The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳12 CdtProblem
↳13 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳14 CdtProblem
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳16 CdtProblem
↳17 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳18 CdtProblem
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳20 CdtProblem
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳22 CdtProblem
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳24 CdtProblem
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳26 CdtProblem
↳27 SIsEmptyProof (⇔)
↳28 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(a(z0)), A(z0))
C(c(z0)) → c3(A(c(a(z0))), C(a(z0)), A(z0))
S tuples:

A(b(z0)) → c2(C(a(z0)), A(z0))
C(c(z0)) → c3(A(c(a(z0))), C(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c3

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

Use narrowing to replace A(b(z0)) → c2(C(a(z0)), A(z0)) by

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

C(c(z0)) → c3(A(c(a(z0))), C(a(z0)), A(z0))
A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(C(b(c(a(z0)))), A(b(z0)))
S tuples:

C(c(z0)) → c3(A(c(a(z0))), C(a(z0)), A(z0))
A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(C(b(c(a(z0)))), A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c2

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

C(c(z0)) → c3(A(c(a(z0))), C(a(z0)), A(z0))
A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
S tuples:

C(c(z0)) → c3(A(c(a(z0))), C(a(z0)), A(z0))
A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c2, c2

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

Use narrowing to replace C(c(z0)) → c3(A(c(a(z0))), C(a(z0)), A(z0)) by

C(c(z0)) → c3(A(c(z0)), C(a(z0)), A(z0))
C(c(b(z0))) → c3(A(c(b(c(a(z0))))), C(a(b(z0))), A(b(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(A(c(z0)), C(a(z0)), A(z0))
C(c(b(z0))) → c3(A(c(b(c(a(z0))))), C(a(b(z0))), A(b(z0)))
S tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(A(c(z0)), C(a(z0)), A(z0))
C(c(b(z0))) → c3(A(c(b(c(a(z0))))), C(a(b(z0))), A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3

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

Removed 2 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(b(z0))), A(b(z0)))
S tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(a(z0)), A(z0))
C(c(b(z0))) → c3(C(a(b(z0))), A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3

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

Use narrowing to replace C(c(z0)) → c3(C(a(z0)), A(z0)) by

C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(C(b(c(a(z0)))), A(b(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(b(z0))) → c3(C(a(b(z0))), A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(C(b(c(a(z0)))), A(b(z0)))
S tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(b(z0))) → c3(C(a(b(z0))), A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(C(b(c(a(z0)))), A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(b(z0))) → c3(C(a(b(z0))), A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
S tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(b(z0))) → c3(C(a(b(z0))), A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3, c3

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

Use narrowing to replace C(c(b(z0))) → c3(C(a(b(z0))), A(b(z0))) by

C(c(b(x0))) → c3(C(b(x0)), A(b(x0)))
C(c(b(z0))) → c3(C(b(c(a(z0)))), A(b(z0)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
C(c(b(x0))) → c3(C(b(x0)), A(b(x0)))
C(c(b(z0))) → c3(C(b(c(a(z0)))), A(b(z0)))
S tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
C(c(b(x0))) → c3(C(b(x0)), A(b(x0)))
C(c(b(z0))) → c3(C(b(c(a(z0)))), A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3, c3

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

Removed 2 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
S tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3, c3

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

C(c(b(z0))) → c3(A(b(z0)))
We considered the (Usable) Rules:none
And the Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x1)) = x12   
POL(C(x1)) = [2]x1   
POL(b(x1)) = [1] + x1   
POL(c(x1)) = x1 + x12   
POL(c2(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
S tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
K tuples:

C(c(b(z0))) → c3(A(b(z0)))
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3, c3

(21) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

A(b(z0)) → c2(C(z0), A(z0))
We considered the (Usable) Rules:none
And the Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x1)) = [2]x12   
POL(C(x1)) = x1   
POL(b(x1)) = [1] + x1   
POL(c(x1)) = x1 + [2]x12   
POL(c2(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
S tuples:

A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
K tuples:

C(c(b(z0))) → c3(A(b(z0)))
A(b(z0)) → c2(C(z0), A(z0))
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3, c3

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

C(c(z0)) → c3(C(z0), A(z0))
We considered the (Usable) Rules:none
And the Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x1)) = [1] + x12   
POL(C(x1)) = x1   
POL(b(x1)) = [1] + x1   
POL(c(x1)) = [2] + x1 + x12   
POL(c2(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
S tuples:

A(b(b(z0))) → c2(A(b(z0)))
K tuples:

C(c(b(z0))) → c3(A(b(z0)))
A(b(z0)) → c2(C(z0), A(z0))
C(c(z0)) → c3(C(z0), A(z0))
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3, c3

(25) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

A(b(b(z0))) → c2(A(b(z0)))
We considered the (Usable) Rules:none
And the Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x1)) = [2]x12   
POL(C(x1)) = [3] + x1   
POL(b(x1)) = [2] + x1   
POL(c(x1)) = [2]x1 + [2]x12   
POL(c2(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → z0
a(b(z0)) → b(c(a(z0)))
c(c(z0)) → b(a(c(a(z0))))
Tuples:

A(b(z0)) → c2(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
C(c(z0)) → c3(C(z0), A(z0))
C(c(b(z0))) → c3(A(b(z0)))
S tuples:none
K tuples:

C(c(b(z0))) → c3(A(b(z0)))
A(b(z0)) → c2(C(z0), A(z0))
C(c(z0)) → c3(C(z0), A(z0))
A(b(b(z0))) → c2(A(b(z0)))
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c3, c3

(27) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(28) BOUNDS(O(1), O(1))