* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ 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(div) = {1},
uargs(minus) = {1},
uargs(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [0]
p(minus) = [1] x1 + [0]
p(p) = [1] x1 + [9]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
p(s(X)) = [1] X + [9]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [0]
>= [0]
= 0()
div(s(X),s(Y)) = [1] X + [0]
>= [1] X + [0]
= s(div(minus(X,Y),s(Y)))
minus(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
minus(s(X),s(Y)) = [1] X + [0]
>= [1] X + [9]
= p(minus(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
- Weak TRS:
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ 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(div) = {1},
uargs(minus) = {1},
uargs(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [8] x2 + [0]
p(minus) = [1] x1 + [6]
p(p) = [1] x1 + [9]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
div(0(),s(Y)) = [8] Y + [16]
> [0]
= 0()
minus(X,0()) = [1] X + [6]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
div(s(X),s(Y)) = [1] X + [8] Y + [18]
>= [1] X + [8] Y + [24]
= s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) = [1] X + [8]
>= [1] X + [15]
= p(minus(X,Y))
p(s(X)) = [1] X + [11]
>= [1] X + [0]
= X
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:
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) -> p(minus(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
minus(X,0()) -> X
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ 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(div) = {1},
uargs(minus) = {1},
uargs(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [12]
p(div) = [1] x1 + [5] x2 + [9]
p(minus) = [1] x1 + [4]
p(p) = [1] x1 + [0]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
minus(s(X),s(Y)) = [1] X + [5]
> [1] X + [4]
= p(minus(X,Y))
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [5] Y + [26]
>= [12]
= 0()
div(s(X),s(Y)) = [1] X + [5] Y + [15]
>= [1] X + [5] Y + [19]
= s(div(minus(X,Y),s(Y)))
minus(X,0()) = [1] X + [4]
>= [1] X + [0]
= X
p(s(X)) = [1] X + [1]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
- Weak TRS:
div(0(),s(Y)) -> 0()
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(div) = {1},
uargs(minus) = {1},
uargs(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(div) = [4] x_1 + [0]
p(minus) = [1] x_1 + [0]
p(p) = [1] x_1 + [1]
p(s) = [1] x_1 + [4]
Following rules are strictly oriented:
div(s(X),s(Y)) = [4] X + [16]
> [4] X + [4]
= s(div(minus(X,Y),s(Y)))
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [8]
>= [2]
= 0()
minus(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
minus(s(X),s(Y)) = [1] X + [4]
>= [1] X + [1]
= p(minus(X,Y))
p(s(X)) = [1] X + [5]
>= [1] X + [0]
= X
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))