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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtInstantiationProof (BOTH BOUNDS(ID, ID))
↳6 CdtProblem
↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtKnowledgeProof (⇔)
↳10 CdtProblem
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳12 CdtProblem
↳13 CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID))
↳14 CdtProblem
↳15 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳16 CdtProblem
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳18 CdtProblem
↳19 CdtKnowledgeProof (⇔)
↳20 CdtProblem
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳22 CdtProblem
↳23 SIsEmptyProof (⇔)
↳24 BOUNDS(O(1), O(1))

(0) Obligation:

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

c(c(b(c(x)))) → b(a(0, c(x)))
c(c(x)) → b(c(b(c(x))))
a(0, x) → c(c(x))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(b(c(z0))), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(b(c(z0))), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A

Compound Symbols:

c1, c2, c3

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
A(0, z0) → c3(C(c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A

Compound Symbols:

c1, c3, c2

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

Use instantiation to replace A(0, z0) → c3(C(c(z0)), C(z0)) by

A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c3(C(c(z0)), C(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c3(C(c(z0)), C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c3(C(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
K tuples:none
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3, c4

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3, c4

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

Use narrowing to replace A(0, c(x0)) → c3(C(c(c(x0))), C(c(x0))) by

A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
A1(0, z0) → c4(C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

(13) CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 of 6 dangling nodes:

A1(0, z0) → c4(C(z0))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(b(c(b(c(z0))))), C(c(z0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

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

Removed 2 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

(17) 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(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
We considered the (Usable) Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
And the Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(A(x1, x2)) = [1] + [5]x1 + x2   
POL(A1(x1, x2)) = [4] + [4]x1 + [5]x2   
POL(C(x1)) = x1   
POL(a(x1, x2)) = [3] + [2]x1 + [4]x2   
POL(b(x1)) = x1   
POL(c(x1)) = [1] + [2]x1   
POL(c1(x1, x2)) = x1 + x2   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
S tuples:

C(c(z0)) → c2(C(z0))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
K tuples:

A1(0, z0) → c4(C(c(z0)))
C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

(19) CdtKnowledgeProof (EQUIVALENT transformation)

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

A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
S tuples:

C(c(z0)) → c2(C(z0))
K tuples:

A1(0, z0) → c4(C(c(z0)))
C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

(21) 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(c(z0)) → c2(C(z0))
We considered the (Usable) Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
And the Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(A(x1, x2)) = [1] + [4]x1 + [2]x2   
POL(A1(x1, x2)) = [3] + [4]x1 + [4]x2   
POL(C(x1)) = [1] + [2]x1   
POL(a(x1, x2)) = [3] + [4]x2   
POL(b(x1)) = x1   
POL(c(x1)) = [1] + [2]x1   
POL(c1(x1, x2)) = x1 + x2   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(b(c(z0)))) → b(a(0, c(z0)))
c(c(z0)) → b(c(b(c(z0))))
a(0, z0) → c(c(z0))
Tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
C(c(z0)) → c2(C(z0))
A1(0, z0) → c4(C(c(z0)))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
S tuples:none
K tuples:

A1(0, z0) → c4(C(c(z0)))
C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
A(0, c(b(c(z0)))) → c3(C(c(b(c(z0)))))
A(0, c(z0)) → c3(C(c(z0)))
C(c(z0)) → c2(C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c4, c3

(23) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(24) BOUNDS(O(1), O(1))