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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtUnreachableProof (⇔, 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 SIsEmptyProof (⇔, 0 ms)
↳8 BOUNDS(O(1), O(1))

(0) Obligation:

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(I, __(P, I))) → U11(tt)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

__(__(z0, z1), z2) → __(z0, __(z1, z2))
__(z0, nil) → z0
__(nil, z0) → z0
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(z0, __(z1, z0))) → U11(tt)
activate(z0) → z0
Tuples:

__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
U11'(tt) → c3(U12'(tt))
ISNEPAL(__(z0, __(z1, z0))) → c5(U11'(tt))
S tuples:

__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
U11'(tt) → c3(U12'(tt))
ISNEPAL(__(z0, __(z1, z0))) → c5(U11'(tt))
K tuples:none
Defined Rule Symbols:

__, U11, U12, isNePal, activate

Defined Pair Symbols:

__', U11', ISNEPAL

Compound Symbols:

c, c3, c5

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
ISNEPAL(__(z0, __(z1, z0))) → c5(U11'(tt))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

__(__(z0, z1), z2) → __(z0, __(z1, z2))
__(z0, nil) → z0
__(nil, z0) → z0
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(z0, __(z1, z0))) → U11(tt)
activate(z0) → z0
Tuples:

U11'(tt) → c3(U12'(tt))
S tuples:

U11'(tt) → c3(U12'(tt))
K tuples:none
Defined Rule Symbols:

__, U11, U12, isNePal, activate

Defined Pair Symbols:

U11'

Compound Symbols:

c3

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

Removed 1 trailing nodes:

U11'(tt) → c3(U12'(tt))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

__(__(z0, z1), z2) → __(z0, __(z1, z2))
__(z0, nil) → z0
__(nil, z0) → z0
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(z0, __(z1, z0))) → U11(tt)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

__, U11, U12, isNePal, activate

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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