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

(0) Obligation:

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

a(a(x1)) → b(b(b(x1)))
a(x1) → d(c(d(x1)))
b(b(b(x1))) → a(f(x1))
b(b(x1)) → c(c(c(x1)))
c(c(x1)) → d(d(d(x1)))
c(d(d(x1))) → f(x1)
f(f(x1)) → f(a(x1))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(b(b(z0)))
a(z0) → d(c(d(z0)))
b(b(b(z0))) → a(f(z0))
b(b(z0)) → c(c(c(z0)))
c(c(z0)) → d(d(d(z0)))
c(d(d(z0))) → f(z0)
f(f(z0)) → f(a(z0))

Tuples:

A(a(z0)) → c1(B(b(b(z0))), B(b(z0)), B(z0))
A(z0) → c2(C(d(z0)))
B(b(b(z0))) → c3(A(f(z0)), F(z0))
B(b(z0)) → c4(C(c(c(z0))), C(c(z0)), C(z0))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

S tuples:

A(a(z0)) → c1(B(b(b(z0))), B(b(z0)), B(z0))
A(z0) → c2(C(d(z0)))
B(b(b(z0))) → c3(A(f(z0)), F(z0))
B(b(z0)) → c4(C(c(c(z0))), C(c(z0)), C(z0))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, b, c, f

Defined Pair Symbols:

A, B, C, F

Compound Symbols:

c1, c2, c3, c4, c6, c7

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

A(a(z0)) → c1(B(b(b(z0))), B(b(z0)), B(z0))
B(b(b(z0))) → c3(A(f(z0)), F(z0))
B(b(z0)) → c4(C(c(c(z0))), C(c(z0)), C(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(b(b(z0)))
a(z0) → d(c(d(z0)))
b(b(b(z0))) → a(f(z0))
b(b(z0)) → c(c(c(z0)))
c(c(z0)) → d(d(d(z0)))
c(d(d(z0))) → f(z0)
f(f(z0)) → f(a(z0))

Tuples:

A(z0) → c2(C(d(z0)))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

S tuples:

A(z0) → c2(C(d(z0)))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, b, c, f

Defined Pair Symbols:

A, C, F

Compound Symbols:

c2, c6, c7

(5) 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(d(d(z0))) → c6(F(z0))

We considered the (Usable) Rules:

a(z0) → d(c(d(z0)))
c(d(d(z0))) → f(z0)
f(f(z0)) → f(a(z0))

And the Tuples:

A(z0) → c2(C(d(z0)))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

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

POL(A(x₁)) = [1] + x₁   
POL(C(x₁)) = x₁   
POL(F(x₁)) = x₁   
POL(a(x₁)) = [2] + [2]x₁ + x₁²   
POL(c(x₁)) = x₁²   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(d(x₁)) = [1] + x₁   
POL(f(x₁)) = [3] + [3]x₁ + x₁²   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(b(b(z0)))
a(z0) → d(c(d(z0)))
b(b(b(z0))) → a(f(z0))
b(b(z0)) → c(c(c(z0)))
c(c(z0)) → d(d(d(z0)))
c(d(d(z0))) → f(z0)
f(f(z0)) → f(a(z0))

Tuples:

A(z0) → c2(C(d(z0)))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

S tuples:

A(z0) → c2(C(d(z0)))
F(f(z0)) → c7(F(a(z0)), A(z0))

K tuples:

C(d(d(z0))) → c6(F(z0))

Defined Rule Symbols:

a, b, c, f

Defined Pair Symbols:

A, C, F

Compound Symbols:

c2, c6, c7

(7) 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.

F(f(z0)) → c7(F(a(z0)), A(z0))

We considered the (Usable) Rules:

a(z0) → d(c(d(z0)))
c(d(d(z0))) → f(z0)
f(f(z0)) → f(a(z0))

And the Tuples:

A(z0) → c2(C(d(z0)))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

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

POL(A(x₁)) = [2] + [2]x₁ + x₁²   
POL(C(x₁)) = x₁²   
POL(F(x₁)) = [3] + x₁²   
POL(a(x₁)) = [2] + [2]x₁ + x₁²   
POL(c(x₁)) = x₁²   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(d(x₁)) = [1] + x₁   
POL(f(x₁)) = [3] + [3]x₁ + x₁²   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(b(b(z0)))
a(z0) → d(c(d(z0)))
b(b(b(z0))) → a(f(z0))
b(b(z0)) → c(c(c(z0)))
c(c(z0)) → d(d(d(z0)))
c(d(d(z0))) → f(z0)
f(f(z0)) → f(a(z0))

Tuples:

A(z0) → c2(C(d(z0)))
C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

S tuples:

A(z0) → c2(C(d(z0)))

K tuples:

C(d(d(z0))) → c6(F(z0))
F(f(z0)) → c7(F(a(z0)), A(z0))

Defined Rule Symbols:

a, b, c, f

Defined Pair Symbols:

A, C, F

Compound Symbols:

c2, c6, c7

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

A(z0) → c2(C(d(z0)))
C(d(d(z0))) → c6(F(z0))

Now S is empty

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