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), 10 ms)
    ↳4 CdtProblem
    ↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 CdtProblem
    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
↳15 CpxTrsMatchBoundsTAProof (⇔, 20 ms)
↳16 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

f(a, f(a, f(b, f(x, y)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
f(a, f(c, f(x, y))) → f(b, f(x, y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:

F(a, f(a, f(b, f(z0, z1)))) → c1(F(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(z0, z1)))))), F(b, f(a, f(a, f(a, f(z0, z1))))), F(a, f(a, f(a, f(z0, z1)))), F(a, f(a, f(z0, z1))), F(a, f(z0, z1)), F(z0, z1))
F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1))
S tuples:

F(a, f(a, f(b, f(z0, z1)))) → c1(F(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(z0, z1)))))), F(b, f(a, f(a, f(a, f(z0, z1))))), F(a, f(a, f(a, f(z0, z1)))), F(a, f(a, f(z0, z1))), F(a, f(z0, z1)), F(z0, z1))
F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

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

Use narrowing to replace F(a, f(a, f(b, f(z0, z1)))) → c1(F(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(z0, z1)))))), F(b, f(a, f(a, f(a, f(z0, z1))))), F(a, f(a, f(a, f(z0, z1)))), F(a, f(a, f(z0, z1))), F(a, f(z0, z1)), F(z0, z1)) by

F(a, f(a, f(b, f(b, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))), F(b, f(z0, z1)))
F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:

F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1))
F(a, f(a, f(b, f(b, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))), F(b, f(z0, z1)))
F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
S tuples:

F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1))
F(a, f(a, f(b, f(b, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))), F(b, f(z0, z1)))
F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c2, c1, c1

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

Use narrowing to replace F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1)) by

F(a, f(c, f(x0, x1))) → c2(F(x0, x1))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:

F(a, f(a, f(b, f(b, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))), F(b, f(z0, z1)))
F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
S tuples:

F(a, f(a, f(b, f(b, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))), F(b, f(z0, z1)))
F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c1, c2

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

Use narrowing to replace F(a, f(a, f(b, f(b, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))), F(b, f(z0, z1))) by

F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:

F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
S tuples:

F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c1, c2, c1

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

Use narrowing to replace F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(a, f(b, f(z0, z1)))))), F(a, f(a, f(a, f(b, f(z0, z1))))), F(a, f(a, f(b, f(z0, z1)))), F(a, f(b, f(z0, z1)))) by

F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(c, f(b, f(a, f(a, f(a, f(x0, x1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))), F(a, f(b, f(x0, x1))))
F(a, f(a, f(b, f(a, f(b, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(a, f(b, f(z0, z1)))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))))
F(a, f(a, f(b, f(a, f(b, f(c, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:

F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(c, f(b, f(a, f(a, f(a, f(x0, x1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))), F(a, f(b, f(x0, x1))))
F(a, f(a, f(b, f(a, f(b, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(a, f(b, f(z0, z1)))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))))
F(a, f(a, f(b, f(a, f(b, f(c, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
S tuples:

F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(c, f(b, f(a, f(a, f(a, f(x0, x1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))), F(a, f(b, f(x0, x1))))
F(a, f(a, f(b, f(a, f(b, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(a, f(b, f(z0, z1)))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))))
F(a, f(a, f(b, f(a, f(b, f(c, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c1, c2, c1, c1

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

Use narrowing to replace F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1))) by

F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(c, f(x0, x1))))) → c1

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:

F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(c, f(b, f(a, f(a, f(a, f(x0, x1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))), F(a, f(b, f(x0, x1))))
F(a, f(a, f(b, f(a, f(b, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(a, f(b, f(z0, z1)))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))))
F(a, f(a, f(b, f(a, f(b, f(c, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(c, f(x0, x1))))) → c1
S tuples:

F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(c, f(b, f(a, f(a, f(a, f(x0, x1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))), F(a, f(b, f(x0, x1))))
F(a, f(a, f(b, f(a, f(b, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(a, f(b, f(z0, z1)))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))))
F(a, f(a, f(b, f(a, f(b, f(c, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
F(a, f(a, f(b, f(c, f(x0, x1))))) → c1
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2, c1, c1, c1, c1

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

Removed 1 trailing nodes:

F(a, f(a, f(b, f(c, f(x0, x1))))) → c1

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:

F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(c, f(b, f(a, f(a, f(a, f(x0, x1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))), F(a, f(b, f(x0, x1))))
F(a, f(a, f(b, f(a, f(b, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(a, f(b, f(z0, z1)))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))))
F(a, f(a, f(b, f(a, f(b, f(c, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
S tuples:

F(a, f(a, f(b, f(x0, x1)))) → c1(F(x0, x1))
F(a, f(c, f(x0, x1))) → c2(F(x0, x1))
F(a, f(a, f(b, f(b, f(b, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))), F(b, f(b, f(z0, z1))))
F(a, f(a, f(b, f(b, f(a, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))), F(b, f(a, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(b, f(c, f(z0, z1)))))) → c1(F(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1))))))))))), F(c, f(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(b, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))), F(b, f(c, f(z0, z1))))
F(a, f(a, f(b, f(b, f(x0, x1))))) → c1(F(c, f(b, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(b, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(c, f(b, f(a, f(a, f(a, f(x0, x1))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))), F(a, f(b, f(x0, x1))))
F(a, f(a, f(b, f(a, f(b, f(b, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(b, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(b, f(z0, z1)))))), F(a, f(a, f(b, f(b, f(z0, z1))))), F(a, f(b, f(b, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(a, f(b, f(z0, z1)))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))))), F(a, f(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1)))))))), F(a, f(a, f(a, f(b, f(a, f(b, f(z0, z1))))))), F(a, f(a, f(b, f(a, f(b, f(z0, z1)))))), F(a, f(b, f(a, f(b, f(z0, z1))))))
F(a, f(a, f(b, f(a, f(b, f(c, f(z0, z1))))))) → c1(F(b, f(c, f(b, f(a, f(a, f(b, f(c, f(b, f(a, f(a, f(b, f(z0, z1)))))))))))), F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1)))))))), F(a, f(a, f(a, f(a, f(b, f(c, f(z0, z1))))))), F(a, f(a, f(a, f(b, f(c, f(z0, z1)))))), F(a, f(a, f(b, f(c, f(z0, z1))))), F(a, f(b, f(c, f(z0, z1)))))
F(a, f(a, f(b, f(a, f(b, f(x0, x1)))))) → c1(F(c, f(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1)))))))), F(b, f(a, f(a, f(a, f(a, f(b, f(x0, x1))))))), F(a, f(a, f(a, f(a, f(b, f(x0, x1)))))), F(a, f(a, f(a, f(b, f(x0, x1))))), F(a, f(a, f(b, f(x0, x1)))))
F(a, f(a, f(b, f(c, f(z0, z1))))) → c1(F(b, f(c, f(b, f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))))), F(c, f(b, f(a, f(a, f(a, f(c, f(z0, z1))))))), F(b, f(a, f(a, f(a, f(c, f(z0, z1)))))), F(a, f(a, f(a, f(c, f(z0, z1))))), F(a, f(a, f(c, f(z0, z1)))), F(a, f(c, f(z0, z1))), F(c, f(z0, z1)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2, c1, c1, c1

(15) 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
b0() → 0
c0() → 0
f0(0, 0) → 1

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