*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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))
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__f/2,mark/1} / {f/2,g/1}
Obligation:
Innermost
basic terms: {a__f,mark}/{f,g}
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(a__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
{}
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.
*** 1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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 DP Rules:
Weak TRS Rules:
mark(f(X1,X2)) -> a__f(mark(X1),X2)
Signature:
{a__f/2,mark/1} / {f/2,g/1}
Obligation:
Innermost
basic terms: {a__f,mark}/{f,g}
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(a__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
{}
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.
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__f(X1,X2) -> f(X1,X2)
mark(g(X)) -> g(mark(X))
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {a__f,mark}/{f,g}
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(a__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
{a__f,mark}
TcT has computed the following interpretation:
p(a__f) = [1 0] x1 + [4]
[0 1] [0]
p(f) = [1 0] x1 + [4]
[0 1] [0]
p(g) = [1 4] x1 + [5]
[0 1] [2]
p(mark) = [1 4] x1 + [0]
[0 1] [0]
Following rules are strictly oriented:
mark(g(X)) = [1 8] X + [13]
[0 1] [2]
> [1 8] X + [5]
[0 1] [2]
= g(mark(X))
Following rules are (at-least) weakly oriented:
a__f(X1,X2) = [1 0] X1 + [4]
[0 1] [0]
>= [1 0] X1 + [4]
[0 1] [0]
= f(X1,X2)
a__f(g(X),Y) = [1 4] X + [9]
[0 1] [2]
>= [1 4] X + [4]
[0 1] [0]
= a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) = [1 4] X1 + [4]
[0 1] [0]
>= [1 4] X1 + [4]
[0 1] [0]
= a__f(mark(X1),X2)
*** 1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__f(X1,X2) -> f(X1,X2)
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {a__f,mark}/{f,g}
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(a__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
{a__f,mark}
TcT has computed the following interpretation:
p(a__f) = [1 0] x1 + [3]
[0 1] [2]
p(f) = [1 0] x1 + [0]
[0 1] [2]
p(g) = [1 6] x1 + [7]
[0 1] [0]
p(mark) = [1 4] x1 + [7]
[0 1] [0]
Following rules are strictly oriented:
a__f(X1,X2) = [1 0] X1 + [3]
[0 1] [2]
> [1 0] X1 + [0]
[0 1] [2]
= f(X1,X2)
Following rules are (at-least) weakly oriented:
a__f(g(X),Y) = [1 6] X + [10]
[0 1] [2]
>= [1 4] X + [10]
[0 1] [2]
= a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) = [1 4] X1 + [15]
[0 1] [2]
>= [1 4] X1 + [10]
[0 1] [2]
= a__f(mark(X1),X2)
mark(g(X)) = [1 10] X + [14]
[0 1] [0]
>= [1 10] X + [14]
[0 1] [0]
= g(mark(X))
*** 1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {a__f,mark}/{f,g}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).