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:

merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

merge(z0, nil) → z0
merge(nil, z0) → z0
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v)))
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))

Tuples:

MERGE(z0, nil) → c
MERGE(nil, z0) → c1
MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))
MERGE(++(z0, z1), ++(u, v)) → c3(MERGE(++(z0, z1), v))

S tuples:

MERGE(z0, nil) → c
MERGE(nil, z0) → c1
MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))
MERGE(++(z0, z1), ++(u, v)) → c3(MERGE(++(z0, z1), v))

K tuples:none
Defined Rule Symbols:

merge

Defined Pair Symbols:

MERGE

Compound Symbols:

c, c1, c2, c3

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

Removed 3 trailing nodes:

MERGE(z0, nil) → c
MERGE(nil, z0) → c1
MERGE(++(z0, z1), ++(u, v)) → c3(MERGE(++(z0, z1), v))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

merge(z0, nil) → z0
merge(nil, z0) → z0
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v)))
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))

Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

S tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

K tuples:none
Defined Rule Symbols:

merge

Defined Pair Symbols:

MERGE

Compound Symbols:

c2

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

merge(z0, nil) → z0
merge(nil, z0) → z0
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v)))
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

S tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

MERGE

Compound Symbols:

c2

(7) 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.

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

We considered the (Usable) Rules:none
And the Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

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

POL(++(x₁, x₂)) = [1] + x₂   
POL(MERGE(x₁, x₂)) = [5]x₁   
POL(c2(x₁)) = x₁   
POL(u) = 0   
POL(v) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

S tuples:none
K tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))

Defined Rule Symbols:none

Defined Pair Symbols:

MERGE

Compound Symbols:

c2

(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))