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

(0) Obligation:

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

f(f(a, x), y) → f(f(x, f(a, y)), a)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(a, z0), z1) → c(F(f(z0, f(a, z1)), a), F(z0, f(a, z1)), F(a, z1))
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(f(a, z0), z1) → c(F(f(z0, f(a, z1)), a), F(z0, f(a, z1)), F(a, z1)) by

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, x0), x1) → c(F(f(x0, f(a, x1)), a), F(x0, f(a, x1)))
S tuples:

F(f(a, x0), x1) → c(F(f(x0, f(a, x1)), a), F(x0, f(a, x1)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use forward instantiation to replace F(f(a, x0), x1) → c(F(f(x0, f(a, x1)), a), F(x0, f(a, x1))) by

F(f(a, a), z1) → c(F(f(a, f(a, z1)), a), F(a, f(a, z1)))
F(f(a, f(a, y0)), z1) → c(F(f(f(a, y0), f(a, z1)), a), F(f(a, y0), f(a, z1)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, a), z1) → c(F(f(a, f(a, z1)), a), F(a, f(a, z1)))
F(f(a, f(a, y0)), z1) → c(F(f(f(a, y0), f(a, z1)), a), F(f(a, y0), f(a, z1)))
S tuples:

F(f(a, a), z1) → c(F(f(a, f(a, z1)), a), F(a, f(a, z1)))
F(f(a, f(a, y0)), z1) → c(F(f(f(a, y0), f(a, z1)), a), F(f(a, y0), f(a, z1)))
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(f(a, a), z1) → c(F(f(a, f(a, z1)), a), F(a, f(a, z1))) by

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, f(a, y0)), z1) → c(F(f(f(a, y0), f(a, z1)), a), F(f(a, y0), f(a, z1)))
F(f(a, a), x0) → c(F(f(a, f(a, x0)), a))
S tuples:

F(f(a, f(a, y0)), z1) → c(F(f(f(a, y0), f(a, z1)), a), F(f(a, y0), f(a, z1)))
F(f(a, a), x0) → c(F(f(a, f(a, x0)), a))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Use narrowing to replace F(f(a, f(a, y0)), z1) → c(F(f(f(a, y0), f(a, z1)), a), F(f(a, y0), f(a, z1))) by

F(f(a, f(a, z0)), x1) → c(F(f(f(z0, f(a, f(a, x1))), a), a), F(f(a, z0), f(a, x1)))
F(f(a, f(a, x0)), x1) → c(F(f(a, x0), f(a, x1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, a), x0) → c(F(f(a, f(a, x0)), a))
F(f(a, f(a, z0)), x1) → c(F(f(f(z0, f(a, f(a, x1))), a), a), F(f(a, z0), f(a, x1)))
F(f(a, f(a, x0)), x1) → c(F(f(a, x0), f(a, x1)))
S tuples:

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Use instantiation to replace F(f(a, a), x0) → c(F(f(a, f(a, x0)), a)) by

F(f(a, a), a) → c(F(f(a, f(a, a)), a))
F(f(a, a), f(a, x1)) → c(F(f(a, f(a, f(a, x1))), a))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, f(a, z0)), x1) → c(F(f(f(z0, f(a, f(a, x1))), a), a), F(f(a, z0), f(a, x1)))
F(f(a, f(a, x0)), x1) → c(F(f(a, x0), f(a, x1)))
F(f(a, a), a) → c(F(f(a, f(a, a)), a))
F(f(a, a), f(a, x1)) → c(F(f(a, f(a, f(a, x1))), a))
S tuples:

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Use instantiation to replace F(f(a, f(a, z0)), x1) → c(F(f(f(z0, f(a, f(a, x1))), a), a), F(f(a, z0), f(a, x1))) by

F(f(a, f(a, z0)), a) → c(F(f(f(z0, f(a, f(a, a))), a), a), F(f(a, z0), f(a, a)))
F(f(a, f(a, z0)), f(a, x1)) → c(F(f(f(z0, f(a, f(a, f(a, x1)))), a), a), F(f(a, z0), f(a, f(a, x1))))
F(f(a, f(a, a)), a) → c(F(f(f(a, f(a, f(a, a))), a), a), F(f(a, a), f(a, a)))
F(f(a, f(a, f(a, x0))), a) → c(F(f(f(f(a, x0), f(a, f(a, a))), a), a), F(f(a, f(a, x0)), f(a, a)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, f(a, x0)), x1) → c(F(f(a, x0), f(a, x1)))
F(f(a, a), a) → c(F(f(a, f(a, a)), a))
F(f(a, a), f(a, x1)) → c(F(f(a, f(a, f(a, x1))), a))
F(f(a, f(a, z0)), a) → c(F(f(f(z0, f(a, f(a, a))), a), a), F(f(a, z0), f(a, a)))
F(f(a, f(a, z0)), f(a, x1)) → c(F(f(f(z0, f(a, f(a, f(a, x1)))), a), a), F(f(a, z0), f(a, f(a, x1))))
F(f(a, f(a, a)), a) → c(F(f(f(a, f(a, f(a, a))), a), a), F(f(a, a), f(a, a)))
F(f(a, f(a, f(a, x0))), a) → c(F(f(f(f(a, x0), f(a, f(a, a))), a), a), F(f(a, f(a, x0)), f(a, a)))
S tuples:

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Use forward instantiation to replace F(f(a, f(a, x0)), x1) → c(F(f(a, x0), f(a, x1))) by

F(f(a, f(a, f(a, y0))), z1) → c(F(f(a, f(a, y0)), f(a, z1)))
F(f(a, f(a, a)), z1) → c(F(f(a, a), f(a, z1)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, a), a) → c(F(f(a, f(a, a)), a))
F(f(a, a), f(a, x1)) → c(F(f(a, f(a, f(a, x1))), a))
F(f(a, f(a, z0)), a) → c(F(f(f(z0, f(a, f(a, a))), a), a), F(f(a, z0), f(a, a)))
F(f(a, f(a, z0)), f(a, x1)) → c(F(f(f(z0, f(a, f(a, f(a, x1)))), a), a), F(f(a, z0), f(a, f(a, x1))))
F(f(a, f(a, a)), a) → c(F(f(f(a, f(a, f(a, a))), a), a), F(f(a, a), f(a, a)))
F(f(a, f(a, f(a, x0))), a) → c(F(f(f(f(a, x0), f(a, f(a, a))), a), a), F(f(a, f(a, x0)), f(a, a)))
F(f(a, f(a, f(a, y0))), z1) → c(F(f(a, f(a, y0)), f(a, z1)))
F(f(a, f(a, a)), z1) → c(F(f(a, a), f(a, z1)))
S tuples:

F(f(a, a), a) → c(F(f(a, f(a, a)), a))
F(f(a, a), f(a, x1)) → c(F(f(a, f(a, f(a, x1))), a))
F(f(a, f(a, z0)), a) → c(F(f(f(z0, f(a, f(a, a))), a), a), F(f(a, z0), f(a, a)))
F(f(a, f(a, z0)), f(a, x1)) → c(F(f(f(z0, f(a, f(a, f(a, x1)))), a), a), F(f(a, z0), f(a, f(a, x1))))
F(f(a, f(a, a)), a) → c(F(f(f(a, f(a, f(a, a))), a), a), F(f(a, a), f(a, a)))
F(f(a, f(a, f(a, x0))), a) → c(F(f(f(f(a, x0), f(a, f(a, a))), a), a), F(f(a, f(a, x0)), f(a, a)))
F(f(a, f(a, f(a, y0))), z1) → c(F(f(a, f(a, y0)), f(a, z1)))
F(f(a, f(a, a)), z1) → c(F(f(a, a), f(a, z1)))
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))