Runtime Complexity (innermost) proof of /tmp/tmpHAOKDV/Ex4_7_37_Bor03

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 87 ms)

↳2 CdtProblem

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

↳4 CdtProblem

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

↳6 CdtProblem

↳7 CdtUnreachableProof (⇔, 0 ms)

↳8 CdtProblem

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

↳10 CdtProblem

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

↳12 CdtProblem

↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 18 ms)

↳14 CdtProblem

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

↳16 CdtProblem

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

↳18 CdtProblem

↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1301 ms)

↳20 CdtProblem

↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 51 ms)

↳22 CdtProblem

↳23 CdtUnreachableProof (⇔, 0 ms)

↳24 CdtProblem

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

↳26 CdtProblem

↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 40 ms)

↳28 CdtProblem

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

↳30 CdtProblem

↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 21 ms)

↳32 CdtProblem

↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 20 ms)

↳34 CdtProblem

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

↳36 CdtProblem

↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 2 ms)

↳38 CdtProblem

↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 20 ms)

↳40 CdtProblem

↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 6 ms)

↳42 CdtProblem

↳43 SIsEmptyProof (⇔, 0 ms)

↳44 BOUNDS(O(1), O(1))

(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 CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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

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

Removed 4 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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

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

Use narrowing to replace ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3)) by

ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(CONS(quot(x0, x2), ok(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(mark(z0), x1), cons(z1, x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(mark(z0), z1), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(z0, x1), cons(mark(z1), x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(z0, mark(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, x1), cons(x2, x3))) → c9

(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(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))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(CONS(quot(x0, x2), ok(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(mark(z0), x1), cons(z1, x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(mark(z0), z1), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(z0, x1), cons(mark(z1), x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(z0, mark(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, x1), cons(x2, x3))) → c9

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(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(CONS(quot(x0, x2), ok(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(mark(z0), x1), cons(z1, x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(mark(z0), z1), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(z0, x1), cons(mark(z1), x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(z0, mark(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, x1), cons(x2, x3))) → c9

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, c6, c9, c9

(7) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(zWquot(cons(mark(z0), x1), cons(z1, x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(mark(z0), z1), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(z0, x1), cons(mark(z1), x3))) → c9(CONS(mark(quot(z0, z1)), zWquot(x1, x3)), QUOT(z0, mark(z1)), ZWQUOT(x1, x3))

(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:

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))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(CONS(quot(x0, x2), ok(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, x1), cons(x2, x3))) → c9
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))

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(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(CONS(quot(x0, x2), ok(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, x1), cons(x2, x3))) → c9

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(zWquot(cons(x0, x1), cons(x2, x3))) → c9

(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))
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:

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))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(CONS(quot(x0, x2), ok(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))

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(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(CONS(quot(x0, x2), ok(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(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))
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:

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))
TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(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(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c48(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)), PROPER(0))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(nil)) → c48(TOP(ok(nil)), PROPER(nil))
TOP(mark(x0)) → c48

(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))
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:

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))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)), PROPER(0))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(nil)) → c48(TOP(ok(nil)), PROPER(nil))
TOP(mark(x0)) → c48

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(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))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)), PROPER(0))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(nil)) → c48(TOP(ok(nil)), PROPER(nil))
TOP(mark(x0)) → c48

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c48

(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))
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:

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))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)), PROPER(0))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(nil)) → c48(TOP(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(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(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))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)), PROPER(0))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(nil)) → c48(TOP(ok(nil)), PROPER(nil))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(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))
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:

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))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(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(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(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))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

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

TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))

We considered the (Usable) Rules:

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

And the Tuples:

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))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(MINUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(QUOT(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(TOP(x₁)) = [4]x₁
POL(ZWQUOT(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁, x₂)) = x₁ + x₂
POL(c17(x₁, x₂)) = x₁ + x₂
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c40(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c41(x₁, x₂)) = x₁ + x₂
POL(c42(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c44(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c45(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c46(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c48(x₁)) = x₁
POL(c48(x₁, x₂)) = x₁ + x₂
POL(c49(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(cons(x₁, x₂)) = [1]
POL(from(x₁)) = [1]
POL(mark(x₁)) = [1]
POL(minus(x₁, x₂)) = [1]
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(quot(x₁, x₂)) = [1]
POL(s(x₁)) = [1]
POL(sel(x₁, x₂)) = [1]
POL(zWquot(x₁, x₂)) = [1]

(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))
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:

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))
TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(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(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(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))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))

K tuples:

TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c49(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(from(z0))) → c49(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c49(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c49(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(minus(z0, 0))) → c49(TOP(mark(0)), ACTIVE(minus(z0, 0)))
TOP(ok(minus(s(z0), s(z1)))) → c49(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(quot(0, s(z0)))) → c49(TOP(mark(0)), ACTIVE(quot(0, s(z0))))
TOP(ok(quot(s(z0), s(z1)))) → c49(TOP(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
TOP(ok(zWquot(z0, nil))) → c49(TOP(mark(nil)), ACTIVE(zWquot(z0, nil)))
TOP(ok(zWquot(nil, z0))) → c49(TOP(mark(nil)), ACTIVE(zWquot(nil, z0)))
TOP(ok(zWquot(cons(z0, z1), cons(z2, z3)))) → c49(TOP(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
TOP(ok(from(z0))) → c49(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c49(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c49(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
TOP(ok(x0)) → c49

(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))
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:

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))
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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))
TOP(ok(from(z0))) → c49(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c49(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c49(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(minus(z0, 0))) → c49(TOP(mark(0)), ACTIVE(minus(z0, 0)))
TOP(ok(minus(s(z0), s(z1)))) → c49(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(quot(0, s(z0)))) → c49(TOP(mark(0)), ACTIVE(quot(0, s(z0))))
TOP(ok(quot(s(z0), s(z1)))) → c49(TOP(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
TOP(ok(zWquot(z0, nil))) → c49(TOP(mark(nil)), ACTIVE(zWquot(z0, nil)))
TOP(ok(zWquot(nil, z0))) → c49(TOP(mark(nil)), ACTIVE(zWquot(nil, z0)))
TOP(ok(zWquot(cons(z0, z1), cons(z2, z3)))) → c49(TOP(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
TOP(ok(from(z0))) → c49(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c49(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c49(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
TOP(ok(x0)) → c49

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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(ok(from(z0))) → c49(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c49(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c49(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(minus(z0, 0))) → c49(TOP(mark(0)), ACTIVE(minus(z0, 0)))
TOP(ok(minus(s(z0), s(z1)))) → c49(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(quot(0, s(z0)))) → c49(TOP(mark(0)), ACTIVE(quot(0, s(z0))))
TOP(ok(quot(s(z0), s(z1)))) → c49(TOP(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
TOP(ok(zWquot(z0, nil))) → c49(TOP(mark(nil)), ACTIVE(zWquot(z0, nil)))
TOP(ok(zWquot(nil, z0))) → c49(TOP(mark(nil)), ACTIVE(zWquot(nil, z0)))
TOP(ok(zWquot(cons(z0, z1), cons(z2, z3)))) → c49(TOP(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
TOP(ok(from(z0))) → c49(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c49(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c49(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))
TOP(ok(x0)) → c49

K tuples:

TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

(23) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2))
ACTIVE(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))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(quot(s(z0), s(z1))) → c6(MINUS(z0, z1), S(z1))
ACTIVE(zWquot(cons(x0, mark(z0)), cons(x2, z1))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(mark(z0), z1))
ACTIVE(zWquot(cons(x0, z0), cons(x2, mark(z1)))) → c9(CONS(quot(x0, x2), mark(zWquot(z0, z1))), QUOT(x0, x2), ZWQUOT(z0, mark(z1)))
ACTIVE(zWquot(cons(ok(z0), x1), cons(ok(z1), x3))) → c9(CONS(ok(quot(z0, z1)), zWquot(x1, x3)), QUOT(ok(z0), ok(z1)), ZWQUOT(x1, x3))
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))
ACTIVE(zWquot(cons(x0, ok(z0)), cons(x2, ok(z1)))) → c9(QUOT(x0, x2), ZWQUOT(ok(z0), ok(z1)))
TOP(mark(from(z0))) → c48(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c48(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c48(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c48(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(minus(z0, z1))) → c48(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(quot(z0, z1))) → c48(TOP(quot(proper(z0), proper(z1))), PROPER(quot(z0, z1)))
TOP(mark(zWquot(z0, z1))) → c48(TOP(zWquot(proper(z0), proper(z1))), PROPER(zWquot(z0, z1)))
TOP(ok(from(z0))) → c49(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(sel(0, cons(z0, z1)))) → c49(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c49(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(minus(z0, 0))) → c49(TOP(mark(0)), ACTIVE(minus(z0, 0)))
TOP(ok(minus(s(z0), s(z1)))) → c49(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(quot(0, s(z0)))) → c49(TOP(mark(0)), ACTIVE(quot(0, s(z0))))
TOP(ok(quot(s(z0), s(z1)))) → c49(TOP(mark(s(quot(minus(z0, z1), s(z1))))), ACTIVE(quot(s(z0), s(z1))))
TOP(ok(zWquot(z0, nil))) → c49(TOP(mark(nil)), ACTIVE(zWquot(z0, nil)))
TOP(ok(zWquot(nil, z0))) → c49(TOP(mark(nil)), ACTIVE(zWquot(nil, z0)))
TOP(ok(zWquot(cons(z0, z1), cons(z2, z3)))) → c49(TOP(mark(cons(quot(z0, z2), zWquot(z1, z3)))), ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))))
TOP(ok(from(z0))) → c49(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c49(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c49(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c49(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(active(z0), z1)), ACTIVE(minus(z0, z1)))
TOP(ok(minus(z0, z1))) → c49(TOP(minus(z0, active(z1))), ACTIVE(minus(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(active(z0), z1)), ACTIVE(quot(z0, z1)))
TOP(ok(quot(z0, z1))) → c49(TOP(quot(z0, active(z1))), ACTIVE(quot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(active(z0), z1)), ACTIVE(zWquot(z0, z1)))
TOP(ok(zWquot(z0, z1))) → c49(TOP(zWquot(z0, active(z1))), ACTIVE(zWquot(z0, z1)))

(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))
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:

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))
TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))
TOP(ok(x0)) → c49

S tuples:

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))
TOP(ok(x0)) → c49

K tuples:

TOP(mark(0)) → c48(TOP(ok(0)))
TOP(mark(nil)) → c48(TOP(ok(nil)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c48, c49

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

Removed 3 trailing nodes:

TOP(ok(x0)) → c49
TOP(mark(nil)) → c48(TOP(ok(nil)))
TOP(mark(0)) → c48(TOP(ok(0)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

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))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))

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

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))

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

POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(MINUS(x₁, x₂)) = [2]x₂
POL(QUOT(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(ZWQUOT(x₁, x₂)) = 0
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

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))
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))

K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

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(ok(z0), ok(z1)) → c29(SEL(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))

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

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))

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

POL(CONS(x₁, x₂)) = [4]x₁ + [5]x₂
POL(FROM(x₁)) = [4]x₁
POL(MINUS(x₁, x₂)) = 0
POL(QUOT(x₁, x₂)) = [2]x₁ + [2]x₂
POL(S(x₁)) = [4]x₁
POL(SEL(x₁, x₂)) = [2]x₁
POL(ZWQUOT(x₁, x₂)) = 0
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = [4] + x₁

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

SEL(z0, mark(z1)) → c28(SEL(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(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))

K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, 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(ok(z0), ok(z1)) → c29(SEL(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))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

SEL(z0, mark(z1)) → c28(SEL(z0, z1))

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

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))

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

POL(CONS(x₁, x₂)) = [4]x₁ + x₂
POL(FROM(x₁)) = 0
POL(MINUS(x₁, x₂)) = 0
POL(QUOT(x₁, x₂)) = x₁ + [2]x₂
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = x₂
POL(ZWQUOT(x₁, x₂)) = 0
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [2] + x₁

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(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))

K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, 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(ok(z0), ok(z1)) → c29(SEL(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))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))

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

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))

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

POL(CONS(x₁, x₂)) = [4]x₁ + [2]x₂
POL(FROM(x₁)) = 0
POL(MINUS(x₁, x₂)) = 0
POL(QUOT(x₁, x₂)) = [2]x₁
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = [2]x₁
POL(ZWQUOT(x₁, x₂)) = x₂
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))

K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, 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(ok(z0), ok(z1)) → c29(SEL(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))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))

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

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))

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

POL(CONS(x₁, x₂)) = [3]x₂
POL(FROM(x₁)) = 0
POL(MINUS(x₁, x₂)) = [4]x₂
POL(QUOT(x₁, x₂)) = 0
POL(S(x₁)) = [4]x₁
POL(SEL(x₁, x₂)) = [4]x₁
POL(ZWQUOT(x₁, x₂)) = 0
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))

K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, 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(ok(z0), ok(z1)) → c29(SEL(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))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

MINUS(mark(z0), z1) → c30(MINUS(z0, z1))

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

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))

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

POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = [2]x₁
POL(MINUS(x₁, x₂)) = x₁ + [4]x₂
POL(QUOT(x₁, x₂)) = [2]x₂
POL(S(x₁)) = [4]x₁
POL(SEL(x₁, x₂)) = x₂
POL(ZWQUOT(x₁, x₂)) = 0
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))

K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, 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(ok(z0), ok(z1)) → c29(SEL(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))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))

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

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))

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

POL(CONS(x₁, x₂)) = [3]x₂
POL(FROM(x₁)) = [4]x₁
POL(MINUS(x₁, x₂)) = 0
POL(QUOT(x₁, x₂)) = [2]x₂
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = [4]x₂
POL(ZWQUOT(x₁, x₂)) = [4]x₂
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = [5] + x₁
POL(ok(x₁)) = [4] + x₁

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:

ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))

K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, 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(ok(z0), ok(z1)) → c29(SEL(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))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

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

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

ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))

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

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))

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

POL(CONS(x₁, x₂)) = [4]x₁
POL(FROM(x₁)) = 0
POL(MINUS(x₁, x₂)) = [2]x₁
POL(QUOT(x₁, x₂)) = [3]x₁
POL(S(x₁)) = [4]x₁
POL(SEL(x₁, x₂)) = [4]x₁ + x₂
POL(ZWQUOT(x₁, x₂)) = x₁ + [2]x₂
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(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:

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))

S tuples:none
K tuples:

MINUS(ok(z0), ok(z1)) → c32(MINUS(z0, 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(ok(z0), ok(z1)) → c29(SEL(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))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
ZWQUOT(z0, mark(z1)) → c37(ZWQUOT(z0, z1))
MINUS(z0, mark(z1)) → c31(MINUS(z0, z1))
MINUS(mark(z0), z1) → c30(MINUS(z0, z1))
ZWQUOT(ok(z0), ok(z1)) → c38(ZWQUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c36(ZWQUOT(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT

Compound Symbols:

c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

(43) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtUnreachableProof (EQUIVALENT transformation)

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

(23) CdtUnreachableProof (EQUIVALENT transformation)

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

(43) SIsEmptyProof (EQUIVALENT transformation)

(44) BOUNDS(O(1), O(1))