### (0) Obligation:

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

w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

w(r(z0)) → r(w(z0))
b(r(z0)) → r(b(z0))
b(w(z0)) → w(b(z0))
Tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
S tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:

w, b

Defined Pair Symbols:

W, B

Compound Symbols:

c, c1, c2

### (3) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

b(r(z0)) → r(b(z0))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(w(z0)) → w(b(z0))
w(r(z0)) → r(w(z0))
Tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
S tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:

b, w

Defined Pair Symbols:

W, B

Compound Symbols:

c, c1, c2

### (5) 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.

B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
We considered the (Usable) Rules:none
And the Tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(B(x1)) = [2]x1
POL(W(x1)) = [5]
POL(b(x1)) = [4]x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(r(x1)) = [3] + x1
POL(w(x1)) = [4] + [4]x1

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(w(z0)) → w(b(z0))
w(r(z0)) → r(w(z0))
Tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
S tuples:

W(r(z0)) → c(W(z0))
K tuples:

B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
Defined Rule Symbols:

b, w

Defined Pair Symbols:

W, B

Compound Symbols:

c, c1, c2

### (7) 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.

W(r(z0)) → c(W(z0))
We considered the (Usable) Rules:

w(r(z0)) → r(w(z0))
b(w(z0)) → w(b(z0))
And the Tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(B(x1)) = [2]x1
POL(W(x1)) = [4] + [4]x1
POL(b(x1)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(r(x1)) = [2] + x1
POL(w(x1)) = [2] + [3]x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(w(z0)) → w(b(z0))
w(r(z0)) → r(w(z0))
Tuples:

W(r(z0)) → c(W(z0))
B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
S tuples:none
K tuples:

B(r(z0)) → c1(B(z0))
B(w(z0)) → c2(W(b(z0)), B(z0))
W(r(z0)) → c(W(z0))
Defined Rule Symbols:

b, w

Defined Pair Symbols:

W, B

Compound Symbols:

c, c1, c2

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

The set S is empty