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 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtRewritingProof (BOTH BOUNDS(ID, ID), 4 ms)
    ↳18 CdtProblem
↳19 CpxTrsMatchBoundsTAProof (⇔, 0 ms)
↳20 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(a, f(f(x, h(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(a, f(f(z0, h(z1)), a))
Tuples:

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Removed 2 trailing nodes:

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(a, f(a, z0)), x1) → c(F(a, f(f(a, f(f(z0, h(h(x1))), a)), a)), F(f(f(a, z0), h(x1)), a), F(f(a, z0), h(x1)))
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)), x1) → c(F(a, f(f(a, f(f(z0, h(h(x1))), a)), a)), F(f(f(a, z0), h(x1)), a), F(f(a, z0), h(x1))) by

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Used rewriting to replace F(f(a, f(a, x0)), x1) → c(F(a, f(a, f(f(f(f(x0, h(h(x1))), a), h(a)), a))), F(f(f(a, x0), h(x1)), a), F(f(a, x0), h(x1))) by F(f(a, f(a, z0)), z1) → c(F(a, f(a, f(f(f(f(z0, h(h(z1))), a), h(a)), a))), F(f(a, f(f(z0, h(h(z1))), a)), a), F(f(a, z0), h(z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(a, f(a, f(a, z0))), x1) → c(F(a, f(f(a, f(f(a, f(f(z0, h(h(h(x1)))), a)), a)), a)), F(f(f(a, f(a, z0)), h(x1)), a), F(f(a, f(a, z0)), h(x1)))
F(f(a, f(a, x0)), x1) → c(F(f(f(a, x0), h(x1)), a), F(f(a, x0), h(x1)))
F(f(a, f(a, z0)), z1) → c(F(a, f(a, f(f(f(f(z0, h(h(z1))), a), h(a)), a))), F(f(a, f(f(z0, h(h(z1))), a)), a), F(f(a, z0), h(z1)))
F(f(a, f(a, a)), a) → c(F(a, f(f(a, f(a, f(f(h(h(a)), h(a)), a))), a)), F(f(f(a, a), h(a)), a), F(f(a, a), h(a)))
F(f(a, f(a, a)), h(x1)) → c(F(a, f(f(a, f(a, f(f(h(h(h(x1))), h(a)), a))), a)), F(f(f(a, a), h(h(x1))), a), F(f(a, a), h(h(x1))))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

Used rewriting to replace F(f(a, f(a, f(a, z0))), x1) → c(F(a, f(f(a, f(f(a, f(f(z0, h(h(h(x1)))), a)), a)), a)), F(f(f(a, f(a, z0)), h(x1)), a), F(f(a, f(a, z0)), h(x1))) by F(f(a, f(a, f(a, z0))), z1) → c(F(a, f(f(a, f(f(a, f(f(z0, h(h(h(z1)))), a)), a)), a)), F(f(a, f(f(f(a, z0), h(h(z1))), a)), a), F(f(a, f(a, z0)), h(z1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(a, f(a, x0)), x1) → c(F(f(f(a, x0), h(x1)), a), F(f(a, x0), h(x1)))
F(f(a, f(a, z0)), z1) → c(F(a, f(a, f(f(f(f(z0, h(h(z1))), a), h(a)), a))), F(f(a, f(f(z0, h(h(z1))), a)), a), F(f(a, z0), h(z1)))
F(f(a, f(a, a)), a) → c(F(a, f(f(a, f(a, f(f(h(h(a)), h(a)), a))), a)), F(f(f(a, a), h(a)), a), F(f(a, a), h(a)))
F(f(a, f(a, a)), h(x1)) → c(F(a, f(f(a, f(a, f(f(h(h(h(x1))), h(a)), a))), a)), F(f(f(a, a), h(h(x1))), a), F(f(a, a), h(h(x1))))
F(f(a, f(a, f(a, z0))), z1) → c(F(a, f(f(a, f(f(a, f(f(z0, h(h(h(z1)))), a)), a)), a)), F(f(a, f(f(f(a, z0), h(h(z1))), a)), a), F(f(a, f(a, z0)), h(z1)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

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

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

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(a, f(a, z0)), z1) → c(F(a, f(a, f(f(f(f(z0, h(h(z1))), a), h(a)), a))), F(f(a, f(f(z0, h(h(z1))), a)), a), F(f(a, z0), h(z1)))
F(f(a, f(a, a)), a) → c(F(a, f(f(a, f(a, f(f(h(h(a)), h(a)), a))), a)), F(f(f(a, a), h(a)), a), F(f(a, a), h(a)))
F(f(a, f(a, a)), h(x1)) → c(F(a, f(f(a, f(a, f(f(h(h(h(x1))), h(a)), a))), a)), F(f(f(a, a), h(h(x1))), a), F(f(a, a), h(h(x1))))
F(f(a, f(a, f(a, z0))), z1) → c(F(a, f(f(a, f(f(a, f(f(z0, h(h(h(z1)))), a)), a)), a)), F(f(a, f(f(f(a, z0), h(h(z1))), a)), a), F(f(a, f(a, z0)), h(z1)))
F(f(a, f(a, z0)), a) → c(F(f(f(a, z0), h(a)), a), F(f(a, z0), h(a)))
F(f(a, f(a, z0)), h(x1)) → c(F(f(f(a, z0), h(h(x1))), a), F(f(a, z0), h(h(x1))))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

(17) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace F(f(a, f(a, a)), a) → c(F(a, f(f(a, f(a, f(f(h(h(a)), h(a)), a))), a)), F(f(f(a, a), h(a)), a), F(f(a, a), h(a))) by F(f(a, f(a, a)), a) → c(F(a, f(f(a, f(a, f(f(h(h(a)), h(a)), a))), a)), F(f(a, f(f(a, h(h(a))), a)), a), F(f(a, a), h(a)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(f(a, f(a, z0)), z1) → c(F(a, f(a, f(f(f(f(z0, h(h(z1))), a), h(a)), a))), F(f(a, f(f(z0, h(h(z1))), a)), a), F(f(a, z0), h(z1)))
F(f(a, f(a, a)), h(x1)) → c(F(a, f(f(a, f(a, f(f(h(h(h(x1))), h(a)), a))), a)), F(f(f(a, a), h(h(x1))), a), F(f(a, a), h(h(x1))))
F(f(a, f(a, f(a, z0))), z1) → c(F(a, f(f(a, f(f(a, f(f(z0, h(h(h(z1)))), a)), a)), a)), F(f(a, f(f(f(a, z0), h(h(z1))), a)), a), F(f(a, f(a, z0)), h(z1)))
F(f(a, f(a, z0)), a) → c(F(f(f(a, z0), h(a)), a), F(f(a, z0), h(a)))
F(f(a, f(a, z0)), h(x1)) → c(F(f(f(a, z0), h(h(x1))), a), F(f(a, z0), h(h(x1))))
F(f(a, f(a, a)), a) → c(F(a, f(f(a, f(a, f(f(h(h(a)), h(a)), a))), a)), F(f(a, f(f(a, h(h(a))), a)), a), F(f(a, a), h(a)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

(19) 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

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