* Step 1: Sum WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ 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(::) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(::) = [1] x2 + [10]
p(append) = [1] x2 + [1]
p(append#1) = [1] x2 + [0]
p(leaf) = [0]
p(nil) = [0]
p(node) = [1] x2 + [1] x3 + [0]
p(subtrees) = [0]
p(subtrees#1) = [0]
p(subtrees#2) = [1] x1 + [0]
p(subtrees#3) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
append(@l1,@l2) = [1] @l2 + [1]
> [1] @l2 + [0]
= append#1(@l1,@l2)
Following rules are (at-least) weakly oriented:
append#1(::(@x,@xs),@l2) = [1] @l2 + [0]
>= [1] @l2 + [11]
= ::(@x,append(@xs,@l2))
append#1(nil(),@l2) = [1] @l2 + [0]
>= [1] @l2 + [0]
= @l2
subtrees(@t) = [0]
>= [0]
= subtrees#1(@t)
subtrees#1(leaf()) = [0]
>= [0]
= nil()
subtrees#1(node(@x,@t1,@t2)) = [0]
>= [0]
= subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [0]
>= [1] @l1 + [0]
= subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [0]
>= [1] @l2 + [11]
= ::(node(@x,@t1,@t2),append(@l1,@l2))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Weak TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ 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(::) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(::) = [1] x2 + [0]
p(append) = [1] x1 + [1] x2 + [0]
p(append#1) = [1] x1 + [1] x2 + [0]
p(leaf) = [0]
p(nil) = [0]
p(node) = [1] x1 + [1] x2 + [1] x3 + [0]
p(subtrees) = [0]
p(subtrees#1) = [0]
p(subtrees#2) = [1] x1 + [0]
p(subtrees#3) = [1] x1 + [1] x2 + [1]
Following rules are strictly oriented:
subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [1]
> [1] @l1 + [1] @l2 + [0]
= ::(node(@x,@t1,@t2),append(@l1,@l2))
Following rules are (at-least) weakly oriented:
append(@l1,@l2) = [1] @l1 + [1] @l2 + [0]
>= [1] @l1 + [1] @l2 + [0]
= append#1(@l1,@l2)
append#1(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [0]
>= [1] @l2 + [1] @xs + [0]
= ::(@x,append(@xs,@l2))
append#1(nil(),@l2) = [1] @l2 + [0]
>= [1] @l2 + [0]
= @l2
subtrees(@t) = [0]
>= [0]
= subtrees#1(@t)
subtrees#1(leaf()) = [0]
>= [0]
= nil()
subtrees#1(node(@x,@t1,@t2)) = [0]
>= [0]
= subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [0]
>= [1] @l1 + [1]
= subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
- Weak TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ 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(::) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(::) = [1] x2 + [0]
p(append) = [1] x2 + [0]
p(append#1) = [1] x2 + [0]
p(leaf) = [0]
p(nil) = [0]
p(node) = [1] x3 + [1]
p(subtrees) = [8]
p(subtrees#1) = [2]
p(subtrees#2) = [1] x1 + [8]
p(subtrees#3) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
subtrees(@t) = [8]
> [2]
= subtrees#1(@t)
subtrees#1(leaf()) = [2]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
append(@l1,@l2) = [1] @l2 + [0]
>= [1] @l2 + [0]
= append#1(@l1,@l2)
append#1(::(@x,@xs),@l2) = [1] @l2 + [0]
>= [1] @l2 + [0]
= ::(@x,append(@xs,@l2))
append#1(nil(),@l2) = [1] @l2 + [0]
>= [1] @l2 + [0]
= @l2
subtrees#1(node(@x,@t1,@t2)) = [2]
>= [16]
= subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [8]
>= [1] @l1 + [8]
= subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [0]
>= [1] @l2 + [0]
= ::(node(@x,@t1,@t2),append(@l1,@l2))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
- Weak TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ 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(::) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(::) = [1] x2 + [0]
p(append) = [1] x1 + [1] x2 + [0]
p(append#1) = [1] x1 + [1] x2 + [0]
p(leaf) = [0]
p(nil) = [0]
p(node) = [1] x2 + [1] x3 + [5]
p(subtrees) = [1] x1 + [0]
p(subtrees#1) = [1] x1 + [0]
p(subtrees#2) = [1] x1 + [1] x3 + [1]
p(subtrees#3) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
subtrees#1(node(@x,@t1,@t2)) = [1] @t1 + [1] @t2 + [5]
> [1] @t1 + [1] @t2 + [1]
= subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [1] @t2 + [1]
> [1] @l1 + [1] @t2 + [0]
= subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
Following rules are (at-least) weakly oriented:
append(@l1,@l2) = [1] @l1 + [1] @l2 + [0]
>= [1] @l1 + [1] @l2 + [0]
= append#1(@l1,@l2)
append#1(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [0]
>= [1] @l2 + [1] @xs + [0]
= ::(@x,append(@xs,@l2))
append#1(nil(),@l2) = [1] @l2 + [0]
>= [1] @l2 + [0]
= @l2
subtrees(@t) = [1] @t + [0]
>= [1] @t + [0]
= subtrees#1(@t)
subtrees#1(leaf()) = [0]
>= [0]
= nil()
subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [0]
>= [1] @l1 + [1] @l2 + [0]
= ::(node(@x,@t1,@t2),append(@l1,@l2))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
- Weak TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ 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(::) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(::) = [1] x2 + [6]
p(append) = [1] x1 + [1] x2 + [0]
p(append#1) = [1] x1 + [1] x2 + [0]
p(leaf) = [1]
p(nil) = [1]
p(node) = [1] x2 + [1] x3 + [6]
p(subtrees) = [1] x1 + [0]
p(subtrees#1) = [1] x1 + [0]
p(subtrees#2) = [1] x1 + [1] x3 + [6]
p(subtrees#3) = [1] x1 + [1] x2 + [6]
Following rules are strictly oriented:
append#1(nil(),@l2) = [1] @l2 + [1]
> [1] @l2 + [0]
= @l2
Following rules are (at-least) weakly oriented:
append(@l1,@l2) = [1] @l1 + [1] @l2 + [0]
>= [1] @l1 + [1] @l2 + [0]
= append#1(@l1,@l2)
append#1(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [6]
>= [1] @l2 + [1] @xs + [6]
= ::(@x,append(@xs,@l2))
subtrees(@t) = [1] @t + [0]
>= [1] @t + [0]
= subtrees#1(@t)
subtrees#1(leaf()) = [1]
>= [1]
= nil()
subtrees#1(node(@x,@t1,@t2)) = [1] @t1 + [1] @t2 + [6]
>= [1] @t1 + [1] @t2 + [6]
= subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [1] @t2 + [6]
>= [1] @l1 + [1] @t2 + [6]
= subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [6]
>= [1] @l1 + [1] @l2 + [6]
= ::(node(@x,@t1,@t2),append(@l1,@l2))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
- Weak TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
append#1(nil(),@l2) -> @l2
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ 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(::) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
{append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}
TcT has computed the following interpretation:
p(::) = [1 0] x_2 + [0]
[0 1] [2]
p(append) = [0 2] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(append#1) = [0 2] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(leaf) = [2]
[4]
p(nil) = [0]
[4]
p(node) = [1 0] x_1 + [1 1] x_2 + [1 3] x_3 + [0]
[0 0] [0 1] [0 1] [2]
p(subtrees) = [4 0] x_1 + [0]
[0 1] [0]
p(subtrees#1) = [4 0] x_1 + [0]
[0 1] [0]
p(subtrees#2) = [1 4] x_1 + [4 5] x_3 + [4 0] x_4 + [0]
[0 1] [0 1] [0 0] [2]
p(subtrees#3) = [1 5] x_1 + [0 2] x_2 + [0]
[0 1] [0 1] [2]
Following rules are strictly oriented:
append#1(::(@x,@xs),@l2) = [1 4] @l2 + [0 2] @xs + [4]
[0 1] [0 1] [2]
> [1 4] @l2 + [0 2] @xs + [0]
[0 1] [0 1] [2]
= ::(@x,append(@xs,@l2))
Following rules are (at-least) weakly oriented:
append(@l1,@l2) = [0 2] @l1 + [1 4] @l2 + [0]
[0 1] [0 1] [0]
>= [0 2] @l1 + [1 4] @l2 + [0]
[0 1] [0 1] [0]
= append#1(@l1,@l2)
append#1(nil(),@l2) = [1 4] @l2 + [8]
[0 1] [4]
>= [1 0] @l2 + [0]
[0 1] [0]
= @l2
subtrees(@t) = [4 0] @t + [0]
[0 1] [0]
>= [4 0] @t + [0]
[0 1] [0]
= subtrees#1(@t)
subtrees#1(leaf()) = [8]
[4]
>= [0]
[4]
= nil()
subtrees#1(node(@x,@t1,@t2)) = [4 4] @t1 + [4 12] @t2 + [4 0] @x + [0]
[0 1] [0 1] [0 0] [2]
>= [4 4] @t1 + [4 5] @t2 + [4 0] @x + [0]
[0 1] [0 1] [0 0] [2]
= subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) = [1 4] @l1 + [4 5] @t2 + [4 0] @x + [0]
[0 1] [0 1] [0 0] [2]
>= [0 2] @l1 + [4 5] @t2 + [0]
[0 1] [0 1] [2]
= subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) = [0 2] @l1 + [1 5] @l2 + [0]
[0 1] [0 1] [2]
>= [0 2] @l1 + [1 4] @l2 + [0]
[0 1] [0 1] [2]
= ::(node(@x,@t1,@t2),append(@l1,@l2))
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
append(@l1,@l2) -> append#1(@l1,@l2)
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
subtrees(@t) -> subtrees#1(@t)
subtrees#1(leaf()) -> nil()
subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x)
subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2
,subtrees#3} and constructors {::,leaf,nil,node}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))