### (0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel(0, cons(z0, z1))) → c1
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(z0, 0)) → c3
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(quot(0, s(z0))) → c5
ACTIVE(quot(s(z0), s(z1))) → c6(S(quot(minus(z0, z1), s(z1))), QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(zWquot(z0, nil)) → c7
ACTIVE(zWquot(nil, z0)) → c8
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c43
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c47
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel(0, cons(z0, z1))) → c1
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(z0, 0)) → c3
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(quot(0, s(z0))) → c5
ACTIVE(quot(s(z0), s(z1))) → c6(S(quot(minus(z0, z1), s(z1))), QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(zWquot(z0, nil)) → c7
ACTIVE(zWquot(nil, z0)) → c8
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c43
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c47
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, minus, quot, zWquot, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, 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, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49

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

Removed 7 trailing nodes:

PROPER(0) → c43
ACTIVE(sel(0, cons(z0, z1))) → c1
ACTIVE(quot(0, s(z0))) → c5
ACTIVE(minus(z0, 0)) → c3
PROPER(nil) → c47
ACTIVE(zWquot(z0, nil)) → c7
ACTIVE(zWquot(nil, z0)) → c8

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c6(S(quot(minus(z0, z1), s(z1))), QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c6(S(quot(minus(z0, z1), s(z1))), QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, minus, quot, zWquot, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c, c2, c4, c6, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c44, c45, c46, c48, c49

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

Removed 4 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, minus, quot, zWquot, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c44, c45, c46, c48, c49, c, c6

### (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(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, minus, quot, zWquot, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c44, c45, c46, c48, c49, 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(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
K tuples:none
Defined Rule Symbols:

active, sel, cons, quot, zWquot, from, s, minus, proper

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c44, c45, c46, c48, c49, c, c1

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

Use narrowing to replace ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, cons, quot, zWquot, from, s, minus, proper

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c44, c45, c46, c48, c49, c, c1, c10

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c11(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, cons, quot, zWquot, from, s, minus, proper

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c44, c45, c46, c48, c49, c, c1, c10, c11

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

Use narrowing to replace ACTIVE(s(z0)) → c12(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, cons, quot, zWquot, from, s, minus, proper

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c44, c45, c46, c48, c49, c, c1, c10, c11, c12

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

Use narrowing to replace PROPER(from(z0)) → c39(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(from(z0))) → c39(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c39(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c39(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c39(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c39(FROM(ok(0)), PROPER(0))
PROPER(from(minus(z0, z1))) → c39(FROM(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(from(quot(z0, z1))) → c39(FROM(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
PROPER(from(zWquot(z0, z1))) → c39(FROM(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
PROPER(from(nil)) → c39(FROM(ok(nil)), PROPER(nil))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
PROPER(from(from(z0))) → c39(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c39(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c39(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c39(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c39(FROM(ok(0)), PROPER(0))
PROPER(from(minus(z0, z1))) → c39(FROM(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(from(quot(z0, z1))) → c39(FROM(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
PROPER(from(zWquot(z0, z1))) → c39(FROM(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
PROPER(from(nil)) → c39(FROM(ok(nil)), PROPER(nil))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
PROPER(from(from(z0))) → c39(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c39(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c39(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c39(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(0)) → c39(FROM(ok(0)), PROPER(0))
PROPER(from(minus(z0, z1))) → c39(FROM(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(from(quot(z0, z1))) → c39(FROM(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
PROPER(from(zWquot(z0, z1))) → c39(FROM(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
PROPER(from(nil)) → c39(FROM(ok(nil)), PROPER(nil))
K tuples:none
Defined Rule Symbols:

active, sel, cons, quot, zWquot, from, s, minus, proper

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c44, c45, c46, c48, c49, c, c1, c10, c11, c12, c39

### (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(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
PROPER(from(from(z0))) → c39(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c39(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c39(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c39(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(minus(z0, z1))) → c39(FROM(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(from(quot(z0, z1))) → c39(FROM(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
PROPER(from(zWquot(z0, z1))) → c39(FROM(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
PROPER(from(0)) → c39(FROM(ok(0)))
PROPER(from(nil)) → c39(FROM(ok(nil)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
PROPER(from(from(z0))) → c39(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c39(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c39(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c39(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(minus(z0, z1))) → c39(FROM(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(from(quot(z0, z1))) → c39(FROM(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
PROPER(from(zWquot(z0, z1))) → c39(FROM(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
PROPER(from(0)) → c39(FROM(ok(0)))
PROPER(from(nil)) → c39(FROM(ok(nil)))
K tuples:none
Defined Rule Symbols:

active, sel, cons, quot, zWquot, from, s, minus, proper

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c44, c45, c46, c48, c49, c, c1, c10, c11, c12, c39, c39

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

Use narrowing to replace PROPER(cons(z0, z1)) → c40(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, from(z0))) → c40(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c40(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c40(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c40(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c40(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, minus(z0, z1))) → c40(CONS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(cons(x0, quot(z0, z1))) → c40(CONS(proper(x0), quot(proper(z0), proper(z1))), PROPER(x0), PROPER(quot(z0, z1)))
PROPER(cons(x0, zWquot(z0, z1))) → c40(CONS(proper(x0), zWquot(proper(z0), proper(z1))), PROPER(x0), PROPER(zWquot(z0, z1)))
PROPER(cons(x0, nil)) → c40(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(from(z0), x1)) → c40(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c40(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c40(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c40(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c40(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(minus(z0, z1), x1)) → c40(CONS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(cons(quot(z0, z1), x1)) → c40(CONS(quot(proper(z0), proper(z1)), proper(x1)), PROPER(quot(z0, z1)), PROPER(x1))
PROPER(cons(zWquot(z0, z1), x1)) → c40(CONS(zWquot(proper(z0), proper(z1)), proper(x1)), PROPER(zWquot(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c40(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
PROPER(from(from(z0))) → c39(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c39(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c39(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c39(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(minus(z0, z1))) → c39(FROM(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(from(quot(z0, z1))) → c39(FROM(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
PROPER(from(zWquot(z0, z1))) → c39(FROM(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
PROPER(from(0)) → c39(FROM(ok(0)))
PROPER(from(nil)) → c39(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c40(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c40(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c40(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c40(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c40(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, minus(z0, z1))) → c40(CONS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(cons(x0, quot(z0, z1))) → c40(CONS(proper(x0), quot(proper(z0), proper(z1))), PROPER(x0), PROPER(quot(z0, z1)))
PROPER(cons(x0, zWquot(z0, z1))) → c40(CONS(proper(x0), zWquot(proper(z0), proper(z1))), PROPER(x0), PROPER(zWquot(z0, z1)))
PROPER(cons(x0, nil)) → c40(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(from(z0), x1)) → c40(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c40(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c40(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c40(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c40(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(minus(z0, z1), x1)) → c40(CONS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(cons(quot(z0, z1), x1)) → c40(CONS(quot(proper(z0), proper(z1)), proper(x1)), PROPER(quot(z0, z1)), PROPER(x1))
PROPER(cons(zWquot(z0, z1), x1)) → c40(CONS(zWquot(proper(z0), proper(z1)), proper(x1)), PROPER(zWquot(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c40(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(from(zWquot(z0, z1))) → c10(FROM(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c11(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c11(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c11(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(minus(z0, 0), x1)) → c11(CONS(mark(0), x1), ACTIVE(minus(z0, 0)))
ACTIVE(cons(minus(s(z0), s(z1)), x1)) → c11(CONS(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(cons(quot(0, s(z0)), x1)) → c11(CONS(mark(0), x1), ACTIVE(quot(0, s(z0))))
ACTIVE(cons(quot(s(z0), s(z1)), x1)) → c11(CONS(mark(s(quot(minus(z0, z1), s(z1)))), x1), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(cons(zWquot(z0, nil), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(z0, nil)))
ACTIVE(cons(zWquot(nil, z0), x1)) → c11(CONS(mark(nil), x1), ACTIVE(zWquot(nil, z0)))
ACTIVE(cons(zWquot(cons(z0, z1), cons(z2, z3)), x1)) → c11(CONS(mark(cons(quot(z0, z2), zWquot(z1, z3))), x1), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(cons(from(z0), x1)) → c11(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c11(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c11(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c11(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(active(z0), z1), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(minus(z0, z1), x1)) → c11(CONS(minus(z0, active(z1)), x1), ACTIVE(minus(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(active(z0), z1), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(quot(z0, z1), x1)) → c11(CONS(quot(z0, active(z1)), x1), ACTIVE(quot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(active(z0), z1), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(cons(zWquot(z0, z1), x1)) → c11(CONS(zWquot(z0, active(z1)), x1), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(from(z0))) → c12(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c12(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c12(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(minus(z0, 0))) → c12(S(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c12(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(quot(0, s(z0)))) → c12(S(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(s(quot(s(z0), s(z1)))) → c12(S(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(s(zWquot(z0, nil))) → c12(S(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(s(zWquot(nil, z0))) → c12(S(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(s(zWquot(cons(z0, z1), cons(z2, z3)))) → c12(S(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(s(from(z0))) → c12(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c12(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c12(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c12(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(s(minus(z0, z1))) → c12(S(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(s(quot(z0, z1))) → c12(S(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
ACTIVE(s(zWquot(z0, z1))) → c12(S(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
PROPER(from(from(z0))) → c39(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c39(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c39(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(sel(z0, z1))) → c39(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(minus(z0, z1))) → c39(FROM(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(from(quot(z0, z1))) → c39(FROM(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
PROPER(from(zWquot(z0, z1))) → c39(FROM(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
PROPER(from(0)) → c39(FROM(ok(0)))
PROPER(from(nil)) → c39(FROM(ok(nil)))
PROPER(cons(x0, from(z0))) → c40(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c40(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c40(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c40(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, 0)) → c40(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, minus(z0, z1))) → c40(CONS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(cons(x0, quot(z0, z1))) → c40(CONS(proper(x0), quot(proper(z0), proper(z1))), PROPER(x0), PROPER(quot(z0, z1)))
PROPER(cons(x0, zWquot(z0, z1))) → c40(CONS(proper(x0), zWquot(proper(z0), proper(z1))), PROPER(x0), PROPER(zWquot(z0, z1)))
PROPER(cons(x0, nil)) → c40(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(from(z0), x1)) → c40(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c40(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c40(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c40(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c40(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(minus(z0, z1), x1)) → c40(CONS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(cons(quot(z0, z1), x1)) → c40(CONS(quot(proper(z0), proper(z1)), proper(x1)), PROPER(quot(z0, z1)), PROPER(x1))
PROPER(cons(zWquot(z0, z1), x1)) → c40(CONS(zWquot(proper(z0), proper(z1)), proper(x1)), PROPER(zWquot(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c40(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, cons, quot, zWquot, from, s, minus, proper

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT, PROPER, TOP

Compound Symbols:

c2, c4, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c41, c42, c44, c45, c46, c48, c49, c, c1, c10, c11, c12, c39, c39, c40

### (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(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(minus(z0, z1)) → minus(active(z0), z1)
active(minus(z0, z1)) → minus(z0, active(z1))
active(quot(z0, z1)) → quot(active(z0), z1)
active(quot(z0, z1)) → quot(z0, active(z1))
active(zWquot(z0, z1)) → zWquot(active(z0), z1)
active(zWquot(z0, z1)) → zWquot(z0, active(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
quot(ok(z0), ok(z1)) → ok(quot(z0, z1))
quot(mark(z0), z1) → mark(quot(z0, z1))
quot(z0, mark(z1)) → mark(quot(z0, z1))
zWquot(mark(z0), z1) → mark(zWquot(z0, z1))
zWquot(z0, mark(z1)) → mark(zWquot(z0, z1))
zWquot(ok(z0), ok(z1)) → ok(zWquot(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))
minus(mark(z0), z1) → mark(minus(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
minus(z0, mark(z1)) → mark(minus(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(0) → ok(0)
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(quot(z0, z1)) → quot(proper(z0), proper(z1))
proper(zWquot(z0, z1)) → zWquot(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(minus(s(z0), s(z1))) → c4(MINUS(z0, z1))
ACTIVE(sel(z0, z1)) → c13(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c14(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(minus(z0, z1)) → c15(MINUS(active(z0), z1), ACTIVE(z0))
ACTIVE(minus(z0, z1)) → c16(MINUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(quot(z0, z1)) → c17(QUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(quot(z0, z1)) → c18(QUOT(z0, active(z1)), ACTIVE(z1))
ACTIVE(zWquot(z0, z1)) → c19(ZWQUOT(active(z0), z1), ACTIVE(z0))
ACTIVE(zWquot(z0, z1)) → c20(ZWQUOT(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c21(FROM(z0))
FROM(ok(z0)) → c22(FROM(z0))
CONS(mark(z0), z1) → c23(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(ok(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c29(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))
QUOT(mark(z0), z1) → c33(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c34(QUOT(z0, z1))
QUOT(ok(z0), ok(z1)) → c35(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
PROPER(s(z0)) → c41(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c42(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(minus(z0, z1)) → c44(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quot(z0, z1)) → c45(QUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(zWquot(z0, z1)) → c46(ZWQUOT(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(CONS(quot(z0, z2), zWquot(z1, z3)))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(QUOT(z0, z2))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c1(ZWQUOT(z1, z3))
ACTIVE(quot(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(quot(s(z0), s(z1))) → c1(S(z1))
ACTIVE(from(from(z0))) → c10(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(sel(0, cons(z0, z1)))) → c10(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c10(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(from(minus(z0, 0))) → c10(FROM(mark(0)), ACTIVE(minus(z0, 0)))
ACTIVE(from(minus(s(z0), s(z1)))) → c10(FROM(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(from(quot(0, s(z0)))) → c10(FROM(mark(0)), ACTIVE(quot(0, s(z0))))
ACTIVE(from(quot(s(z0), s(z1)))) → c10(FROM(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
ACTIVE(from(zWquot(z0, nil))) → c10(FROM(mark(nil)), ACTIVE(zWquot(z0, nil)))
ACTIVE(from(zWquot(nil, z0))) → c10(FROM(mark(nil)), ACTIVE(zWquot(nil, z0)))
ACTIVE(from(zWquot(cons(z0, z1), cons(z2, z3)))) → c10(FROM(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
ACTIVE(from(from(z0))) → c10(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c10(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c10(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(from(sel(z0, z1))) → c10(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
ACTIVE(from(minus(z0, z1))) → c10(FROM(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
ACTIVE(from(quot(z0, z1))) → c10(FROM(quot