*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
Obligation:
Full
basic terms: {a__b,a__f,mark}/{a,b,f}
Applied Processor:
ToInnermost
Proof:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
Obligation:
Innermost
basic terms: {a__b,a__f,mark}/{a,b,f}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, 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) = {2}
Following symbols are considered usable:
{a__b,a__f,mark}
TcT has computed the following interpretation:
p(a) = [0]
p(a__b) = [0]
p(a__f) = [1] x2 + [0]
p(b) = [0]
p(f) = [0]
p(mark) = [8]
Following rules are strictly oriented:
mark(a()) = [8]
> [0]
= a()
mark(b()) = [8]
> [0]
= a__b()
Following rules are (at-least) weakly oriented:
a__b() = [0]
>= [0]
= a()
a__b() = [0]
>= [0]
= b()
a__f(X1,X2,X3) = [1] X2 + [0]
>= [0]
= f(X1,X2,X3)
a__f(a(),X,X) = [1] X + [0]
>= [0]
= a__f(X,a__b(),b())
mark(f(X1,X2,X3)) = [8]
>= [8]
= a__f(X1,mark(X2),X3)
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
Weak DP Rules:
Weak TRS Rules:
mark(a()) -> a()
mark(b()) -> a__b()
Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
Obligation:
Innermost
basic terms: {a__b,a__f,mark}/{a,b,f}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, 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) = {2}
Following symbols are considered usable:
{a__b,a__f,mark}
TcT has computed the following interpretation:
p(a) = [2]
p(a__b) = [2]
p(a__f) = [8] x1 + [1] x2 + [8] x3 + [2]
p(b) = [0]
p(f) = [1] x1 + [1] x2 + [1] x3 + [2]
p(mark) = [8] x1 + [2]
Following rules are strictly oriented:
a__b() = [2]
> [0]
= b()
a__f(a(),X,X) = [9] X + [18]
> [8] X + [4]
= a__f(X,a__b(),b())
mark(f(X1,X2,X3)) = [8] X1 + [8] X2 + [8] X3 + [18]
> [8] X1 + [8] X2 + [8] X3 + [4]
= a__f(X1,mark(X2),X3)
Following rules are (at-least) weakly oriented:
a__b() = [2]
>= [2]
= a()
a__f(X1,X2,X3) = [8] X1 + [1] X2 + [8] X3 + [2]
>= [1] X1 + [1] X2 + [1] X3 + [2]
= f(X1,X2,X3)
mark(a()) = [18]
>= [2]
= a()
mark(b()) = [2]
>= [2]
= a__b()
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__b() -> a()
a__f(X1,X2,X3) -> f(X1,X2,X3)
Weak DP Rules:
Weak TRS Rules:
a__b() -> b()
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
Obligation:
Innermost
basic terms: {a__b,a__f,mark}/{a,b,f}
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) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a) = [6]
p(a__b) = [0]
p(a__f) = [1] x1 + [1] x2 + [1] x3 + [3]
p(b) = [0]
p(f) = [1] x1 + [1] x2 + [1] x3 + [2]
p(mark) = [2] x1 + [3]
Following rules are strictly oriented:
a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [3]
> [1] X1 + [1] X2 + [1] X3 + [2]
= f(X1,X2,X3)
Following rules are (at-least) weakly oriented:
a__b() = [0]
>= [6]
= a()
a__b() = [0]
>= [0]
= b()
a__f(a(),X,X) = [2] X + [9]
>= [1] X + [3]
= a__f(X,a__b(),b())
mark(a()) = [15]
>= [6]
= a()
mark(b()) = [3]
>= [0]
= a__b()
mark(f(X1,X2,X3)) = [2] X1 + [2] X2 + [2] X3 + [7]
>= [1] X1 + [2] X2 + [1] X3 + [6]
= a__f(X1,mark(X2),X3)
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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__b() -> a()
Weak DP Rules:
Weak TRS Rules:
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
Obligation:
Innermost
basic terms: {a__b,a__f,mark}/{a,b,f}
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) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(a) = [4]
p(a__b) = [8]
p(a__f) = [4] x1 + [1] x2 + [3] x3 + [15]
p(b) = [1]
p(f) = [1] x1 + [1] x2 + [1] x3 + [5]
p(mark) = [4] x1 + [4]
Following rules are strictly oriented:
a__b() = [8]
> [4]
= a()
Following rules are (at-least) weakly oriented:
a__b() = [8]
>= [1]
= b()
a__f(X1,X2,X3) = [4] X1 + [1] X2 + [3] X3 + [15]
>= [1] X1 + [1] X2 + [1] X3 + [5]
= f(X1,X2,X3)
a__f(a(),X,X) = [4] X + [31]
>= [4] X + [26]
= a__f(X,a__b(),b())
mark(a()) = [20]
>= [4]
= a()
mark(b()) = [8]
>= [8]
= a__b()
mark(f(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [24]
>= [4] X1 + [4] X2 + [3] X3 + [19]
= a__f(X1,mark(X2),X3)
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(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
Obligation:
Innermost
basic terms: {a__b,a__f,mark}/{a,b,f}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).