```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
u21(ackout(X),Y) -> u22(ackin(Y,X))
- Signature:
{ackin/2,u21/2} / {ackout/1,s/1,u22/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
+ 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(u21) = {1},
uargs(u22) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(ackin) = [1] x2 + [0]
p(ackout) = [1] x1 + [1]
p(s) = [1] x1 + [0]
p(u21) = [1] x1 + [0]
p(u22) = [1] x1 + [0]

Following rules are strictly oriented:
u21(ackout(X),Y) = [1] X + [1]
> [1] X + [0]
= u22(ackin(Y,X))

Following rules are (at-least) weakly oriented:
ackin(s(X),s(Y)) =  [1] Y + [0]
>= [1] Y + [0]
=  u21(ackin(s(X),Y),X)

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:
ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
- Weak TRS:
u21(ackout(X),Y) -> u22(ackin(Y,X))
- Signature:
{ackin/2,u21/2} / {ackout/1,s/1,u22/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
+ 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(u21) = {1},
uargs(u22) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(ackin) = [1] x2 + [0]
p(ackout) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(u21) = [1] x1 + [0]
p(u22) = [1] x1 + [0]

Following rules are strictly oriented:
ackin(s(X),s(Y)) = [1] Y + [1]
> [1] Y + [0]
= u21(ackin(s(X),Y),X)

Following rules are (at-least) weakly oriented:
u21(ackout(X),Y) =  [1] X + [0]
>= [1] X + [0]
=  u22(ackin(Y,X))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
u21(ackout(X),Y) -> u22(ackin(Y,X))
- Signature:
{ackin/2,u21/2} / {ackout/1,s/1,u22/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```