* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(x){x -> add(x,y)} =
mark(add(x,y)) ->^+ a__add(mark(x),mark(y))
= C[mark(x) = mark(x){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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:
all
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.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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:
all
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.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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:
all
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.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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:
all
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.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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:
all
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.
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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:
all
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.
** Step 1.b:7: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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 0] x_1 + [1 4] x_2 + [2]
[0 1] [0 1] [0]
p(a__dbl) = [1 0] x_1 + [4]
[0 1] [2]
p(a__first) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(a__sqr) = [1 0] x_1 + [0]
[0 1] [0]
p(a__terms) = [1 4] x_1 + [0]
[0 1] [0]
p(add) = [1 0] x_1 + [1 4] x_2 + [2]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(dbl) = [1 0] x_1 + [0]
[0 1] [2]
p(first) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[1]
p(recip) = [1 0] x_1 + [0]
[0 1] [0]
p(s) = [0]
[0]
p(sqr) = [1 0] x_1 + [0]
[0 1] [0]
p(terms) = [1 4] x_1 + [0]
[0 1] [0]
Following rules are strictly oriented:
a__dbl(X) = [1 0] X + [4]
[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 + [2]
[0 1] [0 1] [0]
>= [1 0] X1 + [1 4] X2 + [2]
[0 1] [0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [2]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [2]
[0 1] [0]
>= [0]
[0]
= s(add(X,Y))
a__dbl(0()) = [4]
[4]
>= [0]
[2]
= 0()
a__dbl(s(X)) = [4]
[2]
>= [0]
[0]
= s(s(dbl(X)))
a__first(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]
= first(X1,X2)
a__first(0(),X) = [1 4] X + [0]
[0 1] [2]
>= [0]
[1]
= nil()
a__first(s(X),cons(Y,Z)) = [1 4] Y + [0]
[0 1] [0]
>= [1 4] Y + [0]
[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)) = [0]
[0]
>= [0]
[0]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [0]
[0 1] [0]
>= [1 4] N + [0]
[0 1] [0]
= cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= terms(X)
mark(0()) = [8]
[2]
>= [0]
[2]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [2]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 8] X2 + [2]
[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 4] X + [8]
[0 1] [2]
>= [1 4] X + [4]
[0 1] [2]
= a__dbl(mark(X))
mark(first(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__first(mark(X1),mark(X2))
mark(nil()) = [4]
[1]
>= [0]
[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)) = [0]
[0]
>= [0]
[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 + [0]
[0 1] [0]
>= [1 8] X + [0]
[0 1] [0]
= a__terms(mark(X))
** Step 1.b:8: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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 TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(a__dbl) = [1 4] x_1 + [0]
[0 1] [1]
p(a__first) = [1 3] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [2]
p(a__sqr) = [1 0] x_1 + [0]
[0 1] [0]
p(a__terms) = [1 4] x_1 + [0]
[0 1] [0]
p(add) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(dbl) = [1 4] x_1 + [0]
[0 1] [1]
p(first) = [1 3] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [2]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[2]
p(recip) = [1 0] x_1 + [0]
[0 1] [0]
p(s) = [0]
[2]
p(sqr) = [1 0] x_1 + [0]
[0 1] [0]
p(terms) = [1 4] x_1 + [0]
[0 1] [0]
Following rules are strictly oriented:
mark(first(X1,X2)) = [1 7] X1 + [1 8] X2 + [8]
[0 1] [0 1] [2]
> [1 7] X1 + [1 8] X2 + [0]
[0 1] [0 1] [2]
= a__first(mark(X1),mark(X2))
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 + [0]
[0 1] [2]
>= [0]
[2]
= s(add(X,Y))
a__dbl(X) = [1 4] X + [0]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [1]
= dbl(X)
a__dbl(0()) = [0]
[1]
>= [0]
[0]
= 0()
a__dbl(s(X)) = [8]
[3]
>= [0]
[2]
= s(s(dbl(X)))
a__first(X1,X2) = [1 3] X1 + [1 4] X2 + [0]
[0 1] [0 1] [2]
>= [1 3] X1 + [1 4] X2 + [0]
[0 1] [0 1] [2]
= first(X1,X2)
a__first(0(),X) = [1 4] X + [0]
[0 1] [2]
>= [0]
[2]
= nil()
a__first(s(X),cons(Y,Z)) = [1 4] Y + [6]
[0 1] [4]
>= [1 4] Y + [0]
[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]
[0]
>= [0]
[0]
= 0()
a__sqr(s(X)) = [0]
[2]
>= [0]
[2]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [0]
[0 1] [0]
>= [1 4] N + [0]
[0 1] [0]
= cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= 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 + [4]
[0 1] [1]
>= [1 8] X + [0]
[0 1] [1]
= a__dbl(mark(X))
mark(nil()) = [8]
[2]
>= [0]
[2]
= nil()
mark(recip(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [8]
[2]
>= [0]
[2]
= 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 + [0]
[0 1] [0]
>= [1 8] X + [0]
[0 1] [0]
= a__terms(mark(X))
** Step 1.b:9: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
mark(terms(X)) -> a__terms(mark(X))
- Weak TRS:
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(first(X1,X2)) -> a__first(mark(X1),mark(X2))
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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] x_1 + [1 4] x_2 + [1]
[0 1] [0 1] [0]
p(a__dbl) = [1 0] x_1 + [0]
[0 1] [0]
p(a__first) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(a__sqr) = [1 0] x_1 + [0]
[0 1] [0]
p(a__terms) = [1 4] x_1 + [6]
[0 1] [2]
p(add) = [1 4] x_1 + [1 4] x_2 + [1]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(dbl) = [1 0] x_1 + [0]
[0 1] [0]
p(first) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[2]
p(recip) = [1 0] x_1 + [0]
[0 1] [0]
p(s) = [4]
[0]
p(sqr) = [1 0] x_1 + [0]
[0 1] [0]
p(terms) = [1 4] x_1 + [0]
[0 1] [2]
Following rules are strictly oriented:
mark(terms(X)) = [1 8] X + [8]
[0 1] [2]
> [1 8] X + [6]
[0 1] [2]
= a__terms(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 4] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [9]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [5]
[0 1] [0]
>= [4]
[0]
= s(add(X,Y))
a__dbl(X) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [0]
[2]
>= [0]
[2]
= 0()
a__dbl(s(X)) = [4]
[0]
>= [4]
[0]
= s(s(dbl(X)))
a__first(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]
= first(X1,X2)
a__first(0(),X) = [1 4] X + [0]
[0 1] [2]
>= [0]
[2]
= nil()
a__first(s(X),cons(Y,Z)) = [1 4] Y + [4]
[0 1] [0]
>= [1 4] Y + [0]
[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)) = [4]
[0]
>= [4]
[0]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [6]
[0 1] [2]
>= [1 4] N + [0]
[0 1] [0]
= cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) = [1 4] X + [6]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [2]
= terms(X)
mark(0()) = [8]
[2]
>= [0]
[2]
= 0()
mark(add(X1,X2)) = [1 8] X1 + [1 8] X2 + [1]
[0 1] [0 1] [0]
>= [1 8] X1 + [1 8] X2 + [1]
[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 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= a__dbl(mark(X))
mark(first(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__first(mark(X1),mark(X2))
mark(nil()) = [8]
[2]
>= [0]
[2]
= nil()
mark(recip(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[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))
** Step 1.b:10: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(recip(X)) -> recip(mark(X))
mark(sqr(X)) -> a__sqr(mark(X))
- Weak TRS:
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(first(X1,X2)) -> a__first(mark(X1),mark(X2))
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(a__dbl) = [1 0] x_1 + [6]
[0 1] [0]
p(a__first) = [1 0] x_1 + [1 5] x_2 + [0]
[0 1] [0 1] [0]
p(a__sqr) = [1 1] x_1 + [3]
[0 1] [1]
p(a__terms) = [1 7] x_1 + [6]
[0 1] [1]
p(add) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(dbl) = [1 0] x_1 + [6]
[0 1] [0]
p(first) = [1 0] x_1 + [1 5] x_2 + [0]
[0 1] [0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(recip) = [1 2] x_1 + [0]
[0 1] [0]
p(s) = [1]
[1]
p(sqr) = [1 1] x_1 + [0]
[0 1] [1]
p(terms) = [1 7] x_1 + [2]
[0 1] [1]
Following rules are strictly oriented:
mark(sqr(X)) = [1 5] X + [4]
[0 1] [1]
> [1 5] X + [3]
[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] [2]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [1]
[0 1] [1]
>= [1]
[1]
= s(add(X,Y))
a__dbl(X) = [1 0] X + [6]
[0 1] [0]
>= [1 0] X + [6]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [6]
[2]
>= [0]
[2]
= 0()
a__dbl(s(X)) = [7]
[1]
>= [1]
[1]
= 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] [2]
>= [0]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 5] Y + [1]
[0 1] [1]
>= [1 4] Y + [0]
[0 1] [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 1] X + [3]
[0 1] [1]
>= [1 1] X + [0]
[0 1] [1]
= sqr(X)
a__sqr(0()) = [5]
[3]
>= [0]
[2]
= 0()
a__sqr(s(X)) = [5]
[2]
>= [1]
[1]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 7] N + [6]
[0 1] [1]
>= [1 7] N + [5]
[0 1] [1]
= cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) = [1 7] X + [6]
[0 1] [1]
>= [1 7] X + [2]
[0 1] [1]
= terms(X)
mark(0()) = [8]
[2]
>= [0]
[2]
= 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 4] X + [6]
[0 1] [0]
>= [1 4] X + [6]
[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 6] X + [0]
[0 1] [0]
>= [1 6] X + [0]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [5]
[1]
>= [1]
[1]
= s(X)
mark(terms(X)) = [1 11] X + [6]
[0 1] [1]
>= [1 11] X + [6]
[0 1] [1]
= a__terms(mark(X))
** Step 1.b:11: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(recip(X)) -> recip(mark(X))
- Weak TRS:
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(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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(a__dbl) = [1 2] x_1 + [1]
[0 1] [0]
p(a__first) = [1 1] x_1 + [1 4] x_2 + [1]
[0 1] [0 1] [0]
p(a__sqr) = [1 0] x_1 + [1]
[0 1] [0]
p(a__terms) = [1 4] x_1 + [3]
[0 1] [1]
p(add) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [1]
p(dbl) = [1 2] x_1 + [1]
[0 1] [0]
p(first) = [1 1] x_1 + [1 4] x_2 + [1]
[0 1] [0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [2]
[0]
p(recip) = [1 0] x_1 + [2]
[0 1] [0]
p(s) = [2]
[1]
p(sqr) = [1 0] x_1 + [1]
[0 1] [0]
p(terms) = [1 4] x_1 + [0]
[0 1] [1]
Following rules are strictly oriented:
mark(cons(X1,X2)) = [1 4] X1 + [4]
[0 1] [1]
> [1 4] X1 + [0]
[0 1] [1]
= cons(mark(X1),X2)
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] [2]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [2]
[0 1] [1]
>= [2]
[1]
= s(add(X,Y))
a__dbl(X) = [1 2] X + [1]
[0 1] [0]
>= [1 2] X + [1]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [5]
[2]
>= [0]
[2]
= 0()
a__dbl(s(X)) = [5]
[1]
>= [2]
[1]
= s(s(dbl(X)))
a__first(X1,X2) = [1 1] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
>= [1 1] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
= first(X1,X2)
a__first(0(),X) = [1 4] X + [3]
[0 1] [2]
>= [2]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 4] Y + [8]
[0 1] [2]
>= [1 4] Y + [0]
[0 1] [1]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [1]
[0 1] [0]
>= [1 0] X + [1]
[0 1] [0]
= sqr(X)
a__sqr(0()) = [1]
[2]
>= [0]
[2]
= 0()
a__sqr(s(X)) = [3]
[1]
>= [2]
[1]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [3]
[0 1] [1]
>= [1 4] N + [3]
[0 1] [1]
= cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) = [1 4] X + [3]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [1]
= terms(X)
mark(0()) = [8]
[2]
>= [0]
[2]
= 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(dbl(X)) = [1 6] X + [1]
[0 1] [0]
>= [1 6] X + [1]
[0 1] [0]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 5] X1 + [1 8] X2 + [1]
[0 1] [0 1] [0]
>= [1 5] X1 + [1 8] X2 + [1]
[0 1] [0 1] [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [2]
[0]
>= [2]
[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)) = [6]
[1]
>= [2]
[1]
= s(X)
mark(sqr(X)) = [1 4] X + [1]
[0 1] [0]
>= [1 4] X + [1]
[0 1] [0]
= a__sqr(mark(X))
mark(terms(X)) = [1 8] X + [4]
[0 1] [1]
>= [1 8] X + [3]
[0 1] [1]
= a__terms(mark(X))
** Step 1.b:12: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(recip(X)) -> recip(mark(X))
- Weak TRS:
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(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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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) = [2]
[0]
p(a__add) = [1 2] x_1 + [1 2] x_2 + [0]
[0 1] [0 1] [0]
p(a__dbl) = [1 1] x_1 + [6]
[0 1] [0]
p(a__first) = [1 0] x_1 + [1 5] x_2 + [4]
[0 1] [0 1] [1]
p(a__sqr) = [1 0] x_1 + [1]
[0 1] [0]
p(a__terms) = [1 4] x_1 + [5]
[0 1] [2]
p(add) = [1 2] x_1 + [1 2] x_2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [1]
p(dbl) = [1 1] x_1 + [6]
[0 1] [0]
p(first) = [1 0] x_1 + [1 5] x_2 + [3]
[0 1] [0 1] [1]
p(mark) = [1 2] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(recip) = [1 0] x_1 + [4]
[0 1] [1]
p(s) = [4]
[2]
p(sqr) = [1 0] x_1 + [1]
[0 1] [0]
p(terms) = [1 4] x_1 + [2]
[0 1] [2]
Following rules are strictly oriented:
mark(recip(X)) = [1 2] X + [6]
[0 1] [1]
> [1 2] X + [4]
[0 1] [1]
= recip(mark(X))
Following rules are (at-least) weakly oriented:
a__add(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]
= add(X1,X2)
a__add(0(),X) = [1 2] X + [2]
[0 1] [0]
>= [1 2] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 2] Y + [8]
[0 1] [2]
>= [4]
[2]
= s(add(X,Y))
a__dbl(X) = [1 1] X + [6]
[0 1] [0]
>= [1 1] X + [6]
[0 1] [0]
= dbl(X)
a__dbl(0()) = [8]
[0]
>= [2]
[0]
= 0()
a__dbl(s(X)) = [12]
[2]
>= [4]
[2]
= s(s(dbl(X)))
a__first(X1,X2) = [1 0] X1 + [1 5] X2 + [4]
[0 1] [0 1] [1]
>= [1 0] X1 + [1 5] X2 + [3]
[0 1] [0 1] [1]
= first(X1,X2)
a__first(0(),X) = [1 5] X + [6]
[0 1] [1]
>= [0]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 5] Y + [13]
[0 1] [4]
>= [1 2] Y + [0]
[0 1] [1]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [1]
[0 1] [0]
>= [1 0] X + [1]
[0 1] [0]
= sqr(X)
a__sqr(0()) = [3]
[0]
>= [2]
[0]
= 0()
a__sqr(s(X)) = [5]
[2]
>= [4]
[2]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 4] N + [5]
[0 1] [2]
>= [1 2] N + [5]
[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 + [2]
[0 1] [2]
= terms(X)
mark(0()) = [2]
[0]
>= [2]
[0]
= 0()
mark(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]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 2] X1 + [2]
[0 1] [1]
>= [1 2] X1 + [0]
[0 1] [1]
= cons(mark(X1),X2)
mark(dbl(X)) = [1 3] X + [6]
[0 1] [0]
>= [1 3] X + [6]
[0 1] [0]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 2] X1 + [1 7] X2 + [5]
[0 1] [0 1] [1]
>= [1 2] X1 + [1 7] X2 + [4]
[0 1] [0 1] [1]
= a__first(mark(X1),mark(X2))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(s(X)) = [8]
[2]
>= [4]
[2]
= s(X)
mark(sqr(X)) = [1 2] X + [1]
[0 1] [0]
>= [1 2] X + [1]
[0 1] [0]
= a__sqr(mark(X))
mark(terms(X)) = [1 6] X + [6]
[0 1] [2]
>= [1 6] X + [5]
[0 1] [2]
= a__terms(mark(X))
** Step 1.b:13: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
- Weak TRS:
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(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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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] x_1 + [1 1] x_2 + [1]
[0 1] [0 1] [1]
p(a__dbl) = [1 0] x_1 + [0]
[0 1] [2]
p(a__first) = [1 2] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(a__sqr) = [1 0] x_1 + [1]
[0 1] [6]
p(a__terms) = [1 1] x_1 + [1]
[0 1] [6]
p(add) = [1 0] x_1 + [1 1] x_2 + [1]
[0 1] [0 1] [1]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(dbl) = [1 0] x_1 + [0]
[0 1] [2]
p(first) = [1 2] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(mark) = [1 1] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(recip) = [1 0] x_1 + [0]
[0 1] [0]
p(s) = [0]
[2]
p(sqr) = [1 0] x_1 + [0]
[0 1] [6]
p(terms) = [1 1] x_1 + [1]
[0 1] [6]
Following rules are strictly oriented:
mark(add(X1,X2)) = [1 1] X1 + [1 2] X2 + [2]
[0 1] [0 1] [1]
> [1 1] X1 + [1 2] X2 + [1]
[0 1] [0 1] [1]
= a__add(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 1] X2 + [1]
[0 1] [0 1] [1]
>= [1 0] X1 + [1 1] X2 + [1]
[0 1] [0 1] [1]
= add(X1,X2)
a__add(0(),X) = [1 1] X + [1]
[0 1] [1]
>= [1 1] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 1] Y + [1]
[0 1] [3]
>= [0]
[2]
= s(add(X,Y))
a__dbl(X) = [1 0] X + [0]
[0 1] [2]
>= [1 0] X + [0]
[0 1] [2]
= dbl(X)
a__dbl(0()) = [0]
[2]
>= [0]
[0]
= 0()
a__dbl(s(X)) = [0]
[4]
>= [0]
[2]
= s(s(dbl(X)))
a__first(X1,X2) = [1 2] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 2] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
= first(X1,X2)
a__first(0(),X) = [1 4] X + [0]
[0 1] [0]
>= [0]
[0]
= nil()
a__first(s(X),cons(Y,Z)) = [1 4] Y + [4]
[0 1] [2]
>= [1 1] Y + [0]
[0 1] [0]
= cons(mark(Y),first(X,Z))
a__sqr(X) = [1 0] X + [1]
[0 1] [6]
>= [1 0] X + [0]
[0 1] [6]
= sqr(X)
a__sqr(0()) = [1]
[6]
>= [0]
[0]
= 0()
a__sqr(s(X)) = [1]
[8]
>= [0]
[2]
= s(add(sqr(X),dbl(X)))
a__terms(N) = [1 1] N + [1]
[0 1] [6]
>= [1 1] N + [1]
[0 1] [6]
= cons(recip(a__sqr(mark(N))),terms(s(N)))
a__terms(X) = [1 1] X + [1]
[0 1] [6]
>= [1 1] X + [1]
[0 1] [6]
= terms(X)
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(cons(X1,X2)) = [1 1] X1 + [0]
[0 1] [0]
>= [1 1] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(dbl(X)) = [1 1] X + [2]
[0 1] [2]
>= [1 1] X + [0]
[0 1] [2]
= a__dbl(mark(X))
mark(first(X1,X2)) = [1 3] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
>= [1 3] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
= a__first(mark(X1),mark(X2))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(recip(X)) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= recip(mark(X))
mark(s(X)) = [2]
[2]
>= [0]
[2]
= s(X)
mark(sqr(X)) = [1 1] X + [6]
[0 1] [6]
>= [1 1] X + [1]
[0 1] [6]
= a__sqr(mark(X))
mark(terms(X)) = [1 2] X + [7]
[0 1] [6]
>= [1 2] X + [1]
[0 1] [6]
= a__terms(mark(X))
** Step 1.b:14: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
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 runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))