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 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 CdtProblem
    ↳9 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtNarrowingProof (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(x, f(y, a)) → h(f(a, f(f(x, a), y)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

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

F(a, f(z1, a)) → c(F(a, f(f(a, a), z1)), F(f(a, a), z1))
F(f(x0, a), f(z1, a)) → c(F(a, f(f(f(x0, a), a), z1)), F(f(f(x0, a), a), z1))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use instantiation to replace F(f(x0, a), f(z1, a)) → c(F(a, f(f(f(x0, a), a), z1)), F(f(f(x0, a), a), z1)) by

F(f(a, a), f(z1, a)) → c(F(a, f(f(f(a, a), a), z1)), F(f(f(a, a), a), z1))
F(f(f(x0, a), a), f(z1, a)) → c(F(a, f(f(f(f(x0, a), a), a), z1)), F(f(f(f(x0, a), a), a), z1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use forward instantiation to replace F(a, f(z1, a)) → c(F(a, f(f(a, a), z1)), F(f(a, a), z1)) by

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

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

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

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

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

F(a, f(f(z1, a), a)) → c(F(a, h(f(a, f(f(f(a, a), a), z1)))), F(f(a, a), f(z1, a)))
F(a, f(f(x0, a), a)) → c(F(f(a, a), f(x0, a)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(a, a), f(z1, a)) → c(F(a, f(f(f(a, a), a), z1)), F(f(f(a, a), a), z1))
F(f(f(x0, a), a), f(z1, a)) → c(F(a, f(f(f(f(x0, a), a), a), z1)), F(f(f(f(x0, a), a), a), z1))
F(a, f(a, a)) → c(F(a, f(f(a, a), a)))
F(a, f(f(z1, a), a)) → c(F(a, h(f(a, f(f(f(a, a), a), z1)))), F(f(a, a), f(z1, a)))
F(a, f(f(x0, a), a)) → c(F(f(a, a), f(x0, 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, a), f(z1, a)) → c(F(a, f(f(f(a, a), a), z1)), F(f(f(a, a), a), z1)) by

F(f(a, a), f(f(y1, a), a)) → c(F(a, f(f(f(a, a), a), f(y1, a))), F(f(f(a, a), a), f(y1, a)))
F(f(a, a), f(a, a)) → c(F(a, f(f(f(a, a), a), a)), F(f(f(a, a), a), a))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(f(x0, a), a), f(z1, a)) → c(F(a, f(f(f(f(x0, a), a), a), z1)), F(f(f(f(x0, a), a), a), z1))
F(a, f(a, a)) → c(F(a, f(f(a, a), a)))
F(a, f(f(z1, a), a)) → c(F(a, h(f(a, f(f(f(a, a), a), z1)))), F(f(a, a), f(z1, a)))
F(a, f(f(x0, a), a)) → c(F(f(a, a), f(x0, a)))
F(f(a, a), f(f(y1, a), a)) → c(F(a, f(f(f(a, a), a), f(y1, a))), F(f(f(a, a), a), f(y1, a)))
F(f(a, a), f(a, a)) → c(F(a, f(f(f(a, a), a), a)), F(f(f(a, a), a), a))
S tuples:

F(f(f(x0, a), a), f(z1, a)) → c(F(a, f(f(f(f(x0, a), a), a), z1)), F(f(f(f(x0, a), a), a), z1))
F(a, f(a, a)) → c(F(a, f(f(a, a), a)))
F(a, f(f(z1, a), a)) → c(F(a, h(f(a, f(f(f(a, a), a), z1)))), F(f(a, a), f(z1, a)))
F(a, f(f(x0, a), a)) → c(F(f(a, a), f(x0, a)))
F(f(a, a), f(f(y1, a), a)) → c(F(a, f(f(f(a, a), a), f(y1, a))), F(f(f(a, a), a), f(y1, a)))
F(f(a, a), f(a, a)) → c(F(a, f(f(f(a, a), a), a)), F(f(f(a, a), a), a))
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
h0(0) → 0
f0(0, 0) → 1

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