### (0) Obligation:

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

active(2nd(cons(X, cons(Y, Z)))) → mark(Y)
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))
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))
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))
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))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(2nd(cons(z0, cons(z1, z2)))) → c
ACTIVE(from(z0)) → c1(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(2nd(cons(z0, cons(z1, z2)))) → c
ACTIVE(from(z0)) → c1(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(2nd(cons(z0, cons(z1, z2)))) → c

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(from(z0)) → c1(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(from(z0)) → c1(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 2 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
S tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
S tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

### (9) CdtRuleRemovalProof (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)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
We considered the (Usable) Rules:

active(cons(z0, z1)) → cons(active(z0), z1)
s(mark(z0)) → mark(s(z0))
from(mark(z0)) → mark(from(z0))
active(s(z0)) → s(active(z0))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(from(z0)) → mark(cons(z0, from(s(z0))))
proper(s(z0)) → s(proper(z0))
active(2nd(z0)) → 2nd(active(z0))
from(ok(z0)) → ok(from(z0))
active(2nd(cons(z0, cons(z1, z2)))) → mark(z1)
proper(from(z0)) → from(proper(z0))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
s(ok(z0)) → ok(s(z0))
And the Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(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(FROM(x1)) = 0
POL(PROPER(x1)) = 0
POL(S(x1)) = 0
POL(TOP(x1)) = x1
POL(active(x1)) = [4]
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1, x2)) = x1 + x2
POL(c15(x1, x2, x3)) = x1 + x2 + x3
POL(c16(x1, x2)) = x1 + x2
POL(c17(x1, x2)) = x1 + x2
POL(c18(x1, x2)) = x1 + x2
POL(c19(x1, x2)) = x1 + x2
POL(c2(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)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = x1
POL(from(x1)) = x1
POL(mark(x1)) = [2]
POL(ok(x1)) = [5]
POL(proper(x1)) = 0
POL(s(x1)) = x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
S tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(from(z0)) → c1(S(z0))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

### (11) CdtRuleRemovalProof (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(from(z0)) → c1(S(z0))
We considered the (Usable) Rules:

active(cons(z0, z1)) → cons(active(z0), z1)
s(mark(z0)) → mark(s(z0))
from(mark(z0)) → mark(from(z0))
active(s(z0)) → s(active(z0))
2nd(mark(z0)) → mark(2nd(z0))
2nd(ok(z0)) → ok(2nd(z0))
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(from(z0)) → mark(cons(z0, from(s(z0))))
proper(s(z0)) → s(proper(z0))
active(2nd(z0)) → 2nd(active(z0))
from(ok(z0)) → ok(from(z0))
active(2nd(cons(z0, cons(z1, z2)))) → mark(z1)
proper(from(z0)) → from(proper(z0))
proper(2nd(z0)) → 2nd(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
s(ok(z0)) → ok(s(z0))
And the Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(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)) = [3] + x1
POL(CONS(x1, x2)) = 0
POL(FROM(x1)) = 0
POL(PROPER(x1)) = 0
POL(S(x1)) = 0
POL(TOP(x1)) = [4]x1
POL(active(x1)) = 0
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1, x2)) = x1 + x2
POL(c15(x1, x2, x3)) = x1 + x2 + x3
POL(c16(x1, x2)) = x1 + x2
POL(c17(x1, x2)) = x1 + x2
POL(c18(x1, x2)) = x1 + x2
POL(c19(x1, x2)) = x1 + x2
POL(c2(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)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = x1
POL(from(x1)) = [4]x1
POL(mark(x1)) = 0
POL(ok(x1)) = [4] + x1
POL(proper(x1)) = 0
POL(s(x1)) = [4]x1

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
S tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) by

ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
S tuples:

ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

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

ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
S tuples:

ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
S tuples:

ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

### (19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) by

PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

### (23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) by

PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0)))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))
TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0)))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(2nd(cons(z0, cons(z1, z2))))) → c19(TOP(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
TOP(ok(from(z0))) → c19(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(2nd(z0))) → c19(TOP(2nd(active(z0))), ACTIVE(2nd(z0)))
TOP(ok(cons(z0, z1))) → c19(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c19(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(s(z0))) → c19(TOP(s(active(z0))), ACTIVE(s(z0)))

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))
TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(ok(2nd(cons(z0, cons(z1, z2))))) → c19(TOP(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
TOP(ok(from(z0))) → c19(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(2nd(z0))) → c19(TOP(2nd(active(z0))), ACTIVE(2nd(z0)))
TOP(ok(cons(z0, z1))) → c19(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c19(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(s(z0))) → c19(TOP(s(active(z0))), ACTIVE(s(z0)))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))
K tuples:

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c1(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

### (33) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(from(z0)) → c1(S(z0))
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0)))
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0)))
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0)))
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))
TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0)))
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(ok(2nd(cons(z0, cons(z1, z2))))) → c19(TOP(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2)))))
TOP(ok(from(z0))) → c19(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(2nd(z0))) → c19(TOP(2nd(active(z0))), ACTIVE(2nd(z0)))
TOP(ok(cons(z0, z1))) → c19(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c19(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(s(z0))) → c19(TOP(s(active(z0))), ACTIVE(s(z0)))

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (35) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(2nd(cons(z0, cons(z1, z2)))) → mark(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))
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))
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))

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (37) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

CONS(mark(z0), z1) → c8(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(2ND(x1)) = 0
POL(CONS(x1, x2)) = [2]x1
POL(FROM(x1)) = 0
POL(S(x1)) = 0
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c6(x1)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
K tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (39) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(2ND(x1)) = 0
POL(CONS(x1, x2)) = [4]x2
POL(FROM(x1)) = 0
POL(S(x1)) = 0
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c6(x1)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = 0
POL(ok(x1)) = [2] + x1

### (40) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
K tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (41) CdtRuleRemovalProof (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)) → c6(2ND(z0))
S(mark(z0)) → c12(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(2ND(x1)) = [2]x1
POL(CONS(x1, x2)) = x1 + x2
POL(FROM(x1)) = 0
POL(S(x1)) = x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c6(x1)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = x1

### (42) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

2ND(ok(z0)) → c7(2ND(z0))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(ok(z0)) → c13(S(z0))
K tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
2ND(mark(z0)) → c6(2ND(z0))
S(mark(z0)) → c12(S(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (43) CdtRuleRemovalProof (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(ok(z0)) → c7(2ND(z0))
We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(2ND(x1)) = [2]x1
POL(CONS(x1, x2)) = 0
POL(FROM(x1)) = 0
POL(S(x1)) = 0
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c6(x1)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = [3] + x1
POL(ok(x1)) = [4] + x1

### (44) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(ok(z0)) → c13(S(z0))
K tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
2ND(mark(z0)) → c6(2ND(z0))
S(mark(z0)) → c12(S(z0))
2ND(ok(z0)) → c7(2ND(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (45) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

FROM(mark(z0)) → c10(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(2ND(x1)) = 0
POL(CONS(x1, x2)) = [3]x1 + [2]x2
POL(FROM(x1)) = [2]x1
POL(S(x1)) = 0
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c6(x1)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = [3] + x1
POL(ok(x1)) = x1

### (46) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

FROM(ok(z0)) → c11(FROM(z0))
S(ok(z0)) → c13(S(z0))
K tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
2ND(mark(z0)) → c6(2ND(z0))
S(mark(z0)) → c12(S(z0))
2ND(ok(z0)) → c7(2ND(z0))
FROM(mark(z0)) → c10(FROM(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (47) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

FROM(ok(z0)) → c11(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(2ND(x1)) = [4]x1
POL(CONS(x1, x2)) = [2]x1
POL(FROM(x1)) = [3]x1
POL(S(x1)) = 0
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c6(x1)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [2] + x1

### (48) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:

S(ok(z0)) → c13(S(z0))
K tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
2ND(mark(z0)) → c6(2ND(z0))
S(mark(z0)) → c12(S(z0))
2ND(ok(z0)) → c7(2ND(z0))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (49) CdtRuleRemovalProof (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(ok(z0)) → c13(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(2ND(x1)) = [5]x1
POL(CONS(x1, x2)) = [2]x2
POL(FROM(x1)) = [4]x1
POL(S(x1)) = [4]x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c6(x1)) = x1
POL(c7(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [1] + x1

### (50) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

2ND(mark(z0)) → c6(2ND(z0))
2ND(ok(z0)) → c7(2ND(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
S tuples:none
K tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
2ND(mark(z0)) → c6(2ND(z0))
S(mark(z0)) → c12(S(z0))
2ND(ok(z0)) → c7(2ND(z0))
FROM(mark(z0)) → c10(FROM(z0))
FROM(ok(z0)) → c11(FROM(z0))
S(ok(z0)) → c13(S(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

### (51) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty