WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append(l1,l2) -> append#1(l1,l2)
append#1(cons(x,xs),l2) -> cons(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) -> cons(node(x,t1,t2),append(l1,l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/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 {cons,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(cons) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x2 + [8]
p(append#1) = [1] x2 + [0]
p(cons) = [1] x2 + [3]
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 + [8]
> [1] l2 + [0]
= append#1(l1,l2)
Following rules are (at-least) weakly oriented:
append#1(cons(x,xs),l2) = [1] l2 + [0]
>= [1] l2 + [11]
= cons(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]
= cons(node(x,t1,t2),append(l1,l2))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(cons(x,xs),l2) -> cons(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) -> cons(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} / {cons/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 {cons,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(cons) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x1 + [1] x2 + [0]
p(append#1) = [1] x1 + [1] x2 + [0]
p(cons) = [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]
= cons(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(cons(x,xs),l2) = [1] l2 + [1] xs + [0]
>= [1] l2 + [1] xs + [0]
= cons(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 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(cons(x,xs),l2) -> cons(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) -> cons(node(x,t1,t2),append(l1,l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/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 {cons,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(cons) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x2 + [0]
p(append#1) = [1] x2 + [0]
p(cons) = [1] x2 + [0]
p(leaf) = [4]
p(nil) = [0]
p(node) = [1] x3 + [0]
p(subtrees) = [8]
p(subtrees#1) = [11]
p(subtrees#2) = [1] x1 + [0]
p(subtrees#3) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
subtrees#1(leaf()) = [11]
> [0]
= nil()
subtrees#1(node(x,t1,t2)) = [11]
> [8]
= subtrees#2(subtrees(t1),t1,t2,x)
Following rules are (at-least) weakly oriented:
append(l1,l2) = [1] l2 + [0]
>= [1] l2 + [0]
= append#1(l1,l2)
append#1(cons(x,xs),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= cons(x,append(xs,l2))
append#1(nil(),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= l2
subtrees(t) = [8]
>= [11]
= subtrees#1(t)
subtrees#2(l1,t1,t2,x) = [1] l1 + [0]
>= [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]
= cons(node(x,t1,t2),append(l1,l2))
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(cons(x,xs),l2) -> cons(x,append(xs,l2))
append#1(nil(),l2) -> l2
subtrees(t) -> subtrees#1(t)
subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x)
- Weak TRS:
append(l1,l2) -> append#1(l1,l2)
subtrees#1(leaf()) -> nil()
subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x)
subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/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 {cons,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(cons) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x1 + [1] x2 + [1]
p(append#1) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x2 + [1]
p(leaf) = [0]
p(nil) = [1]
p(node) = [1] x1 + [1] x2 + [0]
p(subtrees) = [0]
p(subtrees#1) = [1]
p(subtrees#2) = [1] x1 + [1]
p(subtrees#3) = [1] x1 + [1] x2 + [2]
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 + [1]
>= [1] l1 + [1] l2 + [0]
= append#1(l1,l2)
append#1(cons(x,xs),l2) = [1] l2 + [1] xs + [1]
>= [1] l2 + [1] xs + [2]
= cons(x,append(xs,l2))
subtrees(t) = [0]
>= [1]
= subtrees#1(t)
subtrees#1(leaf()) = [1]
>= [1]
= nil()
subtrees#1(node(x,t1,t2)) = [1]
>= [1]
= subtrees#2(subtrees(t1),t1,t2,x)
subtrees#2(l1,t1,t2,x) = [1] l1 + [1]
>= [1] l1 + [2]
= subtrees#3(subtrees(t2),l1,t1,t2,x)
subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [2]
>= [1] l1 + [1] l2 + [2]
= cons(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(cons(x,xs),l2) -> cons(x,append(xs,l2))
subtrees(t) -> subtrees#1(t)
subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x)
- Weak TRS:
append(l1,l2) -> append#1(l1,l2)
append#1(nil(),l2) -> l2
subtrees#1(leaf()) -> nil()
subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x)
subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/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 {cons,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(cons) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x2 + [0]
p(append#1) = [1] x2 + [0]
p(cons) = [1] x2 + [0]
p(leaf) = [0]
p(nil) = [1]
p(node) = [1] x2 + [1]
p(subtrees) = [0]
p(subtrees#1) = [1]
p(subtrees#2) = [1] x1 + [1]
p(subtrees#3) = [1] x1 + [0]
Following rules are strictly oriented:
subtrees#2(l1,t1,t2,x) = [1] l1 + [1]
> [0]
= subtrees#3(subtrees(t2),l1,t1,t2,x)
Following rules are (at-least) weakly oriented:
append(l1,l2) = [1] l2 + [0]
>= [1] l2 + [0]
= append#1(l1,l2)
append#1(cons(x,xs),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= cons(x,append(xs,l2))
append#1(nil(),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= l2
subtrees(t) = [0]
>= [1]
= subtrees#1(t)
subtrees#1(leaf()) = [1]
>= [1]
= nil()
subtrees#1(node(x,t1,t2)) = [1]
>= [1]
= subtrees#2(subtrees(t1),t1,t2,x)
subtrees#3(l2,l1,t1,t2,x) = [1] l2 + [0]
>= [1] l2 + [0]
= cons(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(cons(x,xs),l2) -> cons(x,append(xs,l2))
subtrees(t) -> subtrees#1(t)
- Weak TRS:
append(l1,l2) -> append#1(l1,l2)
append#1(nil(),l2) -> l2
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) -> cons(node(x,t1,t2),append(l1,l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/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 {cons,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(cons) = {2},
uargs(subtrees#2) = {1},
uargs(subtrees#3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x2 + [0]
p(append#1) = [1] x2 + [0]
p(cons) = [1] x2 + [0]
p(leaf) = [1]
p(nil) = [0]
p(node) = [1] x2 + [1] x3 + [3]
p(subtrees) = [1] x1 + [1]
p(subtrees#1) = [1] x1 + [0]
p(subtrees#2) = [1] x1 + [1] x3 + [2]
p(subtrees#3) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
subtrees(t) = [1] t + [1]
> [1] t + [0]
= subtrees#1(t)
Following rules are (at-least) weakly oriented:
append(l1,l2) = [1] l2 + [0]
>= [1] l2 + [0]
= append#1(l1,l2)
append#1(cons(x,xs),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= cons(x,append(xs,l2))
append#1(nil(),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= l2
subtrees#1(leaf()) = [1]
>= [0]
= nil()
subtrees#1(node(x,t1,t2)) = [1] t1 + [1] t2 + [3]
>= [1] t1 + [1] t2 + [3]
= subtrees#2(subtrees(t1),t1,t2,x)
subtrees#2(l1,t1,t2,x) = [1] l1 + [1] t2 + [2]
>= [1] l1 + [1] t2 + [1]
= subtrees#3(subtrees(t2),l1,t1,t2,x)
subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [0]
>= [1] l2 + [0]
= cons(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: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
append#1(cons(x,xs),l2) -> cons(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) -> cons(node(x,t1,t2),append(l1,l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/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 {cons,leaf,nil,node}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {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(append) = [0 2] x1 + [1 0] x2 + [0]
[0 1] [0 1] [4]
p(append#1) = [0 2] x1 + [1 0] x2 + [0]
[0 1] [0 1] [4]
p(cons) = [1 0] x2 + [0]
[0 1] [1]
p(leaf) = [3]
[0]
p(nil) = [4]
[0]
p(node) = [1 1] x2 + [1 0] x3 + [4]
[0 1] [0 1] [5]
p(subtrees) = [2 0] x1 + [1]
[0 1] [0]
p(subtrees#1) = [2 0] x1 + [1]
[0 1] [0]
p(subtrees#2) = [1 2] x1 + [2 0] x3 + [7]
[0 1] [0 1] [5]
p(subtrees#3) = [1 0] x1 + [0 2] x2 + [3]
[0 1] [0 1] [5]
Following rules are strictly oriented:
append#1(cons(x,xs),l2) = [1 0] l2 + [0 2] xs + [2]
[0 1] [0 1] [5]
> [1 0] l2 + [0 2] xs + [0]
[0 1] [0 1] [5]
= cons(x,append(xs,l2))
Following rules are (at-least) weakly oriented:
append(l1,l2) = [0 2] l1 + [1 0] l2 + [0]
[0 1] [0 1] [4]
>= [0 2] l1 + [1 0] l2 + [0]
[0 1] [0 1] [4]
= append#1(l1,l2)
append#1(nil(),l2) = [1 0] l2 + [0]
[0 1] [4]
>= [1 0] l2 + [0]
[0 1] [0]
= l2
subtrees(t) = [2 0] t + [1]
[0 1] [0]
>= [2 0] t + [1]
[0 1] [0]
= subtrees#1(t)
subtrees#1(leaf()) = [7]
[0]
>= [4]
[0]
= nil()
subtrees#1(node(x,t1,t2)) = [2 2] t1 + [2 0] t2 + [9]
[0 1] [0 1] [5]
>= [2 2] t1 + [2 0] t2 + [8]
[0 1] [0 1] [5]
= subtrees#2(subtrees(t1),t1,t2,x)
subtrees#2(l1,t1,t2,x) = [1 2] l1 + [2 0] t2 + [7]
[0 1] [0 1] [5]
>= [0 2] l1 + [2 0] t2 + [4]
[0 1] [0 1] [5]
= subtrees#3(subtrees(t2),l1,t1,t2,x)
subtrees#3(l2,l1,t1,t2,x) = [0 2] l1 + [1 0] l2 + [3]
[0 1] [0 1] [5]
>= [0 2] l1 + [1 0] l2 + [0]
[0 1] [0 1] [5]
= cons(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(cons(x,xs),l2) -> cons(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) -> cons(node(x,t1,t2),append(l1,l2))
- Signature:
{append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/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 {cons,leaf,nil,node}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))