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), 0 ms)
↳2 CdtProblem
    ↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
    ↳5 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtKnowledgeProof (⇔, 0 ms)
    ↳8 CdtProblem
    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳18 CdtProblem
    ↳19 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
    ↳20 CdtProblem
    ↳21 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳22 CdtProblem
    ↳23 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳24 CdtProblem
    ↳25 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
    ↳26 CdtProblem
    ↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳28 CdtProblem
↳29 CpxTrsMatchBoundsTAProof (⇔, 21 ms)
↳30 BOUNDS(O(1), O(n^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) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace C(c(z0)) → c2(C(b(c(z0))), C(z0)) by

C(c(x0)) → c2(C(x0))

(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(x0)) → c2(C(x0))
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(x0)) → c2(C(x0))
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(x0)) → c2(C(x0))
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(x0)) → c2(C(x0))
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) CdtKnowledgeProof (EQUIVALENT transformation)

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

A1(0, z0) → c3(C(c(z0)), C(z0))

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

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A, A1

Compound Symbols:

c1, c2, c3

(9) 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)))
A(0, c(x0)) → c3

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

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

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c3, c3

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

Removed 1 trailing nodes:

A(0, c(x0)) → c3

(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(x0)) → c2(C(x0))
A1(0, z0) → c3(C(c(z0)), 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(x0)) → c2(C(x0))
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) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c3

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

Use narrowing to replace A(0, c(b(c(z0)))) → c3(C(b(a(0, c(z0)))), C(c(b(c(z0))))) by

A(0, c(b(c(x0)))) → c3(C(b(c(c(c(x0))))), C(c(b(c(x0)))))
A(0, c(b(c(x0)))) → c3(C(c(b(c(x0)))))

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

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

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c3, c3

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

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

A(0, c(x0)) → c3(C(c(x0)))

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

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

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c2, c3, c3

(17) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace C(c(x0)) → c2(C(x0)) by

C(c(c(b(c(y0))))) → c2(C(c(b(c(y0)))))
C(c(c(y0))) → c2(C(c(y0)))

(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))
A1(0, z0) → c3(C(c(z0)), C(z0))
A(0, c(b(c(x0)))) → c3(C(b(c(c(c(x0))))), C(c(b(c(x0)))))
A(0, c(b(c(x0)))) → c3(C(c(b(c(x0)))))
A(0, c(x0)) → c3(C(c(x0)))
C(c(c(b(c(y0))))) → c2(C(c(b(c(y0)))))
C(c(c(y0))) → c2(C(c(y0)))
S tuples:

C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0))
A(0, c(b(c(x0)))) → c3(C(b(c(c(c(x0))))), C(c(b(c(x0)))))
A(0, c(b(c(x0)))) → c3(C(c(b(c(x0)))))
A(0, c(x0)) → c3(C(c(x0)))
C(c(c(b(c(y0))))) → c2(C(c(b(c(y0)))))
C(c(c(y0))) → c2(C(c(y0)))
K tuples:

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c3, c3, c2

(19) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

C(c(c(b(c(y0))))) → c2(C(c(b(c(y0)))))
C(c(c(y0))) → c2(C(c(y0)))

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

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

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c3, c3

(21) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

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

A(0, c(b(c(y0)))) → c3(C(c(b(c(y0)))))

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

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

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

C, A1, A

Compound Symbols:

c1, c3, c3

(23) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace C(c(b(c(z0)))) → c1(A(0, c(z0)), C(z0)) by

C(c(b(c(c(b(c(y0))))))) → c1(A(0, c(c(b(c(y0))))), C(c(b(c(y0)))))
C(c(b(c(b(c(y0)))))) → c1(A(0, c(b(c(y0)))), C(b(c(y0))))

(24) 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:

A1(0, z0) → c3(C(c(z0)), C(z0))
A(0, c(b(c(x0)))) → c3(C(b(c(c(c(x0))))), C(c(b(c(x0)))))
A(0, c(b(c(x0)))) → c3(C(c(b(c(x0)))))
C(c(b(c(c(b(c(y0))))))) → c1(A(0, c(c(b(c(y0))))), C(c(b(c(y0)))))
C(c(b(c(b(c(y0)))))) → c1(A(0, c(b(c(y0)))), C(b(c(y0))))
S tuples:

A(0, c(b(c(x0)))) → c3(C(b(c(c(c(x0))))), C(c(b(c(x0)))))
A(0, c(b(c(x0)))) → c3(C(c(b(c(x0)))))
C(c(b(c(c(b(c(y0))))))) → c1(A(0, c(c(b(c(y0))))), C(c(b(c(y0)))))
C(c(b(c(b(c(y0)))))) → c1(A(0, c(b(c(y0)))), C(b(c(y0))))
K tuples:

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

A1, A, C

Compound Symbols:

c3, c3, c1

(25) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

C(c(b(c(c(b(c(y0))))))) → c1(A(0, c(c(b(c(y0))))), C(c(b(c(y0)))))

(26) 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:

A1(0, z0) → c3(C(c(z0)), C(z0))
A(0, c(b(c(x0)))) → c3(C(b(c(c(c(x0))))), C(c(b(c(x0)))))
A(0, c(b(c(x0)))) → c3(C(c(b(c(x0)))))
C(c(b(c(b(c(y0)))))) → c1(A(0, c(b(c(y0)))), C(b(c(y0))))
S tuples:

A(0, c(b(c(x0)))) → c3(C(b(c(c(c(x0))))), C(c(b(c(x0)))))
A(0, c(b(c(x0)))) → c3(C(c(b(c(x0)))))
C(c(b(c(b(c(y0)))))) → c1(A(0, c(b(c(y0)))), C(b(c(y0))))
K tuples:

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

A1, A, C

Compound Symbols:

c3, c3, c1

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

Use narrowing to replace C(c(b(c(b(c(y0)))))) → c1(A(0, c(b(c(y0)))), C(b(c(y0)))) by

C(c(b(c(b(c(x0)))))) → c1(A(0, c(b(c(x0)))))

(28) 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:

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

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

A1(0, z0) → c3(C(c(z0)), C(z0))
Defined Rule Symbols:

c, a

Defined Pair Symbols:

A1, A, C

Compound Symbols:

c3, c3, c1

(29) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
b0(0) → 0
00() → 0
c0(0) → 1
a0(0, 0) → 2
c1(0) → 3
c1(3) → 2
c2(0) → 6
b2(6) → 5
c2(5) → 4
b2(4) → 2

(30) BOUNDS(O(1), O(n^1))