* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(x){x -> f(x,y)} =
mark(f(x,y)) ->^+ a__f(mark(x),y)
= C[mark(x) = mark(x){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ 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__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__f) = [1] x1 + [0]
p(f) = [1] x1 + [15]
p(g) = [1] x1 + [0]
p(mark) = [1] x1 + [8]
Following rules are strictly oriented:
mark(f(X1,X2)) = [1] X1 + [23]
> [1] X1 + [8]
= a__f(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__f(X1,X2) = [1] X1 + [0]
>= [1] X1 + [15]
= f(X1,X2)
a__f(g(X),Y) = [1] X + [0]
>= [1] X + [8]
= a__f(mark(X),f(g(X),Y))
mark(g(X)) = [1] X + [8]
>= [1] X + [8]
= g(mark(X))
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^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(g(X)) -> g(mark(X))
- Weak TRS:
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ 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__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__f) = [1] x1 + [8]
p(f) = [1] x1 + [9]
p(g) = [1] x1 + [1]
p(mark) = [1] x1 + [0]
Following rules are strictly oriented:
a__f(g(X),Y) = [1] X + [9]
> [1] X + [8]
= a__f(mark(X),f(g(X),Y))
Following rules are (at-least) weakly oriented:
a__f(X1,X2) = [1] X1 + [8]
>= [1] X1 + [9]
= f(X1,X2)
mark(f(X1,X2)) = [1] X1 + [9]
>= [1] X1 + [8]
= a__f(mark(X1),X2)
mark(g(X)) = [1] X + [1]
>= [1] X + [1]
= g(mark(X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
mark(g(X)) -> g(mark(X))
- Weak TRS:
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ 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__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
{a__f,mark}
TcT has computed the following interpretation:
p(a__f) = [1 8] x_1 + [0]
[0 1] [0]
p(f) = [1 8] x_1 + [0]
[0 1] [0]
p(g) = [1 1] x_1 + [3]
[0 1] [1]
p(mark) = [1 1] x_1 + [1]
[0 1] [0]
Following rules are strictly oriented:
mark(g(X)) = [1 2] X + [5]
[0 1] [1]
> [1 2] X + [4]
[0 1] [1]
= g(mark(X))
Following rules are (at-least) weakly oriented:
a__f(X1,X2) = [1 8] X1 + [0]
[0 1] [0]
>= [1 8] X1 + [0]
[0 1] [0]
= f(X1,X2)
a__f(g(X),Y) = [1 9] X + [11]
[0 1] [1]
>= [1 9] X + [1]
[0 1] [0]
= a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) = [1 9] X1 + [1]
[0 1] [0]
>= [1 9] X1 + [1]
[0 1] [0]
= a__f(mark(X1),X2)
** Step 1.b:4: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
- Weak TRS:
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ 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__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
{a__f,mark}
TcT has computed the following interpretation:
p(a__f) = [1 1] x_1 + [0 2] x_2 + [2]
[0 1] [0 0] [2]
p(f) = [1 1] x_1 + [0 2] x_2 + [0]
[0 1] [0 0] [2]
p(g) = [1 8] x_1 + [8]
[0 1] [0]
p(mark) = [1 3] x_1 + [4]
[0 1] [0]
Following rules are strictly oriented:
a__f(X1,X2) = [1 1] X1 + [0 2] X2 + [2]
[0 1] [0 0] [2]
> [1 1] X1 + [0 2] X2 + [0]
[0 1] [0 0] [2]
= f(X1,X2)
Following rules are (at-least) weakly oriented:
a__f(g(X),Y) = [1 9] X + [0 2] Y + [10]
[0 1] [0 0] [2]
>= [1 6] X + [10]
[0 1] [2]
= a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) = [1 4] X1 + [0 2] X2 + [10]
[0 1] [0 0] [2]
>= [1 4] X1 + [0 2] X2 + [6]
[0 1] [0 0] [2]
= a__f(mark(X1),X2)
mark(g(X)) = [1 11] X + [12]
[0 1] [0]
>= [1 11] X + [12]
[0 1] [0]
= g(mark(X))
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))