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), 0 ms)
↳2 CdtProblem
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 93 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (⇔, 0 ms)
↳14 BOUNDS(O(1), O(1))

(0) Obligation:

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

f(a, f(x, a)) → f(a, f(f(a, a), f(a, x)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(z0, a)) → f(a, f(f(a, a), f(a, z0)))
Tuples:

F(a, f(z0, a)) → c(F(a, f(f(a, a), f(a, z0))), F(f(a, a), f(a, z0)), F(a, a), F(a, z0))
S tuples:

F(a, f(z0, a)) → c(F(a, f(f(a, a), f(a, z0))), F(f(a, a), f(a, z0)), F(a, a), F(a, z0))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(a, f(z0, a)) → c(F(a, f(f(a, a), f(a, z0))), F(f(a, a), f(a, z0)), F(a, a), F(a, z0)) by

F(a, f(f(z0, a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))), F(f(a, a), f(a, f(z0, a))), F(a, a), F(a, f(z0, a)))
F(a, f(x0, a)) → c

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(z0, a)) → f(a, f(f(a, a), f(a, z0)))
Tuples:

F(a, f(f(z0, a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))), F(f(a, a), f(a, f(z0, a))), F(a, a), F(a, f(z0, a)))
F(a, f(x0, a)) → c
S tuples:

F(a, f(f(z0, a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))), F(f(a, a), f(a, f(z0, a))), F(a, a), F(a, f(z0, a)))
F(a, f(x0, a)) → c
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 1 trailing nodes:

F(a, f(x0, a)) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(z0, a)) → f(a, f(f(a, a), f(a, z0)))
Tuples:

F(a, f(f(z0, a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))), F(f(a, a), f(a, f(z0, a))), F(a, a), F(a, f(z0, a)))
S tuples:

F(a, f(f(z0, a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))), F(f(a, a), f(a, f(z0, a))), F(a, a), F(a, f(z0, a)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(a, f(f(z0, a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))), F(f(a, a), f(a, f(z0, a))), F(a, a), F(a, f(z0, a))) by

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
F(a, f(f(x0, a), a)) → c(F(f(a, a), f(a, f(x0, a))))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(z0, a)) → f(a, f(f(a, a), f(a, z0)))
Tuples:

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
F(a, f(f(x0, a), a)) → c(F(f(a, a), f(a, f(x0, a))))
S tuples:

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
F(a, f(f(x0, a), a)) → c(F(f(a, a), f(a, f(x0, a))))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 1 trailing nodes:

F(a, f(f(x0, a), a)) → c(F(f(a, a), f(a, f(x0, a))))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(z0, a)) → f(a, f(f(a, a), f(a, z0)))
Tuples:

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
S tuples:

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(11) 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.

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
We considered the (Usable) Rules:

f(a, f(z0, a)) → f(a, f(f(a, a), f(a, z0)))
And the Tuples:

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [2]x2   
POL(a) = 0   
POL(c(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(f(x1, x2)) = [1] + [3]x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(z0, a)) → f(a, f(f(a, a), f(a, z0)))
Tuples:

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
S tuples:none
K tuples:

F(a, f(f(f(z0, a), a), a)) → c(F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, z0))))))), F(f(a, a), f(a, f(f(z0, a), a))), F(a, a), F(a, f(f(z0, a), a)))
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))