Runtime Complexity (innermost) proof of /tmp/tmpjmu5UF/Ex4_7_77_Bor03

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 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

    ↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 590 ms)

    ↳8 CdtProblem

    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳10 CdtProblem

    ↳11 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 267 ms)

    ↳14 CdtProblem

    ↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳16 CdtProblem

    ↳17 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳18 CdtProblem

    ↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳20 CdtProblem

    ↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳22 CdtProblem

    ↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳24 CdtProblem

    ↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳26 CdtProblem

    ↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 664 ms)

    ↳28 CdtProblem

    ↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳30 CdtProblem

    ↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳32 CdtProblem

    ↳33 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳34 CdtProblem

    ↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳36 CdtProblem

    ↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳38 CdtProblem

    ↳39 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳40 CdtProblem

↳41 CpxTrsMatchBoundsTAProof (⇔, 178 ms)

↳42 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(tail(X)) → active(tail(mark(X)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c4(ACTIVE(0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c4(ACTIVE(0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))

K tuples:none
Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11

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

Removed 1 trailing nodes:

MARK(0) → c4(ACTIVE(0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))

K tuples:none
Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c, c1, c2, c3, c5, c6, c7, c8, c9, c10, c11

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))

S tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))

K tuples:none
Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c3, c5, c6, c7, c8, c9, c10, c11, c

(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.

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(tail(z0)) → active(tail(mark(z0)))
mark(0) → active(0)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
tail(active(z0)) → tail(z0)
tail(mark(z0)) → tail(z0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)

And the Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(CONS(x₁, x₂)) = 0
POL(MARK(x₁)) = [4]x₁
POL(TAIL(x₁)) = 0
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = [4]x₁ + [2]x₂
POL(mark(x₁)) = x₁
POL(tail(x₁)) = [1] + [2]x₁
POL(zeros) = 0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c3, c5, c6, c7, c8, c9, c10, c11, c

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

Use narrowing to replace MARK(cons(z0, z1)) → c3(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0)) by

MARK(cons(z0, z1)) → c3(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(z0, z1)) → c3(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c5, c6, c7, c8, c9, c10, c11, c, c3, c3

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

Removed 2 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c5, c6, c7, c8, c9, c10, c11, c, c3, c3, 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.

MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

We considered the (Usable) Rules:

cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(tail(z0)) → active(tail(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
tail(active(z0)) → tail(z0)
tail(mark(z0)) → tail(z0)

And the Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(CONS(x₁, x₂)) = [2]x₂
POL(MARK(x₁)) = [3]x₁
POL(TAIL(x₁)) = 0
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = [4]x₁ + [4]x₂
POL(mark(x₁)) = x₁
POL(tail(x₁)) = [4] + [4]x₁
POL(zeros) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c5, c6, c7, c8, c9, c10, c11, c, c3, c3, c3

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

Use narrowing to replace MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0)) by

MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)), MARK(0))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(x0)) → c5

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)), MARK(0))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(x0)) → c5

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(tail(z0)) → c5(ACTIVE(tail(mark(z0))), TAIL(mark(z0)), MARK(z0))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c3, c5, c5

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

Removed 1 trailing nodes:

MARK(tail(x0)) → c5

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)), MARK(0))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c3, c5

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c3, c5, c5

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

Use narrowing to replace MARK(cons(zeros, x1)) → c3(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros)) by

MARK(cons(zeros, z1)) → c3(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, z1)) → c3(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(zeros, z1)) → c3(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c3, c5, c5, c3

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

Removed 1 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c3, c5, c5, c3

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

Use narrowing to replace MARK(cons(cons(z0, z1), x1)) → c3(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1))) by

MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c3, c5, c5, c3

(27) 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.

MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

We considered the (Usable) Rules:

mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(tail(z0)) → active(tail(mark(z0)))
mark(0) → active(0)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(active(z0)) → tail(z0)
tail(mark(z0)) → tail(z0)

And the Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(CONS(x₁, x₂)) = [5]x₁ + [5]x₂
POL(MARK(x₁)) = [5]x₁
POL(TAIL(x₁)) = [1]
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3) = 0
POL(c3(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = [5]x₁ + [4]x₂
POL(mark(x₁)) = x₁
POL(tail(x₁)) = [1] + [4]x₁
POL(zeros) = 0

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c3, c5, c5, c3

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

Use narrowing to replace MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0))) by

MARK(cons(tail(x0), z1)) → c3(ACTIVE(cons(tail(mark(x0)), z1)), CONS(mark(tail(x0)), z1), MARK(tail(x0)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(z0)), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(tail(zeros), x1)) → c3(ACTIVE(cons(active(tail(active(zeros))), x1)), CONS(mark(tail(zeros)), x1), MARK(tail(zeros)))
MARK(cons(tail(cons(z0, z1)), x1)) → c3(ACTIVE(cons(active(tail(active(cons(mark(z0), z1)))), x1)), CONS(mark(tail(cons(z0, z1))), x1), MARK(tail(cons(z0, z1))))
MARK(cons(tail(0), x1)) → c3(ACTIVE(cons(active(tail(active(0))), x1)), CONS(mark(tail(0)), x1), MARK(tail(0)))
MARK(cons(tail(tail(z0)), x1)) → c3(ACTIVE(cons(active(tail(active(tail(mark(z0))))), x1)), CONS(mark(tail(tail(z0))), x1), MARK(tail(tail(z0))))
MARK(cons(tail(x0), x1)) → c3(CONS(mark(tail(x0)), x1))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(tail(x0), z1)) → c3(ACTIVE(cons(tail(mark(x0)), z1)), CONS(mark(tail(x0)), z1), MARK(tail(x0)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(z0)), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(tail(zeros), x1)) → c3(ACTIVE(cons(active(tail(active(zeros))), x1)), CONS(mark(tail(zeros)), x1), MARK(tail(zeros)))
MARK(cons(tail(cons(z0, z1)), x1)) → c3(ACTIVE(cons(active(tail(active(cons(mark(z0), z1)))), x1)), CONS(mark(tail(cons(z0, z1))), x1), MARK(tail(cons(z0, z1))))
MARK(cons(tail(0), x1)) → c3(ACTIVE(cons(active(tail(active(0))), x1)), CONS(mark(tail(0)), x1), MARK(tail(0)))
MARK(cons(tail(tail(z0)), x1)) → c3(ACTIVE(cons(active(tail(active(tail(mark(z0))))), x1)), CONS(mark(tail(tail(z0))), x1), MARK(tail(tail(z0))))
MARK(cons(tail(x0), x1)) → c3(CONS(mark(tail(x0)), x1))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c5, c5, c3, c3

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

Use narrowing to replace MARK(cons(0, x1)) → c3(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1)) by

MARK(cons(0, z1)) → c3(ACTIVE(cons(0, z1)), CONS(mark(0), z1))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(tail(x0), z1)) → c3(ACTIVE(cons(tail(mark(x0)), z1)), CONS(mark(tail(x0)), z1), MARK(tail(x0)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(z0)), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(tail(zeros), x1)) → c3(ACTIVE(cons(active(tail(active(zeros))), x1)), CONS(mark(tail(zeros)), x1), MARK(tail(zeros)))
MARK(cons(tail(cons(z0, z1)), x1)) → c3(ACTIVE(cons(active(tail(active(cons(mark(z0), z1)))), x1)), CONS(mark(tail(cons(z0, z1))), x1), MARK(tail(cons(z0, z1))))
MARK(cons(tail(0), x1)) → c3(ACTIVE(cons(active(tail(active(0))), x1)), CONS(mark(tail(0)), x1), MARK(tail(0)))
MARK(cons(tail(tail(z0)), x1)) → c3(ACTIVE(cons(active(tail(active(tail(mark(z0))))), x1)), CONS(mark(tail(tail(z0))), x1), MARK(tail(tail(z0))))
MARK(cons(tail(x0), x1)) → c3(CONS(mark(tail(x0)), x1))
MARK(cons(0, z1)) → c3(ACTIVE(cons(0, z1)), CONS(mark(0), z1))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(0, z1)) → c3(ACTIVE(cons(0, z1)), CONS(mark(0), z1))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c5, c5, c3, c3

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

Removed 1 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(tail(x0), z1)) → c3(ACTIVE(cons(tail(mark(x0)), z1)), CONS(mark(tail(x0)), z1), MARK(tail(x0)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(z0)), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(tail(zeros), x1)) → c3(ACTIVE(cons(active(tail(active(zeros))), x1)), CONS(mark(tail(zeros)), x1), MARK(tail(zeros)))
MARK(cons(tail(cons(z0, z1)), x1)) → c3(ACTIVE(cons(active(tail(active(cons(mark(z0), z1)))), x1)), CONS(mark(tail(cons(z0, z1))), x1), MARK(tail(cons(z0, z1))))
MARK(cons(tail(0), x1)) → c3(ACTIVE(cons(active(tail(active(0))), x1)), CONS(mark(tail(0)), x1), MARK(tail(0)))
MARK(cons(tail(tail(z0)), x1)) → c3(ACTIVE(cons(active(tail(active(tail(mark(z0))))), x1)), CONS(mark(tail(tail(z0))), x1), MARK(tail(tail(z0))))
MARK(cons(tail(x0), x1)) → c3(CONS(mark(tail(x0)), x1))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c5, c5, c3, c3

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

Use narrowing to replace MARK(tail(zeros)) → c5(ACTIVE(tail(active(zeros))), TAIL(mark(zeros)), MARK(zeros)) by

MARK(tail(zeros)) → c5(ACTIVE(tail(zeros)), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(zeros)) → c5(ACTIVE(tail(mark(cons(0, zeros)))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(zeros)) → c5

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(tail(x0), z1)) → c3(ACTIVE(cons(tail(mark(x0)), z1)), CONS(mark(tail(x0)), z1), MARK(tail(x0)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(z0)), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(tail(zeros), x1)) → c3(ACTIVE(cons(active(tail(active(zeros))), x1)), CONS(mark(tail(zeros)), x1), MARK(tail(zeros)))
MARK(cons(tail(cons(z0, z1)), x1)) → c3(ACTIVE(cons(active(tail(active(cons(mark(z0), z1)))), x1)), CONS(mark(tail(cons(z0, z1))), x1), MARK(tail(cons(z0, z1))))
MARK(cons(tail(0), x1)) → c3(ACTIVE(cons(active(tail(active(0))), x1)), CONS(mark(tail(0)), x1), MARK(tail(0)))
MARK(cons(tail(tail(z0)), x1)) → c3(ACTIVE(cons(active(tail(active(tail(mark(z0))))), x1)), CONS(mark(tail(tail(z0))), x1), MARK(tail(tail(z0))))
MARK(cons(tail(x0), x1)) → c3(CONS(mark(tail(x0)), x1))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))
MARK(tail(zeros)) → c5(ACTIVE(tail(zeros)), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(zeros)) → c5(ACTIVE(tail(mark(cons(0, zeros)))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(zeros)) → c5

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(mark(z0))), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c5, c5, c3, c3, c5

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

Removed 1 trailing nodes:

MARK(tail(zeros)) → c5

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(tail(x0), z1)) → c3(ACTIVE(cons(tail(mark(x0)), z1)), CONS(mark(tail(x0)), z1), MARK(tail(x0)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(z0)), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(tail(zeros), x1)) → c3(ACTIVE(cons(active(tail(active(zeros))), x1)), CONS(mark(tail(zeros)), x1), MARK(tail(zeros)))
MARK(cons(tail(cons(z0, z1)), x1)) → c3(ACTIVE(cons(active(tail(active(cons(mark(z0), z1)))), x1)), CONS(mark(tail(cons(z0, z1))), x1), MARK(tail(cons(z0, z1))))
MARK(cons(tail(0), x1)) → c3(ACTIVE(cons(active(tail(active(0))), x1)), CONS(mark(tail(0)), x1), MARK(tail(0)))
MARK(cons(tail(tail(z0)), x1)) → c3(ACTIVE(cons(active(tail(active(tail(mark(z0))))), x1)), CONS(mark(tail(tail(z0))), x1), MARK(tail(tail(z0))))
MARK(cons(tail(x0), x1)) → c3(CONS(mark(tail(x0)), x1))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))
MARK(tail(zeros)) → c5(ACTIVE(tail(zeros)), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(zeros)) → c5(ACTIVE(tail(mark(cons(0, zeros)))), TAIL(mark(zeros)), MARK(zeros))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c5, c5, c3, c3

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

Removed 1 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(tail(cons(z0, z1))) → mark(z1)
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(tail(z0)) → active(tail(mark(z0)))
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
tail(mark(z0)) → tail(z0)
tail(active(z0)) → tail(z0)

Tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(tail(z0)) → c5(ACTIVE(tail(z0)), TAIL(mark(z0)), MARK(z0))
MARK(tail(cons(z0, z1))) → c5(ACTIVE(tail(active(cons(mark(z0), z1)))), TAIL(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(tail(tail(z0))) → c5(ACTIVE(tail(active(tail(mark(z0))))), TAIL(mark(tail(z0))), MARK(tail(z0)))
MARK(tail(0)) → c5(ACTIVE(tail(active(0))), TAIL(mark(0)))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(tail(x0), z1)) → c3(ACTIVE(cons(tail(mark(x0)), z1)), CONS(mark(tail(x0)), z1), MARK(tail(x0)))
MARK(cons(tail(z0), x1)) → c3(ACTIVE(cons(active(tail(z0)), x1)), CONS(mark(tail(z0)), x1), MARK(tail(z0)))
MARK(cons(tail(zeros), x1)) → c3(ACTIVE(cons(active(tail(active(zeros))), x1)), CONS(mark(tail(zeros)), x1), MARK(tail(zeros)))
MARK(cons(tail(cons(z0, z1)), x1)) → c3(ACTIVE(cons(active(tail(active(cons(mark(z0), z1)))), x1)), CONS(mark(tail(cons(z0, z1))), x1), MARK(tail(cons(z0, z1))))
MARK(cons(tail(0), x1)) → c3(ACTIVE(cons(active(tail(active(0))), x1)), CONS(mark(tail(0)), x1), MARK(tail(0)))
MARK(cons(tail(tail(z0)), x1)) → c3(ACTIVE(cons(active(tail(active(tail(mark(z0))))), x1)), CONS(mark(tail(tail(z0))), x1), MARK(tail(tail(z0))))
MARK(cons(tail(x0), x1)) → c3(CONS(mark(tail(x0)), x1))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))
MARK(tail(zeros)) → c5(ACTIVE(tail(mark(cons(0, zeros)))), TAIL(mark(zeros)), MARK(zeros))
MARK(tail(zeros)) → c5(TAIL(mark(zeros)), MARK(zeros))

S tuples:

MARK(zeros) → c2(ACTIVE(zeros))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
CONS(z0, mark(z1)) → c7(CONS(z0, z1))
CONS(active(z0), z1) → c8(CONS(z0, z1))
CONS(z0, active(z1)) → c9(CONS(z0, z1))
TAIL(mark(z0)) → c10(TAIL(z0))
TAIL(active(z0)) → c11(TAIL(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
MARK(cons(x0, x1)) → c3(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c3(CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x0)) → c3(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c3
MARK(cons(zeros, z1)) → c3(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c3(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c3(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c3(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c3(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c3(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(x0, x1), x2)) → c3(CONS(mark(cons(x0, x1)), x2))
MARK(cons(0, x0)) → c3(CONS(mark(0), x0))

K tuples:

ACTIVE(tail(cons(z0, z1))) → c1(MARK(z1))
MARK(cons(cons(tail(z0), x1), x2)) → c3(ACTIVE(cons(active(cons(active(tail(mark(z0))), x1)), x2)), CONS(mark(cons(tail(z0), x1)), x2), MARK(cons(tail(z0), x1)))

Defined Rule Symbols:

active, mark, cons, tail

Defined Pair Symbols:

ACTIVE, MARK, CONS, TAIL

Compound Symbols:

c1, c2, c6, c7, c8, c9, c10, c11, c, c3, c3, c5, c5, c3, c3

(41) 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 4.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4]
transitions:
zeros0() → 0
00() → 0
active0(0) → 1
mark0(0) → 2
cons0(0, 0) → 3
tail0(0) → 4
01() → 6
zeros1() → 7
cons1(6, 7) → 5
mark1(5) → 1
zeros1() → 8
active1(8) → 2
01() → 9
active1(9) → 2
02() → 11
zeros2() → 12
cons2(11, 12) → 10
mark2(10) → 2
mark2(6) → 14
cons2(14, 7) → 13
active2(13) → 1
mark3(11) → 16
cons3(16, 12) → 15
active3(15) → 2
02() → 17
active2(17) → 14
cons3(6, 7) → 13
cons3(17, 7) → 13
03() → 18
active3(18) → 16
cons4(11, 12) → 15
cons4(18, 12) → 15

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

(41) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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