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

(0) Obligation:

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

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
PLUS(z0, 0) → c2
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
ACTIVATE(z0) → c4

S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
PLUS(z0, 0) → c2
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
ACTIVATE(z0) → c4

K tuples:none
Defined Rule Symbols:

U11, U12, plus, activate

Defined Pair Symbols:

U11', U12', PLUS, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

PLUS(z0, 0) → c2
ACTIVATE(z0) → c4

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

K tuples:none
Defined Rule Symbols:

U11, U12, plus, activate

Defined Pair Symbols:

U11', U12', PLUS

Compound Symbols:

c, c1, c3

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
activate(z0) → z0

Tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

S tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

K tuples:none
Defined Rule Symbols:

U11, U12, plus, activate

Defined Pair Symbols:

PLUS, U11', U12'

Compound Symbols:

c3, c, c1

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(z0) → z0

Tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

S tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

K tuples:none
Defined Rule Symbols:

activate

Defined Pair Symbols:

PLUS, U11', U12'

Compound Symbols:

c3, c, c1

(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))

We considered the (Usable) Rules:

activate(z0) → z0

And the Tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(PLUS(x₁, x₂)) = [4]x₂   
POL(U11'(x₁, x₂, x₃)) = [2] + [4]x₁ + [4]x₂   
POL(U12'(x₁, x₂, x₃)) = [4]x₂   
POL(activate(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(s(x₁)) = [3] + x₁   
POL(tt) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(z0) → z0

Tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

S tuples:

U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

K tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))

Defined Rule Symbols:

activate

Defined Pair Symbols:

PLUS, U11', U12'

Compound Symbols:

c3, c, c1

(11) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))

Now S is empty

(12) BOUNDS(O(1), O(1))