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:

b(a(a(x1))) → a(b(c(x1)))
c(a(x1)) → a(c(x1))
b(c(a(x1))) → a(b(c(x1)))
c(b(x1)) → d(x1)
a(d(x1)) → d(a(x1))
d(x1) → b(a(x1))
L(a(a(x1))) → L(a(b(c(x1))))
c(R(x1)) → c(b(R(x1)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(a(a(z0))) → a(b(c(z0)))
b(c(a(z0))) → a(b(c(z0)))
c(a(z0)) → a(c(z0))
c(b(z0)) → d(z0)
c(R(z0)) → c(b(R(z0)))
a(d(z0)) → d(a(z0))
d(z0) → b(a(z0))
L(a(a(z0))) → L(a(b(c(z0))))

Tuples:

B(a(a(z0))) → c1(A(b(c(z0))), B(c(z0)), C(z0))
B(c(a(z0))) → c2(A(b(c(z0))), B(c(z0)), C(z0))
C(a(z0)) → c3(A(c(z0)), C(z0))
C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))), B(R(z0)))
A(d(z0)) → c6(D(a(z0)), A(z0))
D(z0) → c7(B(a(z0)), A(z0))
L'(a(a(z0))) → c8(L'(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0))

S tuples:

B(a(a(z0))) → c1(A(b(c(z0))), B(c(z0)), C(z0))
B(c(a(z0))) → c2(A(b(c(z0))), B(c(z0)), C(z0))
C(a(z0)) → c3(A(c(z0)), C(z0))
C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))), B(R(z0)))
A(d(z0)) → c6(D(a(z0)), A(z0))
D(z0) → c7(B(a(z0)), A(z0))
L'(a(a(z0))) → c8(L'(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0))

K tuples:none
Defined Rule Symbols:

b, c, a, d, L

Defined Pair Symbols:

B, C, A, D, L'

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

B(c(a(z0))) → c2(A(b(c(z0))), B(c(z0)), C(z0))
A(d(z0)) → c6(D(a(z0)), A(z0))
L'(a(a(z0))) → c8(L'(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(a(a(z0))) → a(b(c(z0)))
b(c(a(z0))) → a(b(c(z0)))
c(a(z0)) → a(c(z0))
c(b(z0)) → d(z0)
c(R(z0)) → c(b(R(z0)))
a(d(z0)) → d(a(z0))
d(z0) → b(a(z0))
L(a(a(z0))) → L(a(b(c(z0))))

Tuples:

C(R(z0)) → c5(C(b(R(z0))), B(R(z0)))
D(z0) → c7(B(a(z0)), A(z0))
B(a(a(z0))) → c1(A(b(c(z0))), B(c(z0)), C(z0))
C(a(z0)) → c3(A(c(z0)), C(z0))
C(b(z0)) → c4(D(z0))

S tuples:

B(a(a(z0))) → c1(A(b(c(z0))), B(c(z0)), C(z0))
C(a(z0)) → c3(A(c(z0)), C(z0))
C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))), B(R(z0)))
D(z0) → c7(B(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

b, c, a, d, L

Defined Pair Symbols:

C, D, B

Compound Symbols:

c5, c7, c1, c3, c4

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

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(a(a(z0))) → a(b(c(z0)))
b(c(a(z0))) → a(b(c(z0)))
c(a(z0)) → a(c(z0))
c(b(z0)) → d(z0)
c(R(z0)) → c(b(R(z0)))
a(d(z0)) → d(a(z0))
d(z0) → b(a(z0))
L(a(a(z0))) → L(a(b(c(z0))))

Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

S tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

K tuples:none
Defined Rule Symbols:

b, c, a, d, L

Defined Pair Symbols:

C, D, B

Compound Symbols:

c4, c5, c7, c1, c3

(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.

C(a(z0)) → c3(C(z0))

We considered the (Usable) Rules:

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

And the Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

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

POL(B(x₁)) = x₁   
POL(C(x₁)) = [1] + [2]x₁   
POL(D(x₁)) = [1] + [2]x₁   
POL(R(x₁)) = 0   
POL(a(x₁)) = [1] + [2]x₁   
POL(b(x₁)) = x₁   
POL(c(x₁)) = [1] + [2]x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(d(x₁)) = [1] + [2]x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(a(a(z0))) → a(b(c(z0)))
b(c(a(z0))) → a(b(c(z0)))
c(a(z0)) → a(c(z0))
c(b(z0)) → d(z0)
c(R(z0)) → c(b(R(z0)))
a(d(z0)) → d(a(z0))
d(z0) → b(a(z0))
L(a(a(z0))) → L(a(b(c(z0))))

Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

S tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))

K tuples:

C(a(z0)) → c3(C(z0))

Defined Rule Symbols:

b, c, a, d, L

Defined Pair Symbols:

C, D, B

Compound Symbols:

c4, c5, c7, c1, c3

(9) 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.

C(b(z0)) → c4(D(z0))

We considered the (Usable) Rules:

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

And the Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

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

POL(B(x₁)) = x₁   
POL(C(x₁)) = [4] + [4]x₁   
POL(D(x₁)) = [2] + [4]x₁   
POL(R(x₁)) = 0   
POL(a(x₁)) = [1] + [4]x₁   
POL(b(x₁)) = x₁   
POL(c(x₁)) = [1] + [4]x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(d(x₁)) = [1] + [4]x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(a(a(z0))) → a(b(c(z0)))
b(c(a(z0))) → a(b(c(z0)))
c(a(z0)) → a(c(z0))
c(b(z0)) → d(z0)
c(R(z0)) → c(b(R(z0)))
a(d(z0)) → d(a(z0))
d(z0) → b(a(z0))
L(a(a(z0))) → L(a(b(c(z0))))

Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

S tuples:

C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))

K tuples:

C(a(z0)) → c3(C(z0))
C(b(z0)) → c4(D(z0))

Defined Rule Symbols:

b, c, a, d, L

Defined Pair Symbols:

C, D, B

Compound Symbols:

c4, c5, c7, c1, c3

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

D(z0) → c7(B(a(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(a(a(z0))) → a(b(c(z0)))
b(c(a(z0))) → a(b(c(z0)))
c(a(z0)) → a(c(z0))
c(b(z0)) → d(z0)
c(R(z0)) → c(b(R(z0)))
a(d(z0)) → d(a(z0))
d(z0) → b(a(z0))
L(a(a(z0))) → L(a(b(c(z0))))

Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

S tuples:

C(R(z0)) → c5(C(b(R(z0))))
B(a(a(z0))) → c1(B(c(z0)), C(z0))

K tuples:

C(a(z0)) → c3(C(z0))
C(b(z0)) → c4(D(z0))
D(z0) → c7(B(a(z0)))

Defined Rule Symbols:

b, c, a, d, L

Defined Pair Symbols:

C, D, B

Compound Symbols:

c4, c5, c7, c1, c3

(13) 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.

B(a(a(z0))) → c1(B(c(z0)), C(z0))

We considered the (Usable) Rules:

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

And the Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

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

POL(B(x₁)) = x₁   
POL(C(x₁)) = [2] + [2]x₁   
POL(D(x₁)) = [2] + [2]x₁   
POL(R(x₁)) = x₁   
POL(a(x₁)) = [2] + [2]x₁   
POL(b(x₁)) = x₁   
POL(c(x₁)) = [2] + [2]x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(d(x₁)) = [2] + [2]x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(a(a(z0))) → a(b(c(z0)))
b(c(a(z0))) → a(b(c(z0)))
c(a(z0)) → a(c(z0))
c(b(z0)) → d(z0)
c(R(z0)) → c(b(R(z0)))
a(d(z0)) → d(a(z0))
d(z0) → b(a(z0))
L(a(a(z0))) → L(a(b(c(z0))))

Tuples:

C(b(z0)) → c4(D(z0))
C(R(z0)) → c5(C(b(R(z0))))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))
C(a(z0)) → c3(C(z0))

S tuples:

C(R(z0)) → c5(C(b(R(z0))))

K tuples:

C(a(z0)) → c3(C(z0))
C(b(z0)) → c4(D(z0))
D(z0) → c7(B(a(z0)))
B(a(a(z0))) → c1(B(c(z0)), C(z0))

Defined Rule Symbols:

b, c, a, d, L

Defined Pair Symbols:

C, D, B

Compound Symbols:

c4, c5, c7, c1, c3

(15) CdtKnowledgeProof (EQUIVALENT transformation)

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

C(R(z0)) → c5(C(b(R(z0))))
C(b(z0)) → c4(D(z0))

Now S is empty

(16) BOUNDS(O(1), O(1))