### (0) Obligation:

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

c(c(b(c(x)))) → b(a(0, c(x)))
c(c(x)) → b(c(b(c(x))))
a(0, x) → c(c(x))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(b(c(z0))), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(b(c(z0))), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A

Compound Symbols:

c1, c2, c3

### (3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A

Compound Symbols:

c1, c3, c2

### (5) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use instantiation to replace A(0, z0) → c3(C(c(z0)), C(z0)) by

A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c3(C(c(z0)), C(z0))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c3(C(c(z0)), C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c3(C(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3

### (7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3, c4

### (9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

A1(0, z0) → c4(C(z0))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3, c4

### (11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

A1(0, z0) → c4(C(c(z0)))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3, c4

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

Use narrowing to replace A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0))) by

A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

### (15) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

### (17) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
We considered the (Usable) Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
And the Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(A(x1, x2)) = [1] + [5]x1 + x2
POL(A1(x1, x2)) = [4] + [4]x1 + [5]x2
POL(C(x1)) = x1
POL(a(x1, x2)) = [3] + [2]x1 + [4]x2
POL(b(x1)) = x1
POL(c(x1)) = [1] + [2]x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
S tuples:none
K tuples:

A1(0, z0) → c4(C(c(z0)))
C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

### (19) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty