
p(first) = [1 0] x1 + [1 5] x2 + [0]
[0 1] [0 1] [0]
p(mark) = [1 4] x1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(recip) = [1 0] x1 + [0]
[0 1] [0]
p(s) = [2]
[0]
p(sqr) = [1 0] x1 + [0]
[0 1] [1]
p(terms) = [1 4] x1 + [0]
[0 1] [1]
Following rules are strictly oriented:
mark(sqr(X)) = [1 4] X + [4]
[0 1] [1]
> [1 4] X + [1]
[0 1] [1]
= a__sqr(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [2]
[0 1] [0]
>= [2]
[0]
= s(add(X,Y))
a__dbl(X) = [1 4] X + [3]
[0 1] [0]
>= [1 4] X + [3]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [3]
[0]
>= [0]
[0]
= 0()
a__dbl(s(X)) = [5]
[0]
>= [2]
[0]
= s(s(dbl(X)))
a__first(X1,X2) = [1 0] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
>= [1 0] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
= first(X1,X2)
a__first(0(),X) = [1 5] X + [0]
[0 1] [0]
>= [0]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 5] Y + [2]
[0 1] [0]
>= [1 4] Y + [0]
[0 1] [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [1]
[0 1] [1]
>= [1 0] X + [0]
[0 1] [1]
= sqr(X)
a__sqr(0()) = [1]
[1]
>= [0]
[0]
= 0()
a__sqr(s(X)) = [3]
[1]
>= [2]
[0]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [1]
[0 1] [1]
>= [1 4] N + [1]
[0 1] [1]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1 4] X + [1]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [1]
= terms(X)
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 4] X1 + [0]
[0 1] [0]
>= [1 4] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [1 8] X + [3]
[0 1] [0]
>= [1 8] X + [3]
[0 1] [0]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 4] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(recip(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [2]
[0]
>= [2]
[0]
= s(X)
mark(terms(X)) = [1 8] X + [4]
[0 1] [1]
>= [1 8] X + [1]
[0 1] [1]
= a__terms(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(cons(X1,X2)) -> cons(mark(X1),X2)
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(dbl(X)) -> a__dbl(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{a__add,a__dbl,a__first,a__sqr,a__terms,mark}
TcT has computed the following interpretation:
p(0) = [4]
[1]
p(a__add) = [1 0] x1 + [1 4] x2 + [4]
[0 1] [0 1] [1]
p(a__dbl) = [1 0] x1 + [4]
[0 1] [0]
p(a__first) = [1 4] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(a__sqr) = [1 2] x1 + [5]
[0 1] [0]
p(a__terms) = [1 6] x1 + [7]
[0 1] [2]
p(add) = [1 0] x1 + [1 4] x2 + [4]
[0 1] [0 1] [1]
p(cons) = [1 0] x1 + [1]
[0 1] [1]
p(dbl) = [1 0] x1 + [4]
[0 1] [0]
p(first) = [1 4] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(mark) = [1 4] x1 + [0]
[0 1] [0]
p(nil) = [1]
[1]
p(recip) = [1 0] x1 + [0]
[0 1] [0]
p(s) = [5]
[0]
p(sqr) = [1 2] x1 + [5]
[0 1] [0]
p(terms) = [1 6] x1 + [1]
[0 1] [2]
Following rules are strictly oriented:
mark(cons(X1,X2)) = [1 4] X1 + [5]
[0 1] [1]
> [1 4] X1 + [1]
[0 1] [1]
= cons(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [4]
[0 1] [0 1] [1]
>= [1 0] X1 + [1 4] X2 + [4]
[0 1] [0 1] [1]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [8]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [9]
[0 1] [1]
>= [5]
[0]
= s(add(X,Y))
a__dbl(X) = [1 0] X + [4]
[0 1] [0]
>= [1 0] X + [4]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [8]
[1]
>= [4]
[1]
= 0()
a__dbl(s(X)) = [9]
[0]
>= [5]
[0]
= s(s(dbl(X)))
a__first(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]
= first(X1,X2)
a__first(0(),X) = [1 4] X + [8]
[0 1] [1]
>= [1]
[1]
= nil()
a__first(s(X),cons(Y,Z)) = [1 4] Y + [10]
[0 1] [1]
>= [1 4] Y + [1]
[0 1] [1]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 2] X + [5]
[0 1] [0]
>= [1 2] X + [5]
[0 1] [0]
= sqr(X)
a__sqr(0()) = [11]
[1]
>= [4]
[1]
= 0()
a__sqr(s(X)) = [10]
[0]
>= [5]
[0]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 6] N + [7]
[0 1] [2]
>= [1 6] N + [6]
[0 1] [1]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1 6] X + [7]
[0 1] [2]
>= [1 6] X + [1]
[0 1] [2]
= terms(X)
mark(0()) = [8]
[1]
>= [4]
[1]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [8]
[0 1] [0 1] [1]
>= [1 4] X1 + [1 8] X2 + [4]
[0 1] [0 1] [1]
= a__add(mark(X1),mark(X2))
mark(dbl(X)) = [1 4] X + [4]
[0 1] [0]
>= [1 4] X + [4]
[0 1] [0]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 8] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
>= [1 8] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [5]
[1]
>= [1]
[1]
= nil()
mark(recip(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [5]
[0]
>= [5]
[0]
= s(X)
mark(sqr(X)) = [1 6] X + [5]
[0 1] [0]
>= [1 6] X + [5]
[0 1] [0]
= a__sqr(mark(X))
mark(terms(X)) = [1 10] X + [9]
[0 1] [2]
>= [1 10] X + [7]
[0 1] [2]
= a__terms(mark(X))
*** 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(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{a__add,a__dbl,a__first,a__sqr,a__terms,mark}
TcT has computed the following interpretation:
p(0) = [0]
[2]
p(a__add) = [1 4] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(a__dbl) = [1 4] x1 + [6]
[0 1] [0]
p(a__first) = [1 0] x1 + [1 4] x2 + [2]
[0 1] [0 1] [2]
p(a__sqr) = [1 0] x1 + [6]
[0 1] [0]
p(a__terms) = [1 4] x1 + [7]
[0 1] [0]
p(add) = [1 4] x1 + [1 4] x2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x1 + [1]
[0 1] [0]
p(dbl) = [1 4] x1 + [6]
[0 1] [0]
p(first) = [1 0] x1 + [1 4] x2 + [0]
[0 1] [0 1] [2]
p(mark) = [1 4] x1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(recip) = [1 0] x1 + [0]
[0 1] [0]
p(s) = [1]
[1]
p(sqr) = [1 0] x1 + [6]
[0 1] [0]
p(terms) = [1 4] x1 + [7]
[0 1] [0]
Following rules are strictly oriented:
mark(first(X1,X2)) = [1 4] X1 + [1 8] X2 + [8]
[0 1] [0 1] [2]
> [1 4] X1 + [1 8] X2 + [2]
[0 1] [0 1] [2]
= a__first(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__add(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]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [8]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [5]
[0 1] [1]
>= [1]
[1]
= s(add(X,Y))
a__dbl(X) = [1 4] X + [6]
[0 1] [0]
>= [1 4] X + [6]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [14]
[2]
>= [0]
[2]
= 0()
a__dbl(s(X)) = [11]
[1]
>= [1]
[1]
= s(s(dbl(X)))
a__first(X1,X2) = [1 0] X1 + [1 4] X2 + [2]
[0 1] [0 1] [2]
>= [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [2]
= first(X1,X2)
a__first(0(),X) = [1 4] X + [2]
[0 1] [4]
>= [0]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 4] Y + [4]
[0 1] [3]
>= [1 4] Y + [1]
[0 1] [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [6]
[0 1] [0]
>= [1 0] X + [6]
[0 1] [0]
= sqr(X)
a__sqr(0()) = [6]
[2]
>= [0]
[2]
= 0()
a__sqr(s(X)) = [7]
[1]
>= [1]
[1]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [7]
[0 1] [0]
>= [1 4] N + [7]
[0 1] [0]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1 4] X + [7]
[0 1] [0]
>= [1 4] X + [7]
[0 1] [0]
= terms(X)
mark(0()) = [8]
[2]
>= [0]
[2]
= 0()
mark(add(X1,X2)) = [1 8] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
>= [1 8] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 4] X1 + [1]
[0 1] [0]
>= [1 4] X1 + [1]
[0 1] [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [1 8] X + [6]
[0 1] [0]
>= [1 8] X + [6]
[0 1] [0]
= a__dbl(mark(X))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(recip(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [5]
[1]
>= [1]
[1]
= s(X)
mark(sqr(X)) = [1 4] X + [6]
[0 1] [0]
>= [1 4] X + [6]
[0 1] [0]
= a__sqr(mark(X))
mark(terms(X)) = [1 8] X + [7]
[0 1] [0]
>= [1 8] X + [7]
[0 1] [0]
= a__terms(mark(X))
*** 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(recip(X)) -> recip(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
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__add) = {1,2},
uargs(a__dbl) = {1},
uargs(a__first) = {1,2},
uargs(a__sqr) = {1},
uargs(a__terms) = {1},
uargs(cons) = {1},
uargs(recip) = {1}
Following symbols are considered usable:
{a__add,a__dbl,a__first,a__sqr,a__terms,mark}
TcT has computed the following interpretation:
p(0) = [4]
[0]
p(a__add) = [1 0] x1 + [1 4] x2 + [0]
[0 1] [0 1] [2]
p(a__dbl) = [1 2] x1 + [0]
[0 1] [0]
p(a__first) = [1 4] x1 + [1 6] x2 + [1]
[0 1] [0 1] [1]
p(a__sqr) = [1 0] x1 + [0]
[0 1] [0]
p(a__terms) = [1 4] x1 + [5]
[0 1] [2]
p(add) = [1 0] x1 + [1 4] x2 + [0]
[0 1] [0 1] [2]
p(cons) = [1 0] x1 + [0]
[0 1] [0]
p(dbl) = [1 2] x1 + [0]
[0 1] [0]
p(first) = [1 4] x1 + [1 6] x2 + [1]
[0 1] [0 1] [1]
p(mark) = [1 4] x1 + [3]
[0 1] [0]
p(nil) = [5]
[0]
p(recip) = [1 0] x1 + [0]
[0 1] [2]
p(s) = [1]
[1]
p(sqr) = [1 0] x1 + [0]
[0 1] [0]
p(terms) = [1 4] x1 + [3]
[0 1] [2]
Following rules are strictly oriented:
mark(recip(X)) = [1 4] X + [11]
[0 1] [2]
> [1 4] X + [3]
[0 1] [2]
= recip(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [2]
>= [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [2]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [4]
[0 1] [2]
>= [1 4] X + [3]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [1]
[0 1] [3]
>= [1]
[1]
= s(add(X,Y))
a__dbl(X) = [1 2] X + [0]
[0 1] [0]
>= [1 2] X + [0]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [4]
[0]
>= [4]
[0]
= 0()
a__dbl(s(X)) = [3]
[1]
>= [1]
[1]
= s(s(dbl(X)))
a__first(X1,X2) = [1 4] X1 + [1 6] X2 + [1]
[0 1] [0 1] [1]
>= [1 4] X1 + [1 6] X2 + [1]
[0 1] [0 1] [1]
= first(X1,X2)
a__first(0(),X) = [1 6] X + [5]
[0 1] [1]
>= [5]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 6] Y + [6]
[0 1] [2]
>= [1 4] Y + [3]
[0 1] [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= sqr(X)
a__sqr(0()) = [4]
[0]
>= [4]
[0]
= 0()
a__sqr(s(X)) = [1]
[1]
>= [1]
[1]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [5]
[0 1] [2]
>= [1 4] N + [3]
[0 1] [2]
= cons(recip(a__sqr(mark(N)))
,terms(s(N)))
a__terms(X) = [1 4] X + [5]
[0 1] [2]
>= [1 4] X + [3]
[0 1] [2]
= terms(X)
mark(0()) = [7]
[0]
>= [4]
[0]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [11]
[0 1] [0 1] [2]
>= [1 4] X1 + [1 8] X2 + [6]
[0 1] [0 1] [2]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 4] X1 + [3]
[0 1] [0]
>= [1 4] X1 + [3]
[0 1] [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [1 6] X + [3]
[0 1] [0]
>= [1 6] X + [3]
[0 1] [0]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 8] X1 + [1 10] X2 + [8]
[0 1] [0 1] [1]
>= [1 8] X1 + [1 10] X2 + [7]
[0 1] [0 1] [1]
= a__first(mark(X1),mark(X2))
mark(nil()) = [8]
[0]
>= [5]
[0]
= nil()
mark(s(X)) = [8]
[1]
>= [1]
[1]
= s(X)
mark(sqr(X)) = [1 4] X + [3]
[0 1] [0]
>= [1 4] X + [3]
[0 1] [0]
= a__sqr(mark(X))
mark(terms(X)) = [1 8] X + [14]
[0 1] [2]
>= [1 8] X + [8]
[0 1] [2]
= a__terms(mark(X))
*** 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__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__dbl(X) -> dbl(X)
a__dbl(0()) -> 0()
a__dbl(s(X)) -> s(s(dbl(X)))
a__first(X1,X2) -> first(X1,X2)
a__first(0(),X) -> nil()
a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
a__sqr(X) -> sqr(X)
a__sqr(0()) -> 0()
a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) -> terms(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(dbl(X)) -> a__dbl(mark(X))
mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
Signature:
{a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1,s/1,sqr/1,terms/1}
Obligation:
Innermost
basic terms: {a__add,a__dbl,a__first,a__sqr,a__terms,mark}/{0,add,cons,dbl,first,nil,recip,s,sqr,terms}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).