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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(b(z0)) → c1(D(z0))
A(z0) → c2(E(d(z0)), D(z0))
D(d(z0)) → c3(B(e(z0)), E(z0))
B(z0) → c5(D(c(z0)))
E(c(z0)) → c6(D(a(z0)), A(z0))

S tuples:

A(b(z0)) → c1(D(z0))
A(z0) → c2(E(d(z0)), D(z0))
D(d(z0)) → c3(B(e(z0)), E(z0))
B(z0) → c5(D(c(z0)))
E(c(z0)) → c6(D(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, d, b, e

Defined Pair Symbols:

A, D, B, E

Compound Symbols:

c1, c2, c3, c5, c6

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

A(b(z0)) → c1(D(z0))
D(d(z0)) → c3(B(e(z0)), E(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(z0) → c2(E(d(z0)), D(z0))
B(z0) → c5(D(c(z0)))
E(c(z0)) → c6(D(a(z0)), A(z0))

S tuples:

A(z0) → c2(E(d(z0)), D(z0))
B(z0) → c5(D(c(z0)))
E(c(z0)) → c6(D(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, d, b, e

Defined Pair Symbols:

A, B, E

Compound Symbols:

c2, c5, c6

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 3 dangling nodes:

B(z0) → c5(D(c(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(z0) → c2(E(d(z0)), D(z0))
E(c(z0)) → c6(D(a(z0)), A(z0))

S tuples:

A(z0) → c2(E(d(z0)), D(z0))
E(c(z0)) → c6(D(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, d, b, e

Defined Pair Symbols:

A, E

Compound Symbols:

c2, c6

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(z0) → c2(E(d(z0)))
E(c(z0)) → c6(A(z0))

S tuples:

A(z0) → c2(E(d(z0)))
E(c(z0)) → c6(A(z0))

K tuples:none
Defined Rule Symbols:

a, d, b, e

Defined Pair Symbols:

A, E

Compound Symbols:

c2, c6

(9) 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.

E(c(z0)) → c6(A(z0))

We considered the (Usable) Rules:

d(z0) → z0

And the Tuples:

A(z0) → c2(E(d(z0)))
E(c(z0)) → c6(A(z0))

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

POL(A(x₁)) = [2] + [2]x₁²   
POL(E(x₁)) = [2]x₁²   
POL(c(x₁)) = [2] + x₁   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(d(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(z0) → c2(E(d(z0)))
E(c(z0)) → c6(A(z0))

S tuples:

A(z0) → c2(E(d(z0)))

K tuples:

E(c(z0)) → c6(A(z0))

Defined Rule Symbols:

a, d, b, e

Defined Pair Symbols:

A, E

Compound Symbols:

c2, c6

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

A(z0) → c2(E(d(z0)))
E(c(z0)) → c6(A(z0))

Now S is empty

(12) BOUNDS(O(1), O(1))