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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
V(a(c(z0))) → c4(U(b(d(z0))))
W(a(a(z0))) → c6(U(w(z0)), W(z0))
W(a(c(z0))) → c7(U(b(d(z0))))

S tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
V(a(c(z0))) → c4(U(b(d(z0))))
W(a(a(z0))) → c6(U(w(z0)), W(z0))
W(a(c(z0))) → c7(U(b(d(z0))))

K tuples:none
Defined Rule Symbols:

a, u, v, w

Defined Pair Symbols:

V, W

Compound Symbols:

c3, c4, c6, c7

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

Removed 2 trailing nodes:

W(a(c(z0))) → c7(U(b(d(z0))))
V(a(c(z0))) → c4(U(b(d(z0))))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))

S tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))

K tuples:none
Defined Rule Symbols:

a, u, v, w

Defined Pair Symbols:

V, W

Compound Symbols:

c3, c6

(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.

W(a(a(z0))) → c6(U(w(z0)), W(z0))

We considered the (Usable) Rules:

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

And the Tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))

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

POL(U(x₁)) = 0   
POL(V(x₁)) = 0   
POL(W(x₁)) = [2]x₁   
POL(a(x₁)) = [5] + [5]x₁   
POL(b(x₁)) = [3]   
POL(c(x₁)) = [4]   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(d(x₁)) = [5]   
POL(u(x₁)) = [4] + [4]x₁   
POL(v(x₁)) = 0   
POL(w(x₁)) = [4] + [5]x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))

S tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))

K tuples:

W(a(a(z0))) → c6(U(w(z0)), W(z0))

Defined Rule Symbols:

a, u, v, w

Defined Pair Symbols:

V, W

Compound Symbols:

c3, c6

(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.

V(a(a(z0))) → c3(U(v(z0)), V(z0))

We considered the (Usable) Rules:

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

And the Tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))

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

POL(U(x₁)) = [3] + x₁   
POL(V(x₁)) = [2]x₁   
POL(W(x₁)) = x₁   
POL(a(x₁)) = [4] + [3]x₁   
POL(b(x₁)) = 0   
POL(c(x₁)) = 0   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(d(x₁)) = [5]   
POL(u(x₁)) = [4] + [3]x₁   
POL(v(x₁)) = [2] + [2]x₁   
POL(w(x₁)) = [2]x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0))
W(a(a(z0))) → c6(U(w(z0)), W(z0))

S tuples:none
K tuples:

W(a(a(z0))) → c6(U(w(z0)), W(z0))
V(a(a(z0))) → c3(U(v(z0)), V(z0))

Defined Rule Symbols:

a, u, v, w

Defined Pair Symbols:

V, W

Compound Symbols:

c3, c6

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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