Runtime Complexity (innermost) proof of /tmp/tmp3poNyc/Ex6_9_Luc02c

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

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

↳4 CdtProblem

↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1499 ms)

↳6 CdtProblem

↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1526 ms)

↳8 CdtProblem

↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1165 ms)

↳10 CdtProblem

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

↳12 CdtProblem

↳13 CdtUnreachableProof (⇔, 0 ms)

↳14 CdtProblem

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

↳16 CdtProblem

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

↳18 CdtProblem

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

↳20 CdtProblem

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

↳22 CdtProblem

↳23 CdtUnreachableProof (⇔, 0 ms)

↳24 CdtProblem

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

↳26 CdtProblem

↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳28 CdtProblem

↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳30 CdtProblem

↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 8 ms)

↳32 CdtProblem

↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 12 ms)

↳34 CdtProblem

↳35 SIsEmptyProof (⇔, 0 ms)

↳36 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(2nd(cons1(X, cons(Y, Z)))) → mark(Y)
active(2nd(cons(X, X1))) → mark(2nd(cons1(X, X1)))
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

S tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c2

(5) 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(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(S(z0))

We considered the (Usable) Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))

And the Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

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

POL(2ND(x₁)) = 0
POL(2nd(x₁)) = [2]x₁
POL(ACTIVE(x₁)) = [2]
POL(CONS(x₁, x₂)) = 0
POL(CONS1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
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(c2(x₁)) = x₁
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = [2]x₁
POL(cons1(x₁, x₂)) = x₁
POL(from(x₁)) = [5]x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

K tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(S(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c2

(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.

TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))

And the Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

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

POL(2ND(x₁)) = 0
POL(2nd(x₁)) = [4]x₁
POL(ACTIVE(x₁)) = [3]
POL(CONS(x₁, x₂)) = 0
POL(CONS1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
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(c2(x₁)) = x₁
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = [2]x₁
POL(cons1(x₁, x₂)) = x₁
POL(from(x₁)) = [2]x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [4] + x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = [2]x₁

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))

K tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c2

(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.

TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))

And the Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

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

POL(2ND(x₁)) = 0
POL(2nd(x₁)) = x₁
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(CONS1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = [2] + x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
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(c2(x₁)) = x₁
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁
POL(cons1(x₁, x₂)) = x₁ + x₂
POL(from(x₁)) = x₁
POL(mark(x₁)) = [1]
POL(ok(x₁)) = [2] + x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = x₁

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c2

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

Use narrowing to replace ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1)) by

ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
ACTIVE(2nd(cons(mark(z0), z1))) → c1(2ND(mark(cons1(z0, z1))), CONS1(mark(z0), z1))
ACTIVE(2nd(cons(ok(z0), ok(z1)))) → c1(2ND(ok(cons1(z0, z1))), CONS1(ok(z0), ok(z1)))
ACTIVE(2nd(cons(x0, x1))) → c1

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
ACTIVE(2nd(cons(mark(z0), z1))) → c1(2ND(mark(cons1(z0, z1))), CONS1(mark(z0), z1))
ACTIVE(2nd(cons(ok(z0), ok(z1)))) → c1(2ND(ok(cons1(z0, z1))), CONS1(ok(z0), ok(z1)))
ACTIVE(2nd(cons(x0, x1))) → c1

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c2, c1, c1

(13) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(2nd(cons(mark(z0), z1))) → c1(2ND(mark(cons1(z0, z1))), CONS1(mark(z0), z1))
ACTIVE(2nd(cons(ok(z0), ok(z1)))) → c1(2ND(ok(cons1(z0, z1))), CONS1(ok(z0), ok(z1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
ACTIVE(2nd(cons(x0, x1))) → c1
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1, TOP, ACTIVE, PROPER

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c25, c26, c3, c4, c5, c6, c7, c8, c2, c1, c1, c20, c21, c22, c23, c24

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

Removed 1 trailing nodes:

ACTIVE(2nd(cons(x0, x1))) → c1

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1, TOP, ACTIVE, PROPER

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c25, c26, c3, c4, c5, c6, c7, c8, c2, c1, c20, c21, c22, c23, c24

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

Use narrowing to replace TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(x0)) → c25

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(mark(x0)) → c25

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1, TOP, ACTIVE, PROPER

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c26, c3, c4, c5, c6, c7, c8, c2, c1, c20, c21, c22, c23, c24, c25, c25

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c25

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1, TOP, ACTIVE, PROPER

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c26, c3, c4, c5, c6, c7, c8, c2, c1, c20, c21, c22, c23, c24, c25

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

Use narrowing to replace TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(2nd(cons1(z0, cons(z1, z2))))) → c26(TOP(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2)))))
TOP(ok(2nd(cons(z0, z1)))) → c26(TOP(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1))))
TOP(ok(from(z0))) → c26(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(2nd(z0))) → c26(TOP(2nd(active(z0))), ACTIVE(2nd(z0)))
TOP(ok(cons(z0, z1))) → c26(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c26(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(s(z0))) → c26(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))
TOP(ok(x0)) → c26

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(ok(2nd(cons1(z0, cons(z1, z2))))) → c26(TOP(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2)))))
TOP(ok(2nd(cons(z0, z1)))) → c26(TOP(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1))))
TOP(ok(from(z0))) → c26(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(2nd(z0))) → c26(TOP(2nd(active(z0))), ACTIVE(2nd(z0)))
TOP(ok(cons(z0, z1))) → c26(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c26(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(s(z0))) → c26(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))
TOP(ok(x0)) → c26

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0))
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1, ACTIVE, PROPER, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c3, c4, c5, c6, c7, c8, c2, c1, c20, c21, c22, c23, c24, c25, c26, c26

(23) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0))
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(2nd(cons(z0, mark(z1)))) → c1(2ND(mark(cons1(z0, z1))), CONS1(z0, mark(z1)))
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))
TOP(ok(2nd(cons1(z0, cons(z1, z2))))) → c26(TOP(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2)))))
TOP(ok(2nd(cons(z0, z1)))) → c26(TOP(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1))))
TOP(ok(from(z0))) → c26(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(2nd(z0))) → c26(TOP(2nd(active(z0))), ACTIVE(2nd(z0)))
TOP(ok(cons(z0, z1))) → c26(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c26(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(s(z0))) → c26(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1)))
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
TOP(ok(x0)) → c26

S tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c26

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

Removed 1 trailing nodes:

TOP(ok(x0)) → c26

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

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

S(mark(z0)) → c15(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

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

POL(2ND(x₁)) = 0
POL(CONS(x₁, x₂)) = [2]x₂
POL(CONS1(x₁, x₂)) = [3]x₁
POL(FROM(x₁)) = 0
POL(S(x₁)) = [2]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
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(c9(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(ok(z0)) → c16(S(z0))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

S(mark(z0)) → c15(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(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.

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(ok(z0)) → c16(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

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

POL(2ND(x₁)) = [4]x₁
POL(CONS(x₁, x₂)) = [3]x₁ + [4]x₂
POL(CONS1(x₁, x₂)) = 0
POL(FROM(x₁)) = [5]x₁
POL(S(x₁)) = [5]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
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(c9(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = [5] + x₁

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

S(mark(z0)) → c15(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(ok(z0)) → c16(S(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(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.

CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

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

POL(2ND(x₁)) = 0
POL(CONS(x₁, x₂)) = [4]x₂
POL(CONS1(x₁, x₂)) = x₁
POL(FROM(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
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(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))

K tuples:

S(mark(z0)) → c15(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(ok(z0)) → c16(S(z0))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

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

POL(2ND(x₁)) = x₁²
POL(CONS(x₁, x₂)) = [2]x₁ + [3]x₁²
POL(CONS1(x₁, x₂)) = [3]x₂ + x₂²
POL(FROM(x₁)) = 0
POL(S(x₁)) = [2]x₁²
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
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(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(2nd(z0)) → 2nd(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
active(s(z0)) → s(active(z0))
active(cons1(z0, z1)) → cons1(active(z0), z1)
active(cons1(z0, z1)) → cons1(z0, active(z1))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(mark(z0)) → c15(S(z0))
S(ok(z0)) → c16(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:none
K tuples:

S(mark(z0)) → c15(S(z0))
CONS1(mark(z0), z1) → c17(CONS1(z0, z1))
2ND(mark(z0)) → c9(2ND(z0))
2ND(ok(z0)) → c10(2ND(z0))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))
FROM(mark(z0)) → c13(FROM(z0))
FROM(ok(z0)) → c14(FROM(z0))
S(ok(z0)) → c16(S(z0))
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(35) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) 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) CdtUnreachableProof (EQUIVALENT transformation)

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

(23) CdtUnreachableProof (EQUIVALENT transformation)

(24) Obligation:

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

(26) Obligation:

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) 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^2))) transformation)

(34) Obligation:

(35) SIsEmptyProof (EQUIVALENT transformation)

(36) BOUNDS(O(1), O(1))