* Step 1: Sum WORST_CASE(Omega(n^1),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:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ 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:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(x){x -> adx(x)} =
mark(adx(x)) ->^+ a__adx(mark(x))
= C[mark(x) = mark(x){}]
** Step 1.b: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 1.b: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 1.b: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 1.b: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 1.b: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 1.b: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 1.b: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 1.b: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 1.b: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 1.b: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(Omega(n^1),O(n^3))