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:

c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(z0, z1, a) → f(b(b(f(z0), z0), z1))
b(z0, b(z1, a)) → f(b(c(f(a), z0, z1), z1))
f(c(c(z0, a, a), z1, a)) → z0

Tuples:

C(z0, z1, a) → c1(F(b(b(f(z0), z0), z1)), B(b(f(z0), z0), z1), B(f(z0), z0), F(z0))
B(z0, b(z1, a)) → c2(F(b(c(f(a), z0, z1), z1)), B(c(f(a), z0, z1), z1), C(f(a), z0, z1), F(a))

S tuples:

C(z0, z1, a) → c1(F(b(b(f(z0), z0), z1)), B(b(f(z0), z0), z1), B(f(z0), z0), F(z0))
B(z0, b(z1, a)) → c2(F(b(c(f(a), z0, z1), z1)), B(c(f(a), z0, z1), z1), C(f(a), z0, z1), F(a))

K tuples:none
Defined Rule Symbols:

c, b, f

Defined Pair Symbols:

C, B

Compound Symbols:

c1, c2

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

Removed 4 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(z0, z1, a) → f(b(b(f(z0), z0), z1))
b(z0, b(z1, a)) → f(b(c(f(a), z0, z1), z1))
f(c(c(z0, a, a), z1, a)) → z0

Tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))
B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

S tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))
B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

K tuples:none
Defined Rule Symbols:

c, b, f

Defined Pair Symbols:

C, B

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.

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))

We considered the (Usable) Rules:

c(z0, z1, a) → f(b(b(f(z0), z0), z1))
b(z0, b(z1, a)) → f(b(c(f(a), z0, z1), z1))

And the Tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))
B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

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

POL(B(x₁, x₂)) = [4]x₁ + x₂   
POL(C(x₁, x₂, x₃)) = [2] + [5]x₁ + [4]x₂   
POL(a) = [2]   
POL(b(x₁, x₂)) = [4]x₁ + x₂   
POL(c(x₁, x₂, x₃)) = 0   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(z0, z1, a) → f(b(b(f(z0), z0), z1))
b(z0, b(z1, a)) → f(b(c(f(a), z0, z1), z1))
f(c(c(z0, a, a), z1, a)) → z0

Tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))
B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

S tuples:

B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

K tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))

Defined Rule Symbols:

c, b, f

Defined Pair Symbols:

C, B

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.

B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

We considered the (Usable) Rules:

c(z0, z1, a) → f(b(b(f(z0), z0), z1))
b(z0, b(z1, a)) → f(b(c(f(a), z0, z1), z1))

And the Tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))
B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

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

POL(B(x₁, x₂)) = [1] + [2]x₁ + [2]x₂   
POL(C(x₁, x₂, x₃)) = [2] + [5]x₁ + [2]x₂ + [2]x₃   
POL(a) = [2]   
POL(b(x₁, x₂)) = [2] + [4]x₁   
POL(c(x₁, x₂, x₃)) = 0   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(f(x₁)) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(z0, z1, a) → f(b(b(f(z0), z0), z1))
b(z0, b(z1, a)) → f(b(c(f(a), z0, z1), z1))
f(c(c(z0, a, a), z1, a)) → z0

Tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))
B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

S tuples:none
K tuples:

C(z0, z1, a) → c1(B(b(f(z0), z0), z1), B(f(z0), z0))
B(z0, b(z1, a)) → c2(B(c(f(a), z0, z1), z1), C(f(a), z0, z1))

Defined Rule Symbols:

c, b, f

Defined Pair Symbols:

C, B

Compound Symbols:

c1, c2

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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