*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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))
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}/{::,leaf,nil,node}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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 DP Rules:
Weak TRS Rules:
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
basic terms: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}/{::,leaf,nil,node}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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 DP Rules:
Weak TRS Rules:
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
basic terms: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}/{::,leaf,nil,node}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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 DP Rules:
Weak TRS Rules:
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
basic terms: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}/{::,leaf,nil,node}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
append#1(nil(),@l2) -> @l2
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}/{::,leaf,nil,node}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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:
{}
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.
*** 1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2))
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}/{::,leaf,nil,node}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
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:
{append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}
TcT has computed the following interpretation:
p(::) = [1 0] x2 + [0]
[0 1] [2]
p(append) = [0 1] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
p(append#1) = [0 1] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
p(leaf) = [3]
[1]
p(nil) = [4]
[1]
p(node) = [0 4] x1 + [1 4] x2 + [1
3] x3 + [4]
[0 1] [0 1] [0
1] [4]
p(subtrees) = [1 1] x1 + [0]
[0 2] [4]
p(subtrees#1) = [1 1] x1 + [0]
[0 2] [4]
p(subtrees#2) = [1 2] x1 + [1 4] x3 + [0
0] x4 + [0]
[0 1] [0 2] [0
1] [6]
p(subtrees#3) = [1 0] x1 + [0 2] x2 + [0
3] x4 + [0]
[0 1] [0 1] [0
0] [2]
Following rules are strictly oriented:
append#1(::(@x,@xs),@l2) = [1 0] @l2 + [0 1] @xs + [2]
[0 1] [0 1] [2]
> [1 0] @l2 + [0 1] @xs + [0]
[0 1] [0 1] [2]
= ::(@x,append(@xs,@l2))
Following rules are (at-least) weakly oriented:
append(@l1,@l2) = [0 1] @l1 + [1 0] @l2 + [0]
[0 1] [0 1] [0]
>= [0 1] @l1 + [1 0] @l2 + [0]
[0 1] [0 1] [0]
= append#1(@l1,@l2)
append#1(nil(),@l2) = [1 0] @l2 + [1]
[0 1] [1]
>= [1 0] @l2 + [0]
[0 1] [0]
= @l2
subtrees(@t) = [1 1] @t + [0]
[0 2] [4]
>= [1 1] @t + [0]
[0 2] [4]
= subtrees#1(@t)
subtrees#1(leaf()) = [4]
[6]
>= [4]
[1]
= nil()
subtrees#1(node(@x,@t1,@t2)) = [1 5] @t1 + [1 4] @t2 + [0
5] @x + [8]
[0 2] [0 2] [0
2] [12]
>= [1 5] @t1 + [1 4] @t2 + [0
0] @x + [8]
[0 2] [0 2] [0
1] [10]
= subtrees#2(subtrees(@t1)
,@t1
,@t2
,@x)
subtrees#2(@l1,@t1,@t2,@x) = [1 2] @l1 + [1 4] @t2 + [0
0] @x + [0]
[0 1] [0 2] [0
1] [6]
>= [0 2] @l1 + [1 4] @t2 + [0]
[0 1] [0 2] [6]
= subtrees#3(subtrees(@t2)
,@l1
,@t1
,@t2
,@x)
subtrees#3(@l2,@l1,@t1,@t2,@x) = [0 2] @l1 + [1 0] @l2 + [0
3] @t2 + [0]
[0 1] [0 1] [0
0] [2]
>= [0 1] @l1 + [1 0] @l2 + [0]
[0 1] [0 1] [2]
= ::(node(@x,@t1,@t2)
,append(@l1,@l2))
*** 1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3}/{::,leaf,nil,node}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).