### (0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(sel(0, cons(X, Z))) → 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(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
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))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
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(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
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 Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
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(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
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))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
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(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
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(first(0, z0)) → c1
ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c3
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c28
PROPER(nil) → c29
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(first(0, z0)) → c1
ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c3
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c28
PROPER(nil) → c29
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32

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

Removed 4 trailing nodes:

ACTIVE(first(0, z0)) → c1
PROPER(nil) → c29
ACTIVE(sel(0, cons(z0, z1))) → c3
PROPER(0) → c28

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
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(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
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))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
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(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
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(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c, c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32

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

Removed 2 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
S tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c

### (7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1

### (9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1

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

Use narrowing to replace ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1, c5

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1, c5, c6

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

Use narrowing to replace ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1, c5, c6, c7

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

Use narrowing to replace PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)), PROPER(0))
PROPER(from(nil)) → c24(FROM(ok(nil)), PROPER(nil))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)), PROPER(0))
PROPER(from(nil)) → c24(FROM(ok(nil)), PROPER(nil))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)), PROPER(0))
PROPER(from(nil)) → c24(FROM(ok(nil)), PROPER(nil))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24

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

Removed 2 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24

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

Use narrowing to replace PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25

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

Removed 4 trailing tuple parts

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25

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

Use narrowing to replace PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) by

PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)), PROPER(0))
PROPER(s(nil)) → c26(S(ok(nil)), PROPER(nil))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)), PROPER(0))
PROPER(s(nil)) → c26(S(ok(nil)), PROPER(nil))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)), PROPER(0))
PROPER(s(nil)) → c26(S(ok(nil)), PROPER(nil))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(s(nil)) → c26(S(ok(nil)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(s(nil)) → c26(S(ok(nil)))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26

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

Use narrowing to replace PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(s(nil)) → c26(S(ok(nil)))
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c24(FROM(ok(0)))
PROPER(from(nil)) → c24(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(s(nil)) → c26(S(ok(nil)))
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, cons, first, sel, from, s, proper

Defined Pair Symbols:

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

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27

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

Removed 4 trailing tuple parts

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c12(FROM(z0))
FROM(ok(z0)) → c13(FROM(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
FIRST(mark(z0), z1) → c18(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) →