* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
s(X) -> n__s(X)
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
s(X) -> n__s(X)
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__add(x,y)} =
activate(n__add(x,y)) ->^+ add(activate(x),activate(y))
= C[activate(x) = activate(x){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
s(X) -> n__s(X)
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
add(s(X),Y) -> s(n__add(activate(X),Y))
dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
All above mentioned rules can be savely removed.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
s(X) -> n__s(X)
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [2] x1 + [0]
p(add) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(dbl) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__dbl) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__s) = [0]
p(n__sqr) = [1] x1 + [0]
p(n__terms) = [1] x1 + [0]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [0]
p(sqr) = [1] x1 + [11]
p(terms) = [1] x1 + [1]
Following rules are strictly oriented:
sqr(X) = [1] X + [11]
> [1] X + [0]
= n__sqr(X)
sqr(0()) = [11]
> [0]
= 0()
terms(X) = [1] X + [1]
> [1] X + [0]
= n__terms(X)
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [0]
>= [1] X + [0]
= X
activate(n__add(X1,X2)) = [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
= add(activate(X1),activate(X2))
activate(n__dbl(X)) = [2] X + [0]
>= [2] X + [0]
= dbl(activate(X))
activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [0]
>= [0]
= s(X)
activate(n__sqr(X)) = [2] X + [0]
>= [2] X + [11]
= sqr(activate(X))
activate(n__terms(X)) = [2] X + [0]
>= [2] X + [1]
= terms(activate(X))
add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__add(X1,X2)
add(0(),X) = [1] X + [0]
>= [1] X + [0]
= X
dbl(X) = [1] X + [0]
>= [1] X + [0]
= n__dbl(X)
dbl(0()) = [0]
>= [0]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
s(X) = [0]
>= [0]
= n__s(X)
terms(N) = [1] N + [1]
>= [1] N + [11]
= cons(recip(sqr(N)),n__terms(n__s(N)))
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^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
s(X) -> n__s(X)
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
- Weak TRS:
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [8] x1 + [6]
p(add) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(dbl) = [1] x1 + [3]
p(first) = [1] x1 + [1] x2 + [0]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__dbl) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [2]
p(n__s) = [2]
p(n__sqr) = [1] x1 + [0]
p(n__terms) = [1] x1 + [3]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [0]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [3]
Following rules are strictly oriented:
activate(X) = [8] X + [6]
> [1] X + [0]
= X
activate(n__first(X1,X2)) = [8] X1 + [8] X2 + [22]
> [8] X1 + [8] X2 + [12]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [22]
> [0]
= s(X)
activate(n__terms(X)) = [8] X + [30]
> [8] X + [9]
= terms(activate(X))
dbl(X) = [1] X + [3]
> [1] X + [0]
= n__dbl(X)
dbl(0()) = [3]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
activate(n__add(X1,X2)) = [8] X1 + [8] X2 + [6]
>= [8] X1 + [8] X2 + [12]
= add(activate(X1),activate(X2))
activate(n__dbl(X)) = [8] X + [6]
>= [8] X + [9]
= dbl(activate(X))
activate(n__sqr(X)) = [8] X + [6]
>= [8] X + [6]
= sqr(activate(X))
add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__add(X1,X2)
add(0(),X) = [1] X + [0]
>= [1] X + [0]
= X
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [2]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
s(X) = [0]
>= [2]
= n__s(X)
sqr(X) = [1] X + [0]
>= [1] X + [0]
= n__sqr(X)
sqr(0()) = [0]
>= [0]
= 0()
terms(N) = [1] N + [3]
>= [1] N + [5]
= cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) = [1] X + [3]
>= [1] X + [3]
= n__terms(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^1))
+ Considered Problem:
- Strict TRS:
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__sqr(X)) -> sqr(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
s(X) -> n__s(X)
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
- Weak TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__terms(X)) -> terms(activate(X))
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [2] x1 + [0]
p(add) = [1] x1 + [1] x2 + [7]
p(cons) = [1] x1 + [0]
p(dbl) = [1] x1 + [2]
p(first) = [1] x1 + [1] x2 + [0]
p(n__add) = [1] x1 + [1] x2 + [4]
p(n__dbl) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [4]
p(n__s) = [0]
p(n__sqr) = [1] x1 + [0]
p(n__terms) = [1] x1 + [0]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [0]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
activate(n__add(X1,X2)) = [2] X1 + [2] X2 + [8]
> [2] X1 + [2] X2 + [7]
= add(activate(X1),activate(X2))
add(X1,X2) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [4]
= n__add(X1,X2)
add(0(),X) = [1] X + [9]
> [1] X + [0]
= X
first(0(),X) = [1] X + [2]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [0]
>= [1] X + [0]
= X
activate(n__dbl(X)) = [2] X + [0]
>= [2] X + [2]
= dbl(activate(X))
activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [8]
>= [2] X1 + [2] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [0]
>= [0]
= s(X)
activate(n__sqr(X)) = [2] X + [0]
>= [2] X + [0]
= sqr(activate(X))
activate(n__terms(X)) = [2] X + [0]
>= [2] X + [0]
= terms(activate(X))
dbl(X) = [1] X + [2]
>= [1] X + [0]
= n__dbl(X)
dbl(0()) = [4]
>= [2]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [4]
= n__first(X1,X2)
s(X) = [0]
>= [0]
= n__s(X)
sqr(X) = [1] X + [0]
>= [1] X + [0]
= n__sqr(X)
sqr(0()) = [2]
>= [2]
= 0()
terms(N) = [1] N + [0]
>= [1] N + [0]
= cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) = [1] X + [0]
>= [1] X + [0]
= n__terms(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^1))
+ Considered Problem:
- Strict TRS:
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__sqr(X)) -> sqr(activate(X))
first(X1,X2) -> n__first(X1,X2)
s(X) -> n__s(X)
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(0(),X) -> nil()
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(activate) = [4] x1 + [0]
p(add) = [1] x1 + [1] x2 + [6]
p(cons) = [1] x1 + [1] x2 + [0]
p(dbl) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(n__add) = [1] x1 + [1] x2 + [3]
p(n__dbl) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [2]
p(n__s) = [1]
p(n__sqr) = [1] x1 + [1]
p(n__terms) = [1] x1 + [0]
p(nil) = [1]
p(recip) = [1] x1 + [0]
p(s) = [1]
p(sqr) = [1] x1 + [1]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
activate(n__sqr(X)) = [4] X + [4]
> [4] X + [1]
= sqr(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [4] X + [0]
>= [1] X + [0]
= X
activate(n__add(X1,X2)) = [4] X1 + [4] X2 + [12]
>= [4] X1 + [4] X2 + [6]
= add(activate(X1),activate(X2))
activate(n__dbl(X)) = [4] X + [0]
>= [4] X + [0]
= dbl(activate(X))
activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [8]
>= [4] X1 + [4] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [4]
>= [1]
= s(X)
activate(n__terms(X)) = [4] X + [0]
>= [4] X + [0]
= terms(activate(X))
add(X1,X2) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [3]
= n__add(X1,X2)
add(0(),X) = [1] X + [7]
>= [1] X + [0]
= X
dbl(X) = [1] X + [0]
>= [1] X + [0]
= n__dbl(X)
dbl(0()) = [1]
>= [1]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [2]
= n__first(X1,X2)
first(0(),X) = [1] X + [1]
>= [1]
= nil()
s(X) = [1]
>= [1]
= n__s(X)
sqr(X) = [1] X + [1]
>= [1] X + [1]
= n__sqr(X)
sqr(0()) = [2]
>= [1]
= 0()
terms(N) = [1] N + [0]
>= [1] N + [2]
= cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) = [1] X + [0]
>= [1] X + [0]
= n__terms(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^1))
+ Considered Problem:
- Strict TRS:
activate(n__dbl(X)) -> dbl(activate(X))
first(X1,X2) -> n__first(X1,X2)
s(X) -> n__s(X)
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(0(),X) -> nil()
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [4] x1 + [0]
p(add) = [1] x1 + [1] x2 + [1]
p(cons) = [1] x1 + [0]
p(dbl) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(n__add) = [1] x1 + [1] x2 + [1]
p(n__dbl) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(n__sqr) = [1] x1 + [0]
p(n__terms) = [1] x1 + [2]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [4] x1 + [0]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [3]
Following rules are strictly oriented:
terms(N) = [1] N + [3]
> [1] N + [0]
= cons(recip(sqr(N)),n__terms(n__s(N)))
Following rules are (at-least) weakly oriented:
activate(X) = [4] X + [0]
>= [1] X + [0]
= X
activate(n__add(X1,X2)) = [4] X1 + [4] X2 + [4]
>= [4] X1 + [4] X2 + [1]
= add(activate(X1),activate(X2))
activate(n__dbl(X)) = [4] X + [0]
>= [4] X + [0]
= dbl(activate(X))
activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [4] X + [0]
>= [4] X + [0]
= s(X)
activate(n__sqr(X)) = [4] X + [0]
>= [4] X + [0]
= sqr(activate(X))
activate(n__terms(X)) = [4] X + [8]
>= [4] X + [3]
= terms(activate(X))
add(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__add(X1,X2)
add(0(),X) = [1] X + [1]
>= [1] X + [0]
= X
dbl(X) = [1] X + [0]
>= [1] X + [0]
= n__dbl(X)
dbl(0()) = [0]
>= [0]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
s(X) = [4] X + [0]
>= [1] X + [0]
= n__s(X)
sqr(X) = [1] X + [0]
>= [1] X + [0]
= n__sqr(X)
sqr(0()) = [0]
>= [0]
= 0()
terms(X) = [1] X + [3]
>= [1] X + [2]
= n__terms(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(n__dbl(X)) -> dbl(activate(X))
first(X1,X2) -> n__first(X1,X2)
s(X) -> n__s(X)
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(0(),X) -> nil()
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [4] x1 + [0]
p(add) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [4]
p(dbl) = [1] x1 + [1]
p(first) = [1] x1 + [1] x2 + [0]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__dbl) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__s) = [2]
p(n__sqr) = [1] x1 + [0]
p(n__terms) = [1] x1 + [1]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [3]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [4]
Following rules are strictly oriented:
s(X) = [3]
> [2]
= n__s(X)
Following rules are (at-least) weakly oriented:
activate(X) = [4] X + [0]
>= [1] X + [0]
= X
activate(n__add(X1,X2)) = [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]
= add(activate(X1),activate(X2))
activate(n__dbl(X)) = [4] X + [0]
>= [4] X + [1]
= dbl(activate(X))
activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [8]
>= [3]
= s(X)
activate(n__sqr(X)) = [4] X + [0]
>= [4] X + [0]
= sqr(activate(X))
activate(n__terms(X)) = [4] X + [4]
>= [4] X + [4]
= terms(activate(X))
add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__add(X1,X2)
add(0(),X) = [1] X + [0]
>= [1] X + [0]
= X
dbl(X) = [1] X + [1]
>= [1] X + [0]
= n__dbl(X)
dbl(0()) = [1]
>= [0]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
sqr(X) = [1] X + [0]
>= [1] X + [0]
= n__sqr(X)
sqr(0()) = [0]
>= [0]
= 0()
terms(N) = [1] N + [4]
>= [1] N + [4]
= cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) = [1] X + [4]
>= [1] X + [1]
= n__terms(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(n__dbl(X)) -> dbl(activate(X))
first(X1,X2) -> n__first(X1,X2)
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(0(),X) -> nil()
s(X) -> n__s(X)
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [5] x1 + [0]
p(add) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(dbl) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [5]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__dbl) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [1]
p(n__s) = [0]
p(n__sqr) = [1] x1 + [0]
p(n__terms) = [1] x1 + [0]
p(nil) = [5]
p(recip) = [1] x1 + [0]
p(s) = [0]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
first(X1,X2) = [1] X1 + [1] X2 + [5]
> [1] X1 + [1] X2 + [1]
= n__first(X1,X2)
Following rules are (at-least) weakly oriented:
activate(X) = [5] X + [0]
>= [1] X + [0]
= X
activate(n__add(X1,X2)) = [5] X1 + [5] X2 + [0]
>= [5] X1 + [5] X2 + [0]
= add(activate(X1),activate(X2))
activate(n__dbl(X)) = [5] X + [0]
>= [5] X + [0]
= dbl(activate(X))
activate(n__first(X1,X2)) = [5] X1 + [5] X2 + [5]
>= [5] X1 + [5] X2 + [5]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [0]
>= [0]
= s(X)
activate(n__sqr(X)) = [5] X + [0]
>= [5] X + [0]
= sqr(activate(X))
activate(n__terms(X)) = [5] X + [0]
>= [5] X + [0]
= terms(activate(X))
add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__add(X1,X2)
add(0(),X) = [1] X + [0]
>= [1] X + [0]
= X
dbl(X) = [1] X + [0]
>= [1] X + [0]
= n__dbl(X)
dbl(0()) = [0]
>= [0]
= 0()
first(0(),X) = [1] X + [5]
>= [5]
= nil()
s(X) = [0]
>= [0]
= n__s(X)
sqr(X) = [1] X + [0]
>= [1] X + [0]
= n__sqr(X)
sqr(0()) = [0]
>= [0]
= 0()
terms(N) = [1] N + [0]
>= [1] N + [0]
= cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) = [1] X + [0]
>= [1] X + [0]
= n__terms(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(n__dbl(X)) -> dbl(activate(X))
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
s(X) -> n__s(X)
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ 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(add) = {1,2},
uargs(cons) = {1},
uargs(dbl) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(sqr) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [4] x1 + [0]
p(add) = [1] x1 + [1] x2 + [5]
p(cons) = [1] x1 + [1] x2 + [0]
p(dbl) = [1] x1 + [2]
p(first) = [1] x1 + [1] x2 + [1]
p(n__add) = [1] x1 + [1] x2 + [2]
p(n__dbl) = [1] x1 + [2]
p(n__first) = [1] x1 + [1] x2 + [1]
p(n__s) = [1]
p(n__sqr) = [1] x1 + [0]
p(n__terms) = [1] x1 + [1]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [4]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [2]
Following rules are strictly oriented:
activate(n__dbl(X)) = [4] X + [8]
> [4] X + [2]
= dbl(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [4] X + [0]
>= [1] X + [0]
= X
activate(n__add(X1,X2)) = [4] X1 + [4] X2 + [8]
>= [4] X1 + [4] X2 + [5]
= add(activate(X1),activate(X2))
activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [4]
>= [4] X1 + [4] X2 + [1]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [4]
>= [4]
= s(X)
activate(n__sqr(X)) = [4] X + [0]
>= [4] X + [0]
= sqr(activate(X))
activate(n__terms(X)) = [4] X + [4]
>= [4] X + [2]
= terms(activate(X))
add(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [2]
= n__add(X1,X2)
add(0(),X) = [1] X + [5]
>= [1] X + [0]
= X
dbl(X) = [1] X + [2]
>= [1] X + [2]
= n__dbl(X)
dbl(0()) = [2]
>= [0]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__first(X1,X2)
first(0(),X) = [1] X + [1]
>= [0]
= nil()
s(X) = [4]
>= [1]
= n__s(X)
sqr(X) = [1] X + [0]
>= [1] X + [0]
= n__sqr(X)
sqr(0()) = [0]
>= [0]
= 0()
terms(N) = [1] N + [2]
>= [1] N + [2]
= cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) = [1] X + [2]
>= [1] X + [1]
= n__terms(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:10: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
dbl(X) -> n__dbl(X)
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
s(X) -> n__s(X)
sqr(X) -> n__sqr(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
,n__sqr/1,n__terms/1,nil/0,recip/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))