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 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳16 CdtProblem
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 49 ms)
↳18 CdtProblem
↳19 SIsEmptyProof (⇔, 0 ms)
↳20 BOUNDS(O(1), O(1))

(0) Obligation:

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

f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)

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

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

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

F(x0, f(f(z1, a), a)) → c(F(f(f(f(a, a), f(z1, a)), h(a)), x0), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
F(x0, f(x1, a)) → c(F(f(f(f(a, a), x1), h(a)), x0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(x0, f(f(z1, a), a)) → c(F(f(f(f(a, a), f(z1, a)), h(a)), x0), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
F(x0, f(x1, a)) → c(F(f(f(f(a, a), x1), h(a)), x0))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

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

F(x0, f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), x0), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
F(x0, f(f(x1, a), a)) → c(F(f(f(f(f(a, a), x1), h(a)), f(a, a)), h(a)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(x0, f(x1, a)) → c(F(f(f(f(a, a), x1), h(a)), x0))
F(x0, f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), x0), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
F(x0, f(f(x1, a), a)) → c(F(f(f(f(f(a, a), x1), h(a)), f(a, a)), h(a)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 1 trailing nodes:

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(x0, f(x1, a)) → c(F(f(f(f(a, a), x1), h(a)), x0))
F(x0, f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), x0), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

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

F(f(y0, h(a)), f(z1, a)) → c(F(f(f(f(a, a), z1), h(a)), f(y0, h(a))))
F(f(a, a), f(x1, a)) → c(F(f(f(f(a, a), x1), h(a)), f(a, a)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(x0, f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), x0), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
F(f(y0, h(a)), f(z1, a)) → c(F(f(f(f(a, a), z1), h(a)), f(y0, h(a))))
F(f(a, a), f(x1, a)) → c(F(f(f(f(a, a), x1), h(a)), f(a, a)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 2 trailing nodes:

F(f(y0, h(a)), f(z1, a)) → c(F(f(f(f(a, a), z1), h(a)), f(y0, h(a))))
F(f(a, a), f(x1, a)) → c(F(f(f(f(a, a), x1), h(a)), f(a, a)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c

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

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

F(f(y1, h(a)), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(y1, h(a))), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
F(f(a, a), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(a, a)), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(y1, h(a)), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(y1, h(a))), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
F(f(a, a), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(a, a)), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

F(f(y1, h(a)), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(y1, h(a))), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(f(a, a), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(a, a)), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
We considered the (Usable) Rules:

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

F(f(a, a), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(a, a)), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [5]x2   
POL(a) = [5]   
POL(c(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(f(x1, x2)) = [2] + [4]x1 + [5]x2   
POL(h(x1)) = [2] + x1   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, a), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(a, a)), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
S tuples:none
K tuples:

F(f(a, a), f(f(z1, a), a)) → c(F(f(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), f(a, a)), F(f(f(f(f(a, a), z1), h(a)), f(a, a)), h(a)), F(f(a, a), f(z1, a)), F(a, a))
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(19) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(20) BOUNDS(O(1), O(1))