WORST_CASE(?,O(n^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, 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(a__f) = {2}
Following symbols are considered usable:
{a__c,a__f,mark}
TcT has computed the following interpretation:
p(a__c) = [5]
p(a__f) = [1] x2 + [1] x3 + [4]
p(b) = [0]
p(c) = [5]
p(f) = [1] x2 + [1] x3 + [4]
p(mark) = [1] x1 + [6]
Following rules are strictly oriented:
a__c() = [5]
> [0]
= b()
mark(b()) = [6]
> [0]
= b()
mark(c()) = [11]
> [5]
= a__c()
Following rules are (at-least) weakly oriented:
a__c() = [5]
>= [5]
= c()
a__f(X1,X2,X3) = [1] X2 + [1] X3 + [4]
>= [1] X2 + [1] X3 + [4]
= f(X1,X2,X3)
a__f(b(),X,c()) = [1] X + [9]
>= [1] X + [9]
= a__f(X,a__c(),X)
mark(f(X1,X2,X3)) = [1] X2 + [1] X3 + [10]
>= [1] X2 + [1] X3 + [10]
= a__f(X1,mark(X2),X3)
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Weak TRS:
a__c() -> b()
mark(b()) -> b()
mark(c()) -> a__c()
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ 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) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__c) = [0]
p(a__f) = [1] x1 + [1] x2 + [1]
p(b) = [0]
p(c) = [1]
p(f) = [1] x1 + [1] x2 + [2]
p(mark) = [8] x1 + [9]
Following rules are strictly oriented:
mark(f(X1,X2,X3)) = [8] X1 + [8] X2 + [25]
> [1] X1 + [8] X2 + [10]
= a__f(X1,mark(X2),X3)
Following rules are (at-least) weakly oriented:
a__c() = [0]
>= [0]
= b()
a__c() = [0]
>= [1]
= c()
a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [2]
= f(X1,X2,X3)
a__f(b(),X,c()) = [1] X + [1]
>= [1] X + [1]
= a__f(X,a__c(),X)
mark(b()) = [9]
>= [0]
= b()
mark(c()) = [17]
>= [0]
= a__c()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
- Weak TRS:
a__c() -> b()
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ 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) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__c) = [0]
p(a__f) = [1] x1 + [1] x2 + [8]
p(b) = [0]
p(c) = [0]
p(f) = [1] x1 + [1] x2 + [2]
p(mark) = [4] x1 + [4]
Following rules are strictly oriented:
a__f(X1,X2,X3) = [1] X1 + [1] X2 + [8]
> [1] X1 + [1] X2 + [2]
= f(X1,X2,X3)
Following rules are (at-least) weakly oriented:
a__c() = [0]
>= [0]
= b()
a__c() = [0]
>= [0]
= c()
a__f(b(),X,c()) = [1] X + [8]
>= [1] X + [8]
= a__f(X,a__c(),X)
mark(b()) = [4]
>= [0]
= b()
mark(c()) = [4]
>= [0]
= a__c()
mark(f(X1,X2,X3)) = [4] X1 + [4] X2 + [12]
>= [1] X1 + [4] X2 + [12]
= a__f(X1,mark(X2),X3)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__c() -> c()
a__f(b(),X,c()) -> a__f(X,a__c(),X)
- Weak TRS:
a__c() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ 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) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__c) = [0]
p(a__f) = [1] x2 + [1] x3 + [0]
p(b) = [0]
p(c) = [5]
p(f) = [1] x2 + [1] x3 + [0]
p(mark) = [2] x1 + [4]
Following rules are strictly oriented:
a__f(b(),X,c()) = [1] X + [5]
> [1] X + [0]
= a__f(X,a__c(),X)
Following rules are (at-least) weakly oriented:
a__c() = [0]
>= [0]
= b()
a__c() = [0]
>= [5]
= c()
a__f(X1,X2,X3) = [1] X2 + [1] X3 + [0]
>= [1] X2 + [1] X3 + [0]
= f(X1,X2,X3)
mark(b()) = [4]
>= [0]
= b()
mark(c()) = [14]
>= [0]
= a__c()
mark(f(X1,X2,X3)) = [2] X2 + [2] X3 + [4]
>= [2] X2 + [1] X3 + [4]
= a__f(X1,mark(X2),X3)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__c() -> c()
- Weak TRS:
a__c() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ 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) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__c) = [1]
p(a__f) = [1] x1 + [1] x2 + [0]
p(b) = [1]
p(c) = [0]
p(f) = [1] x1 + [1] x2 + [0]
p(mark) = [1] x1 + [11]
Following rules are strictly oriented:
a__c() = [1]
> [0]
= c()
Following rules are (at-least) weakly oriented:
a__c() = [1]
>= [1]
= b()
a__f(X1,X2,X3) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= f(X1,X2,X3)
a__f(b(),X,c()) = [1] X + [1]
>= [1] X + [1]
= a__f(X,a__c(),X)
mark(b()) = [12]
>= [1]
= b()
mark(c()) = [11]
>= [1]
= a__c()
mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [11]
>= [1] X1 + [1] X2 + [11]
= a__f(X1,mark(X2),X3)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))