```* Step 1: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ 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__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__adx) = [1] x1 + [1]
p(a__hd) = [1] x1 + [0]
p(a__incr) = [1] x1 + [0]
p(a__nats) = [0]
p(a__tl) = [1] x1 + [0]
p(a__zeros) = [0]
p(adx) = [0]
p(cons) = [0]
p(hd) = [1] x1 + [0]
p(incr) = [1] x1 + [0]
p(mark) = [0]
p(nats) = [0]
p(s) = [0]
p(tl) = [1] x1 + [0]
p(zeros) = [0]

Following rules are strictly oriented:
a__adx(X) = [1] X + [1]
> [0]
= adx(X)

a__adx(cons(X,Y)) = [1]
> [0]
= a__incr(cons(X,adx(Y)))

Following rules are (at-least) weakly oriented:
a__hd(X) =  [1] X + [0]
>= [1] X + [0]
=  hd(X)

a__hd(cons(X,Y)) =  [0]
>= [0]
=  mark(X)

a__incr(X) =  [1] X + [0]
>= [1] X + [0]
=  incr(X)

a__incr(cons(X,Y)) =  [0]
>= [0]
=  cons(s(X),incr(Y))

a__nats() =  [0]
>= [1]
=  a__adx(a__zeros())

a__nats() =  [0]
>= [0]
=  nats()

a__tl(X) =  [1] X + [0]
>= [1] X + [0]
=  tl(X)

a__tl(cons(X,Y)) =  [0]
>= [0]
=  mark(Y)

a__zeros() =  [0]
>= [0]
=  cons(0(),zeros())

a__zeros() =  [0]
>= [0]
=  zeros()

mark(0()) =  [0]
>= [0]
=  0()

mark(adx(X)) =  [0]
>= [1]
=  a__adx(mark(X))

mark(cons(X1,X2)) =  [0]
>= [0]
=  cons(X1,X2)

mark(hd(X)) =  [0]
>= [0]
=  a__hd(mark(X))

mark(incr(X)) =  [0]
>= [0]
=  a__incr(mark(X))

mark(nats()) =  [0]
>= [0]
=  a__nats()

mark(s(X)) =  [0]
>= [0]
=  s(X)

mark(tl(X)) =  [0]
>= [0]
=  a__tl(mark(X))

mark(zeros()) =  [0]
>= [0]
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ 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__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__adx) = [1] x1 + [15]
p(a__hd) = [1] x1 + [0]
p(a__incr) = [1] x1 + [0]
p(a__nats) = [0]
p(a__tl) = [1] x1 + [0]
p(a__zeros) = [0]
p(adx) = [1] x1 + [15]
p(cons) = [1] x1 + [1] x2 + [9]
p(hd) = [1] x1 + [0]
p(incr) = [1] x1 + [0]
p(mark) = [1] x1 + [0]
p(nats) = [0]
p(s) = [1] x1 + [0]
p(tl) = [1] x1 + [0]
p(zeros) = [9]

Following rules are strictly oriented:
a__hd(cons(X,Y)) = [1] X + [1] Y + [9]
> [1] X + [0]
= mark(X)

a__tl(cons(X,Y)) = [1] X + [1] Y + [9]
> [1] Y + [0]
= mark(Y)

mark(zeros()) = [9]
> [0]
= a__zeros()

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1] X + [15]
>= [1] X + [15]
=  adx(X)

a__adx(cons(X,Y)) =  [1] X + [1] Y + [24]
>= [1] X + [1] Y + [24]
=  a__incr(cons(X,adx(Y)))

a__hd(X) =  [1] X + [0]
>= [1] X + [0]
=  hd(X)

a__incr(X) =  [1] X + [0]
>= [1] X + [0]
=  incr(X)

a__incr(cons(X,Y)) =  [1] X + [1] Y + [9]
>= [1] X + [1] Y + [9]
=  cons(s(X),incr(Y))

a__nats() =  [0]
>= [15]
=  a__adx(a__zeros())

a__nats() =  [0]
>= [0]
=  nats()

a__tl(X) =  [1] X + [0]
>= [1] X + [0]
=  tl(X)

a__zeros() =  [0]
>= [18]
=  cons(0(),zeros())

a__zeros() =  [0]
>= [9]
=  zeros()

mark(0()) =  [0]
>= [0]
=  0()

mark(adx(X)) =  [1] X + [15]
>= [1] X + [15]
=  a__adx(mark(X))

mark(cons(X1,X2)) =  [1] X1 + [1] X2 + [9]
>= [1] X1 + [1] X2 + [9]
=  cons(X1,X2)

mark(hd(X)) =  [1] X + [0]
>= [1] X + [0]
=  a__hd(mark(X))

mark(incr(X)) =  [1] X + [0]
>= [1] X + [0]
=  a__incr(mark(X))

mark(nats()) =  [0]
>= [0]
=  a__nats()

mark(s(X)) =  [1] X + [0]
>= [1] X + [0]
=  s(X)

mark(tl(X)) =  [1] X + [0]
>= [1] X + [0]
=  a__tl(mark(X))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__hd(X) -> hd(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(cons(X,Y)) -> mark(X)
a__tl(cons(X,Y)) -> mark(Y)
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ 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__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__adx) = [1] x1 + [0]
p(a__hd) = [1] x1 + [6]
p(a__incr) = [1] x1 + [0]
p(a__nats) = [0]
p(a__tl) = [1] x1 + [0]
p(a__zeros) = [0]
p(adx) = [1] x1 + [0]
p(cons) = [0]
p(hd) = [0]
p(incr) = [0]
p(mark) = [0]
p(nats) = [0]
p(s) = [0]
p(tl) = [1] x1 + [0]
p(zeros) = [0]

Following rules are strictly oriented:
a__hd(X) = [1] X + [6]
> [0]
= hd(X)

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1] X + [0]
>= [1] X + [0]
=  adx(X)

a__adx(cons(X,Y)) =  [0]
>= [0]
=  a__incr(cons(X,adx(Y)))

a__hd(cons(X,Y)) =  [6]
>= [0]
=  mark(X)

a__incr(X) =  [1] X + [0]
>= [0]
=  incr(X)

a__incr(cons(X,Y)) =  [0]
>= [0]
=  cons(s(X),incr(Y))

a__nats() =  [0]
>= [0]
=  a__adx(a__zeros())

a__nats() =  [0]
>= [0]
=  nats()

a__tl(X) =  [1] X + [0]
>= [1] X + [0]
=  tl(X)

a__tl(cons(X,Y)) =  [0]
>= [0]
=  mark(Y)

a__zeros() =  [0]
>= [0]
=  cons(0(),zeros())

a__zeros() =  [0]
>= [0]
=  zeros()

mark(0()) =  [0]
>= [0]
=  0()

mark(adx(X)) =  [0]
>= [0]
=  a__adx(mark(X))

mark(cons(X1,X2)) =  [0]
>= [0]
=  cons(X1,X2)

mark(hd(X)) =  [0]
>= [6]
=  a__hd(mark(X))

mark(incr(X)) =  [0]
>= [0]
=  a__incr(mark(X))

mark(nats()) =  [0]
>= [0]
=  a__nats()

mark(s(X)) =  [0]
>= [0]
=  s(X)

mark(tl(X)) =  [0]
>= [0]
=  a__tl(mark(X))

mark(zeros()) =  [0]
>= [0]
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__tl(cons(X,Y)) -> mark(Y)
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ 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__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__adx) = [1] x1 + [1]
p(a__hd) = [1] x1 + [0]
p(a__incr) = [1] x1 + [1]
p(a__nats) = [0]
p(a__tl) = [1] x1 + [0]
p(a__zeros) = [1]
p(adx) = [0]
p(cons) = [1]
p(hd) = [0]
p(incr) = [1] x1 + [0]
p(mark) = [1]
p(nats) = [0]
p(s) = [0]
p(tl) = [1] x1 + [0]
p(zeros) = [1]

Following rules are strictly oriented:
a__incr(X) = [1] X + [1]
> [1] X + [0]
= incr(X)

a__incr(cons(X,Y)) = [2]
> [1]
= cons(s(X),incr(Y))

mark(0()) = [1]
> [0]
= 0()

mark(nats()) = [1]
> [0]
= a__nats()

mark(s(X)) = [1]
> [0]
= s(X)

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1] X + [1]
>= [0]
=  adx(X)

a__adx(cons(X,Y)) =  [2]
>= [2]
=  a__incr(cons(X,adx(Y)))

a__hd(X) =  [1] X + [0]
>= [0]
=  hd(X)

a__hd(cons(X,Y)) =  [1]
>= [1]
=  mark(X)

a__nats() =  [0]
>= [2]
=  a__adx(a__zeros())

a__nats() =  [0]
>= [0]
=  nats()

a__tl(X) =  [1] X + [0]
>= [1] X + [0]
=  tl(X)

a__tl(cons(X,Y)) =  [1]
>= [1]
=  mark(Y)

a__zeros() =  [1]
>= [1]
=  cons(0(),zeros())

a__zeros() =  [1]
>= [1]
=  zeros()

mark(adx(X)) =  [1]
>= [2]
=  a__adx(mark(X))

mark(cons(X1,X2)) =  [1]
>= [1]
=  cons(X1,X2)

mark(hd(X)) =  [1]
>= [1]
=  a__hd(mark(X))

mark(incr(X)) =  [1]
>= [2]
=  a__incr(mark(X))

mark(tl(X)) =  [1]
>= [1]
=  a__tl(mark(X))

mark(zeros()) =  [1]
>= [1]
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(tl(X)) -> a__tl(mark(X))
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__tl(cons(X,Y)) -> mark(Y)
mark(0()) -> 0()
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ 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__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(a__adx) = [1] x1 + [0]
p(a__hd) = [1] x1 + [0]
p(a__incr) = [1] x1 + [0]
p(a__nats) = [0]
p(a__tl) = [1] x1 + [0]
p(a__zeros) = [0]
p(adx) = [1] x1 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(hd) = [1] x1 + [0]
p(incr) = [1] x1 + [0]
p(mark) = [1] x1 + [0]
p(nats) = [0]
p(s) = [1] x1 + [0]
p(tl) = [1] x1 + [1]
p(zeros) = [0]

Following rules are strictly oriented:
mark(tl(X)) = [1] X + [1]
> [1] X + [0]
= a__tl(mark(X))

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1] X + [0]
>= [1] X + [0]
=  adx(X)

a__adx(cons(X,Y)) =  [1] X + [1] Y + [0]
>= [1] X + [1] Y + [0]
=  a__incr(cons(X,adx(Y)))

a__hd(X) =  [1] X + [0]
>= [1] X + [0]
=  hd(X)

a__hd(cons(X,Y)) =  [1] X + [1] Y + [0]
>= [1] X + [0]
=  mark(X)

a__incr(X) =  [1] X + [0]
>= [1] X + [0]
=  incr(X)

a__incr(cons(X,Y)) =  [1] X + [1] Y + [0]
>= [1] X + [1] Y + [0]
=  cons(s(X),incr(Y))

a__nats() =  [0]
>= [0]
=  a__adx(a__zeros())

a__nats() =  [0]
>= [0]
=  nats()

a__tl(X) =  [1] X + [0]
>= [1] X + [1]
=  tl(X)

a__tl(cons(X,Y)) =  [1] X + [1] Y + [0]
>= [1] Y + [0]
=  mark(Y)

a__zeros() =  [0]
>= [5]
=  cons(0(),zeros())

a__zeros() =  [0]
>= [0]
=  zeros()

mark(0()) =  [5]
>= [5]
=  0()

mark(adx(X)) =  [1] X + [0]
>= [1] X + [0]
=  a__adx(mark(X))

mark(cons(X1,X2)) =  [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
=  cons(X1,X2)

mark(hd(X)) =  [1] X + [0]
>= [1] X + [0]
=  a__hd(mark(X))

mark(incr(X)) =  [1] X + [0]
>= [1] X + [0]
=  a__incr(mark(X))

mark(nats()) =  [0]
>= [0]
=  a__nats()

mark(s(X)) =  [1] X + [0]
>= [1] X + [0]
=  s(X)

mark(zeros()) =  [0]
>= [0]
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__tl(cons(X,Y)) -> mark(Y)
mark(0()) -> 0()
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ 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__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__adx) = [1] x1 + [0]
p(a__hd) = [1] x1 + [4]
p(a__incr) = [1] x1 + [0]
p(a__nats) = [5]
p(a__tl) = [1] x1 + [4]
p(a__zeros) = [3]
p(adx) = [1] x1 + [0]
p(cons) = [1] x1 + [1] x2 + [1]
p(hd) = [1] x1 + [4]
p(incr) = [1] x1 + [0]
p(mark) = [1] x1 + [5]
p(nats) = [0]
p(s) = [1] x1 + [0]
p(tl) = [1] x1 + [4]
p(zeros) = [0]

Following rules are strictly oriented:
a__nats() = [5]
> [3]
= a__adx(a__zeros())

a__nats() = [5]
> [0]
= nats()

a__zeros() = [3]
> [1]
= cons(0(),zeros())

a__zeros() = [3]
> [0]
= zeros()

mark(cons(X1,X2)) = [1] X1 + [1] X2 + [6]
> [1] X1 + [1] X2 + [1]
= cons(X1,X2)

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1] X + [0]
>= [1] X + [0]
=  adx(X)

a__adx(cons(X,Y)) =  [1] X + [1] Y + [1]
>= [1] X + [1] Y + [1]
=  a__incr(cons(X,adx(Y)))

a__hd(X) =  [1] X + [4]
>= [1] X + [4]
=  hd(X)

a__hd(cons(X,Y)) =  [1] X + [1] Y + [5]
>= [1] X + [5]
=  mark(X)

a__incr(X) =  [1] X + [0]
>= [1] X + [0]
=  incr(X)

a__incr(cons(X,Y)) =  [1] X + [1] Y + [1]
>= [1] X + [1] Y + [1]
=  cons(s(X),incr(Y))

a__tl(X) =  [1] X + [4]
>= [1] X + [4]
=  tl(X)

a__tl(cons(X,Y)) =  [1] X + [1] Y + [5]
>= [1] Y + [5]
=  mark(Y)

mark(0()) =  [5]
>= [0]
=  0()

mark(adx(X)) =  [1] X + [5]
>= [1] X + [5]
=  a__adx(mark(X))

mark(hd(X)) =  [1] X + [9]
>= [1] X + [9]
=  a__hd(mark(X))

mark(incr(X)) =  [1] X + [5]
>= [1] X + [5]
=  a__incr(mark(X))

mark(nats()) =  [5]
>= [5]
=  a__nats()

mark(s(X)) =  [1] X + [5]
>= [1] X + [0]
=  s(X)

mark(tl(X)) =  [1] X + [9]
>= [1] X + [9]
=  a__tl(mark(X))

mark(zeros()) =  [5]
>= [3]
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__tl(X) -> tl(X)
mark(adx(X)) -> a__adx(mark(X))
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 1))), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 1))):

The following argument positions are considered usable:
uargs(a__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
{a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__adx) = [1 0] x_1 + [0]
[0 1]       [0]
p(a__hd) = [1 0] x_1 + [7]
[0 1]       [4]
p(a__incr) = [1 0] x_1 + [0]
[0 1]       [0]
p(a__nats) = [1]
[3]
p(a__tl) = [1 0] x_1 + [3]
[0 1]       [2]
p(a__zeros) = [1]
[0]
p(adx) = [0 0] x_1 + [0]
[0 1]       [0]
p(cons) = [0 2] x_1 + [0 2] x_2 + [0]
[0 1]       [0 1]       [0]
p(hd) = [0 0] x_1 + [0]
[0 1]       [4]
p(incr) = [0 0] x_1 + [0]
[0 1]       [0]
p(mark) = [0 2] x_1 + [1]
[0 1]       [0]
p(nats) = [1]
[3]
p(s) = [1]
[0]
p(tl) = [0 0] x_1 + [0]
[0 1]       [2]
p(zeros) = [0]
[0]

Following rules are strictly oriented:
a__tl(X) = [1 0] X + [3]
[0 1]     [2]
> [0 0] X + [0]
[0 1]     [2]
= tl(X)

mark(hd(X)) = [0 2] X + [9]
[0 1]     [4]
> [0 2] X + [8]
[0 1]     [4]
= a__hd(mark(X))

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1 0] X + [0]
[0 1]     [0]
>= [0 0] X + [0]
[0 1]     [0]
=  adx(X)

a__adx(cons(X,Y)) =  [0 2] X + [0 2] Y + [0]
[0 1]     [0 1]     [0]
>= [0 2] X + [0 2] Y + [0]
[0 1]     [0 1]     [0]
=  a__incr(cons(X,adx(Y)))

a__hd(X) =  [1 0] X + [7]
[0 1]     [4]
>= [0 0] X + [0]
[0 1]     [4]
=  hd(X)

a__hd(cons(X,Y)) =  [0 2] X + [0 2] Y + [7]
[0 1]     [0 1]     [4]
>= [0 2] X + [1]
[0 1]     [0]
=  mark(X)

a__incr(X) =  [1 0] X + [0]
[0 1]     [0]
>= [0 0] X + [0]
[0 1]     [0]
=  incr(X)

a__incr(cons(X,Y)) =  [0 2] X + [0 2] Y + [0]
[0 1]     [0 1]     [0]
>= [0 2] Y + [0]
[0 1]     [0]
=  cons(s(X),incr(Y))

a__nats() =  [1]
[3]
>= [1]
[0]
=  a__adx(a__zeros())

a__nats() =  [1]
[3]
>= [1]
[3]
=  nats()

a__tl(cons(X,Y)) =  [0 2] X + [0 2] Y + [3]
[0 1]     [0 1]     [2]
>= [0 2] Y + [1]
[0 1]     [0]
=  mark(Y)

a__zeros() =  [1]
[0]
>= [0]
[0]
=  cons(0(),zeros())

a__zeros() =  [1]
[0]
>= [0]
[0]
=  zeros()

mark(0()) =  [1]
[0]
>= [0]
[0]
=  0()

mark(adx(X)) =  [0 2] X + [1]
[0 1]     [0]
>= [0 2] X + [1]
[0 1]     [0]
=  a__adx(mark(X))

mark(cons(X1,X2)) =  [0 2] X1 + [0 2] X2 + [1]
[0 1]      [0 1]      [0]
>= [0 2] X1 + [0 2] X2 + [0]
[0 1]      [0 1]      [0]
=  cons(X1,X2)

mark(incr(X)) =  [0 2] X + [1]
[0 1]     [0]
>= [0 2] X + [1]
[0 1]     [0]
=  a__incr(mark(X))

mark(nats()) =  [7]
[3]
>= [1]
[3]
=  a__nats()

mark(s(X)) =  [1]
[0]
>= [1]
[0]
=  s(X)

mark(tl(X)) =  [0 2] X + [5]
[0 1]     [2]
>= [0 2] X + [4]
[0 1]     [2]
=  a__tl(mark(X))

mark(zeros()) =  [1]
[0]
>= [1]
[0]
=  a__zeros()

* Step 8: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
mark(adx(X)) -> a__adx(mark(X))
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ 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__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
{a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__adx) = [1 0] x_1 + [0]
[0 1]       [2]
p(a__hd) = [1 4] x_1 + [6]
[0 1]       [2]
p(a__incr) = [1 0] x_1 + [0]
[0 1]       [0]
p(a__nats) = [0]
[2]
p(a__tl) = [1 4] x_1 + [2]
[0 1]       [2]
p(a__zeros) = [0]
[0]
p(adx) = [1 0] x_1 + [0]
[0 1]       [2]
p(cons) = [1 0] x_1 + [1 0] x_2 + [0]
[0 1]       [0 1]       [0]
p(hd) = [1 4] x_1 + [4]
[0 1]       [2]
p(incr) = [1 0] x_1 + [0]
[0 1]       [0]
p(mark) = [1 4] x_1 + [1]
[0 1]       [0]
p(nats) = [0]
[2]
p(s) = [0 0] x_1 + [0]
[0 1]       [0]
p(tl) = [1 4] x_1 + [2]
[0 1]       [2]
p(zeros) = [0]
[0]

Following rules are strictly oriented:
mark(adx(X)) = [1 4] X + [9]
[0 1]     [2]
> [1 4] X + [1]
[0 1]     [2]
= a__adx(mark(X))

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1 0] X + [0]
[0 1]     [2]
>= [1 0] X + [0]
[0 1]     [2]
=  adx(X)

a__adx(cons(X,Y)) =  [1 0] X + [1 0] Y + [0]
[0 1]     [0 1]     [2]
>= [1 0] X + [1 0] Y + [0]
[0 1]     [0 1]     [2]
=  a__incr(cons(X,adx(Y)))

a__hd(X) =  [1 4] X + [6]
[0 1]     [2]
>= [1 4] X + [4]
[0 1]     [2]
=  hd(X)

a__hd(cons(X,Y)) =  [1 4] X + [1 4] Y + [6]
[0 1]     [0 1]     [2]
>= [1 4] X + [1]
[0 1]     [0]
=  mark(X)

a__incr(X) =  [1 0] X + [0]
[0 1]     [0]
>= [1 0] X + [0]
[0 1]     [0]
=  incr(X)

a__incr(cons(X,Y)) =  [1 0] X + [1 0] Y + [0]
[0 1]     [0 1]     [0]
>= [0 0] X + [1 0] Y + [0]
[0 1]     [0 1]     [0]
=  cons(s(X),incr(Y))

a__nats() =  [0]
[2]
>= [0]
[2]
=  a__adx(a__zeros())

a__nats() =  [0]
[2]
>= [0]
[2]
=  nats()

a__tl(X) =  [1 4] X + [2]
[0 1]     [2]
>= [1 4] X + [2]
[0 1]     [2]
=  tl(X)

a__tl(cons(X,Y)) =  [1 4] X + [1 4] Y + [2]
[0 1]     [0 1]     [2]
>= [1 4] Y + [1]
[0 1]     [0]
=  mark(Y)

a__zeros() =  [0]
[0]
>= [0]
[0]
=  cons(0(),zeros())

a__zeros() =  [0]
[0]
>= [0]
[0]
=  zeros()

mark(0()) =  [1]
[0]
>= [0]
[0]
=  0()

mark(cons(X1,X2)) =  [1 4] X1 + [1 4] X2 + [1]
[0 1]      [0 1]      [0]
>= [1 0] X1 + [1 0] X2 + [0]
[0 1]      [0 1]      [0]
=  cons(X1,X2)

mark(hd(X)) =  [1 8] X + [13]
[0 1]     [2]
>= [1 8] X + [7]
[0 1]     [2]
=  a__hd(mark(X))

mark(incr(X)) =  [1 4] X + [1]
[0 1]     [0]
>= [1 4] X + [1]
[0 1]     [0]
=  a__incr(mark(X))

mark(nats()) =  [9]
[2]
>= [0]
[2]
=  a__nats()

mark(s(X)) =  [0 4] X + [1]
[0 1]     [0]
>= [0 0] X + [0]
[0 1]     [0]
=  s(X)

mark(tl(X)) =  [1 8] X + [11]
[0 1]     [2]
>= [1 8] X + [3]
[0 1]     [2]
=  a__tl(mark(X))

mark(zeros()) =  [1]
[0]
>= [0]
[0]
=  a__zeros()

* Step 9: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):

The following argument positions are considered usable:
uargs(a__adx) = {1},
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
{a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
[0]
p(a__adx) = [1 0 0 2]       [2]
[0 1 0 2] x_1 + [2]
[0 0 1 1]       [3]
[0 0 0 1]       [2]
p(a__hd) = [1 0 1 0]       [0]
[0 1 1 0] x_1 + [1]
[0 0 1 2]       [3]
[0 0 0 1]       [0]
p(a__incr) = [1 0 0 1]       [0]
[0 1 0 1] x_1 + [0]
[0 0 1 0]       [1]
[0 0 0 1]       [0]
p(a__nats) = [2]
[3]
[3]
[2]
p(a__tl) = [1 0 1 0]       [2]
[0 1 1 0] x_1 + [2]
[0 0 1 2]       [3]
[0 0 0 1]       [0]
p(a__zeros) = [0]
[0]
[0]
[0]
p(adx) = [0 0 0 0]       [0]
[0 1 0 1] x_1 + [0]
[0 0 1 1]       [2]
[0 0 0 1]       [2]
p(cons) = [0 1 0 0]       [0 1 0 0]       [0]
[0 1 0 0] x_1 + [0 1 0 0] x_2 + [0]
[0 0 1 0]       [0 0 1 0]       [0]
[0 0 0 1]       [0 0 0 1]       [0]
p(hd) = [0 0 1 0]       [0]
[0 1 1 0] x_1 + [1]
[0 0 1 2]       [3]
[0 0 0 1]       [0]
p(incr) = [0 0 0 0]       [0]
[0 1 0 1] x_1 + [0]
[0 0 1 0]       [1]
[0 0 0 1]       [0]
p(mark) = [0 1 1 0]       [0]
[0 1 1 0] x_1 + [0]
[0 0 1 2]       [0]
[0 0 0 1]       [0]
p(nats) = [0]
[3]
[0]
[2]
p(s) = [0 0 0 0]       [0]
[0 0 0 0] x_1 + [0]
[0 0 0 0]       [0]
[0 0 0 1]       [0]
p(tl) = [0 0 0 0]       [2]
[0 1 1 0] x_1 + [1]
[0 0 1 2]       [3]
[0 0 0 1]       [0]
p(zeros) = [0]
[0]
[0]
[0]

Following rules are strictly oriented:
mark(incr(X)) = [0 1 1 1]     [1]
[0 1 1 1] X + [1]
[0 0 1 2]     [1]
[0 0 0 1]     [0]
> [0 1 1 1]     [0]
[0 1 1 1] X + [0]
[0 0 1 2]     [1]
[0 0 0 1]     [0]
= a__incr(mark(X))

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1 0 0 2]     [2]
[0 1 0 2] X + [2]
[0 0 1 1]     [3]
[0 0 0 1]     [2]
>= [0 0 0 0]     [0]
[0 1 0 1] X + [0]
[0 0 1 1]     [2]
[0 0 0 1]     [2]
=  adx(X)

a__adx(cons(X,Y)) =  [0 1 0 2]     [0 1 0 2]     [2]
[0 1 0 2] X + [0 1 0 2] Y + [2]
[0 0 1 1]     [0 0 1 1]     [3]
[0 0 0 1]     [0 0 0 1]     [2]
>= [0 1 0 1]     [0 1 0 2]     [2]
[0 1 0 1] X + [0 1 0 2] Y + [2]
[0 0 1 0]     [0 0 1 1]     [3]
[0 0 0 1]     [0 0 0 1]     [2]
=  a__incr(cons(X,adx(Y)))

a__hd(X) =  [1 0 1 0]     [0]
[0 1 1 0] X + [1]
[0 0 1 2]     [3]
[0 0 0 1]     [0]
>= [0 0 1 0]     [0]
[0 1 1 0] X + [1]
[0 0 1 2]     [3]
[0 0 0 1]     [0]
=  hd(X)

a__hd(cons(X,Y)) =  [0 1 1 0]     [0 1 1 0]     [0]
[0 1 1 0] X + [0 1 1 0] Y + [1]
[0 0 1 2]     [0 0 1 2]     [3]
[0 0 0 1]     [0 0 0 1]     [0]
>= [0 1 1 0]     [0]
[0 1 1 0] X + [0]
[0 0 1 2]     [0]
[0 0 0 1]     [0]
=  mark(X)

a__incr(X) =  [1 0 0 1]     [0]
[0 1 0 1] X + [0]
[0 0 1 0]     [1]
[0 0 0 1]     [0]
>= [0 0 0 0]     [0]
[0 1 0 1] X + [0]
[0 0 1 0]     [1]
[0 0 0 1]     [0]
=  incr(X)

a__incr(cons(X,Y)) =  [0 1 0 1]     [0 1 0 1]     [0]
[0 1 0 1] X + [0 1 0 1] Y + [0]
[0 0 1 0]     [0 0 1 0]     [1]
[0 0 0 1]     [0 0 0 1]     [0]
>= [0 0 0 0]     [0 1 0 1]     [0]
[0 0 0 0] X + [0 1 0 1] Y + [0]
[0 0 0 0]     [0 0 1 0]     [1]
[0 0 0 1]     [0 0 0 1]     [0]
=  cons(s(X),incr(Y))

a__nats() =  [2]
[3]
[3]
[2]
>= [2]
[2]
[3]
[2]
=  a__adx(a__zeros())

a__nats() =  [2]
[3]
[3]
[2]
>= [0]
[3]
[0]
[2]
=  nats()

a__tl(X) =  [1 0 1 0]     [2]
[0 1 1 0] X + [2]
[0 0 1 2]     [3]
[0 0 0 1]     [0]
>= [0 0 0 0]     [2]
[0 1 1 0] X + [1]
[0 0 1 2]     [3]
[0 0 0 1]     [0]
=  tl(X)

a__tl(cons(X,Y)) =  [0 1 1 0]     [0 1 1 0]     [2]
[0 1 1 0] X + [0 1 1 0] Y + [2]
[0 0 1 2]     [0 0 1 2]     [3]
[0 0 0 1]     [0 0 0 1]     [0]
>= [0 1 1 0]     [0]
[0 1 1 0] Y + [0]
[0 0 1 2]     [0]
[0 0 0 1]     [0]
=  mark(Y)

a__zeros() =  [0]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
=  cons(0(),zeros())

a__zeros() =  [0]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
=  zeros()

mark(0()) =  [0]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
=  0()

mark(adx(X)) =  [0 1 1 2]     [2]
[0 1 1 2] X + [2]
[0 0 1 3]     [6]
[0 0 0 1]     [2]
>= [0 1 1 2]     [2]
[0 1 1 2] X + [2]
[0 0 1 3]     [3]
[0 0 0 1]     [2]
=  a__adx(mark(X))

mark(cons(X1,X2)) =  [0 1 1 0]      [0 1 1 0]      [0]
[0 1 1 0] X1 + [0 1 1 0] X2 + [0]
[0 0 1 2]      [0 0 1 2]      [0]
[0 0 0 1]      [0 0 0 1]      [0]
>= [0 1 0 0]      [0 1 0 0]      [0]
[0 1 0 0] X1 + [0 1 0 0] X2 + [0]
[0 0 1 0]      [0 0 1 0]      [0]
[0 0 0 1]      [0 0 0 1]      [0]
=  cons(X1,X2)

mark(hd(X)) =  [0 1 2 2]     [4]
[0 1 2 2] X + [4]
[0 0 1 4]     [3]
[0 0 0 1]     [0]
>= [0 1 2 2]     [0]
[0 1 2 2] X + [1]
[0 0 1 4]     [3]
[0 0 0 1]     [0]
=  a__hd(mark(X))

mark(nats()) =  [3]
[3]
[4]
[2]
>= [2]
[3]
[3]
[2]
=  a__nats()

mark(s(X)) =  [0 0 0 0]     [0]
[0 0 0 0] X + [0]
[0 0 0 2]     [0]
[0 0 0 1]     [0]
>= [0 0 0 0]     [0]
[0 0 0 0] X + [0]
[0 0 0 0]     [0]
[0 0 0 1]     [0]
=  s(X)

mark(tl(X)) =  [0 1 2 2]     [4]
[0 1 2 2] X + [4]
[0 0 1 4]     [3]
[0 0 0 1]     [0]
>= [0 1 2 2]     [2]
[0 1 2 2] X + [2]
[0 0 1 4]     [3]
[0 0 0 1]     [0]
=  a__tl(mark(X))

mark(zeros()) =  [0]
[0]
[0]
[0]
>= [0]
[0]
[0]
[0]
=  a__zeros()

* Step 10: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__adx(X) -> adx(X)
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> a__adx(a__zeros())
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
{a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))
```