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

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))

S tuples:

D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))

K tuples:none
Defined Rule Symbols:

d, c, v, b

Defined Pair Symbols:

D, C

Compound Symbols:

c2, c3

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 2 dangling nodes:

D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

d, c, v, b

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))