Runtime Complexity (innermost) proof of /tmp/tmpNAVKPz/Ex5_7_Luc97

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 86 ms)

↳2 CdtProblem

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

↳4 CdtProblem

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

↳6 CdtProblem

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

↳8 CdtProblem

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

↳10 CdtProblem

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

↳12 CdtProblem

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

↳14 CdtProblem

↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 3 ms)

↳16 CdtProblem

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

↳18 CdtProblem

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

↳20 CdtProblem

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

↳22 CdtProblem

↳23 CdtUnreachableProof (⇔, 0 ms)

↳24 CdtProblem

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

↳26 CdtProblem

↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 37 ms)

↳28 CdtProblem

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

↳30 CdtProblem

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

↳32 CdtProblem

↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1099 ms)

↳34 CdtProblem

↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1997 ms)

↳36 CdtProblem

↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 79 ms)

↳38 CdtProblem

↳39 CdtUnreachableProof (⇔, 0 ms)

↳40 CdtProblem

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

↳42 CdtProblem

↳43 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))), 36 ms)

↳44 CdtProblem

↳45 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 23 ms)

↳46 CdtProblem

↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 14 ms)

↳48 CdtProblem

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

↳50 CdtProblem

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

↳52 CdtProblem

↳53 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 11 ms)

↳54 CdtProblem

↳55 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 14 ms)

↳56 CdtProblem

↳57 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 18 ms)

↳58 CdtProblem

↳59 SIsEmptyProof (⇔, 0 ms)

↳60 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c1, c3, c5, c7, c8, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))

S tuples:

ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c1, c3, c5, c7, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8

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

Use narrowing to replace ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0)) by

ACTIVE(dbl(s(mark(z0)))) → c1(S(s(mark(dbl(z0)))), S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(s(mark(dbl(z0)))), S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))

S tuples:

ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(s(mark(dbl(z0)))), S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c3, c5, c7, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c1

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

Removed 1 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))

S tuples:

ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c3, c5, c7, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1

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

Use narrowing to replace ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1)) by

ACTIVE(dbls(cons(x0, mark(z0)))) → c3(CONS(dbl(x0), mark(dbls(z0))), DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(CONS(mark(dbl(z0)), dbls(x1)), DBL(mark(z0)), DBLS(x1))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, x1))) → c3

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(CONS(dbl(x0), mark(dbls(z0))), DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(CONS(mark(dbl(z0)), dbls(x1)), DBL(mark(z0)), DBLS(x1))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, x1))) → c3

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(CONS(dbl(x0), mark(dbls(z0))), DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(CONS(mark(dbl(z0)), dbls(x1)), DBL(mark(z0)), DBLS(x1))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, x1))) → c3

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c7, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c3, c3

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

Removed 1 trailing nodes:

ACTIVE(dbls(cons(x0, x1))) → c3

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(CONS(dbl(x0), mark(dbls(z0))), DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(CONS(mark(dbl(z0)), dbls(x1)), DBL(mark(z0)), DBLS(x1))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(CONS(dbl(x0), mark(dbls(z0))), DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(CONS(mark(dbl(z0)), dbls(x1)), DBL(mark(z0)), DBLS(x1))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c7, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c3

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

Removed 2 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c7, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c3, c3

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

Use narrowing to replace ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2)) by

ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(CONS(sel(x0, z1), mark(indx(z0, z1))), SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(CONS(mark(sel(z0, z1)), indx(x1, z1)), SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(CONS(mark(sel(z0, z1)), indx(x1, mark(z1))), SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, x1), x2)) → c7

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(CONS(sel(x0, z1), mark(indx(z0, z1))), SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(CONS(mark(sel(z0, z1)), indx(x1, z1)), SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(CONS(mark(sel(z0, z1)), indx(x1, mark(z1))), SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, x1), x2)) → c7

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(CONS(sel(x0, z1), mark(indx(z0, z1))), SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(CONS(mark(sel(z0, z1)), indx(x1, z1)), SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(CONS(mark(sel(z0, z1)), indx(x1, mark(z1))), SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, x1), x2)) → c7

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c3, c3, c7, c7

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

Removed 1 trailing nodes:

ACTIVE(indx(cons(x0, x1), x2)) → c7

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(CONS(sel(x0, z1), mark(indx(z0, z1))), SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(CONS(mark(sel(z0, z1)), indx(x1, z1)), SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(CONS(mark(sel(z0, z1)), indx(x1, mark(z1))), SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(CONS(sel(x0, z1), mark(indx(z0, z1))), SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(CONS(mark(sel(z0, z1)), indx(x1, z1)), SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(CONS(mark(sel(z0, z1)), indx(x1, mark(z1))), SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c3, c3, c7

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

Removed 3 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c10, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c3, c3, c7, c7

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

Use narrowing to replace ACTIVE(dbl1(s(z0))) → c10(S1(s1(dbl1(z0))), S1(dbl1(z0)), DBL1(z0)) by

ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
ACTIVE(dbl1(s(ok(z0)))) → c10(S1(s1(ok(dbl1(z0)))), S1(dbl1(ok(z0))), DBL1(ok(z0)))
ACTIVE(dbl1(s(x0))) → c10

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
ACTIVE(dbl1(s(ok(z0)))) → c10(S1(s1(ok(dbl1(z0)))), S1(dbl1(ok(z0))), DBL1(ok(z0)))
ACTIVE(dbl1(s(x0))) → c10

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
ACTIVE(dbl1(s(ok(z0)))) → c10(S1(s1(ok(dbl1(z0)))), S1(dbl1(ok(z0))), DBL1(ok(z0)))
ACTIVE(dbl1(s(x0))) → c10

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c59, c60, c61, c62, c63, c8, c1, c3, c3, c7, c7, c10, c10

(23) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(dbl1(s(ok(z0)))) → c10(S1(s1(ok(dbl1(z0)))), S1(dbl1(ok(z0))), DBL1(ok(z0)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
ACTIVE(dbl1(s(x0))) → c10
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
ACTIVE(dbl1(s(x0))) → c10

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP, ACTIVE, PROPER

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c62, c63, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58

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

Removed 1 trailing nodes:

ACTIVE(dbl1(s(x0))) → c10

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP, ACTIVE, PROPER

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c62, c63, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58

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

Use narrowing to replace TOP(mark(z0)) → c62(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c62(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c62(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(01)) → c62(TOP(ok(01)), PROPER(01))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(x0)) → c62

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c62(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c62(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(01)) → c62(TOP(ok(01)), PROPER(01))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(x0)) → c62

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c62(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c62(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(01)) → c62(TOP(ok(01)), PROPER(01))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(x0)) → c62

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP, ACTIVE, PROPER

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c63, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c62, c62

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c62

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c62(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c62(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(01)) → c62(TOP(ok(01)), PROPER(01))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c62(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c62(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(01)) → c62(TOP(ok(01)), PROPER(01))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP, ACTIVE, PROPER

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c63, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c62

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

Removed 3 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP, ACTIVE, PROPER

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c63, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c62, c62

(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.

TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

We considered the (Usable) Rules:

proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
from(ok(z0)) → ok(from(z0))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(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))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))

And the Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

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

POL(0) = 0
POL(01) = 0
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(DBL(x₁)) = 0
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = 0
POL(FROM(x₁)) = 0
POL(INDX(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁, x₂)) = x₁ + x₂
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁, x₂)) = x₁ + x₂
POL(c47(x₁, x₂)) = x₁ + x₂
POL(c48(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c50(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c52(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c53(x₁, x₂)) = x₁ + x₂
POL(c54(x₁, x₂)) = x₁ + x₂
POL(c56(x₁, x₂)) = x₁ + x₂
POL(c57(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c58(x₁, x₂)) = x₁ + x₂
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c62(x₁, x₂)) = x₁ + x₂
POL(c63(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c8(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(dbl(x₁)) = [1]
POL(dbl1(x₁)) = [1]
POL(dbls(x₁)) = [1]
POL(from(x₁)) = [1]
POL(indx(x₁, x₂)) = [1]
POL(mark(x₁)) = [1]
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(quote(x₁)) = [1]
POL(s(x₁)) = [1]
POL(s1(x₁)) = [1]
POL(sel(x₁, x₂)) = [1]
POL(sel1(x₁, x₂)) = [1]

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))

K tuples:

TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP, ACTIVE, PROPER

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c63, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c62, c62

(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.

TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))

We considered the (Usable) Rules:

proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
from(ok(z0)) → ok(from(z0))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(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))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))

And the Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

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

POL(0) = [4]
POL(01) = [3]
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(DBL(x₁)) = 0
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = 0
POL(FROM(x₁)) = 0
POL(INDX(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁)) = x₁
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁, x₂)) = x₁ + x₂
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁, x₂)) = x₁ + x₂
POL(c47(x₁, x₂)) = x₁ + x₂
POL(c48(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c50(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c52(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c53(x₁, x₂)) = x₁ + x₂
POL(c54(x₁, x₂)) = x₁ + x₂
POL(c56(x₁, x₂)) = x₁ + x₂
POL(c57(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c58(x₁, x₂)) = x₁ + x₂
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c62(x₁, x₂)) = x₁ + x₂
POL(c63(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c8(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(dbl(x₁)) = [4]
POL(dbl1(x₁)) = [4]
POL(dbls(x₁)) = [4]
POL(from(x₁)) = [4]
POL(indx(x₁, x₂)) = [4]
POL(mark(x₁)) = [4]
POL(nil) = [4]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(quote(x₁)) = [4]
POL(s(x₁)) = 0
POL(s1(x₁)) = [4]
POL(sel(x₁, x₂)) = [4]
POL(sel1(x₁, x₂)) = [4]

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))

K tuples:

TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP, ACTIVE, PROPER

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c63, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c62, c62

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

Use narrowing to replace TOP(ok(z0)) → c63(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(dbl(0))) → c63(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c63(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(nil))) → c63(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(ok(dbls(cons(z0, z1)))) → c63(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(nil, z0))) → c63(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(indx(cons(z0, z1), z2))) → c63(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c63(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl1(0))) → c63(TOP(mark(01)), ACTIVE(dbl1(0)))
TOP(ok(dbl1(s(z0)))) → c63(TOP(mark(s1(s1(dbl1(z0))))), ACTIVE(dbl1(s(z0))))
TOP(ok(sel1(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c63(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote(s(z0)))) → c63(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(dbl(z0)))) → c63(TOP(mark(dbl1(z0))), ACTIVE(quote(dbl(z0))))
TOP(ok(quote(sel(z0, z1)))) → c63(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(dbl(z0))) → c63(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c63(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c63(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
TOP(ok(dbl1(z0))) → c63(TOP(dbl1(active(z0))), ACTIVE(dbl1(z0)))
TOP(ok(s1(z0))) → c63(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(quote(z0))) → c63(TOP(quote(active(z0))), ACTIVE(quote(z0)))
TOP(ok(x0)) → c63

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))
TOP(ok(dbl(0))) → c63(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c63(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(nil))) → c63(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(ok(dbls(cons(z0, z1)))) → c63(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(nil, z0))) → c63(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(indx(cons(z0, z1), z2))) → c63(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c63(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl1(0))) → c63(TOP(mark(01)), ACTIVE(dbl1(0)))
TOP(ok(dbl1(s(z0)))) → c63(TOP(mark(s1(s1(dbl1(z0))))), ACTIVE(dbl1(s(z0))))
TOP(ok(sel1(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c63(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote(s(z0)))) → c63(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(dbl(z0)))) → c63(TOP(mark(dbl1(z0))), ACTIVE(quote(dbl(z0))))
TOP(ok(quote(sel(z0, z1)))) → c63(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(dbl(z0))) → c63(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c63(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c63(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
TOP(ok(dbl1(z0))) → c63(TOP(dbl1(active(z0))), ACTIVE(dbl1(z0)))
TOP(ok(s1(z0))) → c63(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(quote(z0))) → c63(TOP(quote(active(z0))), ACTIVE(quote(z0)))
TOP(ok(x0)) → c63

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(ok(dbl(0))) → c63(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c63(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(nil))) → c63(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(ok(dbls(cons(z0, z1)))) → c63(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(nil, z0))) → c63(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(indx(cons(z0, z1), z2))) → c63(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c63(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl1(0))) → c63(TOP(mark(01)), ACTIVE(dbl1(0)))
TOP(ok(dbl1(s(z0)))) → c63(TOP(mark(s1(s1(dbl1(z0))))), ACTIVE(dbl1(s(z0))))
TOP(ok(sel1(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c63(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote(s(z0)))) → c63(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(dbl(z0)))) → c63(TOP(mark(dbl1(z0))), ACTIVE(quote(dbl(z0))))
TOP(ok(quote(sel(z0, z1)))) → c63(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(dbl(z0))) → c63(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c63(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c63(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
TOP(ok(dbl1(z0))) → c63(TOP(dbl1(active(z0))), ACTIVE(dbl1(z0)))
TOP(ok(s1(z0))) → c63(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(quote(z0))) → c63(TOP(quote(active(z0))), ACTIVE(quote(z0)))
TOP(ok(x0)) → c63

K tuples:

TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, ACTIVE, PROPER, TOP

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c8, c1, c3, c3, c7, c7, c10, c45, c47, c48, c50, c51, c52, c53, c54, c56, c57, c58, c62, c62, c63, c63

(39) 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))) → c5(SEL(z0, z2))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c12(SEL1(z0, z2))
ACTIVE(quote(s(z0))) → c14(S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(dbl(z0))) → c15(DBL1(z0))
ACTIVE(quote(sel(z0, z1))) → c16(SEL1(z0, z1))
ACTIVE(dbl(z0)) → c17(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c18(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c19(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c20(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c21(INDX(active(z0), z1), ACTIVE(z0))
ACTIVE(dbl1(z0)) → c22(DBL1(active(z0)), ACTIVE(z0))
ACTIVE(s1(z0)) → c23(S1(active(z0)), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c24(SEL1(active(z0), z1), ACTIVE(z0))
ACTIVE(sel1(z0, z1)) → c25(SEL1(z0, active(z1)), ACTIVE(z1))
ACTIVE(quote(z0)) → c26(QUOTE(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(x0))) → c1(S(dbl(x0)), DBL(x0))
ACTIVE(dbl(s(mark(z0)))) → c1(S(dbl(mark(z0))), DBL(mark(z0)))
ACTIVE(dbls(cons(x0, ok(z0)))) → c3(CONS(dbl(x0), ok(dbls(z0))), DBL(x0), DBLS(ok(z0)))
ACTIVE(dbls(cons(ok(z0), x1))) → c3(CONS(ok(dbl(z0)), dbls(x1)), DBL(ok(z0)), DBLS(x1))
ACTIVE(dbls(cons(x0, mark(z0)))) → c3(DBL(x0), DBLS(mark(z0)))
ACTIVE(dbls(cons(mark(z0), x1))) → c3(DBL(mark(z0)), DBLS(x1))
ACTIVE(indx(cons(x0, ok(z0)), ok(z1))) → c7(CONS(sel(x0, ok(z1)), ok(indx(z0, z1))), SEL(x0, ok(z1)), INDX(ok(z0), ok(z1)))
ACTIVE(indx(cons(ok(z0), x1), ok(z1))) → c7(CONS(ok(sel(z0, z1)), indx(x1, ok(z1))), SEL(ok(z0), ok(z1)), INDX(x1, ok(z1)))
ACTIVE(indx(cons(x0, mark(z0)), z1)) → c7(SEL(x0, z1), INDX(mark(z0), z1))
ACTIVE(indx(cons(mark(z0), x1), z1)) → c7(SEL(mark(z0), z1), INDX(x1, z1))
ACTIVE(indx(cons(z0, x1), mark(z1))) → c7(SEL(z0, mark(z1)), INDX(x1, mark(z1)))
ACTIVE(dbl1(s(mark(z0)))) → c10(S1(s1(mark(dbl1(z0)))), S1(dbl1(mark(z0))), DBL1(mark(z0)))
PROPER(dbl(z0)) → c45(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c47(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c48(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c50(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c51(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c52(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c53(FROM(proper(z0)), PROPER(z0))
PROPER(dbl1(z0)) → c54(DBL1(proper(z0)), PROPER(z0))
PROPER(s1(z0)) → c56(S1(proper(z0)), PROPER(z0))
PROPER(sel1(z0, z1)) → c57(SEL1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(quote(z0)) → c58(QUOTE(proper(z0)), PROPER(z0))
TOP(mark(dbl(z0))) → c62(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c62(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c62(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c62(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c62(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c62(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c62(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(dbl1(z0))) → c62(TOP(dbl1(proper(z0))), PROPER(dbl1(z0)))
TOP(mark(s1(z0))) → c62(TOP(s1(proper(z0))), PROPER(s1(z0)))
TOP(mark(sel1(z0, z1))) → c62(TOP(sel1(proper(z0), proper(z1))), PROPER(sel1(z0, z1)))
TOP(mark(quote(z0))) → c62(TOP(quote(proper(z0))), PROPER(quote(z0)))
TOP(ok(dbl(0))) → c63(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c63(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(nil))) → c63(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(ok(dbls(cons(z0, z1)))) → c63(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(nil, z0))) → c63(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(indx(cons(z0, z1), z2))) → c63(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c63(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl1(0))) → c63(TOP(mark(01)), ACTIVE(dbl1(0)))
TOP(ok(dbl1(s(z0)))) → c63(TOP(mark(s1(s1(dbl1(z0))))), ACTIVE(dbl1(s(z0))))
TOP(ok(sel1(0, cons(z0, z1)))) → c63(TOP(mark(z0)), ACTIVE(sel1(0, cons(z0, z1))))
TOP(ok(sel1(s(z0), cons(z1, z2)))) → c63(TOP(mark(sel1(z0, z2))), ACTIVE(sel1(s(z0), cons(z1, z2))))
TOP(ok(quote(0))) → c63(TOP(mark(01)), ACTIVE(quote(0)))
TOP(ok(quote(s(z0)))) → c63(TOP(mark(s1(quote(z0)))), ACTIVE(quote(s(z0))))
TOP(ok(quote(dbl(z0)))) → c63(TOP(mark(dbl1(z0))), ACTIVE(quote(dbl(z0))))
TOP(ok(quote(sel(z0, z1)))) → c63(TOP(mark(sel1(z0, z1))), ACTIVE(quote(sel(z0, z1))))
TOP(ok(dbl(z0))) → c63(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c63(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c63(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c63(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
TOP(ok(dbl1(z0))) → c63(TOP(dbl1(active(z0))), ACTIVE(dbl1(z0)))
TOP(ok(s1(z0))) → c63(TOP(s1(active(z0))), ACTIVE(s1(z0)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(active(z0), z1)), ACTIVE(sel1(z0, z1)))
TOP(ok(sel1(z0, z1))) → c63(TOP(sel1(z0, active(z1))), ACTIVE(sel1(z0, z1)))
TOP(ok(quote(z0))) → c63(TOP(quote(active(z0))), ACTIVE(quote(z0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))
TOP(ok(x0)) → c63

S tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
TOP(ok(x0)) → c63

K tuples:

TOP(mark(0)) → c62(TOP(ok(0)))
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(01)) → c62(TOP(ok(01)))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM, TOP

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61, c62, c63

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

Removed 4 trailing nodes:

TOP(mark(01)) → c62(TOP(ok(01)))
TOP(ok(x0)) → c63
TOP(mark(nil)) → c62(TOP(ok(nil)))
TOP(mark(0)) → c62(TOP(ok(0)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

K tuples:none
Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

(43) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = x₁² + x₁²·x₂
POL(DBL(x₁)) = 0
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = 0
POL(FROM(x₁)) = x₁ + x₁²
POL(INDX(x₁, x₂)) = 0
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = x₁ + x₁²
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [1] + x₁

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))

K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

(45) 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.

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = [2]x₁
POL(DBL(x₁)) = [5]x₁
POL(DBL1(x₁)) = x₁
POL(DBLS(x₁)) = [3]x₁
POL(FROM(x₁)) = 0
POL(INDX(x₁, x₂)) = 0
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = [4]x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [3]x₂
POL(SEL1(x₁, x₂)) = [5]x₁ + [5]x₂
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [5] + x₁

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

SEL(mark(z0), z1) → c31(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))

K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

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

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

INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = [4]x₁ + [3]x₂
POL(DBL(x₁)) = 0
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = 0
POL(FROM(x₁)) = [4]x₁
POL(INDX(x₁, x₂)) = [4]x₂
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = [5]
POL(ok(x₁)) = [2] + x₁

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

SEL(mark(z0), z1) → c31(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))

K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

(49) 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.

INDX(mark(z0), z1) → c34(INDX(z0, z1))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = [4]x₁
POL(DBL(x₁)) = [2]x₁
POL(DBL1(x₁)) = [4]x₁
POL(DBLS(x₁)) = 0
POL(FROM(x₁)) = [3]x₁
POL(INDX(x₁, x₂)) = x₁ + [4]x₂
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = [4]x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = [4]x₂
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

SEL(mark(z0), z1) → c31(SEL(z0, z1))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))

K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

(51) 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(mark(z0), z1) → c31(SEL(z0, z1))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = [2]x₂
POL(DBL(x₁)) = 0
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = [3]x₁
POL(FROM(x₁)) = [2]x₁
POL(INDX(x₁, x₂)) = [4]x₁ + [5]x₂
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = [2]x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = x₁
POL(SEL1(x₁, x₂)) = 0
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [5] + x₁

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))

K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
SEL(mark(z0), z1) → c31(SEL(z0, z1))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

(53) 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.

QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = [2]x₁
POL(DBL(x₁)) = [4]x₁
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = 0
POL(FROM(x₁)) = 0
POL(INDX(x₁, x₂)) = 0
POL(QUOTE(x₁)) = x₁
POL(S(x₁)) = [3]x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [4]x₂
POL(SEL1(x₁, x₂)) = [2]x₂
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [4] + x₁

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))

K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

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

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

S1(mark(z0)) → c38(S1(z0))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = [4]x₁ + [2]x₂
POL(DBL(x₁)) = [4]x₁
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = 0
POL(FROM(x₁)) = [2]x₁
POL(INDX(x₁, x₂)) = x₂
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = [4]x₁
POL(S1(x₁)) = x₁
POL(SEL(x₁, x₂)) = [4]x₁
POL(SEL1(x₁, x₂)) = 0
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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:

S1(ok(z0)) → c39(S1(z0))

K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S1(mark(z0)) → c38(S1(z0))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

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

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

S1(ok(z0)) → c39(S1(z0))

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

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

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

POL(CONS(x₁, x₂)) = [2]x₂
POL(DBL(x₁)) = x₁
POL(DBL1(x₁)) = 0
POL(DBLS(x₁)) = [4]x₁
POL(FROM(x₁)) = [4]x₁
POL(INDX(x₁, x₂)) = [2]x₂
POL(QUOTE(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = [4]x₁
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = [4]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)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl1(0)) → mark(01)
active(dbl1(s(z0))) → mark(s1(s1(dbl1(z0))))
active(sel1(0, cons(z0, z1))) → mark(z0)
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(quote(0)) → mark(01)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(dbl(z0))) → mark(dbl1(z0))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbl1(z0)) → dbl1(active(z0))
active(s1(z0)) → s1(active(z0))
active(sel1(z0, z1)) → sel1(active(z0), z1)
active(sel1(z0, z1)) → sel1(z0, active(z1))
active(quote(z0)) → quote(active(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(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))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
s1(mark(z0)) → mark(s1(z0))
s1(ok(z0)) → ok(s1(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
quote(mark(z0)) → mark(quote(z0))
quote(ok(z0)) → ok(quote(z0))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
proper(dbl1(z0)) → dbl1(proper(z0))
proper(01) → ok(01)
proper(s1(z0)) → s1(proper(z0))
proper(sel1(z0, z1)) → sel1(proper(z0), proper(z1))
proper(quote(z0)) → quote(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))

S tuples:none
K tuples:

S(ok(z0)) → c59(S(z0))
CONS(ok(z0), ok(z1)) → c60(CONS(z0, z1))
FROM(ok(z0)) → c61(FROM(z0))
DBL(mark(z0)) → c27(DBL(z0))
DBL(ok(z0)) → c28(DBL(z0))
DBLS(mark(z0)) → c29(DBLS(z0))
DBLS(ok(z0)) → c30(DBLS(z0))
SEL(z0, mark(z1)) → c32(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c33(SEL(z0, z1))
DBL1(mark(z0)) → c36(DBL1(z0))
DBL1(ok(z0)) → c37(DBL1(z0))
SEL1(mark(z0), z1) → c40(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c41(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c42(SEL1(z0, z1))
INDX(ok(z0), ok(z1)) → c35(INDX(z0, z1))
INDX(mark(z0), z1) → c34(INDX(z0, z1))
SEL(mark(z0), z1) → c31(SEL(z0, z1))
QUOTE(mark(z0)) → c43(QUOTE(z0))
QUOTE(ok(z0)) → c44(QUOTE(z0))
S1(mark(z0)) → c38(S1(z0))
S1(ok(z0)) → c39(S1(z0))

Defined Rule Symbols:

active, dbl, dbls, sel, indx, dbl1, s1, sel1, quote, proper, s, cons, from, top

Defined Pair Symbols:

DBL, DBLS, SEL, INDX, DBL1, S1, SEL1, QUOTE, S, CONS, FROM

Compound Symbols:

c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c59, c60, c61

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(38) Obligation:

(39) CdtUnreachableProof (EQUIVALENT transformation)

(40) Obligation:

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

(42) Obligation:

(43) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

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

(50) Obligation:

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

(52) Obligation:

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

(54) Obligation:

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

(56) Obligation:

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

(58) Obligation:

(59) SIsEmptyProof (EQUIVALENT transformation)

(60) BOUNDS(O(1), O(1))