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 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 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(x, f(y, a)) → f(f(a, f(h(x), y)), a)

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

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F(x0, f(a, a)) → c(F(f(f(a, f(h(a), h(x0))), a), a), F(a, f(h(x0), a)), F(h(x0), a))
F(x0, f(f(z1, a), a)) → c(F(f(a, f(f(a, f(h(h(x0)), z1)), a)), a), F(a, f(h(x0), f(z1, a))), F(h(x0), f(z1, a)))
F(x0, f(x1, 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 2 trailing nodes:

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

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

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

f

Defined Pair Symbols:

F

Compound Symbols:

c, c

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

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

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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