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 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 134 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (⇔, 0 ms)
↳14 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 1 trailing nodes:

F(f(a, x0), a) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 1 trailing nodes:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c

(11) 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, f(a, f(a, z0))), a) → c(F(f(f(f(f(f(f(z0, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), F(f(f(a, f(a, z0)), a), f(a, a)), F(f(a, f(a, z0)), a), F(a, a))
We considered the (Usable) Rules:

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

F(f(a, f(a, f(a, z0))), a) → c(F(f(f(f(f(f(f(z0, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), F(f(f(a, f(a, z0)), a), f(a, a)), F(f(a, f(a, z0)), a), F(a, a))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [4] + [4]x1 + [2]x2   
POL(a) = 0   
POL(c(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(f(x1, x2)) = [2] + [4]x2   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(f(a, f(a, f(a, z0))), a) → c(F(f(f(f(f(f(f(z0, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), F(f(f(a, f(a, z0)), a), f(a, a)), F(f(a, f(a, z0)), a), F(a, a))
S tuples:none
K tuples:

F(f(a, f(a, f(a, z0))), a) → c(F(f(f(f(f(f(f(z0, a), f(a, a)), a), f(a, a)), a), f(a, a)), a), F(f(f(a, f(a, z0)), a), f(a, a)), F(f(a, f(a, z0)), a), F(a, a))
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))