The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(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), 1 ms)
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 SIsEmptyProof (⇔, 0 ms)
↳22 BOUNDS(O(1), O(1))

(0) Obligation:

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

a(f, a(f, x)) → a(x, g)
a(x, g) → a(f, a(g, a(f, x)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(z0, g) → c1(A(f, a(g, a(f, z0))), A(g, a(f, z0)), A(f, z0))
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(z0, g) → c1(A(f, a(g, a(f, z0))), A(g, a(f, z0)), A(f, z0))
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

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

Use narrowing to replace A(z0, g) → c1(A(f, a(g, a(f, z0))), A(g, a(f, z0)), A(f, z0)) by

A(g, g) → c1(A(f, a(g, a(f, a(g, a(f, f))))), A(g, a(f, g)), A(f, g))
A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(x0, g) → c1

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(f, a(g, a(f, a(g, a(f, f))))), A(g, a(f, g)), A(f, g))
A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(x0, g) → c1
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(f, a(g, a(f, a(g, a(f, f))))), A(g, a(f, g)), A(f, g))
A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(x0, g) → c1
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1, c1

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

Removed 1 trailing nodes:

A(x0, g) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(f, a(g, a(f, a(g, a(f, f))))), A(g, a(f, g)), A(f, g))
A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(f, a(g, a(f, a(g, a(f, f))))), A(g, a(f, g)), A(f, g))
A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

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

Use narrowing to replace A(g, g) → c1(A(f, a(g, a(f, a(g, a(f, f))))), A(g, a(f, g)), A(f, g)) by

A(g, g) → c1(A(g, a(f, g)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(g, g) → c1(A(g, a(f, g)))
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(g, g) → c1(A(g, a(f, g)))
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1, c1

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

Use narrowing to replace A(a(f, z0), g) → c1(A(f, a(g, a(z0, g))), A(g, a(f, a(f, z0))), A(f, a(f, z0))) by

A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(a(f, x0), g) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(g, a(f, g)))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(a(f, x0), g) → c1
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(g, a(f, g)))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(a(f, x0), g) → c1
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1, c1, c1

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

Removed 1 trailing nodes:

A(a(f, x0), g) → c1

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(g, a(f, g)))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(g, g) → c1(A(g, a(f, g)))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1, c1

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

Use narrowing to replace A(g, g) → c1(A(g, a(f, g))) by

A(g, g) → c1(A(g, a(f, a(g, a(f, f)))))
A(g, g) → c1

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(g, g) → c1(A(g, a(f, a(g, a(f, f)))))
A(g, g) → c1
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(g, g) → c1(A(g, a(f, a(g, a(f, f)))))
A(g, g) → c1
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1, c1, c1

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

Removed 2 trailing nodes:

A(g, g) → c1(A(g, a(f, a(g, a(f, f)))))
A(g, g) → c1

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(f, a(f, z0)) → c(A(z0, g))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
S tuples:

A(f, a(f, z0)) → c(A(z0, g))
A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c, c1

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

Use forward instantiation to replace A(f, a(f, z0)) → c(A(z0, g)) by

A(f, a(f, a(f, y0))) → c(A(a(f, y0), g))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(f, a(f, a(f, y0))) → c(A(a(f, y0), g))
S tuples:

A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(f, a(f, a(f, y0))) → c(A(a(f, y0), g))
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1, c

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

Removed 2 trailing nodes:

A(a(f, z0), g) → c1(A(f, a(g, a(f, a(g, a(f, z0))))), A(g, a(f, a(f, z0))), A(f, a(f, z0)))
A(f, a(f, a(f, y0))) → c(A(a(f, y0), g))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(f, a(f, z0)) → a(z0, g)
a(z0, g) → a(f, a(g, a(f, z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:none

Compound Symbols:none

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(22) BOUNDS(O(1), O(1))