(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(x0)) → c3

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(x0)) → c3
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(x0)) → c3
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(from(x0)) → c3

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(x0, x1)) → c4

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(x0, x1)) → c4
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(x0, x1)) → c4
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(cons(x0, x1)) → c4

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

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

ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(x0)) → c5

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(x0)) → c5
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(x0)) → c5
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(s(x0)) → c5

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
PROPER(from(x0)) → c17

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
PROPER(from(x0)) → c17
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
PROPER(from(x0)) → c17
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(from(x0)) → c17

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(x0, x1)) → c18

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(x0, x1)) → c18
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(x0, x1)) → c18
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(cons(x0, x1)) → c18

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) by

PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c19

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c19
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c19
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(s(x0)) → c19

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(x0, x1)) → c20

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(x0, x1)) → c20
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(x0, x1)) → c20
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(sel(x0, x1)) → c20

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(x0)) → c22

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(x0)) → c22
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(x0)) → c22
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c22

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

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

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

TOP(mark(0)) → c22(TOP(ok(0)))
We considered the (Usable) Rules:

proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(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))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
And the Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(CONS(x1, x2)) = 0   
POL(FROM(x1)) = 0   
POL(PROPER(x1)) = 0   
POL(S(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(TOP(x1)) = [4]x1   
POL(active(x1)) = x1   
POL(c(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c17(x1, x2)) = x1 + x2   
POL(c18(x1, x2)) = x1 + x2   
POL(c18(x1, x2, x3)) = x1 + x2 + x3   
POL(c19(x1)) = x1   
POL(c19(x1, x2)) = x1 + x2   
POL(c2(x1)) = x1   
POL(c20(x1, x2)) = x1 + x2   
POL(c20(x1, x2, x3)) = x1 + x2 + x3   
POL(c22(x1)) = x1   
POL(c22(x1, x2)) = x1 + x2   
POL(c23(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(cons(x1, x2)) = [2]   
POL(from(x1)) = [2]   
POL(mark(x1)) = [2]   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   
POL(s(x1)) = [2]   
POL(sel(x1, x2)) = [2]   

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(x0)) → c23

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(x0)) → c23
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(x0)) → c23
K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

(51) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(mark(0)) → c22(TOP(ok(0)))
TOP(ok(x0)) → c23
S tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
TOP(ok(x0)) → c23
K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL, TOP

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16, c22, c23

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

Removed 2 trailing nodes:

TOP(ok(x0)) → c23
TOP(mark(0)) → c22(TOP(ok(0)))

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
S tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

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

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

CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x1, x2)) = [2]x1 + [3]x2   
POL(FROM(x1)) = 0   
POL(S(x1)) = [3]x1   
POL(SEL(x1, x2)) = [2]x1 + [4]x2   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
S tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
K tuples:

CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

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

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

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x1, x2)) = [5]x1 + [3]x2   
POL(FROM(x1)) = [2]x1   
POL(S(x1)) = [5]x1   
POL(SEL(x1, x2)) = 0   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [4] + x1   
POL(ok(x1)) = [2] + x1   

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
S tuples:

SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
K tuples:

CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

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

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

SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x1, x2)) = x2   
POL(FROM(x1)) = [4]x1   
POL(S(x1)) = [2]x1   
POL(SEL(x1, x2)) = [2]x2   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = [2] + x1   

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
S tuples:none
K tuples:

CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
SEL(mark(z0), z1) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

(61) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(62) BOUNDS(O(1), O(1))