(0) Obligation:

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

a(a(x)) → c
b(u(x)) → b(d(x))
d(a(x)) → a(d(x))
d(b(x)) → u(a(b(x)))
a(u(x)) → u(a(x))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:

A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(A(b(z0)), B(z0))
S tuples:

A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:

a, b, d

Defined Pair Symbols:

A, B, D

Compound Symbols:

c2, c3, c4, c5

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:

A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
S tuples:

A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
K tuples:none
Defined Rule Symbols:

a, b, d

Defined Pair Symbols:

A, B, D

Compound Symbols:

c2, c3, c4, c5

(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace B(u(z0)) → c3(B(d(z0)), D(z0)) by

B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:

A(u(z0)) → c2(A(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))
S tuples:

A(u(z0)) → c2(A(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))
K tuples:none
Defined Rule Symbols:

a, b, d

Defined Pair Symbols:

A, D, B

Compound Symbols:

c2, c4, c5, c3

(7) CdtUnreachableProof (EQUIVALENT transformation)

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

D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:

A(u(z0)) → c2(A(z0))
S tuples:

A(u(z0)) → c2(A(z0))
K tuples:none
Defined Rule Symbols:

a, b, d

Defined Pair Symbols:

A

Compound Symbols:

c2

(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.

A(u(z0)) → c2(A(z0))
We considered the (Usable) Rules:none
And the Tuples:

A(u(z0)) → c2(A(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A(x1)) = x1   
POL(c2(x1)) = x1   
POL(u(x1)) = [2] + x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:

A(u(z0)) → c2(A(z0))
S tuples:none
K tuples:

A(u(z0)) → c2(A(z0))
Defined Rule Symbols:

a, b, d

Defined Pair Symbols:

A

Compound Symbols:

c2

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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