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 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
↳13 CpxTrsMatchBoundsTAProof (⇔, 0 ms)
↳14 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, b), c), z0) → f(b, f(a, f(c, f(b, z0))))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(f(f(a, b), c), z0) → c1(F(b, f(a, f(c, f(b, z0)))), F(a, f(c, f(b, z0))), F(c, f(b, z0)), F(b, z0))
F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
S tuples:

F(f(f(a, b), c), z0) → c1(F(b, f(a, f(c, f(b, z0)))), F(a, f(c, f(b, z0))), F(c, f(b, z0)), F(b, z0))
F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), 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(f(f(a, b), c), z0) → c1(F(b, f(a, f(c, f(b, z0)))), F(a, f(c, f(b, z0))), F(c, f(b, z0)), F(b, z0)) by

F(f(f(a, b), c), x0) → c1(F(b, f(f(a, c), f(b, x0))), F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, b), c), z0) → f(b, f(a, f(c, f(b, z0))))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), x0) → c1(F(b, f(f(a, c), f(b, x0))), F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1
S tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), x0) → c1(F(b, f(f(a, c), f(b, x0))), F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c2, c1, c1

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

Removed 1 trailing nodes:

F(f(f(a, b), c), x0) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, b), c), z0) → f(b, f(a, f(c, f(b, z0))))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), x0) → c1(F(b, f(f(a, c), f(b, x0))), F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
S tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), x0) → c1(F(b, f(f(a, c), f(b, x0))), F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c2, c1

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

Use narrowing to replace F(f(f(a, b), c), x0) → c1(F(b, f(f(a, c), f(b, x0))), F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0)) by

F(f(f(a, b), c), z2) → c1(F(b, f(f(f(a, c), b), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(f(a, c), f(f(b, z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, b), c), z0) → f(b, f(a, f(c, f(b, z0))))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(f(a, c), b), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(f(a, c), f(f(b, z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)))
S tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(f(a, c), b), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(f(a, c), f(f(b, z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c2, c1, c1

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

Use narrowing to replace F(f(f(a, b), c), z2) → c1(F(b, f(a, f(f(c, b), z2))), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2)) by

F(f(f(a, b), c), z2) → c1(F(b, f(f(a, f(c, b)), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(f(f(c, b), z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, b), c), z0) → f(b, f(a, f(c, f(b, z0))))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(f(a, c), b), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(f(a, c), f(f(b, z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(a, f(c, b)), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(f(f(c, b), z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
S tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(f(a, c), b), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(f(a, c), f(f(b, z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(a, f(c, b)), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(f(f(c, b), z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c2, c1, c1, c1

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

Use narrowing to replace F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(c, f(f(b, z1), z2)))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2))) by

F(f(f(a, b), c), f(x0, x1)) → c1(F(b, f(f(a, c), f(f(b, x0), x1))), F(a, f(c, f(b, f(x0, x1)))), F(c, f(b, f(x0, x1))), F(b, f(x0, x1)))
F(f(f(a, b), c), f(x0, z2)) → c1(F(b, f(a, f(f(c, f(b, x0)), z2))), F(a, f(c, f(b, f(x0, z2)))), F(c, f(b, f(x0, z2))), F(b, f(x0, z2)))
F(f(f(a, b), c), f(f(z1, z2), x1)) → c1(F(b, f(a, f(c, f(f(f(b, z1), z2), x1)))), F(a, f(c, f(b, f(f(z1, z2), x1)))), F(c, f(b, f(f(z1, z2), x1))), F(b, f(f(z1, z2), x1)))
F(f(f(a, b), c), f(x0, x1)) → c1(F(a, f(c, f(b, f(x0, x1)))), F(c, f(b, f(x0, x1))), F(b, f(x0, x1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(f(a, b), c), z0) → f(b, f(a, f(c, f(b, z0))))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), z2) → c1(F(b, f(f(f(a, c), b), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(f(a, c), f(f(b, z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(a, f(c, b)), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(f(f(c, b), z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
F(f(f(a, b), c), f(x0, z2)) → c1(F(b, f(a, f(f(c, f(b, x0)), z2))), F(a, f(c, f(b, f(x0, z2)))), F(c, f(b, f(x0, z2))), F(b, f(x0, z2)))
F(f(f(a, b), c), f(f(z1, z2), x1)) → c1(F(b, f(a, f(c, f(f(f(b, z1), z2), x1)))), F(a, f(c, f(b, f(f(z1, z2), x1)))), F(c, f(b, f(f(z1, z2), x1))), F(b, f(f(z1, z2), x1)))
F(f(f(a, b), c), f(x0, x1)) → c1(F(a, f(c, f(b, f(x0, x1)))), F(c, f(b, f(x0, x1))), F(b, f(x0, x1)))
S tuples:

F(z0, f(z1, z2)) → c2(F(f(z0, z1), z2), F(z0, z1))
F(f(f(a, b), c), z2) → c1(F(b, f(f(f(a, c), b), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(f(a, c), f(f(b, z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)))
F(f(f(a, b), c), z2) → c1(F(b, f(f(a, f(c, b)), z2)), F(a, f(c, f(b, z2))), F(c, f(b, z2)), F(b, z2))
F(f(f(a, b), c), f(z1, z2)) → c1(F(b, f(a, f(f(f(c, b), z1), z2))), F(a, f(c, f(b, f(z1, z2)))), F(c, f(b, f(z1, z2))), F(b, f(z1, z2)))
F(f(f(a, b), c), x0) → c1(F(a, f(c, f(b, x0))), F(c, f(b, x0)), F(b, x0))
F(f(f(a, b), c), f(x0, z2)) → c1(F(b, f(a, f(f(c, f(b, x0)), z2))), F(a, f(c, f(b, f(x0, z2)))), F(c, f(b, f(x0, z2))), F(b, f(x0, z2)))
F(f(f(a, b), c), f(f(z1, z2), x1)) → c1(F(b, f(a, f(c, f(f(f(b, z1), z2), x1)))), F(a, f(c, f(b, f(f(z1, z2), x1)))), F(c, f(b, f(f(z1, z2), x1))), F(b, f(f(z1, z2), x1)))
F(f(f(a, b), c), f(x0, x1)) → c1(F(a, f(c, f(b, f(x0, x1)))), F(c, f(b, f(x0, x1))), F(b, f(x0, x1)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c2, c1, c1, c1

(13) 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

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