Runtime Complexity (innermost) proof of /tmp/tmpjdsl9g/LengthOfFiniteLists_nosorts

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))), 1467 ms)

    ↳8 CdtProblem

    ↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1457 ms)

    ↳10 CdtProblem

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

    ↳12 CdtProblem

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

    ↳14 CdtProblem

    ↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 477 ms)

    ↳16 CdtProblem

    ↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 719 ms)

    ↳18 CdtProblem

    ↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 668 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), 17 ms)

    ↳26 CdtProblem

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

    ↳28 CdtProblem

    ↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 930 ms)

    ↳30 CdtProblem

    ↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 830 ms)

    ↳32 CdtProblem

    ↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1011 ms)

    ↳34 CdtProblem

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

    ↳36 CdtProblem

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

    ↳38 CdtProblem

    ↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1231 ms)

    ↳40 CdtProblem

    ↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 687 ms)

    ↳42 CdtProblem

    ↳43 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 463 ms)

    ↳44 CdtProblem

↳45 CpxTrsMatchBoundsTAProof (⇔, 317 ms)

↳46 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(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
mark(s(X)) → active(s(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)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)
s(mark(X)) → s(X)
s(active(X)) → s(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(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(nil)) → c2(MARK(0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c6(ACTIVE(0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c8(ACTIVE(tt))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c10(ACTIVE(nil))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(nil)) → c2(MARK(0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c6(ACTIVE(0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c8(ACTIVE(tt))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c10(ACTIVE(nil))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

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

Removed 4 trailing nodes:

MARK(0) → c6(ACTIVE(0))
ACTIVE(length(nil)) → c2(MARK(0))
MARK(tt) → c8(ACTIVE(tt))
MARK(nil) → c10(ACTIVE(nil))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c, c1, c3, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3

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

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVE(and(tt, z0)) → c1(MARK(z0))

We considered the (Usable) Rules:

mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(CONS(x₁, x₂)) = [3]x₂
POL(LENGTH(x₁)) = [2]x₁
POL(MARK(x₁)) = [2]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [4]x₁ + [2]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [4]x₁ + [5]x₂
POL(length(x₁)) = [4]x₁
POL(mark(x₁)) = x₁
POL(nil) = [2]
POL(s(x₁)) = [2]x₁
POL(tt) = [4]
POL(zeros) = 0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3

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

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))

We considered the (Usable) Rules:

mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(AND(x₁, x₂)) = x₂
POL(CONS(x₁, x₂)) = 0
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [1] + [2]x₁ + [4]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [4]x₁ + [4]x₂
POL(length(x₁)) = [2]x₁
POL(mark(x₁)) = x₁
POL(nil) = [2]
POL(s(x₁)) = [2]x₁
POL(tt) = 0
POL(zeros) = 0

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3

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

Use narrowing to replace MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0)) by

MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5

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

Removed 4 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5

(15) 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(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))

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(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = [4]x₂
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [2]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [2] + [4]x₁ + [2]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [4]x₁ + [4]x₂
POL(length(x₁)) = [4]x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = [2]x₁
POL(tt) = 0
POL(zeros) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5

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

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

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(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(AND(x₁, x₂)) = [4]x₁
POL(CONS(x₁, x₂)) = [5]x₂
POL(LENGTH(x₁)) = [4]x₁
POL(MARK(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [4]x₁ + [2]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [2]x₁ + [5]x₂
POL(length(x₁)) = [4]x₁
POL(mark(x₁)) = x₁
POL(nil) = [1]
POL(s(x₁)) = [2]x₁
POL(tt) = 0
POL(zeros) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5

(19) 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(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))

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(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(AND(x₁, x₂)) = [2]x₁
POL(CONS(x₁, x₂)) = 0
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [2]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [4]x₁ + [2]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [2]x₁ + [4]x₂
POL(length(x₁)) = [2]x₁
POL(mark(x₁)) = x₁
POL(nil) = [3]
POL(s(x₁)) = [2]x₁
POL(tt) = [1]
POL(zeros) = 0

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5

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

Use narrowing to replace MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0)) by

MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7

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

Removed 3 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7

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

Use narrowing to replace MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0)) by

MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9

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

Removed 3 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9

(29) 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(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))

We considered the (Usable) Rules:

length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
mark(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = x₂
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [5]x₁
POL(S(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [2]x₁ + [4]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9) = 0
POL(c9(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [2]x₁ + [5]x₂
POL(length(x₁)) = [4]x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = [2]x₁
POL(tt) = [2]
POL(zeros) = 0

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9

(31) 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(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

We considered the (Usable) Rules:

length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
mark(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(AND(x₁, x₂)) = x₁ + [2]x₂
POL(CONS(x₁, x₂)) = 0
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [4]x₁ + [4]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9) = 0
POL(c9(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [4]x₁ + [5]x₂
POL(length(x₁)) = [2]x₁
POL(mark(x₁)) = x₁
POL(nil) = [1]
POL(s(x₁)) = [2]x₁
POL(tt) = 0
POL(zeros) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9

(33) 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(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))

We considered the (Usable) Rules:

length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
mark(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(AND(x₁, x₂)) = [4]x₂
POL(CONS(x₁, x₂)) = [2]x₂
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [2]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [2] + [3]x₁ + [4]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9) = 0
POL(c9(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [2]x₁ + [5]x₂
POL(length(x₁)) = [3]x₁
POL(mark(x₁)) = x₁
POL(nil) = [3]
POL(s(x₁)) = [2]x₁
POL(tt) = 0
POL(zeros) = 0

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9

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

Use narrowing to replace MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0)) by

MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9, c11, c11

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

Removed 4 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9, c11, c11, c11

(39) 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(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

We considered the (Usable) Rules:

s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
mark(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(AND(x₁, x₂)) = [3]x₂
POL(CONS(x₁, x₂)) = [5]x₂
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [2]x₁ + [4]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11) = 0
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9) = 0
POL(c9(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [2]x₁ + [5]x₂
POL(length(x₁)) = [4]x₁
POL(mark(x₁)) = x₁
POL(nil) = [1]
POL(s(x₁)) = [2]x₁
POL(tt) = 0
POL(zeros) = 0

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9, c11, c11, c11

(41) 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(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))

We considered the (Usable) Rules:

s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
mark(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(AND(x₁, x₂)) = [1] + [4]x₂
POL(CONS(x₁, x₂)) = [2]x₂
POL(LENGTH(x₁)) = [4]x₁
POL(MARK(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [1] + [5]x₁ + [2]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11) = 0
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9) = 0
POL(c9(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [2]x₁ + [5]x₂
POL(length(x₁)) = [4]x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = [2]x₁
POL(tt) = [1]
POL(zeros) = 0

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9, c11, c11, c11

(43) 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(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))

We considered the (Usable) Rules:

s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
mark(nil) → active(nil)
mark(tt) → active(tt)
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
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(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)

And the Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = [2]x₁
POL(AND(x₁, x₂)) = [1] + x₁ + x₂
POL(CONS(x₁, x₂)) = [2]x₁ + [2]x₂
POL(LENGTH(x₁)) = 0
POL(MARK(x₁)) = [5]x₁
POL(S(x₁)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [1] + [2]x₁ + [4]x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11) = 0
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c9) = 0
POL(c9(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [3]x₁ + [5]x₂
POL(length(x₁)) = [2]x₁
POL(mark(x₁)) = x₁
POL(nil) = [2]
POL(s(x₁)) = [2]x₁
POL(tt) = 0
POL(zeros) = 0

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

S tuples:

MARK(zeros) → c4(ACTIVE(zeros))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c5(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(x0)) → c11
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))

K tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(and(z0, z1))) → c11(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))

Defined Rule Symbols:

active, mark, cons, and, length, s

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c5, c7, c7, c7, c9, c9, c9, c11, c11, c11

(45) 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, 5, 6]
transitions:
zeros0() → 0
00() → 0
tt0() → 0
nil0() → 0
active0(0) → 1
mark0(0) → 2
cons0(0, 0) → 3
and0(0, 0) → 4
length0(0) → 5
s0(0) → 6
01() → 8
zeros1() → 9
cons1(8, 9) → 7
mark1(7) → 1
zeros1() → 10
active1(10) → 2
01() → 11
active1(11) → 2
tt1() → 12
active1(12) → 2
nil1() → 13
active1(13) → 2
02() → 15
zeros2() → 16
cons2(15, 16) → 14
mark2(14) → 2
mark2(8) → 18
cons2(18, 9) → 17
active2(17) → 1
mark3(15) → 20
cons3(20, 16) → 19
active3(19) → 2
02() → 21
active2(21) → 18
cons3(8, 9) → 17
cons3(21, 9) → 17
03() → 22
active3(22) → 20
cons4(15, 16) → 19
cons4(22, 16) → 19

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) 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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

(45) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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