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 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 10 ms)
↳10 CdtProblem
↳11 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 113 ms)
↳22 CdtProblem
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 322 ms)
↳24 CdtProblem
↳25 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 CdtProblem
↳27 CdtUnreachableProof (⇔, 0 ms)
↳28 CdtProblem
↳29 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳30 CdtProblem
↳31 SIsEmptyProof (⇔, 0 ms)
↳32 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(B(z0, b(0, c(z1))), B(0, c(z1)), C(z1))
C(b(z0, c(z1))) → c2(C(c(b(a(0, 0), z0))), C(b(a(0, 0), z0)), B(a(0, 0), z0), A(0, 0))
S tuples:

A(z0, z1) → c1(B(z0, b(0, c(z1))), B(0, c(z1)), C(z1))
C(b(z0, c(z1))) → c2(C(c(b(a(0, 0), z0))), C(b(a(0, 0), z0)), B(a(0, 0), z0), A(0, 0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2

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

Removed 3 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(z0, c(z1))) → c2(C(c(b(a(0, 0), z0))), C(b(a(0, 0), z0)), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(z0, c(z1))) → c2(C(c(b(a(0, 0), z0))), C(b(a(0, 0), z0)), A(0, 0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2

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

Use narrowing to replace C(b(z0, c(z1))) → c2(C(c(b(a(0, 0), z0))), C(b(a(0, 0), z0)), A(0, 0)) by

C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), a(0, 0))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(x0, c(x1))) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), a(0, 0))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(x0, c(x1))) → c2
S tuples:

A(z0, z1) → c1(C(z1))
C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), a(0, 0))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(x0, c(x1))) → c2
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2

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

Removed 1 trailing nodes:

C(b(x0, c(x1))) → c2

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), a(0, 0))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), a(0, 0))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2

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

Use narrowing to replace C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), a(0, 0))))), C(b(a(0, 0), c(z1))), A(0, 0)) by

C(b(c(x0), c(x1))) → c2(C(c(c(b(a(0, 0), b(0, b(0, c(0))))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(a(0, 0), b(0, b(0, c(0))))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(a(0, 0), b(0, b(0, c(0))))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2, c2

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

Use narrowing to replace C(b(0, c(x1))) → c2(C(c(a(0, 0))), C(b(a(0, 0), 0)), A(0, 0)) by

C(b(0, c(x0))) → c2(C(c(b(0, b(0, c(0))))), C(b(a(0, 0), 0)), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(c(b(0, b(0, c(0))))), C(b(a(0, 0), 0)), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(c(b(0, b(0, c(0))))), C(b(a(0, 0), 0)), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2, c2

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

Removed 1 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2, c2

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

Use narrowing to replace C(b(x0, c(x1))) → c2(C(c(b(b(0, b(0, c(0))), x0))), C(b(a(0, 0), x0)), A(0, 0)) by

C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), b(0, b(0, c(0))))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(b(0, b(0, c(0))))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), b(0, b(0, c(0))))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(b(0, b(0, c(0))))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(c(z1), c(x1))) → c2(C(c(c(b(a(0, 0), b(0, b(0, c(0))))))), C(b(a(0, 0), c(z1))), A(0, 0))
C(b(0, c(x1))) → c2(C(c(b(0, b(0, c(0))))), C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2, c2

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2, c2

(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(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
We considered the (Usable) Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
b(z0, 0) → z0
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
And the Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(A(x1, x2)) = [2]x2   
POL(C(x1)) = [2]x1   
POL(a(x1, x2)) = x1 + [4]x2   
POL(b(x1, x2)) = x1   
POL(c(x1)) = [1]   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c2(x1, x2, x3)) = x1 + x2 + x3   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
K tuples:

C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2, c2

(23) 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(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
We considered the (Usable) Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
b(z0, 0) → z0
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
And the Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(A(x1, x2)) = [5]x2   
POL(C(x1)) = [4]x1   
POL(a(x1, x2)) = [4]x1 + [5]x2   
POL(b(x1, x2)) = [4]x1   
POL(c(x1)) = [2]   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c2(x1, x2, x3)) = x1 + x2 + x3   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(z0, z1) → c1(C(z1))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
S tuples:

A(z0, z1) → c1(C(z1))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
K tuples:

C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c2, c2, c2

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

Use instantiation to replace A(z0, z1) → c1(C(z1)) by

A(0, 0) → c1(C(0))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
A(0, 0) → c1(C(0))
S tuples:

C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))
A(0, 0) → c1(C(0))
K tuples:

C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

C, A

Compound Symbols:

c2, c2, c2, c1

(27) CdtUnreachableProof (EQUIVALENT transformation)

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

C(b(c(x0), c(x1))) → c2(C(c(c(b(b(0, b(0, c(0))), a(0, 0))))), C(b(a(0, 0), c(x0))), A(0, 0))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))))
C(b(c(x0), c(x1))) → c2(C(b(a(0, 0), c(x0))), A(0, 0))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)))
C(b(0, c(x0))) → c2(C(b(a(0, 0), 0)), A(0, 0))
C(b(x0, c(x1))) → c2(C(b(a(0, 0), x0)), A(0, 0))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:

A(0, 0) → c1(C(0))
S tuples:

A(0, 0) → c1(C(0))
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:

A

Compound Symbols:

c1

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

Removed 1 trailing nodes:

A(0, 0) → c1(C(0))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0, z1) → b(z0, b(0, c(z1)))
c(b(z0, c(z1))) → c(c(b(a(0, 0), z0)))
b(z0, 0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a, c, b

Defined Pair Symbols:none

Compound Symbols:none

(31) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(32) BOUNDS(O(1), O(1))