The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳4 CdtProblem
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳6 CdtProblem
↳7 SIsEmptyProof (⇔)
↳8 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
u21(ackout(z0), z1) → u22(ackin(z1, z0))
Tuples:

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
S tuples:

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
K tuples:none
Defined Rule Symbols:

ackin, u21

Defined Pair Symbols:

ACKIN, U21

Compound Symbols:

c, c1

(3) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
We considered the (Usable) Rules:

ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
u21(ackout(z0), z1) → u22(ackin(z1, z0))
And the Tuples:

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACKIN(x1, x2)) = [2]   
POL(U21(x1, x2)) = x1   
POL(ackin(x1, x2)) = [2]x1   
POL(ackout(x1)) = [4]   
POL(c(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(s(x1)) = 0   
POL(u21(x1, x2)) = [4]x1   
POL(u22(x1)) = [4]   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
u21(ackout(z0), z1) → u22(ackin(z1, z0))
Tuples:

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
S tuples:

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
K tuples:

U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
Defined Rule Symbols:

ackin, u21

Defined Pair Symbols:

ACKIN, U21

Compound Symbols:

c, c1

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
We considered the (Usable) Rules:

ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
u21(ackout(z0), z1) → u22(ackin(z1, z0))
And the Tuples:

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACKIN(x1, x2)) = [3] + [2]x2   
POL(U21(x1, x2)) = [2]x1   
POL(ackin(x1, x2)) = 0   
POL(ackout(x1)) = [4] + x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(s(x1)) = [4] + x1   
POL(u21(x1, x2)) = [4]x1   
POL(u22(x1)) = [5]   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
u21(ackout(z0), z1) → u22(ackin(z1, z0))
Tuples:

ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
S tuples:none
K tuples:

U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
Defined Rule Symbols:

ackin, u21

Defined Pair Symbols:

ACKIN, U21

Compound Symbols:

c, c1

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))