The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

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

a(b(a(a(x1)))) → c(b(a(b(a(x1)))))
a(c(b(x1))) → a(a(b(c(b(a(x1))))))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(a(a(z0)))) → c(b(a(b(a(z0)))))
a(c(b(z0))) → a(a(b(c(b(a(z0))))))

Tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(a(b(c(b(a(z0)))))), A(b(c(b(a(z0))))), A(z0))

S tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(a(b(c(b(a(z0)))))), A(b(c(b(a(z0))))), A(z0))

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1, c2

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(a(a(z0)))) → c(b(a(b(a(z0)))))
a(c(b(z0))) → a(a(b(c(b(a(z0))))))

Tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(z0))

S tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(z0))

K tuples:none
Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1, c2

(5) 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.

A(c(b(z0))) → c2(A(z0))

We considered the (Usable) Rules:none
And the Tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x₁)) = [2]x₁   
POL(a(x₁)) = [4]x₁   
POL(b(x₁)) = [4] + x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(a(a(z0)))) → c(b(a(b(a(z0)))))
a(c(b(z0))) → a(a(b(c(b(a(z0))))))

Tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(z0))

S tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))

K tuples:

A(c(b(z0))) → c2(A(z0))

Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1, c2

(7) 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.

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))

We considered the (Usable) Rules:none
And the Tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x₁)) = [3] + x₁   
POL(a(x₁)) = [2] + [4]x₁   
POL(b(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(a(a(z0)))) → c(b(a(b(a(z0)))))
a(c(b(z0))) → a(a(b(c(b(a(z0))))))

Tuples:

A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))
A(c(b(z0))) → c2(A(z0))

S tuples:none
K tuples:

A(c(b(z0))) → c2(A(z0))
A(b(a(a(z0)))) → c1(A(b(a(z0))), A(z0))

Defined Rule Symbols:

a

Defined Pair Symbols:

A

Compound Symbols:

c1, c2

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))