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 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
↳17 CpxTrsMatchBoundsTAProof (⇔, 13 ms)
↳18 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

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

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Used rewriting to replace F(a, f(f(a, a), f(f(a, a), x0))) → c(F(f(a, a), f(f(a, a), f(a, f(a, f(a, f(a, x0)))))), F(a, a), F(a, f(a, f(f(a, a), x0))), F(a, f(f(a, a), x0))) by F(a, f(f(a, a), f(f(a, a), z0))) → c(F(f(a, a), f(f(a, a), f(a, f(a, f(a, f(a, z0)))))), F(a, a), F(a, f(f(a, a), f(a, f(a, z0)))), F(a, f(f(a, a), z0)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

(11) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace F(a, f(f(a, a), f(f(a, a), f(f(a, a), z0)))) → c(F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, z0))))))), F(a, a), F(a, f(a, f(f(a, a), f(f(a, a), z0)))), F(a, f(f(a, a), f(f(a, a), z0)))) by F(a, f(f(a, a), f(f(a, a), f(f(a, a), z0)))) → c(F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, z0))))))), F(a, a), F(a, f(f(a, a), f(a, f(a, f(f(a, a), z0))))), F(a, f(f(a, a), f(f(a, a), z0))))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

(13) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace F(a, f(f(a, a), f(f(a, a), x0))) → c(F(a, f(a, f(f(a, a), x0)))) by F(a, f(f(a, a), f(f(a, a), z0))) → c(F(a, f(f(a, a), f(a, f(a, z0)))))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

(15) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace F(a, f(f(a, a), f(f(a, a), f(f(a, a), z0)))) → c(F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, z0))))))), F(a, a), F(a, f(f(a, a), f(a, f(a, f(f(a, a), z0))))), F(a, f(f(a, a), f(f(a, a), z0)))) by F(a, f(f(a, a), f(f(a, a), f(f(a, a), z0)))) → c(F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, z0))))))), F(a, a), F(a, f(f(a, a), f(a, f(f(a, a), f(a, f(a, z0)))))), F(a, f(f(a, a), f(f(a, a), z0))))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

(17) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 0.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
a0() → 0
f0(0, 0) → 1

(18) BOUNDS(O(1), O(n^1))