* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),s(y)) -> f(x,s(c(s(y))))
- Signature:
{f/2} / {c/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {c,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(f) = {2},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(c) = [1] x1 + [14]
p(f) = [1] x2 + [0]
p(s) = [1] x1 + [2]

Following rules are strictly oriented:
f(x,c(y)) = [1] y + [14]
> [1] y + [2]
= f(x,s(f(y,y)))

Following rules are (at-least) weakly oriented:
f(s(x),s(y)) =  [1] y + [2]
>= [1] y + [18]
=  f(x,s(c(s(y))))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(s(x),s(y)) -> f(x,s(c(s(y))))
- Weak TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
- Signature:
{f/2} / {c/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):

The following argument positions are considered usable:
uargs(f) = {2},
uargs(s) = {1}

Following symbols are considered usable:
{f}
TcT has computed the following interpretation:
p(c) = [1 2 0]       [0]
[0 0 2] x_1 + [1]
[0 0 0]       [4]
p(f) = [0 0 3]       [1 2 0]       [0]
[4 0 0] x_1 + [0 0 0] x_2 + [0]
[0 0 0]       [0 0 0]       [0]
p(s) = [1 0 0]       [2]
[0 0 1] x_1 + [0]
[0 0 1]       [4]

Following rules are strictly oriented:
f(s(x),s(y)) = [0 0 3]     [1 0 2]     [14]
[4 0 0] x + [0 0 0] y + [8]
[0 0 0]     [0 0 0]     [0]
> [0 0 3]     [1 0 2]     [12]
[4 0 0] x + [0 0 0] y + [0]
[0 0 0]     [0 0 0]     [0]
= f(x,s(c(s(y))))

Following rules are (at-least) weakly oriented:
f(x,c(y)) =  [0 0 3]     [1 2 4]     [2]
[4 0 0] x + [0 0 0] y + [0]
[0 0 0]     [0 0 0]     [0]
>= [0 0 3]     [1 2 3]     [2]
[4 0 0] x + [0 0 0] y + [0]
[0 0 0]     [0 0 0]     [0]
=  f(x,s(f(y,y)))

* Step 3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),s(y)) -> f(x,s(c(s(y))))
- Signature:
{f/2} / {c/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))